A kinematic vector penalty–projection method for incompressible flow with variable density

Abstract

In this Note, we present a new version of the vector penalty–projection splitting method described in [1] for the fast numerical computation of incompressible flows with variable density and viscosity. We show that the velocity correction can be made completely independent of the mass density ρ. Hence, this step is purely kinematic using the fast Helmholtz–Hodge decompositions proposed in [2]. Then, it is shown that the dynamic step of pressure gradient correction can be fast and locally consistent on edge-based generalized MAC-type unstructured meshes that naturally verify the compatibility condition in the proposed discrete setting. By the way, a new accurate front-tracking Lagrangian-advection technique is also introduced for multiphase flows.This new method preserves the fully vector formulation of both the prediction and correction steps of the original scheme, the primary unknowns being (v,∇p) and ρ by advection, since the pressure Neumann–Poisson problem remains eliminated. The efficiency of the present method is demonstrated through numerical results on sharp test cases.