Diffuse Interface Method for Two-Phase Incompressible Flows
Han, Daozhi (author)
Wang, Xiaoming (professor directing dissertation)
Höflich, Peter (university representative)
Gallivan, Kyle A. (committee member)
Kopriva, David A. (committee member)
Oberlin, Daniel M. (committee member)
Sussman, Mark (committee member)
Florida State University (degree granting institution)
College of Arts and Sciences (degree granting college)
Department of Mathematics (degree granting department)
In this contribution, we focus on the study of multiphase flow using the phase field approach. Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil and gas), cloud and fog (water vapor, water and air). Multiphase flows also play an important role in many engineering and environmental science applications. For two fluids with matched density, the Cahn-Hilliard-Navier-Stokes system (CHNS) is a well accepted phase field model. We propose a novel second order in time numerical scheme for solving the CHNS system. The scheme is based on a second order convex-splitting for the Cahn-Hilliard equation and pressure-projection for the Navier-Stokes equation. We show that the scheme is mass-conservative, satisfies a modified energy law and is therefore unconditionally stable. Moreover, we prove that the scheme is unconditionally uniquely solvable at each time step by exploring the monotonicity associated with the scheme. Thanks to the simple coupling of the scheme, we design an efficient Picard iteration procedure to further decouple the computation of Cahn-Hilliard equation and Navier-Stokes equation. We implement the scheme by the mixed finite element method. Ample numerical experiments are performed to validate the accuracy and efficiency of the numerical scheme. In addition, we propose a novel decoupled unconditionally stable numerical scheme for the simulation of two-phase flow in a Hele-Shaw cell which is governed by the Cahn-Hilliard-Hele-Shaw system (CHHS). The temporal discretization of the Cahn-Hilliard equation is based on a convex-splitting of the associated energy functional. Moreover, the capillary forcing term in the Darcy equation is separated from the pressure gradient at the time discrete level by using an operator-splitting strategy. Thus the computation of the nonlinear Cahn-Hilliard equation is completely decoupled from the update of pressure. Finally, a pressure-stabilization technique is used in the update of pressure so that at each time step one only needs to solve a Poisson equation with constant coefficient. We show that the scheme is unconditionally stable. Numerical results are presented to demonstrate the accuracy and efficiency of our scheme. The CHNS system and CHHS system are two widely used phase field models for two-phase flow in a single domain (either conduit or Hele-Shaw cell/porous media). There are applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, where multiphase flows in conduits and in porous media must be considered together. Geometric configurations that contain both conduit (or vug) and porous media are termed karstic geometry. We present a family of phase field (diffusive interface) models for two phase flow in karstic geometry. These models, the so-called Cahn-Hilliard-Stokes-Darcy system, together with the associated interface boundary conditions are derived by utilizing Onsager's extremum principle. The models derived enjoy physically important energy laws and are consistent with thermodynamics. For the analysis of the Cahn-Hilliard-Stokes-Darcy system, we show that there exists at least a global in time finite energy solution by the compactness argument. A weak-strong uniqueness result is also established, which says that the strong solution, if exists, is unique in the class of weak solutions. Finally, we propose and analyze two unconditionally stable numerical algorithms of first order and second order respectively, for solving the CHSD system. A decoupled numerical procedure for practical implementation of the schemes are also presented. The decoupling is realized through explicit discretization of the velocity in the Cahn-Hilliard equation and extrapolation in time of the interface boundary conditions. At each time step, one only needs to solve a Cahn-Hilliard type equation in the whole domain, a Darcy equation in porous medium, and a Stokes equation in conduit in a separate and sequential fashion. Two numerical experiments, boundary driven and buoyancy driven flows, are performed to illustrate the effectiveness of our scheme. Both numerical simulations are of physical interest for transport processes of two-phase flow in karst geometry.
diffuse interface method, energy law, ground water flow, stability, two-phase flow, well-posedness
June 26, 2015.
A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Includes bibliographical references.
Xiaoming Wang, Professor Directing Dissertation; Peter Hoeflich, University Representative; Kyle Gallivan, Committee Member; David Kopriva, Committee Member; Dan Oberlin, Committee Member; Mark Sussman, Committee Member.
Florida State University
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