UFR 3-31 Test Case

Flow over curved backward-facing step

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Test Case Study

Brief Description of the Study Test Case

The geometry under consideration is shown in Fig. \ref{fig:geometry}. The rounded ramp of height ${H}$ is placed in a high-aspect-ratio duct with upstream height of ${8.52H}$. In the simulations, the flow is assumed to be spanwise homogeneous, with the spanwise slab being ${3.7H}$. The assumption of homogeneity is justified by the fact that the experimental ratio of duct depth to the step height was 38. In the experiment \cite{zhang2010experimental}, tripped boundary layers were allowed to develop on both walls for a distance of about ${25H}$. The Reynolds number, based on $H$ and the inlet free-stream velocity ${U_{in}}$, is 13,700. At ${x/H=-7.4}$, the computational inlet, the momentum-thickness Reynolds number is ${Re_{\theta }=1192}$, and the boundary-layer thickness is ${\delta _{99}=0.83H}$.

The step geometry is based on that used originally by Song and Eaton \cite{song2004reynolds}. In order to enlarge the separated region, the height of the step was increased by a factor of 1.5. This adaptation was undertaken interactively with a parallel experiment by Zhang and Zhong \cite{zhang2010experimental}. The step shape is described by the following three relations, with the origin ${x/H=0}$ being the upstream edge of the ramp:

${\begin{aligned}y_{wall}&=(1-R_{1})+{\sqrt {R_{1}^{2}-x_{2}}}\qquad \mathrm {for} \qquad 0<x/H\leq 2.3\\y_{wall}&=y_{2}-{\sqrt {{\frac {1}{4}}R_{1}^{2}-(x_{2}-x)^{2}}}\qquad \mathrm {for} \qquad 2.3<x/H\leq 2.835\\y_{wall}&=R_{2}-{\sqrt {R_{2}^{2}-(3-x)^{2}}}\qquad \mathrm {for} \qquad 2.835<x/H\leq 2.937\\\end{aligned}}$

with ${R_{1}=4.03}$, ${\ R_{2}=0.333}$, ${\ x_{2}=3.449}$ and ${\ y_{2}=1.936}$ and ${\ y_{wall}=1}$ for ${\ x/H<0}$, ${\ y_{wall}=0}$ for ${\ x/H>2.937}$.

Test Case Experiments

The experimental working section is 5.49m long and 1.2m wide. The ramp height is 31.5mm, with an upstream section 660m long. The incoming flow velocity is fixed at 6.5m/s. Thirty-six pressure tapings are made along the ramp between ${x=-400mm}$ and 90mm with a spacing of 10mm or 20mm. To prevent an interference with the LDA measurement on the central streamwise plane of the ramp, the pressure tapings are located 25mm off this plane. Combined with the existing pressure tapings on the ceiling of the working section downstream of the ramp (150mm in spacing), the coverage of the pressure measurement is extended further up to 600mm. The velocity at the inlet of the test section is monitored using a Pitot tube. The pressure measurement system consists of a low-range pressure transducer and a scani-valve with forty-eight ports. The pressure sensor has a pressure range of -2.5mbar to 2.5mbar and an accuracy of 0.25% full scale span. The system is remotely controlled by a computer equipped with a data acquisition card. The pressure data are sampled at 1kHz and 10,000 data points are used to produce the time-averaged pressure at each tapping.

A three-component Dantec LDA system is used to measure the boundary layer development around the ramp-down section in both the streamwise and the spanwise direction. The laser beam generated by the 5W Argon ion laser is separated by a series of beam separators and colour filters inside the optic transmitter box to produce three pairs of beams with a wavelength of 512nm, 488nm and 476nm respectively. The averaged sampling rate in the freestream is about 1kHz and a lower rate of about 100Hz is obtained in the near-wall region. The number of realisations used to obtain the time-averaged velocity and turbulence fluctuations is set at 10K, and the time threshold for acquiring the data at each point is 30 seconds. The Dantec 3D traverse specially designed for this LDA system has a resolution of 6.25μm in the three directions. The uncertainty of measurements is about 4 ×10^-4 m/s in the freestream and 1.6 ×10^-3 m/s in the near-wall region. All velocity components and all Reynolds stresses have been obtained, and the skin-friction is deduced from the LDA velocity measurements, assuming the first point being in the viscous sub-layer.

CFD Methods

Large-Eddy Simulations

Numerical treatment

The implicitly filtered LES momentum and continuity equations for incompressible flow were solved over a general non-orthogonal, boundary-fitted, multi-block finite volume mesh, using LES-STREAM (e.g. \cite{fishpool2009stability} for more details and related references). In total, the default mesh covering the duct-flow domain contained around 24 million nodes. The sensitivity to the resolution is addressed below. The variables are stored in a co-located manner. The solution is based on a fractional-step time-marching method, with the time derivative approximated by a third-order Gear scheme \cite{fishpool2009stability}. The fluxes are approximated by second-order centred approximations. Within the fractional-step algorithm, a provisional velocity field results from advancing the solution with the flux operators. This is then corrected through the pressure field by projecting the provisional solution onto a divergence-free velocity field. To this end, the pressure is computed as a solution to the pressure-Poisson problem with the aid of a V-cycle multigrid scheme. In order to suppress unphysical oscillations, associated with pressure-velocity decoupling, a practice equivalent to that introduced in \cite{rhie1983numerical} is adopted. Fishpool \& Leschziner \cite{fishpool2009stability} demonstrate that the loss of accuracy associated with the smoothing introduced by this practice is minimal. The subgrid-scale stresses were approximated using either the dynamic Smagorinsky model of \cite{germanoetal91} and \cite{lilly92} or the mixed-time scale model of \cite{inagakietal05}, the latter being local, involving no averaging and hence being computationally cheaper. The mixed-time scale model was adopted in the additional two grid-sensitivity simulations for the baseline case, and it is the one used for the results presented in the database. The rationale of using the simpler mixed-time scale model for the grid-sensitivity study is rooted primarily in the observation that this model gave somewhat lower time-averaged SGS viscosity values than the dynamic model, and thus allowed any grid-sensitivity specifically associated with variations in resolution to emerge with greater clarity.

