UFR 4-16 Evaluation

Flow in a 3D diffuser

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References

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 Jakirlić 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 (U⁺= ln(y⁺) / κ + B 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 (Jakirlić 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_τ , serving here for the normalization  U⁺ = U / U_τ , contributed also to such an outcome (the quantitative information about the U_τ  velocity can be extracted from the friction factor evolution, Fig. 14 in the chapter "Test Case Study").

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 ( U_bulk = 1 m/s ) — its influence on the flow in the diffuser is significant. Unlike with the "anisotropy-blind" k-ε 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⁺ = v dp/dx / (ρU_τ³ ) 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.

Experiment
LES-TUD
LES-NWM UKA-IST
LES-NWR UKA-IST
x/h=2	x/h=5	x/h=8	x/h=12	x/h=15
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.

Figure 30: Diffuser 1 - Evolution of the profiles of the axial velocity components and streamwise turbulence intensity in the vertical plane x-y at the central spanwise locations z/B=1/2 obtained by means of LES

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.

Experiment
LES-TUD
LES-NWM UKA-IST
LES-NWR UKA-IST
x/h=2	x/h=5	x/h=8	x/h=12	x/h=15
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.

Figure 30: Diffuser 1 - Evolution of the profiles of the axial velocity components and streamwise turbulence intensity in the vertical plane x-y at the central spanwise locations z/B=1/2 obtained by means of LES

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.

Experiment
LES-TUD
LES-NWM UKA-IST
LES-NWR UKA-IST
x/h=2	x/h=5	x/h=8	x/h=12	x/h=15
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.

Figure 30: Diffuser 1 - Evolution of the profiles of the axial velocity components and streamwise turbulence intensity in the vertical plane x-y at the central spanwise locations z/B=1/2 obtained by means of LES

Figure 31: Diffuser 1 - Evolution of the profiles of the axial velocity components and streamwise turbulence intensity in the vertical plane x-y at the spanwise locations z/B=7/8 obtained by means of LES

Hybrid LES/RANS (HLR)

These schemes, hybridizing the RANS and LES methods aimed at a reduction of spatial and temporal resolution, have recently experienced growing popularity in the CFD community. Their goal is to combine the advantages of both methods in order to provide a computational procedure that is capable of capturing the large-scale eddy structures with a broader spectrum and the bulk flow unsteadiness - as encountered in the flows involving separation, but at affordable costs. Interested readers are referred to a relevant review about hybrid LES/RANS methods by Fröhlich and von Terzi (2008). We mention here only their general classification into two main groups: the zonal, two-layer schemes - a RANS model resolving the near-wall region is linked at a distinct interface with the conventional LES covering the outer layer (flow core) - and the seamless models, where a RANS-like model formulation, mimicking a sub-scale model, is applied in the entire flow domain.

Diffuser 1

As can be seen from Table 1, four HLR results are available for diffuser 1. Besides the classical DES method (Spalart et al., 1997) based on the one- equation turbulence model by Spalart-Allmaras (1994), TUD has carried out a simulation based on their hybrid method. It relies on the low-Re k-? model due to Launder and Sharma (1974) applied in the near-wall region and the Smagorinsky model in the core flow. HSU applied their own hybrid concept applying an anisotropy-resolving explicit algebraic Reynolds stress model in the near-wall region and a consistent one-equation SGS model in the LES zone (Jaffrezic and Breuer, 2008; Breuer, 2010). In both HLR, the interface between LES and (U)RANS is dynamically determined using different conditions. Finally, KU adopted a non-linear eddy-viscosity model in the RANS region and the SGS model by Inagaki et al. (2005) in the LES part. Since HSU covered a computational domain of [pic] the total number of grid cells is a little bit higher than in the other cases. Otherwise the grids are comparable to each other.

Fig. 32 gives a first impression about the predictive quality of the results obtained showing the distribution of the surface pressure coefficient along the lower wall at the central plane.

Figure 32: HLR-results, pressure coefficient along the bottom flat wall of Diffuser 1

The results of TUD HLR and HSU HLR are found to be in good agreement with the experimental data as well as the DNS data, where the best coincidence is observed for TUD HLR. Obviously, the pressure recovery for DES is too low. It should be emphasized that both latter hybrid simulations were performed using the same grid resolution (see Table 1). Bearing in mind that DES was developed for external aerodynamic flows, it is not unexpected that it fails under the circumstances of an internal separated flow at a fairly low bulk Reynolds number (improved versions of the DES method - Delayed DES and Improved Delayed DES - were not applied presently). Furthermore, the performance of KU HLR is similar to DES. Since this HLR approach is overall similar to TUD HLR and HSU HLR, this non-satisfactory outcome is difficult to explain.

