Stability effect of multidimensional velocity components in numerical flux SLAU

In computational fluid dynamics of compressible fluid flow, the simple low-dissipation advection upstream (SLAU) scheme formulated with multidimensional velocity components (normal and parallel to a cell interface) is a widely employed all-speed scheme. As a variant of SLAU, the mSLAU scheme, which adopts only a velocity component normal to the cell interface instead of multidimensional velocity components, is used for rotorcraft calculations. However, although mSLAU has been claimed to be empirically stable, it has been pointed out that using only the cell-interface-normal velocity component instead of the multidimensional velocity components causes numerical instability. Therefore, to clarify the roles of the multidimensional velocity components for computational stability, we solved some benchmark problems associated with using SLAU or mSLAU. We discovered that the multidimensional velocity components contributed to stability against poor-quality grids by isotropically producing enough amount of numerical dissipation, especially in low-subsonic and supersonic flows. Although mSLAU could practically treat moderate Mach number flows (approximately 0.1 < M < 1.0) when coupled with the minmod limiter, using only the cell-interface-normal velocity component can deteriorate convergence of calculations and lead to susceptibility in the grid geometry.