Formulation of fully implicit method for simulation of flows with interfaces using primitive variables

This paper is concerned with simulation of flows with interface between two incompressible and immiscible fluids on a fixed grid using what is called the single fluid formalism. This formalism views flow of two fluids as that of a single fluid whose density and viscosity change abruptly at the interface. The location of the interface is apriori not known but is to be discovered as part of the solution. Problems of this type are typically solved using an equation for (a) Volume Fraction or, for (b) Level Set function. In the present paper, the governing Navier–Stokes equations in primitive variables are solved on collocated grids using SIMPLE algorithm with a specially derived smoothing pressure correction that satisfies volume conservation. A superficial density is defined and determined from mass conservation equation. It is shown that this equation can be cast in the form of a well-known volume fraction equation. The interface location is determined without interface reconstruction. The convective terms are represented by a TVD scheme to predict less-smeared interface. The surface tension force is evaluated by two methods via geometric and fluid dynamic evaluation of the interface curvature.