Extension of Finite Volume Compressible Flow Solvers to Multi-dimensional, Variable Density Zero Mach Number Flows

Abstract

When attempting to compute unsteady, variable density flows at very small or zero Mach number using a standard finite volume compressible flow solver one faces at least the following difficulties: (i) Spatial pressure variations vanish as the Mach number M→0, but they do affect the velocity field at leading order; (ii) the resulting spatial homogeneity of the leading order pressure implies an elliptic divergence constraint for the energy flux; (iii) violations of this constraint crucially affect the transport of mass, preventing a code to properly advect even a constant density distribution. We overcome these difficulties through a new algorithm for constructing numerical fluxes in the context of multi-dimensional finite volume methods in conservation form. The construction of numerical fluxes involves: (1) An explicit upwind step yielding predictions for the nonlinear convective flux components. (2) A first correction step that introduces pressure gradients which guarantee compliance of the convective fluxes with a divergence constraint. This step requires the solution of a first Poisson-type equation. (3) A second projection step which provides the yet unknown (non-convective) pressure contribution to the total flux of momentum. This second projection requires the solution of another Poisson-type equation and yields the cell centered velocity field at the new time. This velocity field exactly satisfies a divergence constraint consistent with the asymptotic limit. Step (1) can be done by any standard finite volume compressible flow solver. The input to steps (2) and (3) involves solely the fluxes from step (1) and is independent of how these were obtained. Thus, our approach allows any such solver to be extended to compute variable density incompressible flows.