Computation	Grid $(nx,ny,nz)$	$x/H$ (separation)	$x/H$ (reattachment)
Fine	(768,160,192)	0.83	4.36
Coarse	(448,128,192)	0.87	4.21

Table 1: Grid resolution and separation and reattachment positions

To explore the sensitivity to grid resolution, two simulations were performed (Table 1). The separated region (from ${x/H=0}$ to ${x/H=4}$) was resolved using 130 points in the streamwise direction for the coarse mesh, and 280 points in the fine mesh. In both grids, the first grid-cell centre is located below ${y^{+}=1}$ at the bottom wall, whether $y$ is scaled with the local or inflow friction velocity. The maximum streamwise cell distance, ${x^{+}}$, occurs close to the computational inlet and is around 35. This value drops to below 20 and 10 for the coarse and fine mesh, respectively, downstream of ${x/H=0}$, with values of order 5 within the separation zone. The maximum spanwise cell dimension, ${z^{+}}$, is 10 for both the coarse and fine simulations (cf. \cite{bentaleb2012large} for more details).

Downstream of separation, the ratio of maximum cell dimension to the Kolmogorov length (derived from the dissipation rate as part of the budget) is of order 6 for the fine grid and 11 for the coarse grid. The boundary layer on the upper wall was covered by a coarser grid than that on the lower wall, with the conditions between the wall and the wall-nearest ${y^{+}}$ location of around 25 being covered by a log-law wall function, and with about 17 grid nodes covering the boundary-layer thickness. This choice is justified by the fact that the upper boundary layer is far away from the region of interest, is only subjected to a weak adverse pressure gradient and is always fully attached.

Turbulent boundary layers were imposed at the domain inlet (${x/H=-7.4}$) on both the upper and lower walls. To this end, data were taken from a precursor simulation, performed with the aid of a recycling method \cite{lund1998generation}, for a stretch of a boundary layer developing in the actual channel. At a streamwise position at which ${Re_{\theta }=1190}$, every realization of the velocity field was stored over a period of ${1120H/U_{in}}$. This data set was then used to explicitly prescribe the inlet condition. Comparisons between lower-wall boundary-layer data returned by the present precursor simulation and DNS results of Schlatter et al \cite{schlatter2010assessment}, for a similar Reynolds number, ${Re_{\theta }=1400}$, are shown on figure \ref{fig:inflowValidation}. The present plots pertain to a location immediately downstream of the domain inlet, while Fig. 3 in \cite{bentaleb2012large} shows distribution derived from the precursor simulation. The slight difference relative to flat-plate boundary-layer data are due to a non-zero streamwise pressure extending from the curved step upstream towards the inlet.

The time step corresponds to a maximum cell-CFL value of around 0.5. Simulations were run for 300 time units, one time unit being ${H/U_{in}}$, before statistical sampling commenced. The flow was then computed for further 1030 time units during which data for various statistical properties were gathered. Statistical convergence was increased by spanwise averaging.

Data provided

In the database provided for this test case, all quantities have been non-dimensionalized using the step height $H$ and the inflow free-stream velocity ${U_{in}}$. The database contains three types of files, all extracted from the fine grid Large-Eddy simulation:

1. wallQuantities.dat contains boundary-layer data computed on the bottom wall. The file contains the following columns:x, y_wall, C_p, tau_w, delta∗, theta, delta_99where x${\equiv x/H}$, y_wall is given by Eq. \ref{eq:ywall}, and

${\begin{aligned}\mathrm {C\_p} &={\frac {p-p_{0}}{{\frac {1}{2}}\rho U_{in}^{2}}}\\\mathrm {tau\_w} &={\frac {1}{Re_{in}}}\left.{\frac {\partial U}{\partial n}}\right|_{y=0}\\\mathrm {delta*} &=\int _{y_{\min }}^{\infty }\left(1-{\frac {U-U_{\min }}{U_{in}-U_{\min }}}\right)dy\\\mathrm {theta} &=\int _{y_{\min }}^{\infty }{\frac {U-U_{\min }}{U_{in}-U_{\min }}}\left(1-{\frac {U-U_{\min }}{U_{in}-U_{\min }}}\right)dy\end{aligned}}$

where $p_{0}$ is taken at the bottom left corner of the domain and delta_99 is the distance to the wall for which ${U=0.99U_{in}}$. ${U_{\min }}$ is computed at each streamwise location by ${U_{\min }=\min(U(y),0)}$, for ${y_{wall}\leq y\leq y_{\infty }}$.

2. profiles.dat contains the velocity and Reynolds-Stress profiles in the vertical direction:x, y, U, V, W, u', v', w', u'v', u'w', v'w' for 28 different streamwise locations (${x/H}$ = -7.31 (first cell center from the inflow), -7, -6, -5, -4, -3, -2, -1, 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14).
   
   
3. balance.#.dat (with # = k, uu, vv, ww or uv) contains the budgets for $\ k$, ${\overline {uu}}$, ${\overline {vv}}$, ${\overline {ww}}$ and ${\overline {uv}}$ respectively at the same 28 streamwise locations as in profiles.dat.

The different terms in these files are given by:

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Contributed by: Sylvain Lardeau — CD-adapco

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