*3.2. Computational Meshes*

To assess the influence of the mesh resolution and find relatively coarser mesh for the asymmetric planar diffuser, simulations were performed on seven different collocated meshes, as given in Table 1. Here, *uτ* is the wall friction velocity which is found at the inlet dust, and the corresponding Reynolds number based on *uτ* is *Reτ* = 407. In the DNS and from M1 to M5, uniform meshes are applied in the streamwise and spanwise direction for the driver and spatially-developing region, in which the mesh gradually becomes coarser from M1 to M5 in the streamwise direction while the grids remain the same in the other two directions. The nonuniform mesh are used in the streamwise direction for M6 in spatially-developing region and designed such that the spacing gradually increases from the diffuser throat toward the downstream join and the spacing gradually decreases from the inlet dust to the diffuser throat. This kind of design is necessary to resolve the sharp mean gradients in the diffuser throat for LES when the whole grids are relatively coarse. To resolve the boundary layers, the meshes in the wall normal direction distribute non-uniformly more densely near walls for all cases in both the driver and spatially-developing region. What is more, not only the boundary layers upstream but also downstream of the diffuser should be resolved in wall normal direction.

**Table 1.** Mesh resolution of simulations : *Nx*, *Ny*, and *Nz* denote the number of cells in streamwise, wall-normal and spanwise direction, respectively. Δ*x*+, <sup>Δ</sup>*y*+, and Δ*z*<sup>+</sup> are grid spacing used in simulations normalized by *<sup>ν</sup>*/*<sup>u</sup>τ*. Super/subscripts of 1 and 2 represent driver and spatially-developing region, respectively.


## *3.3. Solution Strategy*

A fourth-order central finite-difference discretization scheme is used for the incompressible (filtered) continuity equation and (filtered) Navier–Stokes equation in the (LES) DNS, in which the fractional method is selected for coupling the continuity equation and the pressure field, the second-order Adams–Bashforth method is used to the convective term and viscous term, and the backward Euler method to pressure term. A fourth-order central difference schemes is applied in the Smagorinsky model. For the dynamic procedure, the test filter is used in the streamwise direction and spanwise direction with second-order accuracy and the test-to-grid filter ratio Δ ˜ /Δ = 2. For the transport equation of the SGS kinetic energy, Crank–Nicolson method is utilized to dissipation term, the second-order Adams–Bashforth method to convective and diffusive terms. The initialization data of *ksgs* is solved from *ksgs* = *<sup>ν</sup>sgs*/*Cν*<sup>Δ</sup> 2 using the results of *<sup>ν</sup>sgs* from the dynamic Smagorinsky model. The present numerical method and computer program have been tested extensively in several turbulent flows [5,30–32].

#### **4. Results and Discussion**

All simulations were computed on an NEC SX-8R supercomputer of Cybermedia Center, Osaka University with the time step *dt* = 0.0495 *H*/*Uc*. The grea<sup>t</sup> mass of the total effort of calculation was spent on solving the Poisson equation through the residual cutting method [33]. All simulations were run until the flow fields were fully developed and the first-order and second-order statistics exhibited adequate convergence. All results were collected by time averaging and spatial averaging in the spanwise direction. To allow a good comparison of simulation results and experimental measurements, the data associated with vertical cross-sections *x*/*H* = 9.2, 15.2, 19.2 and 25.2 in the spatially-developing region, which match the location used in previous experiments, are combined into one plot. Note that the results of our DNS for the diffuser agree quite well with the DNS results of Ohta & Kajishima [5]. For the validation discussed in this section, we restrict ourselves to use our own DNS as a comparison with three sets of LESs.

#### *4.1. Comparison of Mesh Resolution*

Figures 3–6 show the axial mean velocity and axial Reynolds stress profiles for the mesh sensitivity study. All simulations used the LES with the standard Smagorinsky model due to its low computational cost. In Figures 3 and 4, profiles of *U* and *<sup>u</sup>*, *u*, agree well with the measurement for the mesh resolution of M1 and M2, while the situation for the mesh resolution of M3 and M4 is reversed, i.e., the deviation between simulations and experiment is large. Thus, the mesh resolution of M1 and M2 rather than M3 and M4 is acceptable for the LES of the diffuser. Furthermore, differences between M1 and M2 are not apparent, although mesh of M2 is relatively coarser than M1. In Figure 4, M1 and M2 both underpredict the peak value of Reynolds stress *<sup>u</sup>*, *u*, and represent a large deviation in the region close to the inclined wall compared with the experimental data. Since this deviation does not decrease much with increasing mesh resolution, we suspect the standard Smagorinsky model itself and the blocking effect of the sidewall boundary layers in the experiment to contribute to this disagreement. Through the previous analysis mesh resolution of M2 and the grid spacing Δ*x*<sup>+</sup> = 50.38 is therefore used as a benchmark mesh and a benchmark axial grid spacing in the inlet of the diffuser, respectively. Based on the benchmark mesh and axial grid spacing, a nonuniform mesh (that is, M6) is designed and tested. In Figures 5 and 6, we compare the profiles of *U* and *<sup>u</sup>*, *u*, from simulation and experiment for the different wall-normal mesh resolution and the nonuniform mesh, which shows a similar trend with M1 and M2 in Figures 3 and 4. With increasing wall-normal mesh refinement from M2 and M5 an increasing agreemen<sup>t</sup> of the LES with the experimental data can be stated, especially in the region close to the walls, while this improvement is quite small. Even the coarsest mesh simulation, i.e., M6, demonstrates a good overall agreemen<sup>t</sup> with the experimental data. Thus, for the study of different SGS model performance for separating flow in the diffuser, the coarsest mesh M6 is used since the influence of the SGS model on the results is largest on this mesh.

