Flow in a 3D diffuser

Confined flows

Underlying Flow Regime 4-16

Evaluation of the results

Both 3D diffuser configurations have served as test cases of the 13th and 14th ERCOFTAC SIG15 Workshops on refined turbulence modelling, Steiner et al. (2009) and Jakirlic et al. (2010b). In addition to different RANS models, the LES and LES-related methods (different seamless and zonal hybrid LES/RANS - HLR - models; DES - Detached Eddy Simulation) were comparatively assessed (visit www.ercoftac.org; under SIG15); the comparative analysis of selected results is presented in the section "Cross- Comparison of CFD calculations with experimental results" of the present contribution. Before starting with the latter, some key physical characteristics illustrated appropriately are discussed as follows.

Physical issues/characteristics of the flow in a 3D diffuser

Here an overview of the most important flow features posing a special challenge to the turbulence modeling is given. Their correct capturing is of decisive importance with respect to the quality of the final results. In order to illustrate these phenomena the experimental and DNS results are used along with some results obtained by LES, hybrid LES/RANS and RANS methods by the groups participating at the SIG15 workshop.

Developed ("equilibrium") flow in the inflow duct / secondary currents

Fig. 20 depicts the linear plot of the axial velocity component across the central plane (z/B=0.5) of the inflow duct at x/h=-2 obtained experimentally indicating a symmetric profile. The inflow conditions correspond clearly to those typical for a fully-developed, equilibrium flow. This is provided by a long inflow duct whose length corresponds to 62.9 channel heights. Fig. 21 shows the semi-log plot of the axial velocity component across the central plane (z/B=0.5) of the inflow duct at x/h=-2. The velocity profile shape obtained by DNS follows closely the logarithmic law, despite a certain departure from it. This departure, expressed in terms of a slight underprediction of the coefficient B in the log-law ([pic] with B=5.2), can also be regarded as a consequence of the back- influence of the adverse pressure gradient evoked by the flow expansion. The pressure coefficient evolution, displayed in Fig. 24, reveals a related pressure increase already in the inflow duct ([pic]). The LES and HLR results (Jakirlic et al., 2010a) exhibit a certain overprediction of the velocity in the logarithmic region. This seems to indicate that the grid may not have been fine enough. On the other hand, the corresponding underprediction of the friction velocity U?, serving here for the normalization - [pic], contributed also to such an outcome (the quantitative information about the U? velocity can be extracted from the friction factor evolution, Fig. 14 in the chapter "Test case studied").

Figure 20: Axial velocity profile corresponding to the "fully-developed" flow in the inflow duct (x/h = -2). Courtesy of J. Eaton (Stanford University)

Figure 21: Axial velocity profile in semi-log coordinates corresponding to the "fully-developed" flow in the inflow duct (x/h = -2). From Jakirlić et al. (2010a)

Unlike the flow through a circular pipe, the flow in a duct with rectangular cross-section is no longer unidirectional. It is characterized by a secondary motion with velocity components perpendicular to the axial direction, Fig. 22. This secondary flow transporting momentum into the duct corners is characterized by jets directed towards the duct walls bisecting each corner with associated vortices at both sides of each jet. This secondary current is Prandtl's flow of the second kind (possible only for turbulent flows) induced by the Reynolds stress anisotropy (which is, as generally known, beyond the reach of the (linear) eddy-viscosity model group in contrast to the Reynolds stress model schemes; corresponding result of the latter model is depicted in Fig. 22c). Indeed, the Reynolds stress gradients cause the generation of forces which induce the normal-to- the-wall velocity components in the secondary flow plane. Accordingly, correct capturing of the anisotropy of turbulence in the inflow duct is an important prerequisite for a successful computation of the diffuser flow (see the Section "Cross-Comparison of CFD calculations with experimental results"). Fig. 22 displays the time-averaged velocity vectors in the cross- plane y-z located at x/h=-2 obtained experimentally and computationally by a zonal Hybrid LES/RANS (HLR) model and by the GL RSM model. Despite the relatively low intensity of the secondary motion - the largest velocity has the magnitude of approximately <(1-2)% of the axial bulk velocity ([pic]) - its influence on the flow in the diffuser is significant. Unlike with the "anisotropy-blind" k-? model (not shown here), the qualitative agreement of the HLR and RSM models achieved with respect to the secondary flow topology discussed above is obvious.

