https://kbwiki.ercoftac.org/w/api.php?action=feedcontributions&user=Munich&feedformat=atom KBwiki - User contributions [en] 2024-03-28T11:33:43Z User contributions MediaWiki 1.39.2 https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=39055 UFR 3-35 Evaluation 2020-11-03T20:31:46Z <p>Munich: /* Datasets for download */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are coloured by the normalized magnitude of the velocity in the symmetry plane, &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 onward and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the reverse flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stress &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stress &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably well both with regard to amplitude and shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not change sign as does the vertical velocity gradient and hence does not follow the eddy viscosity hypothesis. This is due to the complex flow and turbulence behaviour in this region. Consequently, eddy-viscosity models would have difficulties to reproduce the near-wall shear-stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow along the cylinder front face, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and the downflow evolves, is located at the water level in the LES. On the other hand it was shifted downwards in our experiment, due to the plate of acrylic glass placed at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this plate were not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and hence the HV system becomes smaller too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Distributions of the Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; at the height of the horseshoe vortex are shown in Fig. 11 and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three-dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following subsections.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred to mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. When the horseshoe vortex moves towards the cylinder to a region with a downward directed average flow (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces a corresponding positive fluctuation &lt;math&gt; w'&lt;/math&gt;. Therefore, the turbulent transport becomes altogether negative, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in large regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular, around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots of pressure coefficient (Figs. 20 and 21) and friction coefficient (Fig.22) were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress and hence the friction coefficient correctly. A more detailed discussion of this problem can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Figs. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient displayed in Fig. 20 increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum. In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases slower than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the junction vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> |...<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |...<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> |...<br /> |}<br /> <br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> |...<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=39054 UFR 3-35 Evaluation 2020-11-03T20:30:51Z <p>Munich: /* Datasets for download */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are coloured by the normalized magnitude of the velocity in the symmetry plane, &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 onward and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the reverse flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stress &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stress &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably well both with regard to amplitude and shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not change sign as does the vertical velocity gradient and hence does not follow the eddy viscosity hypothesis. This is due to the complex flow and turbulence behaviour in this region. Consequently, eddy-viscosity models would have difficulties to reproduce the near-wall shear-stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow along the cylinder front face, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and the downflow evolves, is located at the water level in the LES. On the other hand it was shifted downwards in our experiment, due to the plate of acrylic glass placed at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this plate were not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and hence the HV system becomes smaller too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Distributions of the Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; at the height of the horseshoe vortex are shown in Fig. 11 and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three-dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following subsections.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred to mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. When the horseshoe vortex moves towards the cylinder to a region with a downward directed average flow (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces a corresponding positive fluctuation &lt;math&gt; w'&lt;/math&gt;. Therefore, the turbulent transport becomes altogether negative, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in large regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular, around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots of pressure coefficient (Figs. 20 and 21) and friction coefficient (Fig.22) were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress and hence the friction coefficient correctly. A more detailed discussion of this problem can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Figs. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient displayed in Fig. 20 increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum. In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases slower than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the junction vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> |...<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |...<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> |...<br /> |}<br /> <br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> |...<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |...<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> |...<br /> |}<br /> <br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> |...<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=39053 UFR 3-35 Evaluation 2020-11-03T20:27:38Z <p>Munich: /* Datasets for download */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are coloured by the normalized magnitude of the velocity in the symmetry plane, &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 onward and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the reverse flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stress &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stress &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably well both with regard to amplitude and shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not change sign as does the vertical velocity gradient and hence does not follow the eddy viscosity hypothesis. This is due to the complex flow and turbulence behaviour in this region. Consequently, eddy-viscosity models would have difficulties to reproduce the near-wall shear-stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow along the cylinder front face, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and the downflow evolves, is located at the water level in the LES. On the other hand it was shifted downwards in our experiment, due to the plate of acrylic glass placed at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this plate were not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and hence the HV system becomes smaller too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Distributions of the Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; at the height of the horseshoe vortex are shown in Fig. 11 and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three-dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following subsections.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred to mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. When the horseshoe vortex moves towards the cylinder to a region with a downward directed average flow (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces a corresponding positive fluctuation &lt;math&gt; w'&lt;/math&gt;. Therefore, the turbulent transport becomes altogether negative, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in large regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular, around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots of pressure coefficient (Figs. 20 and 21) and friction coefficient (Fig.22) were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress and hence the friction coefficient correctly. A more detailed discussion of this problem can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Figs. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient displayed in Fig. 20 increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum. In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases slower than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the junction vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> |...<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |...<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> |...<br /> |}<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> |...<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=39052 UFR 3-35 Evaluation 2020-11-03T20:23:51Z <p>Munich: /* Datasets for download */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are coloured by the normalized magnitude of the velocity in the symmetry plane, &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 onward and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the reverse flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stress &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stress &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably well both with regard to amplitude and shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not change sign as does the vertical velocity gradient and hence does not follow the eddy viscosity hypothesis. This is due to the complex flow and turbulence behaviour in this region. Consequently, eddy-viscosity models would have difficulties to reproduce the near-wall shear-stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow along the cylinder front face, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and the downflow evolves, is located at the water level in the LES. On the other hand it was shifted downwards in our experiment, due to the plate of acrylic glass placed at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this plate were not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and hence the HV system becomes smaller too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Distributions of the Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; at the height of the horseshoe vortex are shown in Fig. 11 and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three-dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following subsections.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred to mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. When the horseshoe vortex moves towards the cylinder to a region with a downward directed average flow (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces a corresponding positive fluctuation &lt;math&gt; w'&lt;/math&gt;. Therefore, the turbulent transport becomes altogether negative, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in large regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular, around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots of pressure coefficient (Figs. 20 and 21) and friction coefficient (Fig.22) were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress and hence the friction coefficient correctly. A more detailed discussion of this problem can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Figs. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient displayed in Fig. 20 increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum. In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases slower than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the junction vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> |...<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |...<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> |...<br /> |}<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> |...<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=39051 UFR 3-35 Evaluation 2020-11-03T20:20:47Z <p>Munich: /* Datasets for download */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are coloured by the normalized magnitude of the velocity in the symmetry plane, &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 onward and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the reverse flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stress &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stress &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably well both with regard to amplitude and shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not change sign as does the vertical velocity gradient and hence does not follow the eddy viscosity hypothesis. This is due to the complex flow and turbulence behaviour in this region. Consequently, eddy-viscosity models would have difficulties to reproduce the near-wall shear-stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow along the cylinder front face, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and the downflow evolves, is located at the water level in the LES. On the other hand it was shifted downwards in our experiment, due to the plate of acrylic glass placed at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this plate were not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and hence the HV system becomes smaller too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Distributions of the Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; at the height of the horseshoe vortex are shown in Fig. 11 and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three-dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following subsections.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred to mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. When the horseshoe vortex moves towards the cylinder to a region with a downward directed average flow (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces a corresponding positive fluctuation &lt;math&gt; w'&lt;/math&gt;. Therefore, the turbulent transport becomes altogether negative, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in large regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular, around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots of pressure coefficient (Figs. 20 and 21) and friction coefficient (Fig.22) were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress and hence the friction coefficient correctly. A more detailed discussion of this problem can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Figs. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient displayed in Fig. 20 increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum. In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases slower than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the junction vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> |...<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |...<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> |...<br /> |}<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> |...<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> |...<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=39050 UFR 3-35 Evaluation 2020-11-03T20:17:48Z <p>Munich: /* Datasets for download */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are coloured by the normalized magnitude of the velocity in the symmetry plane, &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 onward and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the reverse flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stress &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stress &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably well both with regard to amplitude and shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not change sign as does the vertical velocity gradient and hence does not follow the eddy viscosity hypothesis. This is due to the complex flow and turbulence behaviour in this region. Consequently, eddy-viscosity models would have difficulties to reproduce the near-wall shear-stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow along the cylinder front face, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and the downflow evolves, is located at the water level in the LES. On the other hand it was shifted downwards in our experiment, due to the plate of acrylic glass placed at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this plate were not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and hence the HV system becomes smaller too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Distributions of the Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; at the height of the horseshoe vortex are shown in Fig. 11 and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three-dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following subsections.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred to mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. When the horseshoe vortex moves towards the cylinder to a region with a downward directed average flow (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces a corresponding positive fluctuation &lt;math&gt; w'&lt;/math&gt;. Therefore, the turbulent transport becomes altogether negative, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in large regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular, around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots of pressure coefficient (Figs. 20 and 21) and friction coefficient (Fig.22) were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress and hence the friction coefficient correctly. A more detailed discussion of this problem can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Figs. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient displayed in Fig. 20 increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum. In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases slower than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the junction vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=39049 UFR 3-35 Evaluation 2020-11-03T20:16:07Z <p>Munich: /* Datasets for download */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are coloured by the normalized magnitude of the velocity in the symmetry plane, &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field coloured by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 onward and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the reverse flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stress &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stress &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stress &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably well both with regard to amplitude and shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not change sign as does the vertical velocity gradient and hence does not follow the eddy viscosity hypothesis. This is due to the complex flow and turbulence behaviour in this region. Consequently, eddy-viscosity models would have difficulties to reproduce the near-wall shear-stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow along the cylinder front face, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and the downflow evolves, is located at the water level in the LES. On the other hand it was shifted downwards in our experiment, due to the plate of acrylic glass placed at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this plate were not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and hence the HV system becomes smaller too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Distributions of the Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; at the height of the horseshoe vortex are shown in Fig. 11 and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three-dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following subsections.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred to mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. When the horseshoe vortex moves towards the cylinder to a region with a downward directed average flow (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces a corresponding positive fluctuation &lt;math&gt; w'&lt;/math&gt;. Therefore, the turbulent transport becomes altogether negative, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in large regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular, around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots of pressure coefficient (Figs. 20 and 21) and friction coefficient (Fig.22) were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress and hence the friction coefficient correctly. A more detailed discussion of this problem can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Figs. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient displayed in Fig. 20 increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum. In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases slower than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the junction vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38999 UFR 3-35 Evaluation 2020-10-08T19:07:52Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|800px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|800px|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|800px|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|800px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38998 UFR 3-35 Evaluation 2020-10-08T19:06:21Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|550px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|thumb|centre|550px|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|thumb|centre|550px|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|thumb|centre|550px|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38997 UFR 3-35 Evaluation 2020-10-08T19:04:21Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|999px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38996 UFR 3-35 Evaluation 2020-10-08T19:01:44Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|1174px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38995 UFR 3-35 Evaluation 2020-10-08T19:00:56Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> [[File:UFR3-35_U_z.png|thumb|centre|550px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38994 UFR 3-35 Best Practice Advice 2020-10-08T18:53:26Z <p>Munich: /* Recommendations for Future Work */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. <br /> <br /> According to our observation, a subgrid-scale model needs to be able to adapt itself to the laminar-like behaviour of the wall layer under the wall jet. Additionally, we have no hope that explicit wall models relying on the logarithmic law of the wall can give an accurate correlation of the wall shear stress with a velocity value in a wall distance in inner coordinate corresponding to the logarithmic layer (Schanderl et al. 2017a).<br /> <br /> For designing a grid for a DNS, it is important to learn that estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a considerable loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we address the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> Another uncertainty is the roughness of the wall and its effect on the flow. It is our understanding that at the moment little is known on this issue.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2020). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38993 UFR 3-35 Best Practice Advice 2020-10-08T18:48:16Z <p>Munich: /* Application Uncertainties */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. <br /> <br /> According to our observation, a subgrid-scale model needs to be able to adapt itself to the laminar-like behaviour of the wall layer under the wall jet. Additionally, we have no hope that explicit wall models relying on the logarithmic law of the wall can give an accurate correlation of the wall shear stress with a velocity value in a wall distance in inner coordinate corresponding to the logarithmic layer (Schanderl et al. 2017a).<br /> <br /> For designing a grid for a DNS, it is important to learn that estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a considerable loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we address the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> Another uncertainty is the roughness of the wall and its effect on the flow. It is our understanding that at the moment little is known on this issue.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38992 UFR 3-35 Best Practice Advice 2020-10-08T18:41:12Z <p>Munich: /* Application Uncertainties */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. <br /> <br /> According to our observation, a subgrid-scale model needs to be able to adapt itself to the laminar-like behaviour of the wall layer under the wall jet. Additionally, we have no hope that explicit wall models relying on the logarithmic law of the wall can give an accurate correlation of the wall shear stress with a velocity value in a wall distance in inner coordinate corresponding to the logarithmic layer (Schanderl et al. 2017a).<br /> <br /> For designing a grid for a DNS, it is important to learn that estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a considerable loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we address the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38991 UFR 3-35 Best Practice Advice 2020-10-08T18:40:09Z <p>Munich: /* Measurement issues */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. <br /> <br /> According to our observation, a subgrid-scale model needs to be able to adapt itself to the laminar-like behaviour of the wall layer under the wall jet. Additionally, we have no hope that explicit wall models relying on the logarithmic law of the wall can give an accurate correlation of the wall shear stress with a velocity value in a wall distance in inner coordinate corresponding to the logarithmic layer (Schanderl et al. 2017a).<br /> <br /> For designing a grid for a DNS, it is important to learn that estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a considerable loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38990 UFR 3-35 Best Practice Advice 2020-10-08T18:39:39Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. <br /> <br /> According to our observation, a subgrid-scale model needs to be able to adapt itself to the laminar-like behaviour of the wall layer under the wall jet. Additionally, we have no hope that explicit wall models relying on the logarithmic law of the wall can give an accurate correlation of the wall shear stress with a velocity value in a wall distance in inner coordinate corresponding to the logarithmic layer (Schanderl et al. 2017a).<br /> <br /> For designing a grid for a DNS, it is important to learn that estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38989 UFR 3-35 Best Practice Advice 2020-10-08T18:38:41Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. <br /> <br /> According to our observation, a subgrid-scale model needs to be able to adapt itself to the laminar-like behaviour of the wall layer under the wall jet. Additionally, we have no hope that explicit wall models relying on the logarithmic law of the wall can give an accurate correlation of the wall shear stress with a velocity value in a wall distance in inner coordinate corresponding to the logarithmic layer.<br /> <br /> For designing a grid for a DNS, it is important to learn that estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38988 UFR 3-35 Best Practice Advice 2020-10-08T18:32:51Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. Estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38987 UFR 3-35 Best Practice Advice 2020-10-08T18:29:19Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. The turbulence is far from an isotropic state or from equilibrium. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. Estimating the Kolmogorov scale by &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018).<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38986 UFR 3-35 Best Practice Advice 2020-10-08T18:19:02Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. Although the turbulence is far from an isotropic state. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics around the horseshoe vortex in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution was relatively small. Estimating the Kolmogorov scale &lt;math&gt; D/Re^{3/4} &lt;/math&gt; gives a conservative value and is in the correct order of magnitude (Schanderl &amp; Manhart 2018). <br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38985 UFR 3-35 Best Practice Advice 2020-10-08T18:12:49Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. Although the turbulence is far from an isotropic state. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl &amp; Manhart 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution... <br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38984 UFR 3-35 Best Practice Advice 2020-10-08T18:10:33Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. This all indicates that RANS modeling of this flow is a non-trivial task. The turbulent viscosity might be strongly overestimated by standard two equation models, especially in the wall jet.<br /> <br /> For eddy-resolving turbulence modeling strategies, several observations might be of importance. Although the turbulence is far from an isotropic state. However, around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to the dissipation in the same way as gradients in the vertical direction (Schanderl et al. 2018). This implies that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way. We observed a relatively strong grid dependence of the results, although the SGS contribution... <br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38983 UFR 3-35 Best Practice Advice 2020-10-08T15:08:19Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> This flow case represents a highly complex three-dimensional turbulent flow which shows several characteristics making it difficult to be represented by numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations. There is the dynamics of the horseshoe vortex which tumbles in horizontal direction and produces bimodal velocity distributions near the wall (as shown by many authors). The production of turbulent kinetic energy is far from a standard shear stress production observed in canonical boundary layers. In some important parts of the flow (in the wall jet where the largest wall shear stress values can be found) there is even a negative TKE production due to the strong flow acceleration. The levels of streamwise fluctuations under the horseshoe vortex seem to depend on the level of TKE in the wall jet in which a balance between the pressure transport and the negative normal stress production term is observed. In other regions (around the horseshoe vortex) it seems that the pressure and turbulent diffusive transport terms cancel each other and should be modeled as a whole. <br /> <br /> <br /> <br /> The production term is negative in the wall jet due to the strong local acceleration and the transport by turbulent fluctuations and by pressure dominate the TKE balance in the wall jet under the horseshoe vortex. Realistic levels of turbulent kinetic energy appear to depend strongly on those terms. Under the horseshoe vortex extremely large streamwise fluctuations show up while Reynolds shear stress remain very small. The turbulent viscosity might be strongly overestimated by standard two equation models.<br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38982 UFR 3-35 Best Practice Advice 2020-10-08T14:53:39Z <p>Munich: /* Numerical Modelling Issues */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A fully turbulent open channel flow has to be modeled at the inflow plane which should be placed at least 10 diameters upstream of the cylinder. It is essential to have a realistic turbulence structure and secondary flow at the inflow plane. For eddy resolving simulation strategies (DNS, LES and DES) this implies modeling or generating a fully developed open channel flow which is best achieved by a precursor simulation.<br /> <br /> The downstream extent of the domain largely depends on the region of interest and should be placed far enough from the recirculation downstream of the cylinder to avoid instabilities and upstream-propagated waves. According to the authors' estimation this might vary depending on the specific type of the numerical scheme and outflow boundary condition.<br /> <br /> A high spatial resolution is required to capture the horseshoe vortex dynamics and the wall shear stress. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions.<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> For numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations several challenges appear important. The production term is negative in the wall jet due to the strong local acceleration and the transport by turbulent fluctuations and by pressure dominate the TKE balance in the wall jet under the horseshoe vortex. Realistic levels of turbulent kinetic energy appear to depend strongly on those terms. Under the horseshoe vortex extremely large streamwise fluctuations show up while Reynolds shear stress remain very small. The turbulent viscosity might be strongly overestimated by standard two equation models.<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38981 UFR 3-35 Best Practice Advice 2020-10-08T14:42:30Z <p>Munich: /* Physical Modelling */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A high spatial resolution is required to capture the wall shear stress, the streamwise fluctuations and the horseshoe vortex dynamics. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions. <br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> For numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations several challenges appear important. The production term is negative in the wall jet due to the strong local acceleration and the transport by turbulent fluctuations and by pressure dominate the TKE balance in the wall jet under the horseshoe vortex. Realistic levels of turbulent kinetic energy appear to depend strongly on those terms. Under the horseshoe vortex extremely large streamwise fluctuations show up while Reynolds shear stress remain very small. The turbulent viscosity might be strongly overestimated by standard two equation models.<br /> <br /> '''BEEP: Anm. W.Rodi: Numerical Modelling sub-section concerns mainly discretization and grids. Many Wiki contributions have an extra sub-section &quot;Computational Domain and Boundary conditions&quot;. Here advice should be given how far upstream and downstream of the cylinder the inflow and outflow boundaries should be placed, respectively.'''<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> For numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations several challenges appear important. The production term is negative in the wall jet due to the strong local acceleration and the transport by turbulent fluctuations and by pressure dominate the TKE balance in the wall jet under the horseshoe vortex. Realistic levels of turbulent kinetic energy appear to depend strongly on those terms. Under the horseshoe vortex extremely large streamwise fluctuations show up while Reynolds shear stress remain very small. The turbulent viscosity might be strongly overestimated by standard two equation models.<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Best_Practice_Advice&diff=38980 UFR 3-35 Best Practice Advice 2020-10-08T14:31:17Z <p>Munich: /* Key Physics */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __TOC__<br /> = Cylinder-wall junction flow =<br /> == Underlying Flow Regime 3-35 ==<br /> = Best Practice Advice =<br /> == Key Physics ==<br /> The inflow condition is of major relevance for the vortex system. The horseshoe vortex dynamics are driven by the downflow in front of the cylinder. This downward directed flow is caused by a vertical pressure gradient, which in turn depends on the shape of the approaching inflow profile. Therefore, one has to know the flow profile approaching the cylinder in order to be able to interpret the results and compare it to other studies. In the presented study, we took special care to have a fully developed turbulent open-channel flow approaching the cylinder in both experiment and simulation.<br /> <br /> The wall distance of the horseshoe vortex is about &lt;math&gt; 0.06D &lt;/math&gt;, the thickness of the wall jet between the cylinder and the horseshoe vortex center is smaller than &lt;math&gt; 0.01D &lt;/math&gt;. This means that the key physics take place in a very thin layer in front of the cylinder. A simulation or a measurement not resolving this layer might strongly miss some key features such as the levels of the wall shear stress and the large levels of the streamwise velocity fluctuations and their bimodal character under the horseshoe vortex.<br /> <br /> It has been demonstrated that Reynolds shear stresses are small below the velocity maximum of the wall jet between the cylinder and the horseshoe vortex. This indicates a laminar-like behavior of the velocity profile below the jet's velocity maximum. As a consequence, conventional wall models relying on the logarithmic law of the wall will have difficulties to model the wall shear stress in this region.