Grid-point requirement for large eddy simulation: Chapman's estimation revisited

Resolution requirements for large eddy simulation (LES), estimated by Chapman [AIAA J. Vol. 17, p. 1293 (1979)], are modified using accurate formulae for high Reynolds number boundary layer flow. This correction indicates that the number of grid points ($N)$ required for wall-modeled LES is proportional to $Re_L^{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 7}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$7$}} $, but a wall-resolving LES requires $N\sim Re_L^{\raise0.7ex\hbox{${13}$} \!\mathord{\left/ {\vphantom {{13} 7}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$7$}} $, where $L$ is the flat-plate length in the streamwise direction. The number of grid points required for the flow over an aircraft using LES with and without modeling the viscous wall region is estimated: the number of grid points for the wall-modeled LES is one to three orders of magnitude smaller than that for the wall-resolving LES, indicating the practical importance of wall modeling in LES for high Reynolds number flows.