a) Experiment ———————> 0.1 m/s

b) HLR (TU Darmstadt)

c) GL RSM (TU Darmstadt)

Figure 22: Velocity vectors in the y-z plane in the inflow duct (GL RSM – RSM model due to Gibson and Launder, 1978)

Adverse Pressure Gradient (APG) effects

The boundary layer separation is the direct consequence of the Adverse Pressure Gradient imposed on the duct flow by expanding the cross-section area. The following figures should give the potential practitioners insight into the topology and magnitude of the pressure recovery within the diffuser section. Fig. 24-upper displays the non-dimensional pressure gradient p+=? dp/dx / (?U3?) used traditionally to characterize the intensity of the pressure increase in a boundary layers subjected to APG. Accordingly, the displayed results enable a direct comparison with some APG boundary layer experiments. E.g., the range of p+ between 0.01-0.025 was documented in the Nagano et al. (1993) experiments, indicating a much lower level than in the present diffuser. Although the results presented were extracted from the LES and Hybrid LES/RANS simulations their quality is of a fairly high level, keeping in mind good agreement of the near-wall velocity field (see the section "Cross-Comparison of CFD calculations with experimental results"), skin-friction (Fig. 14 in the chapter "Test case studied") and surface pressure (Fig. 24-lower) development with the reference experimental and DNS results.

Figure 23: Development of the pressure field in the diffuser 1. The data are extracted from the HLR-simulation, Jakirlić et al. (2010a) and John-Puthenveettil (2012)

Figure 24: Development of the dimensionless pressure gradient p+ and surface pressure coefficient along the bottom flat wall of the diffuser 1. The data are extracted from the HLR-simulation, Jakirlić et al. (2010a)

Fig. 25 displays the semi-log profile of the axial velocity across the recirculation zone (at x/h=10) indicating a behaviour, typical of a flow affected by an adverse pressure gradient - underprediction of the logarithmic law and strong enhancement of the turbulence intensity (note the modulation of Reynolds stress field towards weakening of the Reynolds stress anisotropy, Fig. 17 in the Chapter "Test Case Studied", corresponding to a substantial mean flow deformation in the streamwise direction, Fig. 15).

Figure 25: Semi-log plots of axial velocity component at a cross-section in the interior of the diffuser 1 (x/h=10) being affected by APG. From Jakirlić et al. (2010a)

See Fig. 18 and associated discussions in Section "Test Case Studied" for the topology of the separated flow in both diffuser configurations.

Cross-comparison of CFD calculations with experimental results

The present cross-comparison of the results obtained by different calculation methods in the DNS, LES, RANS, zonal and seamless Hybrid LES/RANS (including DES) frameworks is based to a large extent on the activity conducted within the two previously-mentioned ERCOFTAC-SIG15 Workshops on Refined Turbulence Modelling, Steiner et al. (2009) and Jakirlic et al. (2010b), see "List of References". A large amount of simulation results along with detailed comparison with the experimentally obtained reference data has been assembled. The diversity of the models/methods applied can be seen from Tables 1 and 2 (Section: Test Case Studied). The specification of the models used as well as further computational details - details about the numerical code used, discretization schemes/code accuracy, grid arrangement/resolution, temporal resolution, details about the inflow (also about fluctuating inflow generation where applicable) and outflow conditions, etc. - are given in the short summaries provided by each computational group, which can be downloaded (see the appropriate link to the "workshop proceedings" at the end of this file).

