## Incompressible channel flow at Re=5200

Turbulent channel flow is a basic validation problem for a WMLES approach. The periodic boundary conditions remove uncertainties and errors due to inflow turbulence, and the geometric simplicity means that one can easily run on a sequence of high-quality grids to really assess grid-convergence.

### Basic set-up

We use the DNS by Lee and Moser (2015) as a validation case. The DNS is performed using the incompressible equations; in the results below, we have solved it at bulk Mach number of 0.2 using compressible solvers. The DNS was solved at specified bulk Reynolds number of $$Re_b = 125,000$$, resulting in a friction Reynolds number of $$Re_\tau \approx 5200$$. The domain used in the DNS was $$(8\pi h, 2h, 3\pi h)$$ in the streamwise, wall-normal and spanwise directions (where $$h$$ is the channel half-width). Since the purpose here is to validate WMLES, we are only interested in the region immediately above the wall-model, say, for $$y/h \lesssim 0.5$$. This near-wall region is less sensitive to the domain size, and we therefore use domans of size $$(12.8h, 2h, 4.8h)$$ here to reduce the number of grid points by about a factor of 4.

For wall-stress models we apply the wall-model over a layer of thickness $$h_{\rm wm}/h = 0.1$$. For hybrid LES/RANS models, the thickness of the RANS layer is taken the same (i.e., the nominal location of the RANS-LES switch is placed at $$y = h_{\rm wm} = 0.1h$$).

### Grids

We use grids at 4 different resolution levels in order to judge grid-convergence. The resolution level is quantified (and labeled) by the spanwise resolution $$\Delta z/h$$: the 4 grids have $$\Delta z/h =$$ 0.100, 0.067, 0.050, and 0.033.

We use grids with 2 different aspect ratios in the wall-parallel plane in order to judge the sensitivity to the grid anisotropy. The base grid has $$\Delta x = \Delta z$$ (which is consistent with the idea that one would not know the flow direction a priori for complex flow problems). Given the known sensitivity of LES to grid anisotropy (cf. Meyers and Sagaut, 2007), we also use grids with $$\Delta x = 2\Delta z$$. While many (most?) channel flow studies using (WM)LES in the literature used a single $$\Delta x / \Delta z$$ ratio and never varied it, it is crucial to appreciate the importance of assessing the sensitivity of a (WM)LES approach to the cell aspect ratio.

There are two versions of every grid: one intended for wall-stress models, one intended for hybrid LES/RANS models. The former have $$\Delta y_w \approx 0.25 \Delta y_c$$ (subscripts $$w$$ and $$c$$ denote the wall and the centerline, respectively) and are refined uniformly and isotropically. The grids for hybrid LES/RANS have $$\Delta y_w / h= 1.5 \! \cdot \! 10^{-4}$$ at all resolution levels; this value produces $$\Delta y^+_w \approx 0.8$$ which is not refined in order to reduce the time-step penalty for compressible solvers. The wall-stress and hybrid LES/RANS grids have identical grid-spacings in the core 69% of the channel (i.e., for $$y/h \gtrsim 0.31$$). At the centerline, $$\Delta y_c = \Delta z$$.

• The lower edge of the LES region (i.e., the location $$y=h_{\rm wm}$$) is marked by a dashed line in the figures. For wall-stress models, the results below this level are not meaningful, and should essentially be ignored (we show them anyway here to be transparent and complete). For hybrid LES/RANS models, the mean velocity below this level is meaningful, but the resolved stresses are not (and should essentially be ignored).
• When it comes to the grid aspect ratio AR ($$= \Delta x / \Delta z$$), the ideal result is (of course) that this should not affect the results at all. Since the spanwise grid-spacing $$\Delta z$$ should be the limiting one for these problems, we show the difference in the results between grids with the same $$\Delta y$$ and $$\Delta z$$ spacings, but with different $$\Delta x$$.