Robust Implicit Direct Discontinuous Galerkin Method for Simulating the Compressible Turbulent Flows

This paper proposes a robust and accurate implicit direct discontinuous Galerkin (DDG) method for simulating laminar and turbulent flows. The Reynolds averaged Navier-Stokes (RANS) equations coupled with negative Spalart-Allmaras (SA) turbulence model are solved by using GMRES method with LU-SGS preconditioner to improve the convergence and efficiency. A distinguished feature of this implicit method is that the exact Jacobian matrix, derived analytically from the Harten-Lax-van Leer-Contact convective flux accompanied with DDG viscous flux and negative SA turbulence source term with no approximation and simplification, is used for the linear system of equations resulted by the implicit temporal discretization. The robustness of the implicit method is greatly enhanced due to using the exact Jacobian matrix. A number of benchmark test cases are selected to demonstrate the performance of the proposed method for simulating laminar and complex turbulent flows. Through the numerical experiments, the implicit DDG method clearly demonstrates its robustness for solving complex turbulent flows with highly stretched and curved grids and thus indicates its potential for real engineering applications.