<br /> <br /> == Numerical Modelling Issues ==<br /> A high spatial resolution is required to capture the wall shear stress, the streamwise fluctuations and the horseshoe vortex dynamics. For the Reynolds number considered, a wall-normal resolution of &lt;math&gt; D/1000&lt;/math&gt; was necessary to obtain converged wall shear stresses. This high resolution was necessary because the wall jet does not follow the law of the wall for turbulent boundary layers and therefore needs to be fully resolved for wall shear stress calculations. A realistic representation of the junction vortex also requires a fine resolution in the horizontal directions. <br /> <br /> From the individual terms contributing to the dissipation of TKE the following observations can be done. The dissipation is far from an isotropic state. Around the horseshoe vortex, the gradients of fluctuations in the main flow direction contribute to dissipation in the same way as gradients in the vertical direction. This means that strongly anisotropic grids are probably not suited to capture the physics in a sufficiently accurate way.<br /> <br /> For numerical methods based on the Reynolds Averaged Navier Stokes (RANS) equations several challenges appear important. The production term is negative in the wall jet due to the strong local acceleration and the transport by turbulent fluctuations and by pressure dominate the TKE balance in the wall jet under the horseshoe vortex. Realistic levels of turbulent kinetic energy appear to depend strongly on those terms. Under the horseshoe vortex extremely large streamwise fluctuations show up while Reynolds shear stress remain very small. The turbulent viscosity might be strongly overestimated by standard two equation models.<br /> <br /> '''BEEP: Anm. W.Rodi: Numerical Modelling sub-section concerns mainly discretization and grids. Many Wiki contributions have an extra sub-section &quot;Computational Domain and Boundary conditions&quot;. Here advice should be given how far upstream and downstream of the cylinder the inflow and outflow boundaries should be placed, respectively.'''<br /> <br /> == Physical Modelling ==<br /> '''BEEP: Am. W.Rodi: Here the modelling in the computations of physical phenomena such as turbulence, combustion, etc was meant and not the modelling of the flow in a laboratory experiment ( see examples in other Wiki contributions). So in your case advice should be given here on turbulence modelling - e.g how necessary it is for your flow to use a scale-resolving method and what SGS model be necessary/suffice. Further any comments on RANS modelling - their applicability and possible problems- would be welcome. For example the already mentioned problems eddy-viscosity models may have in the near-wall jet region, and the apparently large importance of pressure transport which is often neglected in Reynolds Stress RANS models or modelled together with the transport by the turbulent fluctuations. The latter approach seems to be preferable according to your results which indicate that pressure and turbulent transport to a significant extent cancel each other.'''<br /> <br /> == Measurement issues ==<br /> The horseshoe vortex system is a complex three-dimensional flow configuration. Therefore, the two-dimensional data acquisition method is a limitation as the out-of-plane velocity component leads to a corresponding loss of particles. The number of valid samples suffered from this issue in combination of the low seeding density resulting from the large size of the flume. To overcome this issue, we additionally evaluated the PIV images with a &lt;math&gt;32\times32\mathrm{px}&lt;/math&gt; grid. Whenever the instantaneous velocity fields based on a &lt;math&gt;16\times16\mathrm{px}&lt;/math&gt; grid revealed a missing vector, the corresponding vector of the coarser evaluation was taken as a substitute, if possible. In this way, we could improve the number of valid samples and still keep the spatial resolution high.<br /> However, the spatial resolution of the PIV data was too coarse to resolve the velocity gradient correctly. Therefore, a single pixel evaluation is recommended, in order to capture the wall-shear stress correctly.<br /> <br /> == Application Uncertainties ==<br /> When simulating this flow configuration, we adress the largest uncertainties to the inflow conditions of the approach flow and to the representation of the water surface. Both, numerical and experimental approaches, face the challenge in generating a fully developed turbulent boundary layer. Even though we intended to reproduce identical flow conditions and validated both of our methods (PIV and LES) by comparison to results in the literature, we observed differences in our results concerning the size and location of the horseshoe vortex for example (see Fig. 6), which we attribute to the uncertainties in the structure of secondary flows or in modelling the water surface.<br /> <br /> == Recommendations for Future Work ==<br /> Performing a converged Direct Numerical Simulation would end all discussions about models and is - in our opinion - not far out of reach to date (2019). Further, considering surface roughness might give additional insight into the interaction of the wall jet with the wall. <br /> <br /> The experiments could be improved by stereoscopic or tomographic PIV to acquire three dimensional data sets. Furthermore, the temporal resolution could be increased, in order to analyse the time scales of the horseshoe vortex system. The experimental setup can be improved by providing the light sheet from below passing through the transparent bottom plate, while the PIV camera(s) are mounted at the side outside of the flume.<br /> <br /> <br /> &lt;br/&gt;<br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38979 UFR 3-35 Evaluation 2020-10-08T14:12:45Z <p>Munich: /* Pressure coefficient c_{\mathrm{p}}(x) , and of the friction coefficient c_{\mathrm{f}}(x) */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> <br /> <br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38978 UFR 3-35 Evaluation 2020-10-08T14:11:31Z <p>Munich: /* Pressure coefficient c_{\mathrm{p}}(x) , and of the friction coefficient c_{\mathrm{f}}(x) */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> <br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38977 UFR 3-35 Evaluation 2020-10-08T14:01:22Z <p>Munich: /* Diffusive transport of turbulent kinetic energy */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. It is the only positive term in the wall jet between &lt;math&gt; -0.6 &lt; x/D &lt; -0.5 &lt;/math&gt; and therefore essential to understand the TKE balance in the wall jet.<br /> In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38976 UFR 3-35 Evaluation 2020-10-08T13:48:42Z <p>Munich: /* Turbulent kinetic energy */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector are used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38975 UFR 3-35 Evaluation 2020-10-08T13:42:49Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the positions of the horseshoe vortex in the LES and the PIV are slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (see next section). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38974 UFR 3-35 Evaluation 2020-10-08T13:24:36Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (Schanderl etal. 2017). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). In Schanderl etal. (2017) a deeper discussion of the turbulence structure including the turbulent kinetic energy balance terms around the horseshoe vortex system can be found. <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial z &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38973 UFR 3-35 Evaluation 2020-10-08T13:23:10Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The experimental and numerical data agree reasonably both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (Schanderl etal. 2017). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} ~ -1.5 &lt;/math&gt;. Large values of the turbulent kinetic energy can be found around the horseshoe vortex center (due to large &lt;math&gt; \langle w'w'\rangle &lt;/math&gt;) and under the horseshoe vortex near the wall (due to &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;). In Schanderl etal. (2017) a deeper discussion of the turbulence structure including the turbulent kinetic energy balance terms around the horseshoe vortex system can be found. <br /> <br /> There is a specific issue interesting for modeling. In the lower region of the wall jet (between the velocity maximum of the wall jet and the wall) the Reynolds shear stress does not follow the eddy viscosity hypothesis. There is no specific production of turbulent kinetic energy via the term &lt;math&gt; \langle u'w'\rangle \partial \langle u\rangle /\partial x &lt;/math&gt;. This can be partly explained by the low Reynolds number in this layer which leads to a quasi-laminar behavior. Consequently, eddy-viscosity models would have difficulties to reproduce the near wall shear stress distribution. <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38972 UFR 3-35 Evaluation 2020-10-08T12:47:38Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The largest values of &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; are found underneath the horseshoe vortex. This is the foot of the c-shaped distribution of the turbulent kinetic energy. In the wall jet at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt; and &lt;math&gt; -0.5 &lt;/math&gt; the Reynolds stresses are generally small which is explained by the fact that the wall jet is strongly accelerated in upstream direction which leads to a negative production term in the turbulent kinetic energy balance (Schanderl etal. 2017). Large Reynolds shear stress can be found between the horseshoe vortex and the stagnation point S1 at &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;<br /> <br /> <br /> <br /> The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38971 UFR 3-35 Evaluation 2020-10-08T12:33:51Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Reynolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38970 UFR 3-35 Evaluation 2020-10-08T12:24:49Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38969 UFR 3-35 Evaluation 2020-10-08T12:24:20Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical profiles of the streamwise velocity profiles &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38968 UFR 3-35 Evaluation 2020-10-08T12:15:47Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> <br /> <br /> The vertical velcoity profiles &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38967 UFR 3-35 Evaluation 2020-10-08T12:08:43Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|200px|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> The vertical velcoity profiles &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38966 UFR 3-35 Evaluation 2020-10-08T12:05:58Z <p>Munich: /* Horizontal and vertical profiles of the velocity components and Reynolds stresses */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> [[File:UFR3-35_U_z.png|200px|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> The vertical velcoity profiles &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38965 UFR 3-35 Evaluation 2020-10-08T11:55:11Z <p>Munich: /* Location of the characteristic flow structures */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the critical points ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> The vertical velcoity profiles &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38964 UFR 3-35 Evaluation 2020-10-08T08:26:53Z <p>Munich: /* Streamlines */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]].<br /> <br /> == Location of the characteristic flow structures ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> The vertical velcoity profiles &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Evaluation&diff=38963 UFR 3-35 Evaluation 2020-10-08T08:25:51Z <p>Munich: /* Streamlines */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> = Evaluation =<br /> For the evaluation of the numerical and experimental data sets in the symmetry plane upstream of a wall-mounted cylinder the following quantities are presented:<br /> {|<br /> * the streamlines<br /> * the location of the characteristic flow structures<br /> * selected vertical and horizontal profiles of the mean velocity components &lt;math&gt; \langle u\rangle &lt;/math&gt; and &lt;math&gt; \langle w\rangle &lt;/math&gt; as well as the Reynolds stresses &lt;math&gt; \langle u'_iu'_j\rangle &lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt;<br /> * the contour plots of the turbulent kinetic energy and its buget terms such as production, diffusive transport, dissipation, and mean convection<br /> * horizontal profiles of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> == Streamlines == <br /> The streamlines of the PIV and the LES data agree well according to Fig. 6. The plots are superimposed by the normalized magnitude of the velocity field in the symmetry plane, thus &lt;math&gt; ||\vec{U}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt;. The dashed and dash-dotted lines indicate the zero-isoline of the streamwise and vertical velocity component, respectively.<br /> <br /> The approaching turbulent boundary layer is redirected downwards at the flow facing edge of the cylinder caused by a vertical pressure gradient. This downflow reaches the bottom plate of the flume at the stagnation point S3. Here, it is redirected (i) in the out-of-plane direction bending around the cylinder; (ii) towards the cylinder rolling up and forming the corner vortex V3; and (iii) in the upstream direction accelerating and forming a wall-parallel jet. The jet accelerates and exerts a large wall-shear stress on the bottom plate (see Fig. 22). Parts of the downflow form the horseshoe vortex V1. Upstream of this vortex system, the approaching flow is blocked and causes a saddle point S1 with zero velocity magnitude.<br /> <br /> [[File:UFR3-35_PIV_streamlines_mag.png|centre|frame|Fig. 6 a) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{PIV}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]]. The dashed and dash-dotted lines represent locations at which the streamwise and vertical velocity components are zero, repsectively.<br /> [[File:UFR3-35_LES_streamlines_mag.png|centre|frame|Fig. 6 b) Streamlines of time-averaged flow field superimposed by the in-plane velocity magnitude &lt;math&gt; ||\vec{U}_{\mathrm{LES}}|| = \sqrt{\langle u^2\rangle + \langle w^2\rangle}/u_{\mathrm{b}}&lt;/math&gt; ]]. The dashed and dash-dotted lines represent locations at which the streamwise and vertical velocity components are zero, repsectively.<br /> <br /> == Location of the characteristic flow structures ==<br /> The position of the characteristic flow structures highlighted in the streamline plots is listed in the following table<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 5: Position of flow structures<br /> ! <br /> ! PIV<br /> !<br /> ! LES<br /> !<br /> |-<br /> |<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |&lt;math&gt; x/D&lt;/math&gt;<br /> |&lt;math&gt; z/D &lt;/math&gt;<br /> |-<br /> | S1<br /> | &lt;math&gt; -0.788&lt;/math&gt; <br /> | &lt;math&gt; 0.03 &lt;/math&gt; <br /> | &lt;math&gt; -0.843&lt;/math&gt; <br /> | &lt;math&gt; 0.037 &lt;/math&gt; <br /> |-<br /> | S2<br /> | &lt;math&gt; -0.918&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -1.1&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S3<br /> | &lt;math&gt; -0.533&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> | &lt;math&gt; -0.534&lt;/math&gt; <br /> | &lt;math&gt; 0 &lt;/math&gt; <br /> |-<br /> | S4<br /> | &lt;math&gt; -0.507&lt;/math&gt; <br /> | &lt;math&gt; 0.036 &lt;/math&gt; <br /> | &lt;math&gt; -0.50&lt;/math&gt; <br /> | &lt;math&gt; 0.04 &lt;/math&gt; <br /> |-<br /> | V1<br /> | &lt;math&gt; -0.697&lt;/math&gt; <br /> | &lt;math&gt; 0.051 &lt;/math&gt; <br /> | &lt;math&gt; -0.735&lt;/math&gt; <br /> | &lt;math&gt; 0.06 &lt;/math&gt; <br /> |-<br /> | V3<br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.017 &lt;/math&gt; <br /> | &lt;math&gt; -0.513&lt;/math&gt; <br /> | &lt;math&gt; 0.02 &lt;/math&gt; <br /> |}<br /> <br /> == Horizontal and vertical profiles of the velocity components and Reynolds stresses ==<br /> The vertical profiles are extracted at the following positions with respect to the time-averaged position of the horseshoe vortex (see Fig. 7).<br /> [[File:UFR3-35_position_of_profiles.png|thumb|centre|550px|Fig. 7: Position of vertical profiles in the flow]]<br /> <br /> Since the flow structure of the LES and the PIV is slightly different, we use an adjusted &lt;math&gt; x-&lt;/math&gt;coordinate, in order to compare the data at the same position in the flow. The coordinate is defined as follows (Schanderl 2018):<br /> <br /> &lt;math&gt;x_{\mathrm{adj}} = \frac{x - x_{\mathrm{Cyl}}}{x_{\mathrm{Cyl}} - x_{\mathrm{V1}}}&lt;/math&gt;,<br /> <br /> with &lt;math&gt; x_{\mathrm{Cyl}} = -0.5D &lt;/math&gt;, such that &lt;math&gt;x_{\mathrm{adj}} = -1.0 &lt;/math&gt; represents the time-averaged location of the horseshoe vortex centre &lt;math&gt; x_{\mathrm{V1}}&lt;/math&gt;.<br /> <br /> <br /> [[File:UFR3-35_U_z.png|centre|frame|Fig. 8: Vertical profiles of the streamwise velocity component &lt;math&gt; \langle u(z)\rangle /u_{\mathrm{b}}&lt;/math&gt; ]]<br /> <br /> The vertical velcoity profiles &lt;math&gt; u(z) &lt;/math&gt; are presented at the selected locations in Fig. 8. The wall-parallel jet starts to develop from S3 on and the flow accelerates. At &lt;math&gt;x_{\mathrm{adj}} = -0.25 &lt;/math&gt;, a near-wall velocity peak appears, which becomes more pronounced in the upstream direction (see &lt;math&gt;x_{\mathrm{adj}} = -0.5 &lt;/math&gt;). In this region, the wall-shear stress reaches on the one hand its maximum value and on the other hand reveals a plateau-like shape (this is shown and discussed later, see Fig. 22). This means that the largest values of the gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; appear. Underneath the horseshoe vortex, the flow decelerates and the near-wall-peak of the velocity lifts from the bottom plate and becomes less distinct. Further upstream, the near-wall peak of the streamwise velocity disappears as the wall-parallel jet fades out.<br /> Due to the evaluation of the PIV images by an interrogation-window-based cross correlation, the strong gradient &lt;math&gt;\frac{\partial \langle u \rangle}{\partial z}&lt;/math&gt; at the wall cannot be fully resolved, and therefore, the near-wall peak of the streamwise velocity is damped in the experimental data.