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

The optimization of jet engines continues to be a prominent area of interest, particularly with the new environmental standards in place. The need for reduced carbon emissions requires engineers to redesign the rotatery parts with fewer components to reduce weight while maintaining a high level of efficiency. These modifications increase the blade loading resulting in large adverse pressure gradients on the suction side of the airfoil. For instance, under these conditions and given the low Reynolds numbers at the last stages of a low-pressure turbine (LPT) and even more on low-pressure compressor (LPC), the boundary layer may separate on the rear portion of the blade suction side. This phenomenon can have a significant impact on the overall efficiency of the turbomachines. To address this issue during the design process, it is essential to \textit{a priori} assess the size of the recirculation bubble and attempt to minimize it as much as possible. \newline

The industry standard for simulating the stage-scale flow is the Reynolds-Averaged Navier-Stokes (RANS) method. However, RANS frequently fails at off-design conditions due to its inherent modeling assumptions. As an alternative, Large Eddy Simulation (LES) reduces the modeling assumptions by accurately resolving a significant part of the unsteady flow but remains costly at high Reynolds numbers. Wall models reduce the computational cost of LES by modeling the near-wall energetic scales and enabling the application of LES to complex flow configurations of engineering interest. However, most wall models assume that the boundary layer is fully turbulent, at equilibrium, and attached. While these models have proven successful in turbulent boundary layers under moderate adverse pressure gradients, when the adverse pressure gradient becomes too strong, and the boundary layer separates, equilibrium wall models are no longer applicable. To address this limitation, the Mixture Density Network (MDN), originally developed to predict uncertainty, is employed as a wall shear stress model in the context of wall-modeled Large Eddy Simulations (wmLES) of separated flows. Such a network does not predict the mean wall shear stress conditioned by the inputs, but instead predicts the WSS distribution, assuming that any distribution can be approximated by a linear combination of Gaussian distributions. The focus on the accurate prediction of the first two statistical moments is crucial for separated flows. \newline

Before training the MDN, intensive work on the database has been conducted to study the near-wall physics and select suitable model input features. To study the near-wall physics in different flow configurations, the relationship between the instantaneous wall shear stress, velocity field, and pressure gradients is evaluated using space-time correlations. These correlations are extracted from two wall-resolved LES: a channel flow at a friction Reynolds number $Re_\tau$ of 950 and the two-dimensional periodic hill at a bulk Reynolds number $Re_b$ of $10{,}595$. This latter test case features a separation from the top of the hill leading to the development of a free shear layer that reattaches further downstream, creating a large recirculation bubble. The analysis of the correlations highlights that no instantaneous and local correlation is observed in the vicinity of the separation. The domain of high correlation appears to be shifted downstream. Therefore, the model inputs are the velocity field, the pressure gradients, and the curvature extracted at a given wall distance for multiple streamwise positions (taking both upstream and downstream from the prediction location). These inputs are carefully non-dimensionalized using the density, viscosity, and wall-model height, for better generalizability. An 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 model is evaluated \textit{a priori} on synthetic data generated from the law of the wall and shows good generalizability to higher friction Reynolds numbers. The relevance of the MDN-model is evaluated \textit{a posteriori} by performing wmLES using the in-house flow solver Argo-DG, on two channel flows and a separated flow. The novel WSS model outperforms the existing data-driven WSS models in the literature on turbulent channel flows. The prediction is significantly improved compared to the WSS model based on Reichardt's LOTW on a turbulent separated boundary layer. Nonetheless, the size of the recirculation bubble is still underpredicted, indicating a direction for future research.