A wavelet based numerical simulation technique for two-phase flows using the phase field method

In a multiphase flow, bubbles and droplets advect, deform, break, or coalesce. When one simulates such a flow with Computational Fluid Dynamics (CFD) methods, a primary challenge is to resolve the evolution of a transition zone across which steep variations of physical properties occur. The theory of van der Waals describes this transition zone based on a free-energy functional of mass density or molar concentration. The phase-field equation governs the dynamics of such a transition zone. This article presents a wavelet-based approach for phase-field modelling of two-phase flow phenomena using the Allen-Cahn-Navier-Stokes equations. The method is second order in time, which has been verified by the numerical simulation of a moving sharp interface. The average terminal velocity of a rising gas bubble in a liquid demonstrates a good agreement with the terminal velocity found from the data of a laboratory experiment. The calculation of the surface tension by the present method also shows an excellent agreement with the surface tension reported by an experimental study. Up-welling and down-welling disturbances in the Rayleigh-Taylor instability of an interface between a gas and a liquid also demonstrate a good agreement with the corresponding results from a reference numerical simulation. These observations encourage to take advantage of the potential benefits of the wavelet based phase-field method in order to develop efficient CFD simulation techniques for gas-liquid or liquid-liquid two-phase flows.