### 12.1.6. Wall Modeled Large Eddy Simulation (WMLES)

Wall Modeled LES (WMLES) is an alternative to classical LES and reduces the stringent and number-dependent grid resolution requirements of classical wall-resolved LES (Section 12.1.5.1) The principle idea is depicted in Figure 12.13. As described in Section 12.1.5.1, the near-wall turbulence length scales increase linearly with the wall distance, resulting in smaller and smaller eddies as the wall is approached. This effect is limited by molecular viscosity, which damps out eddies inside the viscous sublayer (VS). As the number increases, smaller and smaller eddies appear, since the viscous sublayer becomes thinner. In order to avoid the resolution of these small near-wall scales, RANS and LES models are combined such that the RANS model covers the very near-wall layer, and then switches over to the LES formulation once the grid spacing becomes sufficient to resolve the local scales. This is seen in Figure 12.13 (b), where the RANS layer extends outside of the VS, thus avoiding the need to resolve the inner second row of eddies depicted in the sketch.

Figure 12.13: Concept of WMLES for high Re number flows (a) Wall-resolved LES. (b) WMLES

The WMLES formulation in ANSYS CFD is based on the formulation of Shur et al., 2008 [27]:

| (12–26)

where is the wall distance, is the von Karman constant, is the strain rate and is a near-wall damping function. This formulation was adapted to suit the needs of the ANSYS general purpose CFD codes. Near the wall, the min-function selects the Prandtl mixing length model whereas away from the wall it switches over to the Smagorinsky model. Meshing requirements for the WMLES approach are given in Section 12.2.3.3.

For wall boundary layer flows, the resolution requirements of WMLES depend on the details of the model formulation. In ANSYS Fluent and ANSYS CFX they are (assuming for this estimate that is the streamwise, the wall normal and the spanwise direction as shown in Figure 12.14):

| (12–27)

where , , and are the numbers of cells per boundary layer thickness, , in the streamwise, wall normal, and spanwise directions respectively, (see Figure 12.14). About 6000-8000 cells are needed to cover one boundary layer volume . This is also the minimal resolution for classical LES models at low Reynolds numbers. Actually, for low Reynolds numbers, WMLES turns essentially into classical LES. The advantage of WMLES is that the resolution requirements relative to the boundary layer thickness remain independent of the Reynolds number.

While WMLES is largely Reynolds number-independent for channel and pipe flows (where the boundary layer thickness must be replaced by half of the channel height) there remains a Reynolds number sensitivity for aerodynamic boundary layer flows, where, the ratio of the boundary layer thickness, , to a characteristic body dimension, , is decreasing with increasing Reynolds number. In aerodynamic boundary layer flows, there are more boundary layer volumes to consider at increased Reynolds numbers. It should also be noted that despite the large cost savings of WMLES compared to wall-resolved LES, the cost increase relative to RANS models is still substantial. Typical RANS computations feature only one cell per boundary layer thickness in streamwise and spanwise directions (). In addition, RANS steady-state simulations can be converged in the order of ~102–103 iterations, whereas unsteady simulations typically require ~104–105.

For wall-normal resolution in WMLES, it is recommended that you use grids with at the wall. If this cannot be achieved, the WMLES model is formulated to tolerate coarser values (-insensitive formulation) as well.

Figure 12.14: Sketch of boundary layer profile with thickness , x-streamwise direction, y-normal direction, and z-spanwise direction

For channel and pipe flows, the above resolution requirements for the boundary layer should be applied, only replacing the boundary layer thickness, , with half the channel height, or with the pipe radius in the grid estimation. This estimate would result in a minimum of ~120 cells in the circumferential direction () for a fully developed pipe flow.

It should be noted that reductions in grid resolution similar to WMLES can be achieved with classical LES models when using LES wall functions. However, the generation of suitable grids for LES wall functions is very challenging as the grid spacing normal to the wall and the wall-parallel grid resolution requirements are coupled and strongly dependent on number (unlike RANS where only the wall-normal resolution must be considered).

In ANSYS Fluent, the WMLES formulation can be selected as one of the LES options; in ANSYS CFX it is always activated inside the LES zone of the Zonal Forced LES (ZFLES) method.

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