A wall model for separated flows: embedded learning to improve a posteriori performance

The development of a wall model using machine learning methods for the large-eddy simulation (LES) of separated flows is still an unsolved problem. Our approach is to leverage the significance of separated flow data, for which existing theories are not applicable, and the existing knowledge of wall-bounded flows (such as the law of the wall) along with embedded learning to address this issue. The proposed so-called features-embedded-learning (FEL) wall model comprises two submodels: one for predicting the wall shear stress and another for calculating the eddy viscosity at the first off-wall grid nodes. We train the former using the wall-resolved LES data of the periodic hill flow and the law of the wall. For the latter, we propose a modified mixing length model, with the model coefficient trained using the ensemble Kalman method. The proposed FEL model is assessed using the separated flows with different flow configurations, grid resolutions, and Reynolds numbers. Overall good a posteriori performance is observed for predicting the statistics of the recirculation bubble, wall stresses, and turbulence characteristics. The statistics of the modelled subgrid-scale (SGS) stresses at the first off-wall grids are compared with those calculated using the wall-resolved LES data. The comparison shows that the amplitude and distribution of the SGS stresses obtained using the proposed model agree better with the reference data when compared with the conventional wall model.