Reynolds number dependence of length scales governing turbulent flow separation with application to wall-modeled large-eddy simulations

This article proposes a Reynolds number scaling of the required grid points to perform wall-modeled LES of turbulent flows encountering separation off a solid surface. Based on comparisons between the various time scales in a non-equilibrium (due to the action of an external pressure gradient) turbulent boundary layer, a simple definition of the near-wall ``under-equilibrium"and ``out-of-equilibrium"scales is put forward (where ``under-equilibrium"refers to scales governed by a quasi-balance between the viscous and the pressure gradient terms). It is shown that the former length scale varies with Reynolds number as lp Re^(-2/3). The same scaling is obtained from a simplified Green's function solution of the Poisson equation in the vicinity of the separation point. A-priori analysis demonstrates that the resolution required to reasonably predict the wall-shear stress (for example, errors lower than approximately 10-15% in the entire domain) in several nonequilibrium flows is at least O(10) lp irrespective of the Reynolds number and the Clauser parameter. Further, a series of a-posteriori validation studies are performed to determine the accuracy of this scaling including the flow over the Boeing speed bump, Song-Eaton diffuser, Notre-Dame Ramp, and the backward-facing step. The results suggest that for these flows, scaling the computational grids () such that / lp is independent of the Reynolds number results in accurate predictions of flow separation at the same ``nominal"grid resolution across different Reynolds numbers. Finally, it is suggested that in the vicinity of the separation and reattachment points, the grid-point requirements for wall-modeled large eddy simulations may scale as Re^4/3, which is more restrictive than the previously proposed flat-plate boundary layer-based estimates (Re1) of Choi and Moin (Phys. Fluids, 2012) and Yang and Griffin (Phys. Fluids, 2021).