<br /> <br /> [[File:UFR3-35_uu_z.png|centre|frame|Fig. 9 a) Vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_uw_z.png|centre|frame|Fig. 9 b) Vertical profiles of the Reynolds shear stresses &lt;math&gt; \langle u'w'(z)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; ]] <br /> [[File:UFR3-35_k_z.png|centre|frame|Fig. 9 c) Vertical profiles of the inplane turbulent kinetic energy &lt;math&gt; \langle k(z)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]] <br /> <br /> The vertical profiles of the Reynolds normal stresses &lt;math&gt; \langle u'u'\rangle &lt;/math&gt;, the Renyolds shear stresses &lt;math&gt; \langle u'w'\rangle &lt;/math&gt; and the resulting turbulent kinetic energy &lt;math&gt; \langle k\rangle &lt;/math&gt; comprising only streamwise and vertical fluctuating velocity components (inplane with respect to the symmetry plane) are presented in Fig. 9a) to c), respectively. Note, that the range of the &lt;math&gt;x-&lt;/math&gt;axis is not constant for the sake of better visibility. The accelerating jet is again indicated by a near wall peak of the &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; stress, while the horseshoe vortex leaves its footprint in the stresses &lt;math&gt;\langle u'u'\rangle&lt;/math&gt; and &lt;math&gt;\langle w'w'\rangle&lt;/math&gt; as (local) peak at &lt;math&gt;z_{\mathrm{V1}}/D&lt;/math&gt;. <br /> The shear stress distribution &lt;math&gt;\langle u'w'\rangle&lt;/math&gt; inside the wall-parallel jet is negative (in average) according to the average flow direction. The experimental and numerical data agree with each other both in amplitude as well as in shape. Again, the quality of the PIV data near the wall is undermined by the strong gradients being evaluated using interrogation windows.<br /> <br /> ('''BEEP: Anm. W.Rodi: in the attachment I made a comment on the shear-stress distribution in the wall-jet region. Readers familiar with 2D wall jets may wonder why the shear stress does not change sign around the position of the velocity peak, as it does in such wall jets . Is this a consequence of the 3D nature of the flow? In any case, eddy-viscosity models would predict such a change (because the vertical velocity gradient changessign) and would therefore have difficulties to reproduce the shear stress distribution observed in your study.''')<br /> <br /> <br /> [[File:UFR3-35_W_x_V1.png|centre|frame|Fig. 10: Horizontal profiles of the vertical velocity component &lt;math&gt; \langle w(x)\rangle/u_{\mathrm{b}}&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> Analysing the profile of the vertical velocity component &lt;math&gt;\langle w(x)\rangle &lt;/math&gt; along the &lt;math&gt; x-&lt;/math&gt;axis at the height of the horseshoe vortex (see Fig. 10), reveals on the one hand the clockwise rotation of the vortex, and on the other hand two minima between the horseshoe vortex and the cylinder. At &lt;math&gt;x_{\mathrm{adj}} \approx -0.1 &lt;/math&gt;, the peak in the downwards directed flow stems from the downflow, while the second local minimum at approximately &lt;math&gt;x_{\mathrm{adj}} = -0.65 &lt;/math&gt; represents the downwards rotation of the horseshoe vortex.<br /> Both data sets agree well in shape. However, the LES data indicate higher amplitudes in general, as the stagnation point at the cylinder front at which the approach flow is deflected and evolving the downflow, is located at the water level in the LES. Whereas it was shifted downwards in our experiment, due to the slat of acrylic glass at the water surface. This is, however, not an artefact as a bow wave (surface roller) would evolve in this area if this slat was not present according to e.g. Melville 2008. Therefore, the downwards deflected part of the flow is smaller in the experiment and therefore, the HV system becomes smaller, too.<br /> <br /> [[File:UFR3-35_uiuj_x_V1.png|centre|frame|Fig. 11: Horizontal profiles of the Reynolds stresses &lt;math&gt; \langle u_i'u_j'(x)\rangle/u_{\mathrm{b}}^2&lt;/math&gt; and the turbulent kinetic energy &lt;math&gt; \langle k(x)\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; at the height &lt;math&gt; z_{\mathrm{V1}}/D&lt;/math&gt;]]<br /> <br /> The Reynolds stresses &lt;math&gt;\langle u_i'u_j'\rangle&lt;/math&gt; and the inplane turbulent kinetic energy &lt;math&gt;\langle k\rangle&lt;/math&gt; are shown in Fig. 11 at the height of the horseshoe vortex and reveal similar distributions for the PIV and LES results. At the centre of the vortex, the normal stresses, and consequently the turbulent kinetic energy as well, reveal a peak, which wears off in the up- and downstream direction. In addition, the downflow close to the cylinder surface generates stresses as well.<br /> The shear stress plays a minor role in the region between the cylinder and the horseshoe vortex, while it becomes negative upstream of the horseshoe vortex indicating the interference of the approaching flow with the horseshoe vortex.<br /> <br /> == Distribution of turbulent kinetic energy and its budgets terms: production, diffusive transport, dissipation and convection ==<br /> === Turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_TKE_inplane.png|centre|frame|Fig. 12 a) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_inplane.png|centre|frame|Fig. 12 b) Inplane turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> [[File:UFR3-35_LES_TKE_total.png|centre|frame|Fig. 12 c) Turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;]]<br /> <br /> The spatial distribution of the time-averaged in-plane turbulent kinetic energy &lt;math&gt; \langle k \rangle = 0.5(\langle u'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt; reveals on the one hand the well-known c-shaped structure (e.g. Paik 2007) with the largest amplitude at the centre of the horseshoe vortex stemming from vertical fluctuations induced by the horizontal oscillations of the horseshoe vortex (see Fig. 12 a) and b). The lower branch of the c-shape contains mainly streamwise (horizontal) fluctuations linked to the dynamics of the wall-parallel jet. On the other hand, the experimental and numerical data sets show high similarity and agree well with each other, as the peak amplitude is approximately the same: &lt;math&gt; \langle k_{\mathrm{PIV, inplane}}\rangle = 0.074u_{\mathrm{b}}^2&lt;/math&gt;; &lt;math&gt; \langle k_{\mathrm{LES, inplane}}\rangle = 0.079u_{\mathrm{b}}^2&lt;/math&gt;. The black circle marks the centre of the horseshoe vortex.<br /> <br /> When the fluctutuations of the three dimensional velocity vector is used to determine the turbulent kinetic energy &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.5(\langle u'^2\rangle+\langle v'^2\rangle+\langle w'^2\rangle)/u_{\mathrm{b}}^2&lt;/math&gt;, the additional out-of-plane component does not change the shape of the distribution (see Fig. 12 c). However, the amplitudes at the horseshoe vortex centre and inside the lower branch of the c-shape both increase due to the vortex bending around the cylinder entailing increased out-of-plane fluctuations, e.g. at V1: &lt;math&gt; \langle k_{\mathrm{LES, total}}\rangle = 0.09u_{\mathrm{b}}^2&lt;/math&gt;.<br /> <br /> The budget equation of the turbulent kinetic energy reads as:<br /> <br /> &lt;math&gt; 0 = P + \nabla T - \epsilon + C &lt;/math&gt;, <br /> <br /> and is the balancing sum of the production &lt;math&gt; P &lt;/math&gt;, the diffusive transport term &lt;math&gt; \nabla T &lt;/math&gt;, the dissipation &lt;math&gt; \epsilon &lt;/math&gt;, and the mean convection &lt;math&gt; C &lt;/math&gt;. In case of the LES data, these terms were calculated using the entire three dimensional velocity vector and the corresponding fluctuations. However, the influcence of the out-of-plane component &lt;math&gt; v &lt;/math&gt; is small due to symmetry (Schanderl et al. 2017). The individual terms are determined as follows:<br /> <br /> * production: &lt;math&gt;P = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j}&lt;/math&gt;<br /> * diffusive transport: &lt;math&gt;T = \underbrace{-\frac{1}{2}\langle u_i'u_j'u_j' \rangle}_{\text{turbulent fluctuations}} \underbrace{-\frac{1}{\rho}\langle u_i'p' \rangle}_{\text{pressure transport}} \underbrace{+2\nu\langle u_j's_{ij}\rangle}_{\text{viscous diffusion}}&lt;/math&gt;<br /> * dissipation: &lt;math&gt;\epsilon = 2\nu\langle s_{ij}s_{ij}\rangle &lt;/math&gt;, and &lt;math&gt; s_{ij} = \frac{1}{2}\left(\frac{\partial u_i'}{\partial x_j} + \frac{\partial u_j'}{\partial x_i}\right)&lt;/math&gt; as the fluctuating rate-of-strain tensor<br /> ** in the LES, the dissipation consists of a ''resolved'' and a ''subgrid scale (SGS)'' part:<br /> ** &lt;math&gt;\epsilon_{\mathrm{total}} = \epsilon_{\mathrm{res}} + \epsilon_{\mathrm{SGS}} = 2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle &lt;/math&gt;<br /> * mean convection: &lt;math&gt; C = - \langle u_i\rangle \frac{\partial k}{\partial x_i} &lt;/math&gt; (steady state)<br /> <br /> The individual terms of the budget equation of the TKE are normalized by &lt;math&gt; D/u_{\mathrm{b}}^3&lt;/math&gt; and presented in the following part.<br /> <br /> === Production of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_P.png|centre|frame|Fig. 13 a) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{PIV}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_P.png|centre|frame|Fig. 13 b) Production of turbulent kinetic energy &lt;math&gt; P_{\mathrm{LES}} = -\langle u_i'u_j'\rangle \frac{\partial \langle u_i \rangle}{\partial x_j} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Fig. 13 a) and b) show the TKE production using the PIV and LES results, respecively. The main region of positive production of turbulent kinetic energy can be found upstream of the centre of the hoseshoe vortex (black circle) and inside the wall-parallel jet. The amplitude of both data sets is about &lt;math&gt; 0.3u_{\mathrm{b}}^3/D &lt;/math&gt; in the region of the horseshoe vortex, while the LES data resolve the TKE production in the jet in more detail than the PIV data. The amplitudes here are &lt;math&gt; P_{\mathrm{LES}} \approx 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; and &lt;math&gt; P_{\mathrm{PIV}} \approx 0.2u_{\mathrm{b}}^3/D &lt;/math&gt;. The oscillations of the horseshoe vortex are responsible for causing fluctuations, which in turn produce turbulent kinetic energy.<br /> <br /> From the stagnation point S3 onwards in the upstream direction, the TKE production is negative, meaning that inside the accelerating jet TKE is transferred into mean kinetic energy. When the jet decelerates at about &lt;math&gt; x = -0.7D&lt;/math&gt;, the jet becomes more unstable, fluctuations occur, and therefore, &lt;math&gt; P &lt;/math&gt; becomes positive.<br /> <br /> <br /> === Diffusive transport of turbulent kinetic energy ===<br /> The diffusive transport of TKE is presented in Fig. 14 and can be split up into three individual terms: the transport due to turbulent fluctuations, due to pressure fluctuations, and due to viscous diffusion. Since the pressure in general is not contained in the PIV data, the velocity-pressure correlations, which are required to obtain the transport due to pressure, cannot be calculated, as well. Therefore, we present the three terms individually if available.<br /> <br /> [[File:UFR3-35_PIV_T_turb.png|centre|frame|Fig. 14 a) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, PIV}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_turb.png|centre|frame|Fig. 14 b) Diffusive transport of turbulent kinetic energy due to turbulent fluctuations &lt;math&gt; \nabla T_{\mathrm{turb, LES}} = -\frac{1}{2}\frac{\partial \langle u_i'u_j'u_j' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The distribution of the turbulent transport shows a similar structure as the one of the production. In the region of large positive TKE production, we observe a large negative transport. In particular close to the wall at &lt;math&gt; x = -0.75D&lt;/math&gt;, the large production of &lt;math&gt; 0.4u_{\mathrm{b}}^3/D &lt;/math&gt; is nearly balanced by the turbulent transport &lt;math&gt; T_{\mathrm{turb, LES}} \approx 0.35u_{\mathrm{b}}^3/D &lt;/math&gt;. <br /> <br /> Positive transport indicates that TKE is transported towards these regions. They can be found above the centre of the horseshoe vortex. Accroding to Apsilidis et al. (2015), vertical eruptions from the wall occur here, which coincides with our observations.<br /> <br /> <br /> [[File:UFR3-35_LES_T_press.png|centre|frame|Fig. 15 Diffusive transport of turbulent kinetic energy due to pressure fluctuations &lt;math&gt; \nabla T_{\mathrm{press, LES}} = -\frac{1}{\rho}\frac{\partial \langle u_i'p' \rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_T_visc.png|centre|frame|Fig. 16 Diffusive transport of turbulent kinetic energy due to viscous diffusion &lt;math&gt; \nabla T_{\mathrm{visc, LES}} = 2\nu\frac{\partial \langle u_j's_{ij}\rangle}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The pressure fluctuations (Fig. 15) play an important role in the diffusive TKE-transport. In some regions, the transport terms &lt;math&gt; \nabla T_{\mathrm{turb}}&lt;/math&gt; and &lt;math&gt; \nabla T_{\mathrm{press}}&lt;/math&gt; cancel each other while the horseshoe vortex oscillates in the horizontal direction according to the alternation of the back-flow and zero-flow mode. While the horseshoe vortex moves towards the cylinder to a region with a downwards directed flow in average (&lt;math&gt;\langle w \rangle &lt;0 &lt;/math&gt;), the vortex core with an instantaneous zero &lt;math&gt; w-&lt;/math&gt;velocity induces that the corresponding fluctuation &lt;math&gt; w'&lt;/math&gt; becomes positive. Therefore, the turbulent transport becomes negative in total, and the transport due to pressure fluctuations become positve as &lt;math&gt; p'&lt;0&lt;/math&gt; in the vortex centre (low pressure), resulting in a mutual balance.<br /> <br /> The transport of TKE due to viscous diffusion shows an increased negative transport only close to the wall (see (Fig. 16), which is intuitive as viscous effects increase towards the wall in turbulent flows. In the remaining part, however, the amplitude of &lt;math&gt; \nabla T_{\mathrm{visc}}&lt;/math&gt; is below &lt;math&gt; |0.05|u_{\mathrm{b}}^3/D&lt;/math&gt;, and therefore, the contribution of this term to the TKE transport around the horseshoe vortex is of minor importance.<br /> <br /> === Dissipation of turbulent kinetic energy ===<br /> The dissipation is the sink term in the budget of the TKE transport equation, as the TKE was produced and transported by &lt;math&gt;P&lt;/math&gt; and &lt;math&gt;\nabla T&lt;/math&gt;, respectively, and is dissipated into heat by &lt;math&gt; \epsilon&lt;/math&gt;. The dissipation is positive by definition, and therefore, it appears with a negative sign in the budget equation. In the LES data, the modelled dissipation due to subgird stresses is included and was quantified to be approximately one third of the total dissipation rate. Fig. 17 shows the dissipation rate calculated from PIV and LES data.<br /> <br /> [[File:UFR3-35_PIV_Epsilon.png|centre|frame|Fig. 17 a) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{PIV}} = 2\nu\langle s_{ij}s_{ij}\rangle \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> [[File:UFR3-35_LES_Epsilon.png|centre|frame|Fig. 17 b) Dissipation of turbulent kinetic energy &lt;math&gt; \epsilon_{\mathrm{LES, total}} = (2\nu\langle s_{ij}s_{ij}\rangle + 2\langle \nu_{\mathrm{t}} s_{ij}s_{ij}\rangle)\cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;, i.e. SGS-contribution included]]<br /> <br /> The distribution of the dissipation shows a similar c-shaped structure as the TKE and the production term &lt;math&gt; P&lt;/math&gt; due to the horizontal oscillations of the vortex. Three regions with a high dissipation rate can be located: (i) around the centre of the horseshoe vortex; (ii) underneath the horseshoe vortex inside the wall-parallel jet; and (iii) at the corner vortex V3. The amplitude of the measured dissipation rate exceeds the one stemming from the LES data due to measurement noise undermining the quality of the instantaneous fluctuating velocity gradients. The spatial resolution of the PIV was anyway too coarse to estimate the dissipation correctly such that we did not apply any correction to the experimental data. However, the experimental data serve as a qualitative comparison rather than a quantitative one as the spatial distribution in PIV is similar to the one in the LES. The dissipation rate and the production of the TKE do not cancel each other out as the maximum amplitude in the centre of the horseshoe vortex is &lt;math&gt; \epsilon_{\mathrm{LES}}=0.066u_{\mathrm{b}}^3/D&lt;/math&gt;, which is approximately one third of the production here. Furthermore, the maximum value of the production is located upstream of the horseshoe vortex indicating the role of the transport of small-scale structures between the location of &lt;math&gt; P_{\mathrm{max}}&lt;/math&gt; towards &lt;math&gt; \epsilon_{\mathrm{max}}&lt;/math&gt;.<br /> <br /> === Mean convection of turbulent kinetic energy ===<br /> [[File:UFR3-35_PIV_C.png|centre|frame|Fig. 18 a) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{PIV}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> [[File:UFR3-35_LES_C.png|centre|frame|Fig. 18 b) Mean convection of turbulent kinetic energy &lt;math&gt; C_{\mathrm{LES}} = - \langle u_i\rangle \frac{\partial k}{\partial x_i} \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The approaching flow separates from the bottom wall of the flume, and consequently, becomes unstable with increasing fluctuations. Therefore, the TKE increases along the streamlines, which is indicated by the negative mean convection upstream of the horseshoe vortex (see Fig. 18). The same applies for the wall-parallel jet. After the deflection at S3, the jet accelerates, which decreases TKE in the first place. When the jet starts to decelerate (&lt;math&gt; x \approx -0.63D&lt;/math&gt;) the flow becomes more unstable and the TKE increases. Underneath the horsehsoe vortex, &lt;math&gt; C&lt;/math&gt; changes sign again and the TKE decreases further upstream.<br /> <br /> === Budget of turbulent kinetic energy ===<br /> Finally, the sum of the above mentioned terms is presented in Fig. 19 as the total budget of the TKE. Since the PIV data cannot provide information concerning the pressure, the corresponding residual will not cancel out. In addition, the amplitude of the dissipation rate is uncertain in the experimental data. <br /> <br /> [[File:UFR3-35_PIV_Budget.png|centre|frame|Fig. 19 a) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{PIV}} = P + \nabla T_{\mathrm{turb}} - \epsilon + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> Nevertheless, when analysing the distribution and amplitude of the residual of the TKE budget obtained from PIV, we observe a similar structure to the one of &lt;math&gt; -\nabla T_{\mathrm{press,LES}}&lt;/math&gt;. Therefore, the missing piece in the experimental data of this flow configuration is the contribution of the pressure fluctuations to the transport mechanisms of the TKE (Jenssen 2019).<br /> <br /> [[File:UFR3-35_LES_Budget.png|centre|frame|Fig. 19 b) Residual of turbulent kinetic energy budget &lt;math&gt; R_{\mathrm{LES}} = P + \nabla T - \epsilon_{\mathrm{total}} + C \cdot D/u_{\mathrm{b}}^3 &lt;/math&gt;]]<br /> <br /> The residual of the LES data is small in wide regions &lt;math&gt; &lt;|0.01|u_{\mathrm{b}}^3/D&lt;/math&gt;. In particular around the horseshoe vortex, the residual is close to zero. However, along the cylinder surface and the bottom wall the budget does not fully balance. On the one hand, we assign the large errors to the spatial resolution of the grid in the horizontal direction, which was obviously too coarse to fully resolve the developing boundary layer at the cylinder surface. On the other hand, the sensitivity with respect to the number of samples of the term &lt;math&gt; T_{\mathrm{turb}} = -\frac{1}{2}\langle u_i'u_j'u_j' \rangle &lt;/math&gt; containing triple correlations of the velocity can be responsible <br /> the remaining error in the TKE budget.