Anomaly of multidimensional wave models in viscous flow simulations and its remedy

A cell-centered finite volume upwind scheme using Roe’s multidimensional wave model is applied to solve the two-dimensional compressible Navier-Stokes equations. Unlike the conventional residual splitting approach, a numerical flux function based on the multidimensional wave model at cell interface is directly defined. A higher order spatial accuracy is attained by improving the first order numerical flux function. Calculations are carried out for several benchmark problems, and results are compared with those obtained by the gridaligned upwind scheme. It is found that (i) the first order accurate solutions in the boundary layer region are critically affected by the allowable range for the propagation direction of the primary acoustic wave, and (ii) the higher order accurate solutions are significantly influenced by the adaptability of the wave model for the present flux correction approach. As a result, the multidimensional wave mode-D with a modified shear wave direction performed the best in the viscous flow calculations. Though the multidimensional scheme generally gives the identical solutions with those by the grid-aligned scheme, a better coincidence for the shock position is demonstrated for one example.