**Figure 3.** Comparison of LES with different mesh resolution in the streamwise direction and experimental results of Obi in terms of mean streamwise velocity profiles.

**Figure 4.** Comparison of LES with different mesh resolution in streamwise direction and experimental results of Obi in terms of axial Reynolds stress profiles.

**Figure 5.** Comparison of LES with different mesh resolution and experimental results of Obi in terms of axial mean velocity profiles.

**Figure 6.** Comparison of LES with different mesh resolution and experimental results of Obi in terms of axial Reynolds stress profiles.

#### *4.2. Comparison of Subgrid Modeling*

The DNS as well as five sets of LESs, i.e., SM, DSM, OM, ODM, and VOM, were computed on an NEC SX-8R supercomputer of Cybermedia Center (CMC), Osaka University. The grea<sup>t</sup> mass of the total effort of calculation was spent on solving the Poisson equation through the residual cutting method [33]. The time step used for these computation is about *dt* = 0.0495*H*/*Uc*. On a node of CMC, 13.188 s CPU time are needed to advance the computation one time step for the DNS, 1.071 s CPU time for the LES with SM model, 1.254 s CPU time for the LES with DSM model, 1.278 s CPU time for the LES with OM model, 1.391 s CPU time for the LES with ODM model, and 1.288 s CPU time for the LES with VOM model.

#### 4.2.1. Comparison of Mean Properties

The profiles of the mean velocity for the streamwise direction and the wall-normal direction non-dimensionalized by the mean center velocity at the inlet duct are shown in Figures 7 and 8, respectively, where five sets of LESs corresponding to the SM model, DSM model, OM model, ODM model and our OVM model are compared with DNS and experimental data of Obi. Overall, in Figure 7, the agreemen<sup>t</sup> of profiles of streamwise mean velocity between five LESs and experiment is quite good at all locations. The slight advance of LES velocity profile corresponding to OVM model in comparison with experimental values exists in the vicinity of the flat wall, whereas the situation adjacent to the inclined wall is reversed, i.e., the LES velocity profiles exhibit the slight lag in comparison with the measurement. This situation of velocity profiles for OVM model compared with the experiment is similar to that of DNS, but the latter exhibits more obvious difference. It seems the SM, DSM, OM and ODM models show a better overall agreemen<sup>t</sup> with the measurement than OVM model and DNS. Meanwhile, LESs exhibit better overall match with experiment than DNS. However, as discussed previously, agreemen<sup>t</sup> with the measurement is not a first priority. The results of DNS are should be used as benchmarks for SGS eddy viscosity models as well. From this perspective, the performance of OVM model in the prediction of mean streamwise velocity is slightly better than the four other SGS models since the OVM model agrees well with DNS database. Actually, the blocking effect of the sidewall boundary layers in experiment is much more pronounced for mean velocity in the wall-normal direction and turbulent stresses, which are discussed below. According to the mean velocity in the wall-normal direction from DNS and LESs, see Figure 8, the mean flow is directed from the straight (upper) wall towards the inclined wall in the flow filed except for some small regions adjacent to the inclined wall. Furthermore, the peak velocity *Vmax* of DNS and LESs is located in the center region of the flow filed. However, the strong flows toward the inclined wall and the straight wall are both observed in experimental data. It has been shown [34] that the influence of the side wall of the channel on the flow generated a secondary flow in the laboratory experiment, being markedly so especially in the case of a low Reynolds number. However, the agreemen<sup>t</sup> between simulations and measurement is good in the vicinity of the inclined wall. Compared with DNS, five SGS eddy viscosity models overpredict the mean velocity *V* near the diffuser throat *x*/*H* = 9.2 and underpredict *V* in the other regions *x*/*H* = 15.2, 19.2 and 25.2, except for the location at 25.2 corresponding to the DSM, ODM and OVM models. Clearly, overall, the best agreemen<sup>t</sup> is observed between the OVM model and DNS among five SGS eddy viscosity models.

**Figure 7.** Comparison of LES with different SGS models, DNS and experimental results of Obi in terms of axial mean velocity profiles.

**Figure 8.** Comparison of LES with different SGS models, DNS and experimental results of Obi in terms of wall-normal mean velocity profiles.

