Development of Machine Learning based wall shear stress models for LES in the presence of adverse pressure gradients and separation

The Mixture Density Network (MDN), initially developed to predict uncertainty, is used as a wall shear stress model in wall-modeled Large Eddy Simulations (wmLES) of turbulent separated flows. Separation is a common phenomenon in turbomachinery (e.g., compressor and turbine blades), due to strong adverse pressure gradients and curvature effects. However, most standard wall shear stress (WSS) models are no longer applicable in non-equilibrium conditions because of their inherent modeling assumptions about the boundary layer (i.e., fully turbulent, at equilibrium, and attached). In this study, the MDN is trained on turbulent channel flows at various friction Reynolds numbers and on the two-dimensional periodic hill at the bulk Reynolds number of 10,595. The latter test case is designed to allow separation from the hill crest, followed by a massive recirculation bubble and reattachment of the free shear layer on the flat bottom surface. The model takes the velocity field, instantaneous and mean pressure gradients, and wall curvature as inputs. The model outputs the probability distribution of the two wall-parallel components of the wall shear stress. The databases are carefully non-dimensionalized using the kinematic viscosity and wall-model height for better generalizability. The model was successfully evaluated a priori on synthetic data generated from the law-of-the-wall. The relevance of the MDN-model was evaluated a posteriori by performing wmLES using the in-house flow solver Argo-DG on two channel flows and a separated flow.