Accurate implementation of forcing terms for two-phase flows into SIMPLE algorithm

A consistent numerical method for two-phase flows has been devised, that properly includes generic body forces in the framework of a collocated variable arrangement. We first discuss the accurate implementation of body forces, and then apply this to interfacial forces. Having shown the problems associated with applying forcing terms in a collocated framework, we introduce piezometric pressure into the governing equations for two-phase flow. As a result, the Navier–Stokes equations written for variable property flow, developed by Hong and Walker, are revised to be consistent with the numerical implementation of the level-set function. Consequently, the interfacial forces consist of both surface tension and gravity. We develop appropriate numerical methods to incorporate a generic body force, including the interfacial forces, within the framework of pressure-based algorithms (e.g. SIMPLE algorithm). A suitable pressure–velocity coupling scheme is then proposed for this type of flow solver and compared with other schemes. To validate the formulation, the results of several test cases including the Rayleigh–Taylor instability, a magnetized ferrofluid, and a falling water droplet are presented and compared with the results obtained by a conventional implementation of body forces. The advantages of confining the effect of gravity to the vicinity of an interface are explained, and we demonstrate how this approach enhances numerical solutions.

► We presented a consistent discretization for two-phase flow equations in SIMPLE.
► We imposed no restriction on the form or the distribution of the forcing terms.
► New pressure–velocity coupling, momentum and body force calculations were introduced.