Solving parametric high-Reynolds-number wall-bounded turbulence around airfoils governed by Reynolds-averaged Navier–Stokes equations using time-stepping-oriented neural network

Physics-informed neural networks (PINNs) have recently emerged as popular methods for solving forward and inverse problems governed by partial differential equations. However, PINNs still face significant challenges when solving high-Reynolds-number flows with multi-scale phenomena. In our previous work, we proposed time-stepping-oriented neural network (TSONN), which transforms the ill-conditioned optimization problem of PINNs into a series of well-conditioned sub-problems, successfully solving the three-dimensional laminar flow around a wing at a Reynolds number of 5000. In this paper, we extend TSONN to high-Reynolds-number wall-bounded turbulence around airfoils governed by the Reynolds-Averaged Navier–Stokes (RANS) equations with the Spalart–Allmaras (SA) turbulence model. Specifically, we propose a semi-coupled strategy to address the convergence issues caused by the turbulence model. This strategy updates certain terms in the turbulence model only during the outer iterations while freezing these terms in the inner iterations, thereby avoiding excessive gradients that could jeopardize network optimization. Using this strategy, we successfully solve turbulence around airfoils. Furthermore, we address a parametric problem with respect to the angle of attack. Our experimental results demonstrate that the computational cost of solving this parametric problem using TSONN is comparable to that of solving a single flow problem, highlighting its efficiency in solving parametric problems. To the best of our knowledge, this is the first time that a PINN-like method has been used to solve the RANS equations coupled complex turbulence model, paving the way for fluid-related engineering problems.