Reynolds number dependence of length scales governing turbulent flow separation in wall-modeled LES

In this work, we propose a Reynolds number (Re) scaling for the number of grid points (Ncv) required in wall-modeled LES (WMLES) of separated turbulent boundary layers (TBL). Based on the various time scales in a non-equilibrium TBL, a definition of the near-wall “under-equilibrium" scales is proposed (where “equilibrium" refers to a quasi-balance between the viscous and the pressure gradient terms). This length scale is shown to vary with Reynolds number as l_p ∼ Re^−2/3, which implies that for a flow solver with nested grids, the grid point requirements for capturing regions of flow separation in WMLES employing equilibrium wall closures scale as Ncv ∼ Re^4/3. A-priori analysis demonstrates that the resolution (Δ) required to reasonably predict the wall stress in several non-equilibrium flows is at least O(10) l_p, irrespective of the Reynolds number and Clauser parameter. Three flows are considered a-posteriori: the Boeing speed bump (Uzun & Malik, 2022), an arc shaped diffuser (Song & Eaton 2004), and a smooth ramp (Simmons et al., 2017); our results corroborate the a-priori estimates to demonstrate that the grid point requirements for capturing flow separation are more stringent than the estimates (Ncv ∼ Re¹) of Choi and Moin. Finally, we also show that an appropriate non-equilibrium wall model can reduce this cost scaling to Ncv ∼ Re¹ even in separated flow regions.