A novel multigrid method for the solution of the steady Reynolds-averaged Navier–Stokes equations is presented, that gives convergence speeds similar to laminar flow multigrid solvers. The method is applied to Menter’s one-equation turbulence model. New aspects of the method are the combination of nonlinear Gauss–Seidel smoothing on the finest grid with linear coarse-grid corrections, and local damping in the initial stages of the computation, to keep the solution stable; the damping needed is estimated with the nonlinear smoother. Efficiency on the finest grid is increased with full multigrid, second-order accuracy is obtained with defect correction. Tests on boundary layers and airfoil flows show the efficiency of the method.
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Journal of Computational Physics
Development of a State-of-the Art Surface-Capturing Method for Two-Fluid Flow
Scientific Computing

Wackers, J, & Koren, B. (2007). Multigrid solution for the steady RANS equations. Journal of Computational Physics, 226(2), 1784–1807.