An Artificial Compressibility Formulation for Phase-field Model and its Application to Two-Phase Flows

Abstract

An Artificial Compressibility Formulation for Phase-field Model and its Application to Two-phase Flows With the advent of high-speed computers, computational methods have become a very useful tool for solving problems in science and engineering along with analytical and ex perimental approaches. The starting point of computational methods is the mathematical model, the form, and origin of which depends on the particular field of study. Many im portant physical processes in nature are governed by partial differential equations (PDE’s). For this reason, it is important to understand the physical behavior of the PDE’s. Also, the knowledge of mathematical character, properties, and solution of the equations are re quired. A proper mathematical model and a good numerical method can provide realistic answerstocomplexphysicalphenomenaforwhichanalyticalsolutionmaynotbeavailable in a finite time. Two-phase flow occurs in nature and many areas of physical and biological sciences like oil recovery processes (water and oil), blood flow (plasma and blood cell), mud-flow (wa ter and suspended particles), atmosphere and ocean system (air and water), cloud and fog (water and air). In dealing with the two-phase flow, an important consideration is how to modelthemovinginterface/surface. Nevertheless,thePDEsdescribingthetwo-phaseflow are highly nonlinear and stiff so it is difficult to solve them analytically and a numerical simulation is an alternate option. The numerical solution obtained may only approximate that of original problem or at least within some required tolerance of the true solution. However,accuratesimulationofmovinginterfacepresentsaproblemofconsiderablediffi cultyandisthereforeverychallengingfordevelopingnumericalmethodsusinglarge-scale computation. Also, the boundary conditions need a particular treatment near the moving interface during numerical simulations. Shocks in the compressible flows, vortex sheets in inviscid flows, and boundaries between immiscible fluids are some of the very known examples. Mathematicalmodelsadoptedinbothanalyticalandnumericalstudiesforavarietyoftwo phase flow with moving interface are classified into two types, i.e., sharp interface models and diffuse interface models. Sharp interface models like level set method assume that the interface has zero thickness. However, in the phase transition, the existence of a transition region introduces the idea of the diffuse interface that allows the interface to have finite thickness. Onetypeofdiffuse-interfacemodelsofparticularinterestisaphase-fieldmodel by an introduction of a phase-field variable that represents the interface. In this approach, the phase-field variable is a continuous function in space and time. Phase-field models are numerically attractive for not tracking the interface explicitly but can be obtained as a part of the solution processes. InordertosolveunsteadyincompressibleNavier-Stokesequations,severalnumericalmeth ods are developed, including the artificial compressibility method. In this research work, a numerical algorithm based on artificial compressibility formulation of the phase-field modelisusedforsimulatingtwo-phaseflowsproblems. Thecoupledhydrodynamicalsys tem consists of the incompressible Navier-Stokes equations and volume preserving Allen Cahntypephase-fieldequationarerecastintoconservativeformswithsourceterms,which are suited for implementing high-order and high-resolution discretization schemes. The Boussinesq approximation is used for buoyancy effects in the flow with moderately differ ent densities. The performance of the numerical method is demonstrated by its application to some benchmark two-phase flow problems.