4.2.2. Comparison of Turbulent Stresses and Resolved Turbulent Kinetic Energy

In Figures 9–12, we compare the corresponding profiles of the turbulent shear stress *<sup>u</sup>*,*v*,, and turbulent normal stresses *<sup>u</sup>*,*u*,, *<sup>v</sup>*,*v*,, *<sup>w</sup>*,*w*, from simulations and experiment, where the turbulent stresses normalized by the square of mean center velocity *Uc*, <> refers to time averaging and spatial averaging in the homogenous direction. Note that the simulation results of three SGS eddy viscosity models are the summation of the resolvable portion and SGS portion, which coincides with the DNS and experiment. It has been shown [6] that experimental errors are higher for turbulent stresses than the mean flow velocities, in particular at the regions in which measurement volumes are

large in comparison with the local gradients of turbulent stresses. However, overall, good agreemen<sup>t</sup> between the computations and experiment is observed at locations in the vicinity of either walls. As shown in Figure 9, turbulent shear stresses from three SGS eddy viscosity models, the DNS and experiment all exhibit a characteristic shape with a double peak. The locations of the peak value of all three SGS eddy viscosity models agree well with the DNS, whereas they are higher compared with measurement. All five SGS eddy viscosity models underpredict the peak value of turbulent shear stress compared with DNS, in which the performance of the DSM and ODM models are better than that of the SM and OM models, the OVM model demonstrates the best agreement. This can be seen especially at the location of *x*/*H* = 15.2 and 19.2. Compared with measurement, all five SGS eddy viscosity models show a better agreemen<sup>t</sup> in the region close to the straight wall than the inclined wall. In the center region of the diffuser, the SM, OM, DSM and ODM models underpredict the turbulent shear stress in comparison with experiment, whereas the performance of the OVM model is a little bit of opposite. As shown in Figure 10, profiles of *<sup>u</sup>*,*u*, for the five SGS eddy viscosity models exhibit more prominent difference in the center region of the diffuser than adjacent to either walls, where the deviation between the SM model and the DNS or measurement is largest. Compared with the DNS as well as measurement, the SM, OM and DSM models underpredict the value of *<sup>u</sup>*,*u*, at all locations, except for some of small regions near the diffuser throat and further downstream. The slight advance of *<sup>u</sup>*,*u*, from the OVM model compared with experiment is observed in the region of flow interior. Obviously, not only the overall but also the local agreemen<sup>t</sup> between the OVM and ODM models and DNS / experiment is better than other three SGS eddy viscosity models. With respect to profiles of *<sup>v</sup>*,*v*, and *<sup>w</sup>*,*w*,, an overall better agreemen<sup>t</sup> between the LESs and the DNS is observed compared with that of turbulent shear stress and normal stress *<sup>u</sup>*,*u*,. A characteristic double-peak shape can still be seen for profiles in Figures 11 and 12. The location of the peak value moves away from the wall into the flow interior with increasing distance from the diffuser throat. While the peak value of the turbulent stress *<sup>v</sup>*,*v*, from DNS as well as LESs gravely deviates from the measurement, locations of the peak value are close to each other and coincide with the measurement, except for the location at *x*/*H* = 9.2. In comparison with the DNS, SM model, DSM model, OM model and ODM model underpredict *<sup>v</sup>*,*v*, and *<sup>w</sup>*,*w*, at all locations, except for some of small regions adjacent to straight wall, whereas the OVM model slightly overpredicts both of them in the vicinity of the inclined wall. Overall, the OVM model agree better with the DNS than other four SGS eddy viscosity models. Figure 13 shows the profiles of the resolved turbulent kinetic energy *K* non-dimensionalized by the square of the mean center velocity. All five SGS eddy viscosity models capture the locations as well as the magnitude of the double-peak in *K*. Compared with the DNS, the five SGS models accurately predict the near-wall peak of the straight wall except for the location at *x*/*H* = 25.2, but underpredict the larger peak value at all location. In particular, the SM model and the OM model do not correctly capture the location of the larger peak for the locations at *x*/*H* = 15.2. The OVM model agrees well with the DNS with respect to the location as well as the value of the double-peak in the resolved turbulent kinetic energy.

**Figure 9.** Comparison of LES with different SGS models, DNS and experimental results of Obi in terms of Reynolds shear stress profiles.

**Figure 10.** Comparison of LES with different SGS models, DNS and experimental results of Obi in terms of Reynolds stress *<sup>u</sup>*,*u*, /*U*2*c*profiles.

**Figure 11.** Comparison of LES with different SGS models, DNS and experimental results of Obi in terms of Reynolds stress *<sup>v</sup>*,*v*, /*U*2*c* profiles.

**Figure 12.** Comparison of LES with different SGS models, DNS and experimental results of Obi in terms of Reynolds stress *<sup>w</sup>*,*w*, /*U*2*c* profiles.

**Figure 13.** Comparison of LES with different SGS models, DNS and experimental results of Obi in terms of resolved turbulent kinetic energy profiles.
