## Wall-stress models

There are two major components to the implementation of a wall-stress model in an LES code:

1. The wall-stress model itself links an instantaneous wall-parallel velocity magnitude (i.e., a scalar value) $$U_\|$$ at some wall-normal distance $$h_{\rm wm}$$ to the resulting instantaneous wall-parallel shear stress magnitude $$\tau_{w, \|}$$. For compressible solvers, one would also need the pressure and temperature at distance $$h_{\rm wm}$$, and one would also produce the wall temperature $$T_w$$ or the wall heat flux $$q_w$$, depending on the boundary condition. In any case, this wall-stress model can (and should) be implemented as a separate module (or class, or function) in the code, since it exchanges very little information with the actual LES code (a few scalar values per wall location).
2. The coupling of the wall-stress model with the LES code requires more intrusive changes in the LES code. One must find the location corresponding to a distance $$h_{\rm wm}$$ from the wall, interpolate the velocity to that location, decompose the velocity vector into components, and then (after having called the wall-model function), assemble the wall-stress vector and use this as the boundary condition in the LES code.

Step 2, the coupling with the LES code, is code-specific. We will describe a typical implementation below, but this should be viewed as an illustration only.

Step 1 is where the details of any specific wall-stress model go. This step is specific to the wall-stress model, but not to the actual LES code; therefore, a properly written wall-stress model function should be transferable between LES codes with essentially no changes needed. We describe a few wall-stress models at this site, including an algebraic one and an ODE-based one.

The wall-stress model (Step 1) is highly encapsulated, and thus it could (and should!) be verified in isolation before coupling with the LES code. Some such code-verification cases are provided here.

The single most important factor in producing accurate WMLES solutions is the LES grid. The fact that the LES grid is more important than the wall-stress model itself is perhaps counter-intuitive, but is nevertheless true. Therefore, it is crucial to understand the grid requirements for WMLES.