High-order linear and non-linear residual distribution schemes for turbulent compressible flows

Abstract

A high-order Residual Distribution scheme for the solution of the compressible RANS equations is developed. The one-equation Spalart–Allmaras turbulence model is solved together with the RANS equations as a fully coupled system, in which both the mean flow equations and the turbulence equation are solved with high-order of accuracy. A continuous approximation of the solution is adopted and standard Lagrangian basis functions are used to construct the discrete space. Since viscous terms involve the gradient of the numerical solution which has a discontinuous normal component across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same basis functions used for the solution. The non-linear system of equations resulting from the numerical discretization is solved with the non-linear LU-SGS method. Several numerical experiments are performed to verify the accuracy of the numerical method, these are based on both subsonic and transonic flows in two and three spatial dimensions.