Phase-Field Models for Two-Phase Flows Using the Least-Squares Method
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The correct representation of the interface dynamics in two–phase flows plays an important role in the understanding of multiphase flow phenomena. There are many numerical methods available to describe the interface dynamics, such as volume of fluid, level set method and combination of these two, but they are not adequate to handle flow phenomena with complex topological changes without the use of artificial manipulations of the interface behavior. On the other hand, phase–field methods can provide a robust and accurate numerical description of the interface dynamics based on thermodynamic principles without requiring ad–hoc rules. The Cahn–Hilliard equation and the Korteweg type–of equation are the two most representative models within the phase–field methods. Solving these equations numerically is faced with difficulties though – 1) they are high–order, non– linear and mixed type–of partial differential equations, and 2) a high spatial resolution is required in a thin interfacial region to reproduce the correct physics. In particular, the Korteweg–type equation is a mixed hyperbolic–elliptic type of system, as a result of the non–convex part of the van der Waals equation of state. This fact requires the introduction of auxiliary variables or addition of artificial viscosities to deal with instability in energy. In addition, the governing equations for both the Cahn–Hilliard and the Korteweg models need to be recast to reduce their differential order when used with low order numerical methods, increasing the number of unknowns and discrete system size significantly. These two equations can be coupled with the equations for hydrodynamics, resulting in the Navier–Stokes–Cahn–Hilliard (NSCH) system and the Navier–Stokes Korteweg (NSK) system, respectively. The main objective of this dissertation is to develop a numerical model capable of handling the complexities related to the phase–field models applied to two-phase flows. In this work, the NSCH system is considered for isothermal incompressible binary fluids and the NSK system is for thermal compressible two–phase flow. The numerical solution is implemented by using the least–squares method that always results in a symmetric postive–definite system even for non–self–adjoint operators and can circumvent the LBB condition. For the discretization, the spectral element method with C1 higher–order approximations is used to provide the necessary differentiability of the solution across elements without having to reduce the order of the differential equations to first order. An h–adaptive mesh technique is implemented into the least–squares approach with maintaining the C1 continuity between non–conformal elements. The numerical solver uses a space–time coupled formulation and an element–by–element solution technique. The solver is implemented in an in–house Matlab MPI code. The developed solver is verified with a convergence analysis. Various multiphase flow phenomena are handled by our solver, ranging from coalescence, spinodal decomposition, falling droplet by gravity, thermocapillary convection to evaporation and condensation of a bubble. Our results are evaluated by conducting parametric studies, by validating with experimental observations and by comapring with numerical results from the literature. One of the main novelties of this work is the use of the least–squares method to solve the NSCH and NSK systems, making it possible to solve these phase–field models without any special numerical treatment. The model can simulate the interfacial dynamics such as coalescence of droplets and bubbles based on thermodynamic principles, instead of using ad–hoc rules. Furthermore, regarding the C1 continuous h–adaptive refinement technique, this work presents the first implementation of a high–order non–conformal discretization into the least–squares approach. We focus on the application of this adaptive scheme to the phase–field models by comparing two refinement criteria to provide the optimal refined grid at each time step for a transient problem. In addition to the cases covered in this thesis, our scheme can be applicable in many other examples of interfacial dynamics without requiring any special numerical scheme. The high–order adaptive least–squares scheme can also be applicable to many other situations where high–order non–conformal grid is needed.