Fig. 33 depicts the time-averaged streamwise velocity contours at five cross-sections. The bold line indicates zero-streamwise-velocity and thus encloses the recirculation region. As visible from the experimental data the recirculation starts at the upper-right corner, i.e. the corner between the two diverging walls. At x/h=5, the separation bubble remains in the corner, both in the experiments and in the simulations by TUD HLR and HSU HLR. However, both predictions show an inaccurate pressure distribution, i.e. TUD DES and KU HLR deliver a completely separated flow region along the entire upper wall. For DES the flow is even separated along the side wall. At the next cross-section (x/h=8), it can be seen that the recirculation region has started to spread across the top of the diffuser. Again, TUD DES and KU HLR predict enlarged separation regions compared to the experiment, whereas the other two approaches perform well. Further downstream, at x/h=12 and 15, a massive separation region can be observed covering the entire top wall of the diffuser. Overall an excellent agreement between the hybrid predictions and the measurements is found, except for DES which yields a too small separation region (note that the same grid was used for TUD-DES as for the TUD-HLR computations).

Experiment	HSU-HLR	TUD-HLR	KU-HLR	TUD-DES
Figure 33: HLR-results, contours of streamwise velocity at cross-sections x/h = 2, 5, 8, 12 and 15 of Diffuser 1

Contours of the streamwise velocity fluctuations are depicted in Fig. 34 for three cross-sections. As can be seen, both TUD HLR and HSU HLR deliver a reasonable agreement with the measurements. The level and location of the maxima are well captured. That is not the case for TUD DES and KU HLR which strongly overpredict the level of the r.m.s. values. Again, both simulations show a large coincidence. For a more detailed comparison, profiles of the mean and r.m.s. velocities were extracted at various locations in the flow field (see workshop proceedings at www.ercoftac.org, under SIG15). They support the trends found in the contour plots and are thus not reproduced here.

The DES method used by TUD is the one of Spalart et al., 1997 (denoted by DES97 in some references). The reasons for such a poor result could be an inappropriate position of interface between the near-wall RANS region (covered by the Spalart-Allmaras one-equation model) and the flow core simulated by LES depending solely on the numerical grid applied. In the DES- upgrades - Delayed DES and Improved Delayed DES - this issue is further elaborated, Spalart (2009). As already emphasized, the grid used presently is the same used also in TUD-HLR. No attempt to modify the grid for DES appropriately was undertaken.

Experiment	HSU-HLR	TUD-HLR	KU-HLR	TUD-DES
Figure 34: HLR-results, contours of streamwise rms velocity (urms/Ub × 100) at cross-sections x/h = 5, 8 and 12 of Diffuser 1

In conclusion, hybrid methods perform generally well for the separated flow in diffuser 1. DES was not expected to work well for such an internal flow and thus fulfills the expectations. Nevertheless, it remains unclear why the results of KU HLR strongly deviate from the other two hybrid approaches although, on first sight, the methods seem to be similar.

Diffuser 2

For the diffuser 2 the same hybrid methods were applied as for diffuser 1, except for DES (see Table 2). Furthermore, the grids applied are comparable to those compared for diffuser 1. Since neither experimental nor DNS data are available for the pressure distribution, the discussion starts with the contours of the time-averaged streamwise velocity at five cross-sections depicted in Fig. 35. As expected based on the experimental results, the shape of the separation bubble in diffuser 2 differs fundamentally from the recirculation zone found in diffuser 1. In contrast to diffuser 1, where the reverse-flow region spreads across the top wall, in diffuser 2, it remains localized near the sharp corner and the side wall. This feature is correctly reproduced by all three hybrid simulations. Nevertheless, the extensions of the recirculation regions differ. TUD HLR yields slightly too small zones compared to the measurements, whereas the zones predicted by HSU HLR are slightly too large and KU HLR shows no unique trend.

Experiment
HSU-HLR
TUD-HLR
KU-HLR
x/h=2	x/h=5	x/h=8	x/h=12	x/h=15
Figure 35: Figure 35: HLR-results, contours of streamwise velocity at cross-sections x/h = 2, 5, 8, 12 and 15 of diffuser 2

In comparison to diffuser 1, similar distributions of the rms values of the streamwise stress component (see cross-comparison in the files whose links is given at the end of this file) are found for HSU HLR and KU HLR. In contrast, for TUD HLR a strong reduction of the velocity fluctuations is observed in the streamwise direction which is clearly visible in the backmost cross-sections. The reason for this behavior is unclear, since the same method shows a different trend for diffuser 1. Unfortunately, higher- order statistics were not measured for this case and thus a final evaluation is difficult.

