The grid (for the LES) is just as important for wall-modeled LES (WMLES) as it is for traditional LES — the only difference is that the grid requirements are different. Failure to satisfy the grid requirements means that significant errors will be produced in the LES itself — these errors are not the fault of the wall-model, and make it impossible to say anything conclusive about the performance of the wall-model being tested. Thus it is absolutely crucial to remove or reduce these errors when assessing or validating or investigating wall-models. In engineering situations, one may have a higher tolerance for errors and thus decide to use slightly coarser grids; this is of course perfectly fine, but one should make this decision conciously. In academic situations, it is hard to think of any circumstance under which it would be OK to violate these grid criteria.

In the following we discuss the grid requirements for WMLES using the wall-stress modeling approach. We discuss only the requirements for the LES solver; the requirements for the wall-stress model itself are discussed in the sections on the specific wall-stress models.

The grid requirements for the LES stem from the need for the grid to resolve the energetic scales of turbulence in the outer layer. Since those eddies have a size that scales with \(\delta\) (the boundary layer thickness), the grid requirements are necessarily in terms of \(\delta\). In addition, the grid must resolve the energetic eddies in the log-layer all the way down to height \(h_{\rm wm}\) where the wall-model takes information from; since the size of those log-layer eddies scales with wall-distance, we get additional grid requirements in terms of \(h_{\rm wm}\) (the lowest wall-distance at which we need accurate turbulence, and thus the most stringent grid requirement).

Satisfying the criteria in both the outer layer and in the log-layer yields

\[

\Delta x \lesssim \left\{

\begin{array}{ll}

0.05\delta – 0.10\delta &\,, \ {\rm outer\ layer} \\

0.6h_{\rm wm} – 1.0h_{\rm wm} &\,, \ {\rm log-layer}

\end{array} \right.

\]

\[

\Delta y \lesssim \left\{

\begin{array}{ll}

0.01\delta – 0.04\delta &\,, \ {\rm outer\ layer} \\

0.2h_{\rm wm} – 0.3h_{\rm wm} &\,, \ {\rm log-layer}

\end{array} \right.

\]

\[

\Delta z \lesssim \left\{

\begin{array}{ll}

0.04\delta – 0.08\delta &\,, \ {\rm outer\ layer} \\

0.4h_{\rm wm} – 0.8h_{\rm wm} &\,, \ {\rm log-layer}

\end{array} \right.

\]

where the ranges of the critical values come from tests in 4 different LES codes: (i) the *Hybrid* code which is a nominally 6th-order accurate finite-difference code with low numerical dissipation; (ii) a modified version of the *FDL3DI* code which uses high-order “compact” (Pade) schemes and dealiasing filters (cf. Kawai and Larsson, 2012); (iii) the unstructured finite-volume code *CharlesX* code which is between 1st- and 3rd-order accurate depending on the quality of the grid (cf. Larsson et al, 2015); (iv) a 2nd-order, incompressible, staggered-grid finite-difference code (Lee et al, 2013).

A corollary to these grid criteria is that the cross-over point happens for \(h_{\rm wm} \approx 0.1\delta\), and thus this is the value that we typically target.

There has been some confusion about these criteria in the published literature, with some authors referring to them as the “third grid-point rule”. This stems from the log-layer criterion on \(\Delta y\), but it is crucial to note that this is just one of six criteria that must be satisfied! In our view, it is better to view these criteria as, for given \(\delta\) and \(h_{\rm wm}\), these are the grid-spacings below which the WMLES solution will not change by much (i.e., will be essentially grid-converged, in a statistical sense).

### References

- Kawai, S., and Larsson, J., 2012, “Wall-Modeling in Large Eddy Simulation: Length Scales, Grid Resolution, and Accuracy,” Phys. Fluids, p. 015105.
- Larsson, J., Laurence, S., Bermejo-Moreno, I., Bodart, J., Karl, S., and Vicquelin, R., 2015, “Incipient Thermal Choking and Stable Shock-Train Formation in the Heat-Release Region of a Scramjet Combustor. Part II: Large Eddy Simulations,” Combustion and Flame, pp. 907–920.
- Lee, J., Cho, M., and Choi, H., 2013, “Large Eddy Simulations of Turbulent Channel and Boundary Layer Flows at High Reynolds Number with Mean Wall Shear Stress Boundary Condition,” Physics of Fluids, p. 110808.