A new high-resolution spatial discretization scheme is presented for use within the volume-ofluid (VOF) method. This scheme is an adaptation of the previously published boundary gradient maximization (BGM) scheme, with an extension for time-dependent solutions. The scheme does not explicitly reconstruct the interface, but rather resolves the values of the volume fraction in the area of the interface. The scheme is upwind biased to provide numerical stability, and the face values are limited to meet boundedness criteria and prevent variable overshoot. Unlike most other high-resolution schemes found in the literature, the proposed scheme does not use upwind-biased and downwind-biased “switching” to maintain stability. This thesis presents a number of test cases including 2-D and 3-D cases on both structured and unstructured grids. The results display the method’s ability to predict good shape of the volume fraction interface and resolve the volume fraction discontinuity.

In this thesis we investigate the accuracy of high-order Extended Finite Element Methods (XFEMs) for the solution of discontinuous problems, with a view to computing high-order solutions to a two-phase flow problem. We start by demonstrating optimal exponential rates of convergence for a spectral/hp element method applied to a smooth problem. We then consider an immersed method on a fixed background mesh that uses level sets to capture the location of a discontinuity and the XFEM to characterise this discontinuity on element interiors. We present an improvement to the modified XFEM of [Moes et. al., 2003] and then use it to solve both a Poisson problem and a linear elasticity problem with discontinuities modelled independently of the mesh. Very close to optimal rates of convergence are recovered for the Poisson problem with both straight and quadratically curved interfaces for approximations up to order p=4. These rates are better than those published in the literature for the XFEM with a curved weak discontinuity, and they are also the first optimally convergent results achieved using the modified XFEM for any problem with approximations of order p>1. Almost optimal rates of convergence are then also recovered for an elastic problem with a circular discontinuity for approximations up to order p=4.The use of the XFEM for time-dependent problems is investigated, and a novel level set update method that retains the signed distance property without need for reinitialisation is also presented. Finally we apply these methods to the time-dependent simulation of a two-phase flow problem. We validate the method against both an analytic dispersion relation for relaxation under small interface perturbations and an existing implementation for large interface perturbations. We then present a proof-of-concept implementation of a high-order immersed method for an oscillating tank flow problem and demonstrate the ability of our implementation to simulate problems with large amplitude interface deformations.

Dans l'industrie du pétrole et des moteurs, les écoulements de fluides non-miscibles sont fréquemment rencontrés : écoulements d'hydrocarbures dans les conduites, séparation en production, injection de carburant dans les moteurs, procédés de raffinage, etc.Pour modéliser ce type d'écoulement, deux approches sont possibles. Soit l'écoulement est décrit de façon macroscopique et les phénomènes locaux (rupture et coalescence des gouttes, glissement des phases, compaction locale, etc.) sont modélisés à l'aide de lois de fermeture analytiques ou empiriques. Soit l'écoulement est modélisé de manière directe à l'échelle de la goutte et on s'attache à décrire précisément l'interface et les interactions entre les phases. C'est cette dernière approche que nous avons proposé d'adopter pour étudier des écoulements à phase dispersée liquide-liquide, et plus particulièrement les phénomènes de rupture et coalescence, collision ou déformation de gouttes. Ainsi, le but principal de ce travail de thèse a été le développement d'un code de simulation numérique directe capable de modéliser un écoulement diphasique liquide-liquide, afin d'étudier en détail les effets de coalescence et de rupture entre les gouttes. Ce travail a nécessité l’utilisation d’une technique de suivi d’interface appropriée et le développement d’un solveur des équations de Navier-Stokes incompressible pour calculer le champ de vitesse, ainsi qu’une méthode de couplage entre ces deux solveurs pour la simulation des écoulements diphasiques. Notre outil numérique a été validé sur de nombreux cas tests académiques et appliqué à l'étude du processus de séparation liquide-liquide. / The flow of immiscible fluids is a frequent issue in the petroleum industry: hydrocarbon in pipelines, separation process for production, fuel injection in engines, refinery treatment processes, etc.There are two possible approaches to model this type of flow. In the first one, the flow is described macroscopically. In this case, local phenomena (breakage or coalescence of droplets, phase slip, local compaction) are modeled thanks to analytic closure laws or empiric laws. In the second approach, the flow is simulated indirectly on a scale of droplet and we want to describe precisely the interface and the interactions between phases. We propose here to consider the second method to study liquid/liquid dispersed flows and especially the phenomena of breakage or coalescence and collision or distortion of the droplets.Thus, the main purpose of this work was the development of a direct numerical simulation code that is capable to model a liquid-liquid two-phase flow, in order to study in detail the effects of droplets coalescence and breakage. To model a two-phase flow, it is necessary to choose an appropriate interface tracking method and to develop a solver for Navier-Stokes incompressible equations to compute the velocity and pressure values. Also, a coupling method that is able to handle the discontinuous quantities at the interface has to be implemented. Our numerical tool has been validated on numerous academic test cases and applied to study the process of liquid-liquid separation.