RANS

Numerous RANS models were applied ranging from some standard eddy-viscosity and full Reynolds stress models (e.g., standard k-? model, k-? SST, the basic differential Reynolds stress model due to Gibson and Launder, 1978, and a relevant quadratic version due to Speziale et al., 1991) to some explicit algebraic Reynolds stress model versions (EARSM) and linear/non- linear, EVM and RSM models based on the Durbin's elliptic relaxation method (ERM, 1991), see Tables 1 and 2 for detailed specification.

As important prerequisite for the successful computation is the correct capturing of the flow in the inflow duct which features secondary motion characterized by jets directed towards the channel walls bisecting each corner and associated vortices at both sides of each jet, see Fig. 22. These secondary currents are 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. That the latter model groups yields a qualitatively correct behaviour is shown in Fig. 22c

Fig. 36 shows the contour plots of the axial velocity component in two characteristic streamwise cross-sectional areas of diffuser 1 obtained by a selection of different RANS model versions, being representative of all applied model formulations. Whereas the initial separation zone development (x/h=5) follows qualitatively the reference results, its subsequent evolution exhibits different patterns depending on the model concept applied. The k-? SST model and the ?-f model (a numerically robust version of Durbin's v2-f model proposed by Hanjalic et al., 2004; the separation pattern obtained by both UoM versions of the v2-f model - Laurence et al., 2004 - follows closely the  ?-f results) resulted in a flow separating completely at the deflected side wall contrary to the experimental findings indicating the separation zone along the upper deflected wall. Similar results were obtained with all eddy-viscosity-based models listed in Table 1. Keeping in mind the inherent incapability of this model group to correctly represent the afore-mentioned secondary motion across the inflow duct, this outcome represents no surprise. The RSM model group returned the flow topology in much better agreement with the experimental results. Whereas the basic RSM model (denoted by GLRSM) resulted in a separation pattern occupying both upper corners (similar behaviour was documented in the case of the ANSYS BSL-RSM model) the application of both EARSM model versions (applied by ANSYS) and the Elliptic Blending RSM (a near-wall differential model based on the ERM method; Manceau and Hanjalic, 2002) returned the 3D separation pattern occupying entirely the upper sloped wall in good agreement with experiment.

	Experiment, x/h=5		Experiment, x/h=15
ANSYS SST	IUS ZETA-F	TUD-GLRSM	UoM RIJEBM	ANSYS-EARSM
Figure 36: Iso-contours of the axial velocity field in the cross planes y-z at two selected streamwise locations within the Diffuser 1 section obtained by different RANS models (thick line denotes the zero-velocity line) in comparison with experiments

Figs. 37 a) and b) illustrate the pressure coefficient development along the lower flat wall. The poor agreement (underprediction of the Cp- magnitude in the diffuser section) obtained by the eddy-viscosity model group in Fig. 37 a) (here only the predictions of application of the ERM- based EVM models are depicted) is consistent with the incorrect prediction of the velocity field. Apart from some unexpectedly large departures (pertinent especially to ANSYS-WJ model, which returned correctly the shape of the separation region, and to the UoM-RIJSSG model, application of which led to similar results as the TUD-GLRSM model; the absence of a wall reflection term in the latter can be blamed for the latter deviation) all other results agree reasonably well with the reference data. The differences between the results pertain partially to different grid resolutions. According to Table 1 most RANS computations were performed with grids comprising a comparable number of grid cells (between 1.6-1.9 Mio. cells; exceptions are UoM and OPU contributions). However, the solution domains adopted were of different lengths: e.g. the ANSYS colleagues adopted the straight outlet duct being almost 30h long - in all other cases this length amounted up to 10h; accordingly the resolution in the diffuser section - despite the total number of the grid cells being about 1.6 Mio. - was somewhat lower). The large departures obtained by two advanced models - Non-linear k-? model and Two-Component Limit RSM model in conjunction with analytical wall functions, Table 1 - applied by OPU/UniOs cannot be plausibly discussed here due to the extremely coarse grid containing only 0.2 Mio. cells. Unfortunately the contributors didn't make an attempt to refine the grid appropriately. The streamwise fluctuation intensity field predicted by the RANS models will not be discussed here. It can only be said that the results obtained by the three RSM models (see iso- contours of the axial velocity field displayed in Fig. 36) are in qualitatively good agreement with the experiment indicating intensified turbulence production in the regions bordering the separation zone characterized by large velocity gradients.

