On robust boundary treatments for wall-modeled LES with high-order discontinuous finite element methods

To robustly and accurately simulate wall-bounded turbulent flows at high Reynolds numbers, we propose suitable boundary treatments for wall-modeled large-eddy simulation (WMLES) coupled with a high-order flux reconstruction (FR) method. First, we show the need to impose an auxiliary boundary condition on auxiliary variables (solution gradients) that are commonly introduced in high-order discontinuous finite element methods (DFEMs). Auxiliary boundary conditions are introduced in WMLES, where the grid resolution is too coarse to resolve the inner layer of a turbulent boundary layer. Another boundary treatment to further enhance stability with under-resolved grids, is the use of a modal filter only in the wall-normal direction of wall-adjacent cells to remove the oscillations. A grid convergence study of turbulent channel flow with a high Reynolds number ($R{e}_{\tau }\approx 5200$) shows that the present WMLES framework accurately predicts velocity profiles, Reynolds shear stress, and skin friction coefficients at the grid resolutions recommended in the literature. It was confirmed that a small amount of filtering is sufficient to stabilize computation, with negligible influence on prediction accuracy. In addition, non-equilibrium periodic hill flow with a curved wall, including flow separation, reattachment, and acceleration at a high Reynolds number ($R{e}_h\approx 37,000$), is reported. Considering stability and the prediction accuracy, we recommend a loose auxiliary wall boundary conditions with a less steep velocity gradient for WMLES using high-order DFEMs.