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

In multiphase flow phenomena, bubbles and droplets are advected, deformed, break up into smaller ones, and coalesce with each other. A primary challenge of classical computational fluid dynamics (CFD) methods for such flows is to effectively describe a transition zone between phases across which physical properties vary steeply but continuously. Based on the van der Waals theory, Allen-Cahn phase field method describes the face-to-face existence of two fluids with a free-energy functional of mass density or molar concentration, without imposing topological constraints on interface as phase boundary. In this article, a CFD simulation methodology is described by solving the Allen-Cahn-Navier-Stokes equations using a wavelet collocation method. The second order temporal accuracy is verified by simulating a moving sharp interface. The average terminal velocity of a rising gas bubble in a liquid that is computed by the present method has agreed with that computed by a laboratory experiment. The calculation of the surface tension force by the present method also shows an excellent agreement with what was obtained by an experiment. The up-welling and down-welling disturbances in a Rayleigh-Taylor instability are computed and compared with that from a reference numerical simulation. These results show that the wavelet based phase-field method is an efficient CFD simulation technique for gas-liquid or liquid-liquid flows.