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 not performed well in unsteady environments since the preconditioning parameters that are optimal for efficiency are detrimental to the level of spatial dissipation necessary for accuracy. A unified flux formulation is presented where the optimal scaling required for spatial accuracy is independent of the preconditioning required for convergence thus providing a framework that is valid over a broad range of flow conditions. Both upwind flux-difference and AUSM-type schemes are investigated. In both cases, the use of unsteady preconditioning scaling in the flux formulation is shown to be critical for preserving unsteady accuracy. In the flux-difference case, the formulation is based on a generalized blending of the steady and unsteady preconditioning terms. In the AUSM case, the formulation introduces two modifications to the standard AUSM+up scheme, designated as AUSM+up' wherein the pressure dissipation is scaled using unsteady preconditioning and AUSM+u'p' wherein both the pressure and velocity dissipation terms are scaled by the unsteady preconditioning. 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 in terms of vortex shape preservation and control of odd-even splitting errors.