Analysis of a constant-coefficient pressure equation method for fast computations of two-phase flows at high density ratios

An analysis of a modified pressure-correction formulation for fast simulations of fully resolved incompressible two-phase flows has been carried out. By splitting of the density weighted pressure gradient, the pressure equation is reduced to a constant-coefficient Poisson equation, for which efficient linear solvers can be used. While the gain in speed-up is well documented, the error introduced by the temporal extrapolation of the pressure gradient requires further investigations. In this paper it is shown that the modified pressure equation can lead to unphysical pressure oscillations and large errors. By appropriately combining the extrapolated pressure gradient with a matching volume fraction gradient grid convergence at high density ratios could be recovered. The cases of a one-dimensional front and a sphere translating at uniform velocity were first considered, allowing to decouple the pressure equation from the momentum equation. Subsequently, the case of a rising bubble in an upflow is analysed for which the full set of governing equations is solved. The pressure jump extrapolation error has been found dependent on the density ratio and the CFL number. Ultimately, the gain in the computational time, made possible by the use of fast Poisson solvers, should be weighted by the additional computational time the reduction of the aforementioned error may require.