Method for evaluating spatial accuracy order of CFD and applications to WCNS scheme on four typically distorted meshes

Accuracy order of computational fluid dynamics represents the grid convergence rate of numerical results towards exact physical results. Accuracy order may be affected by numerical schemes, grid quality, boundary treatments, etc. Fifthorder Weighted Compact Nonlinear Scheme (WCNS-E-5) is a high resolution finite difference scheme and the robustness in capturing strong shock waves and high endurance in low quality grids has been shown. However, the adaptive weighted methods with smoothness indicators and the grid quality may degrade numerical accuracy. In order to find out the ultimate accuracy when grid distortion and boundary treatments are involved, two inversion formulas are derived by analyzing numerical solutions of partial difference equations. One formula is suited to the condition where exact solutions are available; the other is suited to the condition where exact solutions are unknown. The accuracy order of WCNS-E-5 is evaluated on four typically distorted meshes: oblique crossed meshes, stretched meshes, cornered meshes and skewed meshes. WCNS-E-5 shows high order property on the four distorted meshes for the tested benchmark problems including linear waves, an isentropic vortex, and a hypersonic boundary layer.