Most wall shear stress models assume the boundary layer to be fully turbulent, at equilibrium, and attached. Under these strong assumptions, that are often not verified in industrial applications, these models predict an averaged behavior. To address the instantaneous and non-equilibrium phenomenon of separation, the mixture density network (MDN), the neural network implementation of a Gaussian Mixture Model, initially deployed for uncertainty prediction, is employed as a wall shear stress model in the context of wall-modeled large eddy simulations (wmLES) of turbulent separated flows. The MDN is trained to estimate the conditional probability \(p(\varvec{\tau }_w\vert \textbf{x})\) , knowing certain entries \(\textbf{x}\) , to better predict the instantaneous wall shear stress \(\varvec{\tau }_w\) (which is then sampled from the distribution). In this work, an MDN is trained on a turbulent channel flow at the friction Reynolds number \(Re_{\tau}\) of 1000 and on the two-dimensional periodic hill at the bulk Reynolds number of 10,595. The latter test case is known to feature a massive separation from the hill crest. By construction, the model outputs the probability distribution of the two wall-parallel components of the wall shear stress, conditioned by the model inputs: the instantaneous velocity field, the instantaneous and mean pressure gradients, and the wall curvature. Generalizability is ensured by carefully non-dimensionalizing databases with the kinematic viscosity and wall-model height. The relevance of the MDN model is evaluated a posteriori by performing wmLES using the in-house high-order discontinuous Galerkin (DG) flow solver, named Argo-DG, on a turbulent channel flow at \(Re_{\tau} =2000\) and on the same periodic hill flow. The data-driven WSS model significantly improves the prediction of the wall shear stress on both the upper and lower walls of the periodic hill compared to quasi-analytical WSS models.
