Algebraic Wall-Modeled LES Model (WMLES)

While widely used in the academic community, LES has had very limited impact on industrial simulations.

The reason lies in the excessively high resolution requirements for wall boundary layers. Near the wall, the largest scales in the turbulent spectrum are nevertheless geometrically very small and require a very fine grid and a small time step. In addition, unlike RANS, the grid cannot only be refined in the wall normal direction, but also must resolve turbulence in the wall parallel plane. This can only be achieved for flows at very low Reynolds number and on very small geometric scales (the extent of the LES domain cannot be much larger than 10-100 times the boundary layer thickness parallel to the wall).

For this reason, using LES is only recommended for flows where wall boundary layers are not relevant and need not be resolved, or for flows where the boundary layers are laminar due to the low Reynolds number.

However, there are only very few such flows and other approaches need to be employed. A promising approach to overcome the Reynolds number scaling limitations of LES is the algebraic Wall-Modeled LES (WMLES) approach [421] (p. 782). In WMLES, the RANS portion of the model is only activated in the inner part of the logarithmic layer and the outer part of the boundary layer is covered by a modified LES formulation. Since the inner portion of the boundary layer is responsible for the Reynolds number dependency of the LES model, the WMLES approach can be applied at the same grid resolution to an ever increasing Reynolds number for channel flow simulations.

Turbulence

Note that for wall boundary layers, the Reynolds number scaling is not entirely avoided, as the thickness of the boundary layer declines relative to body dimensions with increasing Reynolds number. Assuming a certain number of grid nodes per ‘boundary layer volume’, the overall grid spacing will decrease and the overall number of cells will increase with the Reynolds number.

Another interesting option, and possibly the best use of the WMLES option, is when it is combined with the Embedded LES (ELES) implementation in ANSYS Fluent. ELES limits the LES zone to small regions of the domain and covers the less critical areas with RANS. WMLES extends this functionality to high Reynolds numbers in the LES domain. To apply this option, refer to Setting Up the Large Eddy Simulation Model and Setting Up the Embedded Large Eddy Simulation (ELES) Model in the User’s Guide.

4.14.2.4.1. Algebraic WMLES Model Formulation

The original Algebraic WMLES formulation was proposed in the works of Shur et.al.[421] (p. 782). It combines a mixing length model with a modified Smagorinsky model [430] (p. 782) and with the wall-damping function of Piomelli [366] (p. 779).

In the Shur et al. model [421] (p. 782), the eddy viscosity is calculated with the use of a hybrid length scale:

(4.284) where is the wall distance, is the strain rate, and are constants, and is the normal to the wall inner scaling. The LES model is based on a modified grid scale to account for the grid anisotropies in wall-modeled flows:

(4.285) Here, is the maximum edge length for a rectilinear hexahedral cell (for other cell types and/or

conditions an extension of this concept is used). is the wall-normal grid spacing, and is a constant.

The above model has been optimized for use in ANSYS Fluent but remains within the spirit of the ori-ginal model.

4.14.2.4.1.1. Reynolds Number Scaling

The main advantage of the WMLES formulation is the improved Reynolds number scaling. The classical resolution requirements for wall resolved LES is

(4.286) where

(4.287) and is the number of cells across the boundary layer, h, (or half channel height h) and ν is the kin-ematic viscosity. The wall friction velocity is defined as

(4.288) where is the wall shear stress and is the density.

The resolution requirement for WMLES is

Large Eddy Simulation (LES) Model

(4.289)

Again with for a channel (where H is the channel height), or (where is the boundary layer thickness).

Considering a periodic channel (in the span and streamwise direction) with the geometry dimensions:

(4.290) you obtain the following cell numbers ( total number of cells in channel) for the wall-resolved LES and WMLES:

Wall-Resolved LES:

Table 4.1: Wall-Resolved Grid Size as a Function of Reynolds Number 500

Table 4.2: WMLES Grid Size as a Function of Reynolds Number 500

The Reynolds number is based on the friction velocity:

(4.291) From the above tables, it can be seen that the quadratic Reynolds number dependence of the

Wall-Resolved LES has been avoided by WMLES and that the CPU effort is substantially reduced.

However, it is still important to remember that WMLES is much more computationally expensive than RANS. Considering a wall boundary layer flow, with WMLES one must cover every boundary layer volume

with 10 x (30-40) x 20 = 6000-8000 cells in the stream-wise, normal, and span-wise direction.

RANS can be estimated to 1 x (30-40) x 1 =30-40 cells. In addition, RANS can be computed steady-state whereas WMLES requires an unsteady simulation with CFL~0.3.

4.14.2.4.2. Algebraic WMLES S-Omega Model Formulation

One of the deficiencies of the WMLES approach is that the model does not provide zero eddy-viscosity for flows with constant shear when you use a modified Smagorinsky model in the LES zone. For this reason, the WMLES model does not allow the computation of transitional effects, and can produce overly large eddy-viscosities in separating shear layers (like from a backstep).

Turbulence

A way to enhance the WMLES formulation given in Equation 4.284 (p. 107) is to compute the LES portion of the model using the difference instead of , where is the strain rate and is the vorticity magnitude. This enhancement is referred to as the WMLES - formulation.