RANS Solutions Using High Order Discontinuous Galerkin Methods

We present a practical approach for the numerical solution of the Reynolds averaged Navier-Stokes (RANS) equations using high-order discontinuous Galerkin methods. Turbulence is modeled by the Spalart-Allmaras (SA) one-equation model. We introduce an artificial viscosity model for SA equation which is aimed at accommodating high-order RANS approximations on grids which would otherwise be too coarse. Generally, the model term is only active at the edge of the boundary layer, where the grid resolution is insufficient to capture the abrupt change in curvature required for the eddy viscosity profile to match its free-stream value. Furthermore, the amount of viscosity required decreases with the grid resolution and vanishes when the resolution is sufficiently high. For transonic computations, an additional shock-capturing artificial viscosity model term is required. Numerical predictions for turbulent flows past a flat plate and a NACA 0012 airfoil are presented via comparison with the experimental measurements. In the flat plate case, grid refinement studies are performed in order to assess the convergence properties and demonstrate the effectiveness of high-order approximations.