Improved Flux Formulations for Unsteady Low Mach Number Flows

Abstract

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.

Godunov-type schemes with an inertia term for unsteady full Mach number range flow calculations

“…A particular problem with the AUSM + -up method is that the damping by the pressure difference term in the mass flux expression which is appropriate for steady low Mach number flow is too high for propagation of smooth acoustic signals in unsteady low Mach number flows. On the other hand, as observed by Sachdev et al [19], the dissipation is too low for propagation of acoustic discontinuities (low Mach number Riemann problems). So, it becomes very delicate to tune the pressure dissipation such that it functions properly for the different types of low Mach number flows.…”

“…However, we noted that the quality of the momentum interpolation, if properly defined for unsteady calculations in a Rhie-Chow-like manner (see [15][16][17]), was not reached for some tests at low Mach number. Improvement of predictions for unsteady low Mach number flows by the AUSM + -up scheme and the related SLAU scheme (Simple Low Dissipative AUSM) by introduction of Strouhal number dependence in the coefficient of the pressure dissipation term in the mass flux expression was also obtained by Sachdev et al [19]. These authors demonstrated that the scaling of the coupling coefficient has to be quite different for steady low Mach number problems and for unsteady low Mach number problems.…”

A fully coupled turbulent low-speed preconditioner for harmonic balance applications

“…Originally, Venkateswaran and Merkle [8] proposed an unsteady preconditioning mechanism for implicit and dual-time stepping methods in which an unsteady Mach number is added to the low-speed preconditioner (LSP). Potsdam et al [33] have presented an improvement for unsteady scaling of the preconditioning matrix while more recently, Sachdev et al [34] have extended the work of Potsdam et al [33] to utilize the mixed unsteady and steady preconditioning parameter. The basic idea is that an unsteady Mach number scaling is added to the controlling parameter of the LSP such that…”

Development of the generalized MacCormack scheme and its extension to low Mach number flows

“…On the other hand, schemes like the convective upwind and split pressure (CUSP) [20,21] and the advection upwind split method (AUSM) [22] decompose the flux into convective (or advective) and pressure fluxes. CUSP and AUSM schemes have also been extended to low Mach number flows using preconditioning [1,17,[24][25][26][27]. The CUSP scheme behavior depends on the local Mach number.…”

“…The AUSM scheme is based on the same principles but differs in the implementation of the dissipation term [23]. CUSP and AUSM schemes have also been extended to low Mach number flows using preconditioning [1,17,[24][25][26][27].…”