Unified formulation of the momentum-weighted interpolation for collocated variable arrangements

Momentum-weighted interpolation (MWI) is a widely used discretisation method to prevent pressure-velocity decoupling in simulations of incompressible and low Mach number flows on meshes with a collocated variable arrangement. Despite its popularity, a unified and consistent formulation of the MWI is not available at present. In this work, a discretisation procedure is devised following an in-depth analysis of the individual terms of the MWI, derived from physically consistent arguments, based on which a unified formulation of the MWI for flows on structured and unstructured meshes is proposed, including extensions for discontinuous source terms in the momentum equations as well as discontinuous changes of density. As shown by the presented analysis and numerical results, the MWI enforces a low-pass filter on the pressure field, which suppresses oscillatory solutions. Furthermore, the numerical dissipation of kinetic energy introduced by the MWI is shown to converge with third order in space and is independent of the time-step, if the MWI is derived consistently from the momentum equations. In the presence of source terms, the low-pass filter on the pressure field can be shaped by a careful choice of the interpolation coefficients to ensure the filter only acts on the driving pressure gradient that is associated with the fluid motion, which is shown to be vitally important for the accuracy of the numerical solution. To this end, a force-balanced discretisation of the source terms is proposed, that precisely matches the discretisation of the pressure gradients and preserves the force applied to the flow. Using representative test cases of incompressible and low Mach number flows, including flows with discontinuous source terms and two-phase flows with large density ratios, the newly proposed formulation of the MWI is favourably compared against existing formulations and is shown to significantly reduce, or even eliminate, solution errors, with an increased stability for flows with large density ratios.