<br /> <br /> == Pressure coefficient &lt;math&gt; c_{\mathrm{p}}(x) &lt;/math&gt;, and of the friction coefficient &lt;math&gt; c_{\mathrm{f}}(x) &lt;/math&gt; ==<br /> <br /> The pressure coefficient &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt; is computed as:<br /> <br /> &lt;math&gt; c_{\mathrm{p}} = \frac{\langle p \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;,<br /> <br /> while the friction coefficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is determined as:<br /> <br /> &lt;math&gt; c_{\mathrm{f}} = \frac{\langle \tau_{\mathrm{w}} \rangle}{\frac{\rho}{2} u_{\mathrm{b}}^2} &lt;/math&gt;.<br /> <br /> The following plots were taken from the LES data only since the pressure is not contained in the PIV data and the spatial resolution in the experiment appeared to be too coarse to calculate the wall-shear stress, thus the friction coefficient, correctly. A more detailed discussion of this artefact can be found in Schanderl et al. (2017). The first data point in the experimental data could be obtained at &lt;math&gt; z_1 \approx 0.0036D \approx 10 \mathrm{px}&lt;/math&gt;. In the LES, the first grid point was at &lt;math&gt; z_1 \approx 0.0005D &lt;/math&gt;, which is about a factor of 7 finer than the experimental results.<br /> <br /> A streamline plot in light grey is given for the sake of better orientation in a qualitative sense in Fig. 21 and 22 showing the positions in the &lt;math&gt; z-&lt;/math&gt;direction at which the profiles were extracted. The red symbols refer to the profiles near the wall indicated by the red solid line, whereas the blue symbols and the blue solid line refer to the profile at the height of the horseshoe vortex.<br /> <br /> [[File:UFR3-35_LES_Pressure.png|centre|frame|Fig. 20: Spatial distribution of the pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]]<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cp.png|centre|frame|Fig. 21: Pressure coefficient &lt;math&gt; c_{\mathrm{p}}&lt;/math&gt; ]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_cf.png|centre|frame|Fig. 22: Friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The time-averaged pressure coefficient increases in the horizontal direction when the flow approaches the cylinder. The horseshoe vortex is indicated by a local minimum (see Fig. 20). In the vertical direction towards the bottom wall of the flume, the pressure increases following the down-flow and has its local maximum at the stagnation point S3. At this point, the downflow is deflected in all directions forming an accelerating wall-parallel jet especially in the uptream direction. The acceleration is indicated by the decrease of the pressure coefficient along the local flow direction, which points in the upstream direction. At &lt;math&gt; x_{\mathrm{adj}}= -1.0 &lt;/math&gt;, underneath the horseshoe vortex, the pressure coefficient shows a kink and decreases less significantly than before (see Fig. 21).<br /> <br /> The accelerating wall-parallel jet is also perceptible in the distribution of the friction ceofficient &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; (see Fig. 22), which was extracted at the first grid point. From the stagnation point S3 on, with zero wall-shear stress, the friction coefficient reveals a sharp peak underneath the foot vortex V3. In the upstream direction, the wall-shear stress increases due to the accelerating jet until a plateau is reached. Here, the friction coefficient is almost constant at &lt;math&gt; |c_{\mathrm{f}}| = 0.01 &lt;/math&gt;, which represents the maximum absolute value. It should be noted that the maximum is not located underneahth the horseshoe vortex. The distribution of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is dominated by the wall-parallel jet rather than by the horseshoe vortex. Further upstream, the value of &lt;math&gt; c_{\mathrm{f}} &lt;/math&gt; is small but negative, indicating the fading near-wall jet pointing still in the upstream direction. Unlike Apsilidis et al. (2015), we could not observe a second vortex here.<br /> <br /> = Datasets for download = <br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_data.zip]]<br /> * LES: friction coefficient &lt;math&gt; c_{\mathrm{f}}&lt;/math&gt; [[Media:UFR3-35_C_cf.txt]]<br /> * How to read in the data as an example (MatLab): [[Media:UFR3_35_read_data.m]]<br /> |}<br /> <br /> The datasets are structured as follows:<br /> <br /> {|<br /> * 2D plots of the PIV have &lt;math&gt; 50 \times 171 (n \times m) &lt;/math&gt; data points<br /> * 2D plots of the LES have &lt;math&gt; 143 \times 131 (n \times m) &lt;/math&gt; data points<br /> * The .txt files are reshaped such as each column has &lt;math&gt; n \cdot m &lt;/math&gt; entries<br /> * The first 11 lines of the .txt files belong to the header, which are indicated by the #-symbol<br /> * Each column corresponds to the following data and is comma separated<br /> |}<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 6: Structure of the datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> | 8<br /> | 9<br /> | 10<br /> | 11<br /> | 12<br /> | 13<br /> | 14<br /> | 15<br /> | 16<br /> | 17<br /> | 18<br /> | 19<br /> |-<br /> | '''PIV'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> | &lt;math&gt; \epsilon\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; - &lt;/math&gt;<br /> |-<br /> | '''LES'''<br /> | &lt;math&gt; x_{\mathrm{adj}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{x}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'v'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle v'w'\rangle}{u_{\mathrm{b}}^2} &lt;/math&gt;<br /> | &lt;math&gt; P\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; C\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{turb}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{press}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \nabla T_{\mathrm{visc}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; \epsilon_{\mathrm{total}}\frac{D}{u_{\mathrm{b}}^3} &lt;/math&gt;<br /> | &lt;math&gt; c_{\mathrm{p}} &lt;/math&gt;<br /> |}<br /> <br /> <br /> <br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_References&diff=38962 UFR 3-35 References 2020-10-08T08:20:27Z <p>Munich: /* References */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __NOTOC__<br /> =Cylinder-wall junction flow=<br /> == Underlying Flow Regime 3-35 ==<br /> = References =<br /> {|<br /> * Apsilidis, N., Diplas, P., Dancey, C. L., and Bouratsis, P. (2015). Time-resolved flow dynamics and Reynolds number effects at a wall-cylinder junction. ''Journal of Fluid Mechanics'' 776, 475-511.<br /> * Baghbadorani, D. A., Beheshti, A. &amp; Ataie-Ashtiani, B. (2017) Scour hole depth prediction around pile groups: review, comparison of existing methods, and proposition of a new approach. ''Natural Hazards'' 88(2), 977-1001.<br /> * Baker, C. J. (1980) The turbulent horseshoe vortex. ''Journal of wind engineering and industrial aerodynamics'' 6, 9-23.<br /> * Bruns, J., Dengel, P. &amp; Fernholz, H. H. (1992). Mean flow and turbulence measurements in an incompressible two-dimensional turbulent boundary layer. Part I: data. Tech. Rep., Herman-Föttinger-Institut für Thermo- und Fluiddynamik, TU Berlin.<br /> * Clauser, F.H. (1954). Turbulent boundary layer in adverse pressure gradients. ''J. Aero. Sci.'' 21:91–108.<br /> * Dargahi, B. (1989). The turbulent flow field around a circular cylinder. ''Experiments in Fluids'' 8(1-2):1-12.<br /> * Devenport, W. J. and Simpson, R. L. (1990). Timedependent and time-averaged turbulence structure near the nose of a wing-body junction. ''Journal of Fluid Mechanics'' 210:23-55.<br /> * Escauriaza, C. and Sotiropoulos, F. (2011). Reynolds Number Effects on the Coherent Dynamics of the Turbulent Horseshoe Vortex System. ''Flow, Turbulence and Combustion'' 86(2):231-262. <br /> * Ettema, R., Kirkil, G. &amp; Muste, M. (2006). Similitude of Large-Scale Turbulence in Experiments on Local Scour at Cylinders. ''Journal of Hydraulic Engineering'' 132(1),33-40.<br /> * Jenssen, U. (2019). Experimental Study of the Flow Around a Scouring Bridge Pier. PhD thesis, Technical University of Munich, Germany.<br /> * Kirkil, G. and Constantinescu, G. (2015). Effects of cylinder Reynolds number on the turbulent horseshoe vortex system and near wake of a surface-mounted circular cylinder. ''Physics of Fluids'' 27(7), 075102.<br /> * Laursen, E. M. &amp; Toch, A. (1956). Scour around bridge piers and abutements. Tech. Rep. ''Iowa Institute of Hydraulic Research''.<br /> * Link, O., Pfleger, F. &amp; Zanke, U. (2008). Characteristics of developping scour-holes at sand-embedded cylinder. ''International Journal of Sediment Research'' 23, 258-266.<br /> * Manhart, M. (2004). A zonal grid algorithm for DNS of turbulent boundary layers. ''Computers and Fluids'' 33(3):435–461.<br /> * Martinuzzi, R. &amp; Tropea, C. (1993). The Flow Around Surface-Mounted, Prismatic Obstacles Placed in a Fully Developed Channel Flow. ''Journal of Fluids Engineering'' 115(1),85-92.<br /> * Melville, B. W. (2008). The physics of local scour at bridge piers. ''Fourth International Conference on Scour and Erosion (ICSE-4)'', Tokyo, Japan <br /> * Melville, B. W. &amp; Raudkivi, A. J. (1977). Flow characteristics in local scour at bridge piers. ''Journal of Hydraulic Research'' 15(4), 373-380.<br /> * Nicoud, F. &amp; Ducros, F. (1999). Subgrid-scale stress modelling based on the square of the velocity gradient tensor. ''Flow, Turbulence and Combustion'' 62(3):183–200.<br /> * Paik, J., Escauriaza, C., and Sotiropoulos, F. (2007). On the bimodal dynamics of the turbulent horseshoe vortex system in a wing-body junction. ''Physics of Fluids'' (19):045107.<br /> * Peller, N. (2010). Numerische Simulation turbulenter Strömungen mit Immersed Boundaries. PhD thesis, Technische Universität München, München.<br /> * Peller, N., Duc, A. L., Tremblay, F. &amp; Manhart, M. (2006). High-order stable interpolations for immersed boundary methods. ''International Journal of Numerical Methods in Fluids'' 52:1175–1193. <br /> * Pfleger, F. (2011). Experimentelle Untersuchung der Auskolkung um einen zylindrischen Brückenpfeiler. PhD thesis in german, Technical University of Munich, Germany.<br /> * Roulund, A., Mutlu Sumer, B., Fredsoe, J. &amp; Michelsen, J. (2005). Numerical and experimental investigation of flow and scour around a circular pile. ''Journal of Fluid Mechanics'' 534, 351-401.<br /> * Schanderl, W. (2018). Large-Eddy Simulation of the flow around a wall-mounted cylinder. PhD thesis, Technical University of Munich, Germany. <br /> * Schanderl, W., Jenssen, U., and Manhart, M. (2017a). Near-wall stress balance in front of a wall-mounted cylinder. ''Flow, Turbulence and Combustion'' 99(3-4):665–684.<br /> * Schanderl, W., Jenssen, U., Strobl, C., and Manhart, M. (2017b). The structure and budget of turbulent kinetic energy in front of a wall-mounted cylinder. ''Journal of Fluid Mechanics'' 827:285-321. <br /> * Schanderl, W. and Manhart, M. (2016). Reliability of wall shear stress estimations of the flow around a wall-mounted cylinder. ''Computers and Fluids'' 128:16-29. <br /> * Schanderl, W. and Manhart, M. (2018). Dissipation of Turbulent Kinetic Energy in a Cylinder Wall Junction Flow. ''Flow, Turbulence and Combustion'' 101(2):499–519.<br /> * Simpson, R. L. (2001). Junction Flows. ''Annual Review of Fluid Mechanics'', 33:415-443.<br /> |}<br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_References&diff=38961 UFR 3-35 References 2020-10-08T08:16:57Z <p>Munich: /* References */</p> <hr /> <div>{{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> __NOTOC__<br /> =Cylinder-wall junction flow=<br /> == Underlying Flow Regime 3-35 ==<br /> = References =<br /> {|<br /> * Apsilidis, N., Diplas, P., Dancey, C. L., and Bouratsis, P. (2015). Time-resolved flow dynamics and Reynolds number effects at a wall-cylinder junction. ''Journal of Fluid Mechanics'' 776:475-511.<br /> * Baghbadorani, D. A., Beheshti, A. &amp; Ataie-Ashtiani, B. (2017) Scour hole depth prediction around pile groups: review, comparison of existing methods, and proposition of a new approach. ''Natural Hazards'' 88(2), 977-1001.<br /> * Baker, C. J. (1980) The turbulent horseshoe vortex. ''Journal of wind engineering and industrial aerodynamics'' 6, 9-23.<br /> * Bruns, J., Dengel, P. &amp; Fernholz, H. H. (1992). Mean flow and turbulence measurements in an incompressible two-dimensional turbulent boundary layer. Part I: data. Tech. Rep., Herman-Föttinger-Institut für Thermo- und Fluiddynamik, TU Berlin.<br /> * Clauser, F.H. (1954). Turbulent boundary layer in adverse pressure gradients. ''J. Aero. Sci.'' 21,<br /> 91–108.<br /> * Dargahi, B. (1989). The turbulent flow field around a circular cylinder. ''Experiments in Fluids'' 8(1-2):1-12.<br /> * Devenport, W. J. and Simpson, R. L. (1990). Timedependent and time-averaged turbulence structure near the nose of a wing-body junction. ''Journal of Fluid Mechanics'' 210:23-55.<br /> * Escauriaza, C. and Sotiropoulos, F. (2011). Reynolds Number Effects on the Coherent Dynamics of the Turbulent Horseshoe Vortex System. ''Flow, Turbulence and Combustion'' 86(2):231-262. <br /> * Ettema, R., Kirkil, G. &amp; Muste, M. (2006) Similitude of Large-Scale Turbulence in Experiments on Local Scour at Cylinders. ''Journal of Hydraulic Engineering'' 132(1),33-40.<br /> * Jenssen, U. (2019). Experimental Study of the Flow Around a Scouring Bridge Pier. PhD thesis, Technical University of Munich, Germany.<br /> * Kirkil, G. and Constantinescu, G. (2015). Effects of cylinder Reynolds number on the turbulent horseshoe vortex system and near wake of a surface-mounted circular cylinder. ''Physics of Fluids'' 27(7).<br /> * Laursen, E. M. &amp; Toch, A. (1956) Scour around bridge piers and abutements. Tech. Rep. ''Iowa Institute of Hydraulic Research''.<br /> * Link, O., Pfleger, F. &amp; Zanke, U. (2008) Characteristics of developping scour-holes at sand-embedded cylinder. ''International Journal of Sediment Research'' 23, 258-266.<br /> * Manhart, M. (2004) A zonal grid algorithm for DNS of turbulent boundary layers. ''Computers and Fluids'' 33(3):435–461.<br /> * Martinuzzi, R. &amp; Tropea, C. (1993) The Flow Around Surface-Mounted, Prismatic Obstacles Placed in a Fully Developed Channel Flow. ''Journal of Fluids Engineering'' 115(1),85-92.<br /> * Melville, B. W. (2008) The physics of local scour at bridge piers. ''Fourth International Conference on Scour and Erosion (ICSE-4)'', Tokyo, Japan <br /> * Melville, B. W. &amp; Raudkivi, A. J. (1977) Flow characteristics in local scour at bridge piers. ''Journal of Hydraulic Research'' 15(4), 373-380.<br /> * Nicoud, F. &amp; Ducros, F. (1999). Subgrid-scale stress modelling based on the square of the velocity gradient tensor. ''Flow, Turbulence and Combustion'' 62(3):183–200.<br /> * Paik, J., Escauriaza, C., and Sotiropoulos, F. (2007). On the bimodal dynamics of the turbulent horseshoe vortex system in a wing-body junction. ''Physics of Fluids'' (19):045107.<br /> * Peller, N. (2010). Numerische Simulation turbulenter Strömungen mit Immersed Boundaries. PhD thesis, Technische Universität München, München.<br /> * Peller, N., Duc, A. L., Tremblay, F. &amp; Manhart, M. (2006) High-order stable interpolations for immersed boundary methods. ''International Journal of Numerical Methods in Fluids'' 52:1175–1193. <br /> * Pfleger, F. (2011) Experimentelle Untersuchung der Auskolkung um einen zylindrischen Brückenpfeiler. PhD thesis in german, Technical University of Munich, Germany.<br /> * Roulund, A., Mutlu Sumer, B., Fredsoe, J. &amp; Michelsen, J. (2005) Numerical and experimental investigation of flow and scour around a circular pile. ''Journal of Fluid Mechanics'' 534, 351-401.<br /> * Schanderl, W. (2018). Large-Eddy Simulation of the flow around a wall-mounted cylinder. PhD thesis, Technical University of Munich, Germany. <br /> * Schanderl, W., Jenssen, U., and Manhart, M. (2017a). Near-wall stress balance in front of a wall-mounted cylinder. ''Flow, Turbulence and Combustion'' 99(3-4):665–684.<br /> * Schanderl, W., Jenssen, U., Strobl, C., and Manhart, M. (2017b). The structure and budget of turbulent kinetic energy in front of a wall-mounted cylinder. ''Journal of Fluid Mechanics'' 827:285-321. <br /> * Schanderl, W. and Manhart, M. (2016). Reliability of wall shear stress estimations of the flow around a wall-mounted cylinder. ''Computers and Fluids'' 128:16-29. <br /> * Schanderl, W. and Manhart, M. (2018). Dissipation of Turbulent Kinetic Energy in a Cylinder Wall Junction Flow. ''Flow, Turbulence and Combustion'' 101(2):499–519.<br /> * Simpson, R. L. (2001). Junction Flows. ''Annual Review of Fluid Mechanics'', 33:415-443.<br /> |}<br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Test_Case&diff=38960 UFR 3-35 Test Case 2020-10-08T08:13:11Z <p>Munich: /* Inflow condition */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> = General Remark =<br /> The experimental and numerical setups applied in this study were described in detail by Schanderl et al. (2017b) (PIV and LES). The experiment is further described by Jenssen (2019), the numerics in Schanderl &amp; Manhart (2016), Schanderl et al. (2017a) and Schanderl &amp; Manhart (2018). Thus, the following shall provide a brief overview only.<br /> <br /> = Test Case Experiments =<br /> In order to provide both numerical and experimental data acquired for the same flow configuration under identical (as good as possible) boundary conditions, we performed a large eddy simulation and a particle image velocimetry experiment. We studied the flow around a wall-mounted slender circular cylinder at a height larger than the flow depth an a flow depth of &lt;math&gt; z_0 = 1.5D&lt;/math&gt;. The width of the rectangular channel was &lt;math&gt; 11.7D&lt;/math&gt; (see Fig. 1). The investigated Reynolds number was approximately &lt;math&gt; Re_D = \frac{u_{\mathrm{b}}D}{\nu} = 39{,}000&lt;/math&gt;, the Froude number was in the subcritical region. As inflow condition we applied a fully-developed open-channel flow. The particular flow conditions were chosen to be as close as possible to the conditions of Dargahi (1989).