Robust Multigrid Solution of Rans Equations with Two-equation Turbulence Models

The design of a new, truly robust multigrid framework for the solution of steady-state Reynolds-averaged Navier-Stokes (RANS) equations with two-equation turbulence models is presented. While the mean-flow equations and the turbulence model equations are advanced in time in a loosely-coupled manner, their multigrid cycling is strongly coupled (FC-MG). Thanks to the loosely-coupled approach, the unconditionally positive-convergent implicit time integration scheme for two-equation turbulence models (UPC) is used. An extension of the UPC scheme within the multigrid method is proposed. The resulting novel FC-MG-UPC algorithm is nearly free of artificial stabilizing techniques, leading to increased multigrid efficiency. To demonstrate the robustness of the proposed algorithm, it is applied to linear and non-linear two-equation turbulence models. Numerical experiments are conducted, simulating separated flow about the NACA4412 airfoil and transonic flow about the RAE2822 airfoil. Results obtained from numerical simulations demonstrate the strong consistency and case-independence of the method.