Note on RANS, Hybrid RANS/LES, WRLES, WMLES and SGS Models

 In the computation of turbulent flow, there are three main approaches: Reynolds averaged Navier-Stokes (RANS), large eddy simulation (LES), and direct numerical simulation (DNS). LES and DNS belong to the scale-resolving methods, in which some turbulent scales (or eddies) are resolved rather than modeled. In contrast to LES, all turbulent scales are modeled in RANS.

Another scale-resolving method is the hybrid RANS/LES approach, in which the boundary layer is computed with a RANS approach while some turbulent scales outside the boundary layer are resolved, as shown in Figure 1. In this figure, the red arrows denote resolved turbulent eddies and their relative size. 

Depending on whether near-wall eddies are resolved or modeled, LES can be further divided into two types: wall-resolved LES (WRLES) and wall-modeled LES (WMLES). To resolve the near-wall eddies, the mesh needs to have enough resolution in both the wall-normal (y+ ~ 1) and wall-parallel directions (x+ and z+ ~ 10-50) in terms of the wall viscous scale as shown in Figure 1. For high-Reyolds number flows, the cost of resolving these near-wall eddies can be prohibitively high because of their small size. 

In WMLES, the eddies in the outer part of the boundary layer are resolved while the near-wall eddies are modeled as shown in Figure 1. The near-wall mesh size in both the wall-normal and wall-parallel directions is on the order of a fraction of the boundary layer thickness. Wall-model data in the form of velocity, density, and viscosity are obtained from the eddy-resolved region of the boundary layer and used to compute the wall shear stress. The shear stress is then used as a boundary condition to update the flow variables.    

Figure 1. An illustration of RANS, hybrid RANS/LES, WRLES, and WMLES approaches

I discussed the role of sub-grid scale (SGS) models for WRLES in an earlier blog post in 2017. For adaptive high-order methods such as the discontinuous Galerkin (DG), spectral difference (SD), and flux reconstruction/correction procedure via reconstruction (FR/CPR) methods, the best results are obtained without any explicit SGS models (called implicit LES or ILES) because of the embedded numerical dissipation. With enough mesh resolution, the physical and numerical dissipations appear sufficient to stabilize WRLES.

In WMLES, the near-wall turbulent scales are severely under-resolved. As a result, ILES is usually not stable without other stabilizing techniques such as filtering, limiting et al. We recently experimented with explicit SGS models such as the Smagorinsky, WALE, and Vreman models for WMLES. Through comparison with channel flow DNS data, the Vreman model was found to achieve the most accurate and consistent results. 

In summary, we advocate ILES for WRLES and the Vreman model for WMLES with adaptive high-order methods such as DG, SD, and FR/CPR.