A generic balanced-force algorithm for finite volume method on polyhedral unstructured grids with non-orthogonality

Abstract

A generic balanced-force algorithm is proposed to solve incompressible multiphase flows with complex interfaces and large density ratios on the polyhedral unstructured grids of large non-orthogonality, skewness and non-uniformity. This algorithm combines the finite volume method based on the volume of fluid (VOF) approach with the fractional step method in a collocated framework, which retains a complete balance between the gravity, surface tension with exact curvature, and the resulting pressure gradients. In the original scheme, the gravity and surface tension terms are represented as gradients of the volume fraction and then discretized in the identical fashion as the pressure gradient, which results in a balanced formulation on the structured grids. However, this scheme cannot be readily applied to unstructured grids, despite that it has been misused for more than one decade. As demonstrated in the paper, it is insufficient to achieve force balance merely by the identical discretization of scalar gradients due to the non-orthogonality of unstructured grids, which will generate severe spurious currents that intensify with larger non-orthogonality angle and higher fluid density ratio. Therefore, we present a novel balanced-force algorithm to treat the non-orthogonal term in a different manner, which significantly suppresses, or even eliminates, the spurious velocity for two-phase flows with large density ratio on arbitrary mesh topology. Various numerical examples have demonstrated that the present algorithm and computational framework offer a promising platform to provide high-fidelity and robust predictions for practical multiphase flow simulations involving highly complex geometry.