Stochastic forcing for sub-grid scale models in wall-modeled large-eddy simulation

In the framework of wall-modeled large-eddy simulation (WMLES), the problem of combining sub-grid scale (SGS) models with the standard wall law is commonly acknowledged and expressed through multiple undesired near-wall behaviors. In this work, it is first observed that the static Smagorinsky model predicts efficiently the wall shear stress in a wall-modeled context, while more advanced static models like wall-adapting local eddy (WALE) viscosity or Sigma with proper cubic damping fail. It is, however, known that Smagorinsky is overall too dissipative in the bulk flow and in purely sheared flows, whereas the two other models are better suited for near-wall flows. The observed difficulty comes from the fact that the SGS model relies on the filtered velocity gradient tensor that necessarily comes with large errors in the near-wall region in the context of WMLES. Since the first off-wall node is usually located in the turbulent zone of the boundary layer, the turbulent structures within the first cell are neither resolved by the grid nor represented by the SGS model, which results in a lack of turbulent activity. In order to account for these subgrid turbulent structures, a stochastic forcing method derived from Reynolds-averaged Navier–Stokes (RANS) turbulence models is proposed and applied to the velocity gradients to better estimate the near-wall turbulent viscosity while providing the missing turbulent activity usually resulting from the WMLES approach. Based on such corrections, it is shown that the model significantly improves the wall shear stress prediction when used with the WALE and Sigma models.