A unified method of data assimilation and turbulence modeling for separated flows at high Reynolds numbers

In recent years, machine learning methods represented by deep neural networks (DNNs) have been a new paradigm of turbulence modeling. However, in the scenario of high Reynolds numbers, there are still some bottlenecks, including the lack of high-fidelity data and the stability problem in the coupling process of turbulence models and the Reynolds-averaged Navier–Stokes (RANS) solvers. In this paper, we propose an improved ensemble Kalman inversion method as a unified approach of data assimilation and turbulence modeling for separated flows at high Reynolds numbers. A novel ensemble design method based on transfer learning and a regularizing strategy are proposed to improve the method. The trainable parameters of DNN are optimized according to the given experimental surface pressure coefficients in the framework of mutual coupling between the RANS solvers and DNN eddy viscosity models. In this way, data assimilation and model training are integrated into one step to get the high-fidelity turbulence models agree well with experiments directly. The effectiveness of the method is verified by cases of flows around S809 airfoil at high Reynolds numbers. Through assimilation of few experimental states, we can get turbulence models generalizing well to both attached and separated flows at different angles of attack, which also perform well in stability and robustness. The errors of lift coefficients at high angles of attack are significantly reduced by more than three times compared with the traditional Spalart–Allmaras model.