  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.