<br /> <br /> [[File:UFR3-35_configuration.png|centre|frame|Fig. 1: Sketch of flow configuration]]<br /> <br /> = Experimental set-up = <br /> The experimental set-up is shown in Fig. 2. A high-level water tank fed the flume with a constant energy head. After the inlet, a flow straightener, a surface waves damper and vortex generators as recommended by (Counihan 1969) were installed such that the turbulent open-channel flow developed along the entry length of &lt;math&gt;200D&lt;/math&gt; corresponding to 42 hydraulic diameters of the open channel flow. A sluice gate at the end of the flume controlled the water depth before the water recirculated driven by a pump. The experimental parameters are listed in Table 1:<br /> <br /> [[File:UFR3-35_Flume.png|thumb|centre|800px|Fig. 2: Experimental set-up]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 1: Experimental parameters<br /> ! Description<br /> ! Value<br /> ! Unit<br /> |-<br /> | Cylinder diameter &lt;math&gt;D&lt;/math&gt;<br /> | &lt;math&gt;0.1&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow depth &lt;math&gt;z_0&lt;/math&gt;<br /> | &lt;math&gt;0.15&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Channel width &lt;math&gt;b&lt;/math&gt;<br /> | &lt;math&gt;1.17&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow rate &lt;math&gt;Q&lt;/math&gt;<br /> | &lt;math&gt;0.069&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^3 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Depth-averaged velocity of approach flow &lt;math&gt;u_{\mathrm{b}}&lt;/math&gt;<br /> | &lt;math&gt;0.3986&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}\, \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Kinematic viscosity &lt;math&gt;\nu&lt;/math&gt;<br /> | &lt;math&gt;1.0502\cdot 10^{-6}&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^2 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_D&lt;/math&gt;<br /> | &lt;math&gt;37{,}954&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{z_0} = \frac{u_\mathrm{b} \cdot 4R_{\mathrm{hyd}}}{\nu} = \frac{u_\mathrm{b} \cdot 4(b\cdot z_0)/(2z_0+b) }{\nu} &lt;/math&gt;<br /> | &lt;math&gt;181{,}162&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{\tau} = \frac{u_{\tau} \cdot z_0}{\nu}&lt;/math&gt;<br /> | &lt;math&gt;2571&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |}<br /> <br /> = Measurement technique =<br /> The experimental data were acquired by conducting planar monoscopic 2D-2C PIV in the vertical symmetry plane upstream of the cylinder. The PIV snapshots were evaluated by the standard interrogation window based cross-correlation of &lt;math&gt;16\times16\mathrm{px} &lt;/math&gt;. Doing so, we achieved instantaneous velocity fields of the streamwise (&lt;math&gt;u&lt;/math&gt;) and the wall-normal (&lt;math&gt;w&lt;/math&gt;) velocity component. From these data the time-averaged turbulent statistics were calculated in the post-processing.<br /> We used a CCD-camera with a &lt;math&gt;2048\times2048\mathrm{px} &lt;/math&gt; square sensor. The size of a pixel was &lt;math&gt;36.86 \mu\mathrm{m}&lt;/math&gt;, therefore the spatial resolution of the images was &lt;math&gt;2712 \mathrm{px}/D &lt;/math&gt;. The size of the interrogation windows was &lt;math&gt;5.8976\cdot 10^{-3} D&lt;/math&gt;. The temporal resolution was &lt;math&gt;7.25\mathrm{Hz}&lt;/math&gt;, which is approximately twice the macro time scale &lt;math&gt;u_{\mathrm{b}}/D = 3.9 \mathrm{Hz}&lt;/math&gt;.<br /> The light sheet was approximately 2mm thick provided by a &lt;math&gt;532\mathrm{nm}&lt;/math&gt; Nd:YAG laser. The f-number and the focal length of the lens were &lt;math&gt;2.8&lt;/math&gt; and &lt;math&gt;105\mathrm{mm}&lt;/math&gt;, respectively. <br /> <br /> <br /> The PIV set-up is shown in detail in Fig. 3, including the qualitative size of the field-of-views (FOV) for investigating the approaching ''boundary layer'' as well as the flow in front of the wall-mounted ''cylinder''.<br /> <br /> [[File:UFR3-35_PIV_setup.png|thumb|centre|600px|Fig. 3: PIV set-up]]<br /> <br /> At the measurement section, the flume had transparent walls. Therefore, the laser light, which entered the flow from above could pass with a minimum amount of surface reflections through the bottom wall. However, an acrylic glass plate had to be mounted at the water-air interface to suppress the bow waves of the cylinder and let the light sheet enter the water body perpendicularly (see Fig. 3). The influence of this device at the water surface was tested and considered to be insignificant at the cylinder-wall junction.<br /> <br /> Hollow glass spheres were used as seeding and had a diameter of &lt;math&gt;10 \mu\mathrm{m}&lt;/math&gt;. The corresponding Stokes number was &lt;math&gt;4.7\cdot 10^{-3}&lt;/math&gt;, and therefore, the particles were considered to follow the flow precisely.<br /> <br /> The total number of recorded double frames was &lt;math&gt;27{,}000&lt;/math&gt;, the time-delay between two image frames was &lt;math&gt;700 \mu\mathrm{s}&lt;/math&gt;. Therefore, the total sampling time was &lt;math&gt;27{,}000/7.25 = 3724\mathrm{s}&lt;/math&gt; or &lt;math&gt;1484D&lt;/math&gt;. During the experiment seeding and other particles accumulated along the bottom plate, which undermined the image quality by increasing the surface reflection. Therefore, the data acquisition was stopped after &lt;math&gt;1500&lt;/math&gt; images to allow surface cleaning and to empty the limited capacity of the laboratory PC's RAM. The sampling time of such a batch was &lt;math&gt;1500/7.25 = 207 \mathrm{s}&lt;/math&gt; or &lt;math&gt;82D/u_b&lt;/math&gt;.<br /> <br /> The data acquisition time and number of valid vectors was validated by the convergence of statistical moments. In the centre of the HV the number valid samples had its minimum. Therefore, the time-series at the centre of the HV was analysed as a reference for the entire flow field. The standard error of the mean was &lt;math&gt;0.0065&lt;/math&gt; times the standard deviation, the corresponding error in the fourth central moment is &lt;math&gt;0.0545&lt;/math&gt;.<br /> <br /> <br /> The standard error of the mean value of the measured velocities was determined as follows:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) = \frac{\sigma(u)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_2(u)}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;,<br /> <br /> the standard error of the higher central moments &lt;math&gt; M_n = \langle u'^n \rangle &lt;/math&gt; was obtained likewise:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}\left( \langle u'^n \rangle \right) = \frac{\sigma\left(u'^n\right)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_{\mathrm{2n}}-M_{\mathrm{n}}^2}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;.<br /> <br /> <br /> The standard errors with respect to the standard deviation &lt;math&gt; \sigma &lt;/math&gt; were quantified as follows:<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 2: Standard errors of extimating selected central moments using the experimental data (PIV)<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) / \sigma &lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_2 \right) / \sigma^2&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_3 \right) / \sigma^3&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_4 \right) / \sigma^4&lt;/math&gt;<br /> |-<br /> | &lt;math&gt; 0.0065 &lt;/math&gt;<br /> | &lt;math&gt; 0.0089 &lt;/math&gt;<br /> | &lt;math&gt; 0.0237 &lt;/math&gt;<br /> | &lt;math&gt; 0.0545 &lt;/math&gt;<br /> |}<br /> <br /> = CFD Code and Methods =<br /> &lt;br/&gt;<br /> We applied our in-house finite-volume code MGLET with a staggered Cartesian grid. The grid was equidistant in the horizontal directions and stretched away from the wall in the vertical direction by a factor smaller than &lt;math&gt;1.01&lt;/math&gt;. The horizontal grid spacing was four times as large as the vertical one. The time integration was done by applying a third order Runge-Kutta sheme, the spatial approximation by second order central differences and the maximum of the CFL number was in the range of 0.55 to 0.82. To model the cylindrical body, a second order immersed boundary method was applied (Peller et al.2006; Peller 2010). The sub-grid scales were modelled using the Wall-Adapting Local Eddy-Viscosity (WALE) model (Nicoud &amp; Ducros 1999). Around the cylinder, the grid was refined by three locally embedded grids (Manhart 2004), each reducing the grid spacing by a factor of two (see Fig. 4). A grid study by Schanderl &amp; Manhart 2016 demonstrated that the number of grid refinements was sufficient. The resulting grid resolution in the vertical direction at the bottom plate around the cylinder was smaller than approximately 1.9 wall units based on the wall-shear stress of the approaching flow in the precursor, averaged over the span &lt;math&gt; -1.25 &lt; y/D &lt; 1.25&lt;/math&gt; (Schanderl &amp; Manhart 2016). The distance of the first grid point in the finest grid level was less than 1.6 wall units based on the local wall shear stress. The fraction of the modelled dissipation is about 30% of the total dissipation rate (Schanderl &amp; Manhart 2018).<br /> <br /> The setup simulated was intended to be identical to the experimental one. To model the bottom and side walls, we applied no-slip boundary conditions, whereas the free surface was modelled by a slip boundary condition. Therefore, the Froude number in the LES was infinitesimal and no surface waves occurred. By conducting a precursor simulation a fully-developed turbulent open-channel flow was achieved as inflow condition. The streamwise boundary conditions were periodic, and the precursor domain had a length of 30D (see Fig. 4). The wall-nearest point of the precursor grid had a distance of 7.5 wall units. <br /> <br /> [[File:UFR3-35_gridLVL.PNG|centre|frame|Fig. 4: Grid arrangement of the LES (Schanderl 2018)]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 3: Applied grids in the LES (Schanderl &amp; Manhart 2016)<br /> ! Grid<br /> ! Level of refinement<br /> ! Cells per diameter<br /> horizontal / vertical<br /> <br /> ! Grid spacing<br /> &lt;math&gt;\Delta{x}^{+}/\Delta{y}^{+}/\Delta{z}_{\mathrm{wall}}^{+}&lt;/math&gt;<br /> <br /> ! Number of grid cells<br /> |-<br /> | Precursor <br /> | 0<br /> | <br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;44\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Base<br /> | 0 <br /> | &lt;math&gt;31.25/125&lt;/math&gt;<br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;35\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 1<br /> | 1 <br /> | &lt;math&gt;62.5/250&lt;/math&gt;<br /> | &lt;math&gt;30/30/7.5&lt;/math&gt;<br /> | &lt;math&gt;80\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 2<br /> | 2 <br /> | &lt;math&gt;125/500&lt;/math&gt;<br /> | &lt;math&gt;15/15/3.7&lt;/math&gt;<br /> | &lt;math&gt;64\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 3<br /> | 3 <br /> | &lt;math&gt;250/1000&lt;/math&gt;<br /> | &lt;math&gt;7.5/7.5/1.9&lt;/math&gt;<br /> | &lt;math&gt;177\cdot 10^6 &lt;/math&gt;<br /> |}<br /> <br /> = Inflow condition = <br /> There is a strong influence of the oncoming flow on the flow around the cylinder (Schanderl &amp; Manhart 2016). In this section we provide information on the inflow condition as obtained in the symmetry plane of the channel. The simulation data were taken from the precursor simulation and the experimental data were measured by planar PIV in the empty channel at the position where later the cylinder was placed. We provide only in-plane quantities which were measured by the PIV. The data are made dimensionless by the friction velocity in the symmetry plane of the undisturbed flow. In case of the LES it was determined by the velocity gradient at the wall and in case of the PIV it was determined by the method of Clauser (1954). <br /> <br /> Figure 5 shows the vertical time-averaged profiles along &lt;math&gt; z^+ = \frac{z\cdot u_{\tau}}{\nu}&lt;/math&gt; of the<br /> <br /> &lt;ol style=&quot;list-style-type:lower-alpha&quot;&gt;<br /> &lt;li&gt;streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}.&lt;/math&gt;&lt;/li&gt;<br /> &lt;/ol&gt;<br /> For comparison, the data of Bruns et al. (1992) are included at a comparable Reynolds number based on the momentum thickness &lt;math&gt; \delta_2 &lt;/math&gt;.<br /> <br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_log_law_inflow.png|thumb|centre|450px|Fig. 5 a) streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uu_inflow.png|thumb|centre|450px|Fig. 5 b) Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_ww_inflow.png|thumb|centre|450px|Fig. 5 c) Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uw_inflow.png|thumb|centre|450px|Fig. 5 d) Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The corresponding datasets can be downloaded from below. The first 11 lines belong to the header and are indicated by the #-symbol. For both PIV and LES, each column refers to the data listed in Table 3 and is comma separated. For MatLab users, we provide a script at the end of the section [[UFR 3-35 Evaluation|Evaluation]] of this document as an example of reading the data, which can be used as a template to modify for reading the inflow data as well.<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 4: Structure of the inflow datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> |-<br /> | '''PIV'''/'''LES'''<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; z^+ &lt;/math&gt;<br /> | &lt;math&gt; u^+ &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> |}<br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_inflow_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_inflow_data.txt]]<br /> |}<br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Test_Case&diff=38959 UFR 3-35 Test Case 2020-10-08T08:08:31Z <p>Munich: /* Inflow condition */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> = General Remark =<br /> The experimental and numerical setups applied in this study were described in detail by Schanderl et al. (2017b) (PIV and LES). The experiment is further described by Jenssen (2019), the numerics in Schanderl &amp; Manhart (2016), Schanderl et al. (2017a) and Schanderl &amp; Manhart (2018). Thus, the following shall provide a brief overview only.<br /> <br /> = Test Case Experiments =<br /> In order to provide both numerical and experimental data acquired for the same flow configuration under identical (as good as possible) boundary conditions, we performed a large eddy simulation and a particle image velocimetry experiment. We studied the flow around a wall-mounted slender circular cylinder at a height larger than the flow depth an a flow depth of &lt;math&gt; z_0 = 1.5D&lt;/math&gt;. The width of the rectangular channel was &lt;math&gt; 11.7D&lt;/math&gt; (see Fig. 1). The investigated Reynolds number was approximately &lt;math&gt; Re_D = \frac{u_{\mathrm{b}}D}{\nu} = 39{,}000&lt;/math&gt;, the Froude number was in the subcritical region. As inflow condition we applied a fully-developed open-channel flow. The particular flow conditions were chosen to be as close as possible to the conditions of Dargahi (1989).<br /> <br /> [[File:UFR3-35_configuration.png|centre|frame|Fig. 1: Sketch of flow configuration]]<br /> <br /> = Experimental set-up = <br /> The experimental set-up is shown in Fig. 2. A high-level water tank fed the flume with a constant energy head. After the inlet, a flow straightener, a surface waves damper and vortex generators as recommended by (Counihan 1969) were installed such that the turbulent open-channel flow developed along the entry length of &lt;math&gt;200D&lt;/math&gt; corresponding to 42 hydraulic diameters of the open channel flow. A sluice gate at the end of the flume controlled the water depth before the water recirculated driven by a pump. The experimental parameters are listed in Table 1:<br /> <br /> [[File:UFR3-35_Flume.png|thumb|centre|800px|Fig. 2: Experimental set-up]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 1: Experimental parameters<br /> ! Description<br /> ! Value<br /> ! Unit<br /> |-<br /> | Cylinder diameter &lt;math&gt;D&lt;/math&gt;<br /> | &lt;math&gt;0.1&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow depth &lt;math&gt;z_0&lt;/math&gt;<br /> | &lt;math&gt;0.15&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Channel width &lt;math&gt;b&lt;/math&gt;<br /> | &lt;math&gt;1.17&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow rate &lt;math&gt;Q&lt;/math&gt;<br /> | &lt;math&gt;0.069&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^3 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Depth-averaged velocity of approach flow &lt;math&gt;u_{\mathrm{b}}&lt;/math&gt;<br /> | &lt;math&gt;0.3986&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}\, \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Kinematic viscosity &lt;math&gt;\nu&lt;/math&gt;<br /> | &lt;math&gt;1.0502\cdot 10^{-6}&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^2 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_D&lt;/math&gt;<br /> | &lt;math&gt;37{,}954&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{z_0} = \frac{u_\mathrm{b} \cdot 4R_{\mathrm{hyd}}}{\nu} = \frac{u_\mathrm{b} \cdot 4(b\cdot z_0)/(2z_0+b) }{\nu} &lt;/math&gt;<br /> | &lt;math&gt;181{,}162&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{\tau} = \frac{u_{\tau} \cdot z_0}{\nu}&lt;/math&gt;<br /> | &lt;math&gt;2571&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |}<br /> <br /> = Measurement technique =<br /> The experimental data were acquired by conducting planar monoscopic 2D-2C PIV in the vertical symmetry plane upstream of the cylinder. The PIV snapshots were evaluated by the standard interrogation window based cross-correlation of &lt;math&gt;16\times16\mathrm{px} &lt;/math&gt;. Doing so, we achieved instantaneous velocity fields of the streamwise (&lt;math&gt;u&lt;/math&gt;) and the wall-normal (&lt;math&gt;w&lt;/math&gt;) velocity component. From these data the time-averaged turbulent statistics were calculated in the post-processing.<br /> We used a CCD-camera with a &lt;math&gt;2048\times2048\mathrm{px} &lt;/math&gt; square sensor. The size of a pixel was &lt;math&gt;36.86 \mu\mathrm{m}&lt;/math&gt;, therefore the spatial resolution of the images was &lt;math&gt;2712 \mathrm{px}/D &lt;/math&gt;. The size of the interrogation windows was &lt;math&gt;5.8976\cdot 10^{-3} D&lt;/math&gt;. The temporal resolution was &lt;math&gt;7.25\mathrm{Hz}&lt;/math&gt;, which is approximately twice the macro time scale &lt;math&gt;u_{\mathrm{b}}/D = 3.9 \mathrm{Hz}&lt;/math&gt;.<br /> The light sheet was approximately 2mm thick provided by a &lt;math&gt;532\mathrm{nm}&lt;/math&gt; Nd:YAG laser. The f-number and the focal length of the lens were &lt;math&gt;2.8&lt;/math&gt; and &lt;math&gt;105\mathrm{mm}&lt;/math&gt;, respectively. <br /> <br /> <br /> The PIV set-up is shown in detail in Fig. 3, including the qualitative size of the field-of-views (FOV) for investigating the approaching ''boundary layer'' as well as the flow in front of the wall-mounted ''cylinder''.<br /> <br /> [[File:UFR3-35_PIV_setup.png|thumb|centre|600px|Fig. 3: PIV set-up]]<br /> <br /> At the measurement section, the flume had transparent walls. Therefore, the laser light, which entered the flow from above could pass with a minimum amount of surface reflections through the bottom wall. However, an acrylic glass plate had to be mounted at the water-air interface to suppress the bow waves of the cylinder and let the light sheet enter the water body perpendicularly (see Fig. 3). The influence of this device at the water surface was tested and considered to be insignificant at the cylinder-wall junction.<br /> <br /> Hollow glass spheres were used as seeding and had a diameter of &lt;math&gt;10 \mu\mathrm{m}&lt;/math&gt;. The corresponding Stokes number was &lt;math&gt;4.7\cdot 10^{-3}&lt;/math&gt;, and therefore, the particles were considered to follow the flow precisely.<br /> <br /> The total number of recorded double frames was &lt;math&gt;27{,}000&lt;/math&gt;, the time-delay between two image frames was &lt;math&gt;700 \mu\mathrm{s}&lt;/math&gt;. Therefore, the total sampling time was &lt;math&gt;27{,}000/7.25 = 3724\mathrm{s}&lt;/math&gt; or &lt;math&gt;1484D&lt;/math&gt;. During the experiment seeding and other particles accumulated along the bottom plate, which undermined the image quality by increasing the surface reflection. Therefore, the data acquisition was stopped after &lt;math&gt;1500&lt;/math&gt; images to allow surface cleaning and to empty the limited capacity of the laboratory PC's RAM. The sampling time of such a batch was &lt;math&gt;1500/7.25 = 207 \mathrm{s}&lt;/math&gt; or &lt;math&gt;82D/u_b&lt;/math&gt;.