On the computation of compressible flows on unstructured hybrid grids

Computation of compressible flows using an unstructured hybrid grid method is presented in this paper. The spatial discretization is carried out using a cell-centered finite volume approximation. The spatially discretized governing equations are integrated in time using a linearized implicit scheme. A fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. Three open issues related to the cell-centered finite volume formulation: gradient approximation, viscous discretization, and interface value estimation are examined, analyzed, and discussed. It is proven mathematically and demonstrated numerically that an inappropriate treatment of these issues may result in inaccurate and inconsistent numerical solutions to the Navier-Stokes equations on non-uniform grids. An inverse-distance weighted average is proposed to evaluate the flow variables at the cell interfaces, and found to be effective to achieve a consistent convergence on non-uniform grids for the solution of the Navier-Stokes equations on non-uniform and irregular grids.