In this section, a short summary of some specific outcomes and the most important conclusions are given. The presentation of results and corresponding discussion is given separately for DNS/LES, hybrid LES/RANS (HLR) and RANS methods. The analysis of the results obtained was conducted with respect to the size and shape of the flow separation pattern and associated mean flow and turbulence features: pressure redistribution along the lower non-deflected wall, axial velocity contours, axial velocity and Reynolds stress component profiles at selected streamwise and spanwise positions.

Here, just a selection of the results obtained will be shown and discussed. At the end of the section links are given to the files (the "workshop proceedings") comprising among others the detailed descriptions of the numerical methods and turbulence models used by all participating groups as well as the complete cross-comparisons of all reference and computational results concerning the mean velocity and turbulence fields at vertical planes at two spanwise positions (z/B=1/2 and z/B=7/8) and fifteen streamwise positions.

In the meantime a number of additional computational studies dealing with the flow in the present 3D diffuser configurations have been published. Brief information on these hs been given in the "Relevant Studies" Section.

DNS and LES

Typical LES results for the two diffusers are shown in Fig. 26. The iso- contours of the zero mean streamwise velocity component are plotted and the mean separation lines are highlighted by white dashed lines. Three highly complex shaped regions of mean reverse flow can be discerned: SB1, SB2 and SB3. SB1 is an artefact of the specific setup shown here, where the rounded corners at the inlet of the diffuser were replaced by sharp edges. SB2 and SB3 match, within experimental uncertainties, the reference data of Cherry et al. (2008) and, according to Ohlsson et al. (2010), are even closer to the DNS data than the experiments.

Figure 26: Mean streamwise velocity iso-contours illustrating the three-dimensional mean separation patterns in both considered configurations, diffuser 1 (left) and diffuser 2 (right), obtained by LES (ITS). From Schneider et al. (2010)

For diffusers 1 and 2, there are nine and four LES results, respectively (see Tables 1 and 2). These were obtained on various grids ranging from 1.1 to 42.9 million cells, by employing different SGS models, wall models and numerical methods, and by varying the size of the computational domain. As opposed to the DNS, for all LES, unsteady inlet data were generated using a periodic duct setup as a precursor simulation. The various contributions varied always in some aspects, but also had sufficient commonalities, such that a careful analysis allowed drawing conclusions on the following aspects: inflow data generation, placement of inflow and outflow boundaries, relevance of the near-wall region, role of the numerical method and SGS model, and resolution requirements.

In Fig. 27, the pressure recovery predictions of the various LES for diffuser 1 are compared with the experimental and DNS data. All but one LES performed well. A similar outcome can be seen in the mean and r.m.s. velocity profiles, Figs. 30 and 31. The inadequate LES was performed on the coarsest grid that was designed for RANS calculations, i.e. most cells were placed near the walls. A more detailed analysis showed that, for LES, it is important to have sufficient resolution in the core area of the diffusers. This resolution is needed to accurately compute the production of large coherent structures that exchange momentum and kinetic energy in the flow and, therefore, promote reattachment. Other factors, like numerical method and SGS models, played a minor role. Even the near-wall region could be bridged by wall-function models (see also Schneider et al., 2010). In addition, it was verified that the precursor simulation of a periodic duct flow can produce accurate unsteady inlet data, hence leading to substantial savings in grid points compared to computing the complete inlet duct (as for the DNS). Also an outflow boundary with a buffer layer placed in the straight part of the outlet duct turned out to be sufficient compared to computing also the outlet contraction far downstream (see Figs. 3 and 4 in the section "Test case studied").

Figure 27: Pressure coefficient along the bottom flat wall of Diffuser 1 obtained by experiment, DNS and various LES (streamwise distance normalized by diffuser length)

