Analysis of multi-timescale wall modeling for turbulent wall-bounded flows with separation

The multi-timescale wall model has demonstrated successful performance for various non-equilibrium flows (Fowler et al., JFM 2022, 2023) induced by rapid temporal changes in the flow forcing. The results confirmed that temporally-induced non-equilibrium near-wall dynamics can be modeled accurately using the laminar nonequilibrium stress evolution derived from a laminar Stokes-layer solution. To account for spatially imposed accelerations for flows such as periodic hill or separation bubbles, the model was extended using the laminar Stokes solution with Lagrangian filtered input values to account for the spatial acceleration. In addition, the classic equilibrium wall model assumes that the wall shear stress is in the same general direction as the input LES velocity at the wall model matching location. This limitation compromises the capability of the wall model to predict the separation point when the matching location is outside of the initial backflow region. To address this limitation, a reversal velocity profile was incorporated into the fitting function (Meneveau 2020, JoT) for the equilibrium portion of the wall model. The equilibrium wall model with the reversal velocity profile and the laminar nonequilibrium component resulted in the improved equilibrium multi-timescale wall model (EQMTS). The equilibrium and EQMTS wall models were tested on the periodic hill case and flow separation induced by a contoured, free-slip top wall. The equilibrium wall model overpredicts the peak in the spatially accelerating region with favorable pressure gradients. In contrast, the EQMTS wall model shows improved agreement with the reference DNS and WRLES results. However, the streamwise velocity profiles are found to be inaccurate in general when coarse resolutions are employed. Once the grid spacing is sufficiently refined to yield accurate velocity profiles, the predictions from various wall models become similar also for the mean wall stress distribution. Results demonstrate that while multi-time scale wall modeling yields significant improvements for temporal non-equilibrium, predicting flow separation remains as an open challenge.