Grid-point and time-step requirements for direct numerical simulation and large-eddy simulation

We revisit the grid-point requirement estimates in Choi and Moin [Phys. Fluid, 24, 011702 (2012)] and establish more general grid-point requirements for direct numerical simulations (DNS) and large-eddy simulations (LES) of a spatially developing turbulent boundary layer. We show that, by allowing the local grid spacing to scale with the local Kolmogorov length scale, the grid-point requirement for DNS of a spatially developing turbulent boundary layer is $N\sim Re_{L_x}^{2.05}$ rather than $N\sim Re_{L_x}^{2.64}$ as suggested by Choi and Moin, where $N$ is the number of grid points and $L_x$ is the length of the plate. In addition to the grid-point requirement, we estimate the time-step requirement for DNS and LES. We show that, for a code that treats the convective term explicitly, the time steps required to get converged statistics are $N_t\sim Re_{L_x}/Re_{x_0}^{6/7}$ for wall-modeled LES and $N_t\sim Re_{L_x}/Re_{x_0}^{1/7}$ for wall-resolved LES and DNS (with different prefactors), where $Re_{x_0}$ is the inlet Reynolds number. The grid-point and time-step requirement estimates allow us to estimate the overall cost of DNS and LES. According to present estimates, the costs of DNS, wall-resolved LES and wall-modeled LES scale as $Re_{L_x}^{2.91}$, $Re_{L_x}^{2.72}$, and $Re_{L_x}^{1.14}$.
Comment: 11 pages, 4 figures