In Fig. 28, mean streamwise velocity contours at five cross-sections (x/h=2, 5, 8, 12 and 15) of diffuser 1 are shown using the same contour levels for the experimental reference data and three selected LES. Overall, the agreement is fairly good and all three LES deliver results of similar quality if compared to the DNS (see Fig. 13 in Section "Test case studied"). While the DNS uses 172 million cells and a high-order accurate flow solver, HSU LES DSM and TUD LES DSM use both sophisticated SGS models (dynamic version of the Smagorinsky model) and wall-resolving grids with 17.6 million cells (HSU) and 4 million cells (TUD) and ITS LES SM employs even only 1.6 million cells, the Smagorinsky model and a simple equidistant grid in conjunction with an adaptive wall-function. To discern differences more clearly, the zero velocity line is marked by a thicker line to highlight the reverse flow region. In the LES results for x/h=12, a bump in this line can be seen, whereas the experimental data suggests a horizontal line. Therefore, at first glance, this bump appears to be unnatural. However, the DNS data reveal the same feature. Considering the uncertainty in determining the zero-velocity line, the bump may possibly be present even in the experiments. Moreover, a recent study (Schneider et al., 2011) demonstrates that the strength of secondary flow patterns in the inlet duct has a strong impact on the existence of this bump and how pronounced it will be. Even a complete change in the location of the reverse flow region can be attained, for cases where the sense of rotation of the secondary flow was altered.

Experiment	LES-TUD	LES-HSU	LES-UKA-ITS






Figure 28: Diffuser 1 — Mean streamwise velocity contours at five selected streamwise positions within diffuser section obtained by LES (see Fig. 13 in Section "Test case studied" for the DNS results)

Fig. 29 displays mean streamwise velocity contours at five cross-sections (x/h=2, 5, 8, 12 and 15) of diffuser 2 illustrating the LES capability to capture the influence of the geometry modifications on the three- dimensional separation pattern. The similarity between the two latter LES result sets obtained by UKA-ITS, LES-NWM (wall-modelled using wall functions) and LES-NWR (wall-resolved), is obvious despite significant difference in grid size: 2 Mio. cells in total for wall-modelled LES and 42.9 Mio. cells in total for wall-resolved LES.

x/h=2	x/h=5	x/h=8	x/h=12	x/h=15
Experiment

LES-TUD

LES-NWM UKA-IST

LES-NWR UKA-IST

Figure 29: Diffuser 2 — Mean streamwise velocity contours at five selected streamwise positions within diffuser section obtained by LES (LES-NWM — Wall-Modelled (wall functions) LES; LES-NWR — Wall-Resolving LES)

An open issue is the asymmetry in the streamwise velocity profile of the diffuser inlet as found by the experiments (Fig. 20). This could neither be reproduced by DNS with the complete inlet channel nor by LES with inflow data generators. The origin of this asymmetry remains unclear. In addition, DNS and LES data exhibit a higher velocity at the lower wall than experiments. Otherwise, eddy-resolving strategies, like DNS and LES, could capture the separated flow in the 3d-diffusers and the geometric sensitivity of the flow sufficiently well, as long as the secondary motion in the inlet duct and the generation of the large coherent structures in the free shear layers inside the diffuser were resolved sufficiently.

For diffuser 1, Figs. 30 and 31 compare calculated and measured mean velocity and streamwise turbulence intensity profiles at fourteen selected locations within the inflow duct, diffuser section and straight outlet duct in two vertical planes, the one coinciding with the central spanwise position z/B=1/2 and the second positioned closer to the deflected side wall at z/B=7/8. The overall agreement of the results obtained by LES by three groups (HSU, UKA-IST and TUD) with the experimental database is very good. The most important differences are found in the early stage of the separation process at the upper deflected wall (Figs. 30-upper and 31- upper) as well as in the core region of the diffuser section. The most consistent agreement was obtained by the UKA-ITS group despite a fairly moderate number of grid cells (only 1.6 Mio. in total); the (significant) differences in the grid resolution are given in Table 1 (Section "Test Case Studied"). The UKA-ITS group applied uniform grid cells distribution in the y-direction using a wall function method for the wall treatment. The other two LES-simulations were performed using a much finer near-wall grid resolution (integration up to the wall has been applied), but a somewhat coarser grid in the core flow. The grid and wall modelling issues are discussed in the introductory part of this section.

Contributed by: Suad Jakirlić, Gisa John-Puthenveettil — Technische Universität Darmstadt