# Channel grids

The domain size is taken as $$(12.8h, 2h, 4.8h)$$, which is smaller than in the DNS by about factors of 2 in the wall-parallel directions. This is done to reduce the computational cost, and will not affect the assessment of the wall-modeling (provided we interpret the results correctly, e.g., accepting differences in the wake region).

The central 80% of the channel constitute the nominal LES region. 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, we place the nominal switch from LES to RANS at the same height.

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, $$\Delta y_c = \Delta z$$.

Details on the grids are given in the figure and table below. Note that the number of grid points in the table refers to the vertices (or nodes) — i.e., the number of cells in a finite-volume grid would be one less in each direction.

\ \\ \hspace*{5mm} {\rm Grids\, for\, WMLES\, with\, a\, wall-stress\, model} \\ \begin{array}{|l|c|c|c|c|} \hline {\rm Class\,1\ (LES+stress\, model)} & N_x \ (\Delta x) & N_y & N_z \ (\Delta z) & AR (\Delta x/\Delta z) \\ \hline {\rm Channel\_AR1\_129X33} & 129 \ (0.100 h) & 33 & 49 \ (0.100 h) & 1.0 \\ \hline {\rm Channel\_AR1\_193X49} & 193 \ (0.067 h) & 49 & 73 \ (0.067 h) & 1.0 \\ \hline {\rm Channel\_AR1\_257X65} & 257 \ (0.050 h) & 65 & 97 \ (0.050 h) & 1.0 \\ \hline {\rm Channel\_AR1\_385X97} & 385 \ (0.033 h) & 97 & 145 \ (0.033 h) & 1.0 \\ \hline {\rm Channel\_AR2\_65X33} & 65 \ (0.200 h) & 33 & 49 \ (0.100 h) & 2.0 \\ \hline {\rm Channel\_AR2\_97X49} & 97 \ (0.134 h) & 49 & 73 \ (0.067 h) & 2.0 \\ \hline {\rm Channel\_AR2\_129X65} & 129 \ (0.100 h) & 65 & 97 \ (0.050 h) & 2.0 \\ \hline {\rm Channel\_AR2\_193X97} & 193 \ (0.067 h) & 97 & 145 \ (0.033 h) & 2.0 \\ \hline \end{array}
\ \\ \hspace*{5mm} {\rm Grids\, for\, hybrid\, RANS/LES} \\ \begin{array}{|l|c|c|c|c|} \hline {\rm Class\, 2\ (hybrid\, RANS/LES)} & N_x \ (\Delta x) & N_y & N_z \ (\Delta z) & AR \ (\Delta x/\Delta z) \\ \hline {\rm Channel\_AR1\_129X77} & 129 \ (0.100 h) & 77 & 49 \ (0.100 h) & 1.0 \\ \hline {\rm Channel\_AR1\_193X109} & 193 \ (0.067 h) & 109 & 73 \ (0.067 h) & 1.0 \\ \hline {\rm Channel\_AR1\_257X141} & 257 \ (0.050 h) & 141 & 97 \ (0.050 h) & 1.0 \\ \hline {\rm Channel\_AR1\_385X197} & 385 \ (0.033 h) & 197 & 145 \ (0.033 h) & 1.0 \\ \hline {\rm Channel\_AR2\_65X77} & 65 \ (0.200 h) & 77 & 49 \ (0.100 h) & 2.0 \\ \hline {\rm Channel\_AR2\_97X109} & 97 \ (0.134 h) & 109 & 73 \ (0.067 h) & 2.0 \\ \hline {\rm Channel\_AR2\_129X141} & 129 \ (0.100 h) & 141 & 97 \ (0.050 h) & 2.0 \\ \hline {\rm Channel\_AR2\_193X197} & 193 \ (0.067 h) & 197 & 145 \ (0.033 h) & 2.0 \\ \hline \end{array}

These grids are made available here in two different ways. For people who want to generate their own grids (e.g., in some research codes), the $$y$$-coordinates for the grids are given in the following ASCII files:

For people who instead want to use a full grid file (e.g., in a more commercial-like code), the full grids can be created in PLOT3D format in the following way. First, download $$x-y$$ slices of the grids in PLOT3D format:

These $$x-y$$ slices can then be extruded to form the full 3D grid by using the following extrude.f90 code. Currently, the only output format option is PLOT3D (64-bit unformatted). However, the subroutine output_p3d can easily be replaced with a user-defined subroutine to provide grid files in some other format. Note that this utility is set up to produce “double-precision” floating point values by default (as a result the it should NOT be compiled with options to up-convert floating point values, i.e., -r8 is not needed). The extrude.f90 code exports the grid as a PLOT3D unformatted file, i.e., as

write(unit_num) num_blocks
write(unit_num) (ni(n), nj(n), nk(n), n=1, num_blocks)
do n = 1, num_blocks
write(unit_num)
( ( ( x(i,j,k), i=1,ni(n) ) , j=1,nj(n) ) , k=1,nk(n) ) ,
( ( ( y(i,j,k), i=1,ni(n) ) , j=1,nj(n) ) , k=1,nk(n) ) ,
( ( ( z(i,j,k), i=1,ni(n) ) , j=1,nj(n) ) , k=1,nk(n) )
end do

but this can easily be modified to output the final 3D grid in other formats. To use this utility, simply compile the fortran source code, e.g.,

\> gfortran -o extrude.out extrude.f90

and run the code and answer the prompts.