Figure 37a: Pressure coefficient evolution in diffuser 1 along the bottom flat wall obtained by advanced eddy-viscosity model schemes: elliptic-relaxation-method-based models and a non-linear model

Figure 37b: Pressure coefficient evolution in diffuser 1 along the bottom flat wall obtained by the Reynolds-stress model concept

The velocity field characterizing the flow structure in the diffuser 2 is illustrated in Fig. 38 by displaying the iso-contours of the axial velocity field in the cross planes y-z at three selected streamwise locations. As discussed previously (sections devoted to DNS/LES and HLR results) the separation zone here is completely situated at the deflected side wall. The application of the most RANS models to the diffuser 1 configuration has incorrectly resulted in a separation pattern pertinent to diffuser 2 (see Fig. 36). The results depicted in Fig. 38 exhibiting, at least qualitatively, the experimentally determined flow structure are therefore "expected". However, the differences in the shape and size of the recirculation zone are obvious. The linear ?-f model (UniRo contributions) resulted in a by far too large recirculation zone leading to intensive flow acceleration in the through-flow portion, i.e. positive-velocity region. This result can be regarded as representative for all other linear EVM models used. Important improvement was obtained by applying a non-linear formulation of the ?-f model. The recirculation region is substantially reduced to yield a much better quantitative agreement with the experimental findings. The separation bubble obtained by selected RSM models shows a shape that is closest to the experimental results, but there are important differences concerning some details. All three RSM models resulted in the separation region to be situated in the corners between two deflected walls and between the sloped side wall and the lower flat wall, contrary to the experimental finding. However, these tiny separated regions are of low backflow intensity, so that the quantitative agreement can be regarded as reasonable (this statement is valid also for the diffuser 1). For a more quantitative comparison the readers are referred to the "workshop proceedings".

It should also be emphasized that the near-wall treatment was not of decisive importance. In this configuration the flow unsteadiness was introduced into the wall boundary layer from the core flow in accordance with the so-called "top-to-bottom" process (communication with M. Leschziner). This fact justified the use of wall functions in conjunction with some RANS models (it is also valid for some LES simulations, see e.g. ITS contribution) enabling a coarser grid resolution in the near-wall regions.

Experiment, x/h=5		Experiment, x/h=8	Experiment, x/h=15
UniRo-ZF	UniRo ZF-NL	TUD-GLRSM	UoM RIJEBM	ANSYS-EARSM
Figure 38: Selection of RANS-results, contours of streamwise velocity at cross-sections x/h = 5, 8 and 15 of Diffuser 2 (the dark blue area in the ANSYS-EARSM computation represents the reverse flow region)

Available CFD results: ERCOFTAC SIG15 Workshop Proceedings

The following documents are freely available (see also www.ercoftac.org under SIG15):

• Short summaries comprising the computational method descriptions used by participating computational groups from the 13th Workshop (Technical University Graz, Austria, September, 2008) - 13th-Workshop_summaries.pdf
• Short summaries comprising the computational method descriptions used by participating computational groups from the 14th Workshop ("La Sapienza" University of Rome, Italy, September, 2009) - 14th-Workshop_summaries.pdf
• Diffuser 1: List of the participants and computational models applied - Diffuser1_contributers.pdf
• Diffuser 1: cross-comparison of the results obtained by LES and Hybrid LES/RANS methods - Diffuser1_LES-and-HLR.pdf
• Diffuser 1: cross-comparison of the results obtained by RANS method: Eddy-viscosity models (only linear model formulations) - Diffuser1_RANS-EVM.pdf
• Diffuser 1: cross-comparison of the results obtained by RANS method: Eddy-viscosity models (formulations based on the Elliptic-relaxation method) - Diffuser1_RANS-EVM-ERM.pdf
• Diffuser 1: cross-comparison of the results obtained by RANS method: Reynolds stress models - Diffuser1_RANS-RSM.pdf
• Diffuser 2: List of the participants and computational models applied - Diffuser2_contributers.pdf
• Diffuser 2: cross-comparison of the results obtained by LES and Hybrid LES/RANS methods - Diffuser2_LES-and-HLR.pdf
• Diffuser 2: cross-comparison of the results obtained by RANS method: Eddy-viscosity models - Diffuser2_RANS-EVM.pdf
• Diffuser 2: cross-comparison of the results obtained by RANS method: Reynolds stress models - Diffuser2_RANS-RSM.pdf

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

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