<br /> <br /> The data acquisition time and number of valid vectors was validated by the convergence of statistical moments. In the centre of the HV the number valid samples had its minimum. Therefore, the time-series at the centre of the HV was analysed as a reference for the entire flow field. The standard error of the mean was &lt;math&gt;0.0065&lt;/math&gt; times the standard deviation, the corresponding error in the fourth central moment is &lt;math&gt;0.0545&lt;/math&gt;.<br /> <br /> <br /> The standard error of the mean value of the measured velocities was determined as follows:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) = \frac{\sigma(u)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_2(u)}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;,<br /> <br /> the standard error of the higher central moments &lt;math&gt; M_n = \langle u'^n \rangle &lt;/math&gt; was obtained likewise:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}\left( \langle u'^n \rangle \right) = \frac{\sigma\left(u'^n\right)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_{\mathrm{2n}}-M_{\mathrm{n}}^2}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;.<br /> <br /> <br /> The standard errors with respect to the standard deviation &lt;math&gt; \sigma &lt;/math&gt; were quantified as follows:<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 2: Standard errors of extimating selected central moments using the experimental data (PIV)<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) / \sigma &lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_2 \right) / \sigma^2&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_3 \right) / \sigma^3&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_4 \right) / \sigma^4&lt;/math&gt;<br /> |-<br /> | &lt;math&gt; 0.0065 &lt;/math&gt;<br /> | &lt;math&gt; 0.0089 &lt;/math&gt;<br /> | &lt;math&gt; 0.0237 &lt;/math&gt;<br /> | &lt;math&gt; 0.0545 &lt;/math&gt;<br /> |}<br /> <br /> = CFD Code and Methods =<br /> &lt;br/&gt;<br /> We applied our in-house finite-volume code MGLET with a staggered Cartesian grid. The grid was equidistant in the horizontal directions and stretched away from the wall in the vertical direction by a factor smaller than &lt;math&gt;1.01&lt;/math&gt;. The horizontal grid spacing was four times as large as the vertical one. The time integration was done by applying a third order Runge-Kutta sheme, the spatial approximation by second order central differences and the maximum of the CFL number was in the range of 0.55 to 0.82. To model the cylindrical body, a second order immersed boundary method was applied (Peller et al.2006; Peller 2010). The sub-grid scales were modelled using the Wall-Adapting Local Eddy-Viscosity (WALE) model (Nicoud &amp; Ducros 1999). Around the cylinder, the grid was refined by three locally embedded grids (Manhart 2004), each reducing the grid spacing by a factor of two (see Fig. 4). A grid study by Schanderl &amp; Manhart 2016 demonstrated that the number of grid refinements was sufficient. The resulting grid resolution in the vertical direction at the bottom plate around the cylinder was smaller than approximately 1.9 wall units based on the wall-shear stress of the approaching flow in the precursor, averaged over the span &lt;math&gt; -1.25 &lt; y/D &lt; 1.25&lt;/math&gt; (Schanderl &amp; Manhart 2016). The distance of the first grid point in the finest grid level was less than 1.6 wall units based on the local wall shear stress. The fraction of the modelled dissipation is about 30% of the total dissipation rate (Schanderl &amp; Manhart 2018).<br /> <br /> The setup simulated was intended to be identical to the experimental one. To model the bottom and side walls, we applied no-slip boundary conditions, whereas the free surface was modelled by a slip boundary condition. Therefore, the Froude number in the LES was infinitesimal and no surface waves occurred. By conducting a precursor simulation a fully-developed turbulent open-channel flow was achieved as inflow condition. The streamwise boundary conditions were periodic, and the precursor domain had a length of 30D (see Fig. 4). The wall-nearest point of the precursor grid had a distance of 7.5 wall units. <br /> <br /> [[File:UFR3-35_gridLVL.PNG|centre|frame|Fig. 4: Grid arrangement of the LES (Schanderl 2018)]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 3: Applied grids in the LES (Schanderl &amp; Manhart 2016)<br /> ! Grid<br /> ! Level of refinement<br /> ! Cells per diameter<br /> horizontal / vertical<br /> <br /> ! Grid spacing<br /> &lt;math&gt;\Delta{x}^{+}/\Delta{y}^{+}/\Delta{z}_{\mathrm{wall}}^{+}&lt;/math&gt;<br /> <br /> ! Number of grid cells<br /> |-<br /> | Precursor <br /> | 0<br /> | <br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;44\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Base<br /> | 0 <br /> | &lt;math&gt;31.25/125&lt;/math&gt;<br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;35\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 1<br /> | 1 <br /> | &lt;math&gt;62.5/250&lt;/math&gt;<br /> | &lt;math&gt;30/30/7.5&lt;/math&gt;<br /> | &lt;math&gt;80\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 2<br /> | 2 <br /> | &lt;math&gt;125/500&lt;/math&gt;<br /> | &lt;math&gt;15/15/3.7&lt;/math&gt;<br /> | &lt;math&gt;64\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 3<br /> | 3 <br /> | &lt;math&gt;250/1000&lt;/math&gt;<br /> | &lt;math&gt;7.5/7.5/1.9&lt;/math&gt;<br /> | &lt;math&gt;177\cdot 10^6 &lt;/math&gt;<br /> |}<br /> <br /> = Inflow condition = <br /> There is a strong influence of the oncoming flow on the flow around the cylinder (Schanderl &amp; Manhart 2016). In this section we provide information on the inflow condition as obtained in the symmetry plane of the channel. The simulation data were taken from the precursor simulation and the experimental data were measured by planar PIV in the empty channel at the position where later the cylinder was placed. <br /> <br /> Figure 5 shows the vertical time-averaged profiles along &lt;math&gt; z^+ = \frac{z\cdot u_{\tau}}{\nu}&lt;/math&gt; of the<br /> <br /> &lt;ol style=&quot;list-style-type:lower-alpha&quot;&gt;<br /> &lt;li&gt;streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}.&lt;/math&gt;&lt;/li&gt;<br /> &lt;/ol&gt;<br /> For comparison, the data of Bruns et al. (1992) are included at a comparable Reynolds number based on the momentum thickness &lt;math&gt; \delta_2 &lt;/math&gt;.<br /> <br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_log_law_inflow.png|thumb|centre|450px|Fig. 5 a) streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uu_inflow.png|thumb|centre|450px|Fig. 5 b) Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_ww_inflow.png|thumb|centre|450px|Fig. 5 c) Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uw_inflow.png|thumb|centre|450px|Fig. 5 d) Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The corresponding datasets can be downloaded from below. The first 11 lines belong to the header and are indicated by the #-symbol. For both PIV and LES, each column refers to the data listed in Table 3 and is comma separated. For MatLab users, we provide a script at the end of the section [[UFR 3-35 Evaluation|Evaluation]] of this document as an example of reading the data, which can be used as a template to modify for reading the inflow data as well.<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 4: Structure of the inflow datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> |-<br /> | '''PIV'''/'''LES'''<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; z^+ &lt;/math&gt;<br /> | &lt;math&gt; u^+ &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> |}<br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_inflow_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_inflow_data.txt]]<br /> |}<br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Test_Case&diff=38958 UFR 3-35 Test Case 2020-10-08T08:05:30Z <p>Munich: /* Inflow condition */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> = General Remark =<br /> The experimental and numerical setups applied in this study were described in detail by Schanderl et al. (2017b) (PIV and LES). The experiment is further described by Jenssen (2019), the numerics in Schanderl &amp; Manhart (2016), Schanderl et al. (2017a) and Schanderl &amp; Manhart (2018). Thus, the following shall provide a brief overview only.<br /> <br /> = Test Case Experiments =<br /> In order to provide both numerical and experimental data acquired for the same flow configuration under identical (as good as possible) boundary conditions, we performed a large eddy simulation and a particle image velocimetry experiment. We studied the flow around a wall-mounted slender circular cylinder at a height larger than the flow depth an a flow depth of &lt;math&gt; z_0 = 1.5D&lt;/math&gt;. The width of the rectangular channel was &lt;math&gt; 11.7D&lt;/math&gt; (see Fig. 1). The investigated Reynolds number was approximately &lt;math&gt; Re_D = \frac{u_{\mathrm{b}}D}{\nu} = 39{,}000&lt;/math&gt;, the Froude number was in the subcritical region. As inflow condition we applied a fully-developed open-channel flow. The particular flow conditions were chosen to be as close as possible to the conditions of Dargahi (1989).<br /> <br /> [[File:UFR3-35_configuration.png|centre|frame|Fig. 1: Sketch of flow configuration]]<br /> <br /> = Experimental set-up = <br /> The experimental set-up is shown in Fig. 2. A high-level water tank fed the flume with a constant energy head. After the inlet, a flow straightener, a surface waves damper and vortex generators as recommended by (Counihan 1969) were installed such that the turbulent open-channel flow developed along the entry length of &lt;math&gt;200D&lt;/math&gt; corresponding to 42 hydraulic diameters of the open channel flow. A sluice gate at the end of the flume controlled the water depth before the water recirculated driven by a pump. The experimental parameters are listed in Table 1:<br /> <br /> [[File:UFR3-35_Flume.png|thumb|centre|800px|Fig. 2: Experimental set-up]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 1: Experimental parameters<br /> ! Description<br /> ! Value<br /> ! Unit<br /> |-<br /> | Cylinder diameter &lt;math&gt;D&lt;/math&gt;<br /> | &lt;math&gt;0.1&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow depth &lt;math&gt;z_0&lt;/math&gt;<br /> | &lt;math&gt;0.15&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Channel width &lt;math&gt;b&lt;/math&gt;<br /> | &lt;math&gt;1.17&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow rate &lt;math&gt;Q&lt;/math&gt;<br /> | &lt;math&gt;0.069&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^3 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Depth-averaged velocity of approach flow &lt;math&gt;u_{\mathrm{b}}&lt;/math&gt;<br /> | &lt;math&gt;0.3986&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}\, \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Kinematic viscosity &lt;math&gt;\nu&lt;/math&gt;<br /> | &lt;math&gt;1.0502\cdot 10^{-6}&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^2 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_D&lt;/math&gt;<br /> | &lt;math&gt;37{,}954&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{z_0} = \frac{u_\mathrm{b} \cdot 4R_{\mathrm{hyd}}}{\nu} = \frac{u_\mathrm{b} \cdot 4(b\cdot z_0)/(2z_0+b) }{\nu} &lt;/math&gt;<br /> | &lt;math&gt;181{,}162&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{\tau} = \frac{u_{\tau} \cdot z_0}{\nu}&lt;/math&gt;<br /> | &lt;math&gt;2571&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |}<br /> <br /> = Measurement technique =<br /> The experimental data were acquired by conducting planar monoscopic 2D-2C PIV in the vertical symmetry plane upstream of the cylinder. The PIV snapshots were evaluated by the standard interrogation window based cross-correlation of &lt;math&gt;16\times16\mathrm{px} &lt;/math&gt;. Doing so, we achieved instantaneous velocity fields of the streamwise (&lt;math&gt;u&lt;/math&gt;) and the wall-normal (&lt;math&gt;w&lt;/math&gt;) velocity component. From these data the time-averaged turbulent statistics were calculated in the post-processing.<br /> We used a CCD-camera with a &lt;math&gt;2048\times2048\mathrm{px} &lt;/math&gt; square sensor. The size of a pixel was &lt;math&gt;36.86 \mu\mathrm{m}&lt;/math&gt;, therefore the spatial resolution of the images was &lt;math&gt;2712 \mathrm{px}/D &lt;/math&gt;. The size of the interrogation windows was &lt;math&gt;5.8976\cdot 10^{-3} D&lt;/math&gt;. The temporal resolution was &lt;math&gt;7.25\mathrm{Hz}&lt;/math&gt;, which is approximately twice the macro time scale &lt;math&gt;u_{\mathrm{b}}/D = 3.9 \mathrm{Hz}&lt;/math&gt;.<br /> The light sheet was approximately 2mm thick provided by a &lt;math&gt;532\mathrm{nm}&lt;/math&gt; Nd:YAG laser. The f-number and the focal length of the lens were &lt;math&gt;2.8&lt;/math&gt; and &lt;math&gt;105\mathrm{mm}&lt;/math&gt;, respectively. <br /> <br /> <br /> The PIV set-up is shown in detail in Fig. 3, including the qualitative size of the field-of-views (FOV) for investigating the approaching ''boundary layer'' as well as the flow in front of the wall-mounted ''cylinder''.<br /> <br /> [[File:UFR3-35_PIV_setup.png|thumb|centre|600px|Fig. 3: PIV set-up]]<br /> <br /> At the measurement section, the flume had transparent walls. Therefore, the laser light, which entered the flow from above could pass with a minimum amount of surface reflections through the bottom wall. However, an acrylic glass plate had to be mounted at the water-air interface to suppress the bow waves of the cylinder and let the light sheet enter the water body perpendicularly (see Fig. 3). The influence of this device at the water surface was tested and considered to be insignificant at the cylinder-wall junction.<br /> <br /> Hollow glass spheres were used as seeding and had a diameter of &lt;math&gt;10 \mu\mathrm{m}&lt;/math&gt;. The corresponding Stokes number was &lt;math&gt;4.7\cdot 10^{-3}&lt;/math&gt;, and therefore, the particles were considered to follow the flow precisely.<br /> <br /> The total number of recorded double frames was &lt;math&gt;27{,}000&lt;/math&gt;, the time-delay between two image frames was &lt;math&gt;700 \mu\mathrm{s}&lt;/math&gt;. Therefore, the total sampling time was &lt;math&gt;27{,}000/7.25 = 3724\mathrm{s}&lt;/math&gt; or &lt;math&gt;1484D&lt;/math&gt;. During the experiment seeding and other particles accumulated along the bottom plate, which undermined the image quality by increasing the surface reflection. Therefore, the data acquisition was stopped after &lt;math&gt;1500&lt;/math&gt; images to allow surface cleaning and to empty the limited capacity of the laboratory PC's RAM. The sampling time of such a batch was &lt;math&gt;1500/7.25 = 207 \mathrm{s}&lt;/math&gt; or &lt;math&gt;82D/u_b&lt;/math&gt;.<br /> <br /> The data acquisition time and number of valid vectors was validated by the convergence of statistical moments. In the centre of the HV the number valid samples had its minimum. Therefore, the time-series at the centre of the HV was analysed as a reference for the entire flow field. The standard error of the mean was &lt;math&gt;0.0065&lt;/math&gt; times the standard deviation, the corresponding error in the fourth central moment is &lt;math&gt;0.0545&lt;/math&gt;.<br /> <br /> <br /> The standard error of the mean value of the measured velocities was determined as follows:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) = \frac{\sigma(u)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_2(u)}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;,<br /> <br /> the standard error of the higher central moments &lt;math&gt; M_n = \langle u'^n \rangle &lt;/math&gt; was obtained likewise:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}\left( \langle u'^n \rangle \right) = \frac{\sigma\left(u'^n\right)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_{\mathrm{2n}}-M_{\mathrm{n}}^2}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;.<br /> <br /> <br /> The standard errors with respect to the standard deviation &lt;math&gt; \sigma &lt;/math&gt; were quantified as follows:<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 2: Standard errors of extimating selected central moments using the experimental data (PIV)<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) / \sigma &lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_2 \right) / \sigma^2&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_3 \right) / \sigma^3&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_4 \right) / \sigma^4&lt;/math&gt;<br /> |-<br /> | &lt;math&gt; 0.0065 &lt;/math&gt;<br /> | &lt;math&gt; 0.0089 &lt;/math&gt;<br /> | &lt;math&gt; 0.0237 &lt;/math&gt;<br /> | &lt;math&gt; 0.0545 &lt;/math&gt;<br /> |}<br /> <br /> = CFD Code and Methods =<br /> &lt;br/&gt;<br /> We applied our in-house finite-volume code MGLET with a staggered Cartesian grid. The grid was equidistant in the horizontal directions and stretched away from the wall in the vertical direction by a factor smaller than &lt;math&gt;1.01&lt;/math&gt;. The horizontal grid spacing was four times as large as the vertical one. The time integration was done by applying a third order Runge-Kutta sheme, the spatial approximation by second order central differences and the maximum of the CFL number was in the range of 0.55 to 0.82. To model the cylindrical body, a second order immersed boundary method was applied (Peller et al.2006; Peller 2010). The sub-grid scales were modelled using the Wall-Adapting Local Eddy-Viscosity (WALE) model (Nicoud &amp; Ducros 1999). Around the cylinder, the grid was refined by three locally embedded grids (Manhart 2004), each reducing the grid spacing by a factor of two (see Fig. 4). A grid study by Schanderl &amp; Manhart 2016 demonstrated that the number of grid refinements was sufficient. The resulting grid resolution in the vertical direction at the bottom plate around the cylinder was smaller than approximately 1.9 wall units based on the wall-shear stress of the approaching flow in the precursor, averaged over the span &lt;math&gt; -1.25 &lt; y/D &lt; 1.25&lt;/math&gt; (Schanderl &amp; Manhart 2016). The distance of the first grid point in the finest grid level was less than 1.6 wall units based on the local wall shear stress. The fraction of the modelled dissipation is about 30% of the total dissipation rate (Schanderl &amp; Manhart 2018).