Data-driven Reynolds stress models based on the frozen treatment of Reynolds stress tensor and Reynolds force vector

For developing a reliable data-driven Reynold stress tensor (RST) model, successful reconstruction of the mean velocity field based on high-fidelity information (i.e., direct numerical simulations or large-eddy simulations) is crucial and challenging, considering the ill-conditioning problem of Reynolds-averaged Navier-Stokes (RANS) equations. It is shown that the frozen treatment of the Reynolds force vector (RFV) reduced the ill-conditioning problem even for the cases with a very high Reynolds number; therefore, it has a better potential to be used in the data-driven development of the RANS models. In this study, we compare the algebraic RST correction models that are trained based on the frozen treatment of both RFV and RST for the aforementioned potential. We derive a vector-based framework for the RFV similar to the tensor-based framework for the RST. Regarding the complexity of the models, we compare sparse regression on a set of candidate functions and a multi-layer perceptron network. The training process is applied to the high-fidelity data of three cases, including square-duct secondary flow, roughness-induced secondary flow, and periodic hills flow. The results showed that using the RFV discrepancy values, instead of the RST discrepancy values, generally does not improve the reconstruction of the mean velocity field despite the fact that the propagation of the RFV discrepancy data shows lower errors in the propagation process of all three cases. Regarding the complexity, using multi-layer perceptron improves the prediction of the cases with secondary flows, but it shows similar performance in the case of periodic hills.