Improved flux formulations for unsteady low Mach number flows

Preconditioning techniques that are used to alleviate numerical stiffness due to low Mach numbers in steady flows have typically not performed well for unsteady low Mach problems because the preconditioning scaling requirements for preserving discrete accuracy in time-accurate flows are very different from those for steady flows. Specifically, distinct scalings are necessary for the velocity and pressure fields under the low-Mach, high-Strouhal conditions characteristic of acoustic wave problems. In this article, a unified flux formulation is presented where the optimal scaling required for spatial accuracy is maintained over a broad range of flow conditions. Both upwind flux-difference and AUSM-type schemes are investigated and, in both cases, the judicious use of "steady" and "unsteady" preconditioning scalings in the flux formulation is shown to be critical for preserving accuracy. Low Mach number vortex propagation and acoustic problems are used to demonstrate the strengths of the formulation. These studies show that the AUSM family generally performs better than the blended flux-difference schemes especially in terms of vortex shape preservation.