abstract: The Volume-of-Fluid method is a popular method for interface tracking in Multiphase applications within Computational Fluid Dynamics. To date there exists several algorithms for reconstruction of a geometric interface surface. Of these are the Finite Difference algorithm, Least Squares Volume-of-Fluid Interface Reconstruction Algorithm, LVIRA, and the Efficient Least Squares Volume-of-Fluid Interface Reconstruction Algorithm, ELVIRA. Along with these geometric interface reconstruction algorithms, there exist several volume-of-fluid transportation algorithms. This paper will discuss two operator-splitting advection algorithms and an unsplit advection algorithm. Using these three interface reconstruction algorithms, and three advection algorithms, a comparison will be drawn to see how different combinations of these algorithms perform with respect to accuracy as well as computational expense. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2015

abstract: Many physical phenomena and industrial applications involve multiphase fluid flows and hence it is of high importance to be able to simulate various aspects of these flows accurately. The Dynamic Contact Angles (DCA) and the contact lines at the wall boundaries are a couple of such important aspects. In the past few decades, many mathematical models were developed for predicting the contact angles of the inter-face with the wall boundary under various flow conditions. These models are used to incorporate the physics of DCA and contact line motion in numerical simulations using various interface capturing/tracking techniques. In the current thesis, a simple approach to incorporate the static and dynamic contact angle boundary conditions using the level set method is developed and implemented in multiphase CFD codes, LIT (Level set Interface Tracking) (Herrmann (2008)) and NGA (flow solver) (Desjardins et al (2008)). Various DCA models and associated boundary conditions are reviewed. In addition, numerical aspects such as the occurrence of a stress singularity at the contact lines and grid convergence of macroscopic interface shape are dealt with in the context of the level set approach. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2015

The level set method offers a simple and robust approach to modeling liquid-vapor interfaces that arise in boiling and condensing flows. The current liquid-vapor phase-transition techniques used with the level set method are not able to account for different thermal conductivities and specific heats in each respective phase, nor are they able to accurately account for latent heat absorption and release. This paper presents a new level set based technique for liquid-vapor phase-transition that accounts for different material properties in each respective phase, such as thermal conductivity and specific heat, while maintaining the interface at the saturation temperature. The phase-transition technique is built on the ghost fluid framework coupled with the standard level set method. A new technique is presented for constructing ghost nodes that implicitly captures the immersed boundary conditions and is second order accurate. The method is tested against analytical solutions, and it is used to model film boiling. The new phase-transition technique will greatly assist efforts to accurately capture the physics of boiling and condensing flows. In addition to presenting a new phase transition technique, a coupled level set volume of fluid advection scheme is developed for phase transition flows. The new scheme resolves the mass loss problem associated with the level set method, and the method provides an easy way to accurately calculate the curvature of an interface, which can be difficult with the volume of fluid method. A film boiling simulation is performed to illustrate the superior performance of the coupled level set volume of fluid approach over the level set method and the volume of fluid method.