Accurate and efficient discretization of Navier-Stokes equations on mixed grids

The discretization of Navier-Stokes equations on mixed unstructured grids is discussed. Because mixed grids consist of different cell types, the question arises as to how the discretization should treat these cell types in order to result in a stable and accurate solution method. This issue is addressed in relation to the discretization of inviscid and viscous fluxes. The discretization of the inviscid fluxes is carried out with both a centered and an upwind scheme. For the centered scheme, problems exist on mixed grids with the damping properties of the fourth-difference operator. The problems with the fourth-difference operator are investigated for two stencils by both truncation error and Fourier analysis. It is shown that one form exhibits superior damping for high frequencies, which is corroborated using numerical experiments. For the upwind scheme, a commonly used gradient-reconstruction method based on the Green-Gauss theorem does not show satisfactory behavior on mixed grids. Another gradient reconstruction method based on a Taylor series expansion previously derived in two dimensions is extended to three dimensions. This method is more accurate on mixed grids but requires more storage.