<br /> <br /> The setup simulated was intended to be identical to the experimental one. To model the bottom and side walls, we applied no-slip boundary conditions, whereas the free surface was modelled by a slip boundary condition. Therefore, the Froude number in the LES was infinitesimal and no surface waves occurred. By conducting a precursor simulation a fully-developed turbulent open-channel flow was achieved as inflow condition. The streamwise boundary conditions were periodic, and the precursor domain had a length of 30D (see Fig. 4). The wall-nearest point of the precursor grid had a distance of 7.5 wall units. <br /> <br /> [[File:UFR3-35_gridLVL.PNG|centre|frame|Fig. 4: Grid arrangement of the LES (Schanderl 2018)]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 3: Applied grids in the LES (Schanderl &amp; Manhart 2016)<br /> ! Grid<br /> ! Level of refinement<br /> ! Cells per diameter<br /> horizontal / vertical<br /> <br /> ! Grid spacing<br /> &lt;math&gt;\Delta{x}^{+}/\Delta{y}^{+}/\Delta{z}_{\mathrm{wall}}^{+}&lt;/math&gt;<br /> <br /> ! Number of grid cells<br /> |-<br /> | Precursor <br /> | 0<br /> | <br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;44\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Base<br /> | 0 <br /> | &lt;math&gt;31.25/125&lt;/math&gt;<br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;35\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 1<br /> | 1 <br /> | &lt;math&gt;62.5/250&lt;/math&gt;<br /> | &lt;math&gt;30/30/7.5&lt;/math&gt;<br /> | &lt;math&gt;80\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 2<br /> | 2 <br /> | &lt;math&gt;125/500&lt;/math&gt;<br /> | &lt;math&gt;15/15/3.7&lt;/math&gt;<br /> | &lt;math&gt;64\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 3<br /> | 3 <br /> | &lt;math&gt;250/1000&lt;/math&gt;<br /> | &lt;math&gt;7.5/7.5/1.9&lt;/math&gt;<br /> | &lt;math&gt;177\cdot 10^6 &lt;/math&gt;<br /> |}<br /> <br /> = Inflow condition = <br /> There is a strong influence of the oncoming flow on the flow around the cylinder (Schanderl &amp; Manhart 2016).<br /> <br /> Figure 5 shows the vertical time-averaged profiles along &lt;math&gt; z^+ = \frac{z\cdot u_{\tau}}{\nu}&lt;/math&gt; of the<br /> <br /> &lt;ol style=&quot;list-style-type:lower-alpha&quot;&gt;<br /> &lt;li&gt;streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}.&lt;/math&gt;&lt;/li&gt;<br /> &lt;/ol&gt;<br /> For comparison, the data of Bruns et al. (1992) are included at a comparable Reynolds number based on the momentum thickness &lt;math&gt; \delta_2 &lt;/math&gt;.<br /> <br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_log_law_inflow.png|thumb|centre|450px|Fig. 5 a) streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uu_inflow.png|thumb|centre|450px|Fig. 5 b) Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_ww_inflow.png|thumb|centre|450px|Fig. 5 c) Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uw_inflow.png|thumb|centre|450px|Fig. 5 d) Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The corresponding datasets can be downloaded from below. The first 11 lines belong to the header and are indicated by the #-symbol. For both PIV and LES, each column refers to the data listed in Table 3 and is comma separated. For MatLab users, we provide a script at the end of the section [[UFR 3-35 Evaluation|Evaluation]] of this document as an example of reading the data, which can be used as a template to modify for reading the inflow data as well.<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 4: Structure of the inflow datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> |-<br /> | '''PIV'''/'''LES'''<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; z^+ &lt;/math&gt;<br /> | &lt;math&gt; u^+ &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> |}<br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_inflow_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_inflow_data.txt]]<br /> |}<br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich https://kbwiki.ercoftac.org/w/index.php?title=UFR_3-35_Test_Case&diff=38957 UFR 3-35 Test Case 2020-10-08T07:58:59Z <p>Munich: /* CFD Code and Methods */</p> <hr /> <div>=Cylinder-wall junction flow=<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> = General Remark =<br /> The experimental and numerical setups applied in this study were described in detail by Schanderl et al. (2017b) (PIV and LES). The experiment is further described by Jenssen (2019), the numerics in Schanderl &amp; Manhart (2016), Schanderl et al. (2017a) and Schanderl &amp; Manhart (2018). Thus, the following shall provide a brief overview only.<br /> <br /> = Test Case Experiments =<br /> In order to provide both numerical and experimental data acquired for the same flow configuration under identical (as good as possible) boundary conditions, we performed a large eddy simulation and a particle image velocimetry experiment. We studied the flow around a wall-mounted slender circular cylinder at a height larger than the flow depth an a flow depth of &lt;math&gt; z_0 = 1.5D&lt;/math&gt;. The width of the rectangular channel was &lt;math&gt; 11.7D&lt;/math&gt; (see Fig. 1). The investigated Reynolds number was approximately &lt;math&gt; Re_D = \frac{u_{\mathrm{b}}D}{\nu} = 39{,}000&lt;/math&gt;, the Froude number was in the subcritical region. As inflow condition we applied a fully-developed open-channel flow. The particular flow conditions were chosen to be as close as possible to the conditions of Dargahi (1989).<br /> <br /> [[File:UFR3-35_configuration.png|centre|frame|Fig. 1: Sketch of flow configuration]]<br /> <br /> = Experimental set-up = <br /> The experimental set-up is shown in Fig. 2. A high-level water tank fed the flume with a constant energy head. After the inlet, a flow straightener, a surface waves damper and vortex generators as recommended by (Counihan 1969) were installed such that the turbulent open-channel flow developed along the entry length of &lt;math&gt;200D&lt;/math&gt; corresponding to 42 hydraulic diameters of the open channel flow. A sluice gate at the end of the flume controlled the water depth before the water recirculated driven by a pump. The experimental parameters are listed in Table 1:<br /> <br /> [[File:UFR3-35_Flume.png|thumb|centre|800px|Fig. 2: Experimental set-up]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 1: Experimental parameters<br /> ! Description<br /> ! Value<br /> ! Unit<br /> |-<br /> | Cylinder diameter &lt;math&gt;D&lt;/math&gt;<br /> | &lt;math&gt;0.1&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow depth &lt;math&gt;z_0&lt;/math&gt;<br /> | &lt;math&gt;0.15&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Channel width &lt;math&gt;b&lt;/math&gt;<br /> | &lt;math&gt;1.17&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}]&lt;/math&gt;<br /> |-<br /> | Flow rate &lt;math&gt;Q&lt;/math&gt;<br /> | &lt;math&gt;0.069&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^3 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Depth-averaged velocity of approach flow &lt;math&gt;u_{\mathrm{b}}&lt;/math&gt;<br /> | &lt;math&gt;0.3986&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}\, \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Kinematic viscosity &lt;math&gt;\nu&lt;/math&gt;<br /> | &lt;math&gt;1.0502\cdot 10^{-6}&lt;/math&gt;<br /> | &lt;math&gt;[\mathrm{m}^2 \mathrm{s}^{-1}]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_D&lt;/math&gt;<br /> | &lt;math&gt;37{,}954&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{z_0} = \frac{u_\mathrm{b} \cdot 4R_{\mathrm{hyd}}}{\nu} = \frac{u_\mathrm{b} \cdot 4(b\cdot z_0)/(2z_0+b) }{\nu} &lt;/math&gt;<br /> | &lt;math&gt;181{,}162&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |-<br /> | Reynolds number &lt;math&gt;Re_{\tau} = \frac{u_{\tau} \cdot z_0}{\nu}&lt;/math&gt;<br /> | &lt;math&gt;2571&lt;/math&gt;<br /> | &lt;math&gt;[-]&lt;/math&gt;<br /> |}<br /> <br /> = Measurement technique =<br /> The experimental data were acquired by conducting planar monoscopic 2D-2C PIV in the vertical symmetry plane upstream of the cylinder. The PIV snapshots were evaluated by the standard interrogation window based cross-correlation of &lt;math&gt;16\times16\mathrm{px} &lt;/math&gt;. Doing so, we achieved instantaneous velocity fields of the streamwise (&lt;math&gt;u&lt;/math&gt;) and the wall-normal (&lt;math&gt;w&lt;/math&gt;) velocity component. From these data the time-averaged turbulent statistics were calculated in the post-processing.<br /> We used a CCD-camera with a &lt;math&gt;2048\times2048\mathrm{px} &lt;/math&gt; square sensor. The size of a pixel was &lt;math&gt;36.86 \mu\mathrm{m}&lt;/math&gt;, therefore the spatial resolution of the images was &lt;math&gt;2712 \mathrm{px}/D &lt;/math&gt;. The size of the interrogation windows was &lt;math&gt;5.8976\cdot 10^{-3} D&lt;/math&gt;. The temporal resolution was &lt;math&gt;7.25\mathrm{Hz}&lt;/math&gt;, which is approximately twice the macro time scale &lt;math&gt;u_{\mathrm{b}}/D = 3.9 \mathrm{Hz}&lt;/math&gt;.<br /> The light sheet was approximately 2mm thick provided by a &lt;math&gt;532\mathrm{nm}&lt;/math&gt; Nd:YAG laser. The f-number and the focal length of the lens were &lt;math&gt;2.8&lt;/math&gt; and &lt;math&gt;105\mathrm{mm}&lt;/math&gt;, respectively. <br /> <br /> <br /> The PIV set-up is shown in detail in Fig. 3, including the qualitative size of the field-of-views (FOV) for investigating the approaching ''boundary layer'' as well as the flow in front of the wall-mounted ''cylinder''.<br /> <br /> [[File:UFR3-35_PIV_setup.png|thumb|centre|600px|Fig. 3: PIV set-up]]<br /> <br /> At the measurement section, the flume had transparent walls. Therefore, the laser light, which entered the flow from above could pass with a minimum amount of surface reflections through the bottom wall. However, an acrylic glass plate had to be mounted at the water-air interface to suppress the bow waves of the cylinder and let the light sheet enter the water body perpendicularly (see Fig. 3). The influence of this device at the water surface was tested and considered to be insignificant at the cylinder-wall junction.<br /> <br /> Hollow glass spheres were used as seeding and had a diameter of &lt;math&gt;10 \mu\mathrm{m}&lt;/math&gt;. The corresponding Stokes number was &lt;math&gt;4.7\cdot 10^{-3}&lt;/math&gt;, and therefore, the particles were considered to follow the flow precisely.<br /> <br /> The total number of recorded double frames was &lt;math&gt;27{,}000&lt;/math&gt;, the time-delay between two image frames was &lt;math&gt;700 \mu\mathrm{s}&lt;/math&gt;. Therefore, the total sampling time was &lt;math&gt;27{,}000/7.25 = 3724\mathrm{s}&lt;/math&gt; or &lt;math&gt;1484D&lt;/math&gt;. During the experiment seeding and other particles accumulated along the bottom plate, which undermined the image quality by increasing the surface reflection. Therefore, the data acquisition was stopped after &lt;math&gt;1500&lt;/math&gt; images to allow surface cleaning and to empty the limited capacity of the laboratory PC's RAM. The sampling time of such a batch was &lt;math&gt;1500/7.25 = 207 \mathrm{s}&lt;/math&gt; or &lt;math&gt;82D/u_b&lt;/math&gt;.<br /> <br /> The data acquisition time and number of valid vectors was validated by the convergence of statistical moments. In the centre of the HV the number valid samples had its minimum. Therefore, the time-series at the centre of the HV was analysed as a reference for the entire flow field. The standard error of the mean was &lt;math&gt;0.0065&lt;/math&gt; times the standard deviation, the corresponding error in the fourth central moment is &lt;math&gt;0.0545&lt;/math&gt;.<br /> <br /> <br /> The standard error of the mean value of the measured velocities was determined as follows:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) = \frac{\sigma(u)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_2(u)}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;,<br /> <br /> the standard error of the higher central moments &lt;math&gt; M_n = \langle u'^n \rangle &lt;/math&gt; was obtained likewise:<br /> <br /> &lt;math&gt; \varepsilon_{\mathrm{std}}\left( \langle u'^n \rangle \right) = \frac{\sigma\left(u'^n\right)}{\sqrt{N_{\mathrm{samples}}}} = \frac{\sqrt{M_{\mathrm{2n}}-M_{\mathrm{n}}^2}}{\sqrt{N_{\mathrm{samples}}}} &lt;/math&gt;.<br /> <br /> <br /> The standard errors with respect to the standard deviation &lt;math&gt; \sigma &lt;/math&gt; were quantified as follows:<br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 2: Standard errors of extimating selected central moments using the experimental data (PIV)<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}(\langle u \rangle) / \sigma &lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_2 \right) / \sigma^2&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_3 \right) / \sigma^3&lt;/math&gt;<br /> | &lt;math&gt; \varepsilon_{\mathrm{std}}\left( M_4 \right) / \sigma^4&lt;/math&gt;<br /> |-<br /> | &lt;math&gt; 0.0065 &lt;/math&gt;<br /> | &lt;math&gt; 0.0089 &lt;/math&gt;<br /> | &lt;math&gt; 0.0237 &lt;/math&gt;<br /> | &lt;math&gt; 0.0545 &lt;/math&gt;<br /> |}<br /> <br /> = CFD Code and Methods =<br /> &lt;br/&gt;<br /> We applied our in-house finite-volume code MGLET with a staggered Cartesian grid. The grid was equidistant in the horizontal directions and stretched away from the wall in the vertical direction by a factor smaller than &lt;math&gt;1.01&lt;/math&gt;. The horizontal grid spacing was four times as large as the vertical one. The time integration was done by applying a third order Runge-Kutta sheme, the spatial approximation by second order central differences and the maximum of the CFL number was in the range of 0.55 to 0.82. To model the cylindrical body, a second order immersed boundary method was applied (Peller et al.2006; Peller 2010). The sub-grid scales were modelled using the Wall-Adapting Local Eddy-Viscosity (WALE) model (Nicoud &amp; Ducros 1999). Around the cylinder, the grid was refined by three locally embedded grids (Manhart 2004), each reducing the grid spacing by a factor of two (see Fig. 4). A grid study by Schanderl &amp; Manhart 2016 demonstrated that the number of grid refinements was sufficient. The resulting grid resolution in the vertical direction at the bottom plate around the cylinder was smaller than approximately 1.9 wall units based on the wall-shear stress of the approaching flow in the precursor, averaged over the span &lt;math&gt; -1.25 &lt; y/D &lt; 1.25&lt;/math&gt; (Schanderl &amp; Manhart 2016). The distance of the first grid point in the finest grid level was less than 1.6 wall units based on the local wall shear stress. The fraction of the modelled dissipation is about 30% of the total dissipation rate (Schanderl &amp; Manhart 2018).<br /> <br /> The setup simulated was intended to be identical to the experimental one. To model the bottom and side walls, we applied no-slip boundary conditions, whereas the free surface was modelled by a slip boundary condition. Therefore, the Froude number in the LES was infinitesimal and no surface waves occurred. By conducting a precursor simulation a fully-developed turbulent open-channel flow was achieved as inflow condition. The streamwise boundary conditions were periodic, and the precursor domain had a length of 30D (see Fig. 4). The wall-nearest point of the precursor grid had a distance of 7.5 wall units. <br /> <br /> [[File:UFR3-35_gridLVL.PNG|centre|frame|Fig. 4: Grid arrangement of the LES (Schanderl 2018)]]<br /> <br /> <br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 3: Applied grids in the LES (Schanderl &amp; Manhart 2016)<br /> ! Grid<br /> ! Level of refinement<br /> ! Cells per diameter<br /> horizontal / vertical<br /> <br /> ! Grid spacing<br /> &lt;math&gt;\Delta{x}^{+}/\Delta{y}^{+}/\Delta{z}_{\mathrm{wall}}^{+}&lt;/math&gt;<br /> <br /> ! Number of grid cells<br /> |-<br /> | Precursor <br /> | 0<br /> | <br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;44\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Base<br /> | 0 <br /> | &lt;math&gt;31.25/125&lt;/math&gt;<br /> | &lt;math&gt;60/60/15&lt;/math&gt;<br /> | &lt;math&gt;35\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 1<br /> | 1 <br /> | &lt;math&gt;62.5/250&lt;/math&gt;<br /> | &lt;math&gt;30/30/7.5&lt;/math&gt;<br /> | &lt;math&gt;80\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 2<br /> | 2 <br /> | &lt;math&gt;125/500&lt;/math&gt;<br /> | &lt;math&gt;15/15/3.7&lt;/math&gt;<br /> | &lt;math&gt;64\cdot 10^6 &lt;/math&gt;<br /> |-<br /> | Grid 3<br /> | 3 <br /> | &lt;math&gt;250/1000&lt;/math&gt;<br /> | &lt;math&gt;7.5/7.5/1.9&lt;/math&gt;<br /> | &lt;math&gt;177\cdot 10^6 &lt;/math&gt;<br /> |}<br /> <br /> = Inflow condition = <br /> Figure 5 shows the vertical time-averaged profiles along &lt;math&gt; z^+ = \frac{z\cdot u_{\tau}}{\nu}&lt;/math&gt; of the<br /> <br /> &lt;ol style=&quot;list-style-type:lower-alpha&quot;&gt;<br /> &lt;li&gt;streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;&lt;/li&gt;<br /> &lt;li&gt;Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}.&lt;/math&gt;&lt;/li&gt;<br /> &lt;/ol&gt;<br /> For comparison, the data of Bruns et al. (1992) are included at a comparable Reynolds number based on the momentum thickness &lt;math&gt; \delta_2 &lt;/math&gt;.<br /> <br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_log_law_inflow.png|thumb|centre|450px|Fig. 5 a) streamwise velocity &lt;math&gt;\langle u^+\rangle = \langle u \rangle / u_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uu_inflow.png|thumb|centre|450px|Fig. 5 b) Reynolds normal stresses &lt;math&gt; \langle u'u' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> &lt;div&gt;&lt;ul&gt; <br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_ww_inflow.png|thumb|centre|450px|Fig. 5 c) Reynolds normal stresses &lt;math&gt; \langle w'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;li style=&quot;display: inline-block;&quot;&gt; [[File:UFR3-35_uw_inflow.png|thumb|centre|450px|Fig. 5 d) Reynolds shear stresses &lt;math&gt; \langle u'w' \rangle / u^2_{\tau}&lt;/math&gt;]] &lt;/li&gt;<br /> &lt;/ul&gt;&lt;/div&gt;<br /> <br /> The corresponding datasets can be downloaded from below. The first 11 lines belong to the header and are indicated by the #-symbol. For both PIV and LES, each column refers to the data listed in Table 3 and is comma separated. For MatLab users, we provide a script at the end of the section [[UFR 3-35 Evaluation|Evaluation]] of this document as an example of reading the data, which can be used as a template to modify for reading the inflow data as well.<br /> {| class=&quot;wikitable&quot; style=&quot;text-align: center;&quot; border=&quot;1&quot; style=&quot;margin: auto;&quot;<br /> |+ style=&quot;caption-side:bottom;&quot;|Tab. 4: Structure of the inflow datasets<br /> | Column number<br /> | 1<br /> | 2<br /> | 3<br /> | 4<br /> | 5<br /> | 6<br /> | 7<br /> |-<br /> | '''PIV'''/'''LES'''<br /> | &lt;math&gt; \frac{z}{D} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u\rangle}{u_{\mathrm{b}}} &lt;/math&gt;<br /> | &lt;math&gt; z^+ &lt;/math&gt;<br /> | &lt;math&gt; u^+ &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'u'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle w'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> | &lt;math&gt; \frac{\langle u'w'\rangle}{u^2_{\tau}} &lt;/math&gt;<br /> |}<br /> <br /> {|<br /> * PIV data: [[Media:UFR3-35_X_inflow_data.txt]]<br /> * LES data: [[Media:UFR3-35_C_inflow_data.txt]]<br /> |}<br /> <br /> ----<br /> {{ACContribs<br /> |authors=Ulrich Jenssen, Wolfgang Schanderl, Michael Manhart<br /> |organisation=Technical University Munich<br /> }}<br /> {{UFRHeader<br /> |area=3<br /> |number=35<br /> }}<br /> <br /> <br /> © copyright ERCOFTAC 2019</div> Munich