A Spalart–Allmaras turbulence model implementation in a discontinuous Galerkin solver for incompressible flows

In this paper the artificial compressibility flux Discontinuous Galerkin (DG) method for the solution of the incompressible Navier–Stokes equations has been extended to deal with the Reynolds-Averaged Navier–Stokes (RANS) equations coupled with the Spalart–Allmaras (SA) turbulence model. DG implementations of the RANS and SA equations for compressible flows have already been reported in the literature, including the description of limiting or stabilization techniques adopted in order to prevent the turbulent viscosity ν ˜ from becoming negative. In this paper we introduce an SA model implementation that deals with negative ν ˜ values by modifying the source and diffusion terms in the SA model equation only when the working variable or one of the model closure functions become negative. This results in an efficient high-order implementation where either stabilization terms or even additional equations are avoided. We remark that the proposed implementation is not DG specific and it is well suited for any numerical discretization of the RANS-SA governing equations. The reliability, robustness and accuracy of the proposed implementation have been assessed by computing several high Reynolds number turbulent test cases: the flow over a flat plate ( Re = 10 7 ), the flow past a backward-facing step ( Re = 37400 ) and the flow around a NACA 0012 airfoil at different angles of attack ( α = 0 ° , 10 ° , 15 ° ) and Reynolds numbers ( Re = 2.88 × 10 6 , 6 × 10 6 ).