Direct numerical simulation of multi-phase flow in complex media

Tesi en modalitat de compendi de publicacionsIn numerous applications, two-phase liquid-gas transport at sub-millimeter length scales plays a substantial role in the determination of the behavior of the system at hand. As its main application, the present work focuses on the polymer electrolyte membrane (PEM) fuel cells. Desirable performance and operational life-time of this class of high-throughput energy conversion devices requires an effective water management, which per se relies on proper prediction of the water-air transport mechanisms. Such two-phase flow involves interfacial forces and phenomena, like hysteresis, that are associated with the physicochemical properties the liquid, gas, and if present, the solid substrate. In this context, numerical modeling is a viable means to obtain valuable predictive understanding of the transport mechanisms, specially for cases that experimental analyses are complicated and/or prohibitively expensive. In this work, an efficient finite element/level-set framework is developed for three-dimensional simulation of two-phase flow. In order to achieve a robust solver for practical applications, the physical complexities are consistently included and the involved numerical issues are properly tackled; the pressure discontinuity at the liquid-gas interface is consistently captured by utilizing an enriched finite element space. The method is stabilized within the framework of variational multiscale stabilization technique. A novel treatment is further proposed for the small-cut instability problem. It is shown that the proposed model can provide accurate results minimizing the spurious currents. A robust technique is also developed in order to filter out the possible noises in the level-set field. It is shown that it is a key to prevent irregularities caused by the persistent remnant of the spurious currents. It is shown how the well-established contact-line models can be incorporated into the variational formulation. The importance of the inclusion of the sub-elemental hydrodynamics is also elaborated. The results presented in the present work rely on the combination of the linearized molecular kinetic and the hydrodynamic theories. Recalling the realistic behavior of liquids in contact with solid substrates, the contact--angle hysteresis phenomenon is taken into account by imposing a consistent pinning/unpinning mechanism developed within the framework of the level-set method. Aside from the main developments, a novel technique is also proposed to significantly improve the accuracy and minimize the the loss in the geometrical features of the interface during the level-set convection based on the back and forth error compensation correction (BFECC) algorithm. Within the context of this thesis, the numerical model is validated for various cases of gas bubble in a liquid and liquid droplets in a gas. For the latter scenario, besides free droplets, the accuracy of the proposed numerical method is assessed for capturing the dynamics droplets spreading on solid substrates. The performance of the model is then analyzed for the capturing the configuration of a water droplet on an inclined substrate in the presence the contact--angle hysteresis. The proposed method is finally employed to simulate the dynamics of a water droplet confined in a gas channel and exposed to air-flow.Existen numerosas aplicaciones industriales en las que transporte bifásico (líquido-gas) a escalas submilimétricas resulta crucial para la determinación del comportamiento del sistema en cuestión. Entre todas ellas, el presente trabajo se centra en las pilas de combustible con membrana de electrolito polimérico (PEMFC). El rendimiento deseable y la vida útil operativa de esta clase de dispositivos de conversión de energía de alto rendimiento requieren una gestión eficaz del agua (conocida como “water management”), que per se depende de la predicción adecuada de los mecanismos de transporte de agua y aire. Así pues, el análisis del flujo microfluídico de dos fases obliga considerar fuerzas y fenómenos interfaciales, tales como la histéresis, que están asociados con las propiedades fisicoquímicas del líquido, el gas y, si está presente, el sustrato sólido. En este contexto, la modelización numérica es una alternativa viable para obtener una predicción precisa de los mecanismos de transporte, especialmente en aquellos casos en los que los análisis experimentales son prohibitivos, ya sea por su complejidad o coste económico. En este trabajo, se desarrolla un marco eficiente, basado en la combinación del método de elementos finitos y el método de “level-set”, para la simulación tridimensional de flujos bifásicos. Con el fin de lograr una herramienta numérica robusta para aplicaciones prácticas, las complejidades físicas se incluyen consistentemente y los problemas numéricos involucrados se abordan adecuadamente. Concretamente, la discontinuidad de la presión en la interfaz líquido-gas se captura consistentemente utilizando un espacio de elementos finitos enriquecido. La estabilización del método se consigue mediante la introducción de la técnica de multiescalas variacionales. Asimismo, se propone también un tratamiento novedoso para el problema de la inestabilidad de tipo “small-cut”. Se muestra que el modelo propuesto puede proporcionar resultados precisos minimizando las corrientes espurias en la interfaz liquido-gas. Complementariamente, se presenta una nueva metodología para filtrar el ruido en el campo de “level-set”. Esta metodología resulta ser crucial para prevenir las irregularidades provocadas por el remanente persistente de las corrientes espurias. El comportamiento de la línea de contacto es considerado a través de la inclusión los modelos correspondientes en la formulación variacional. A este respecto, el presente trabajo aborda la importancia de la inclusión de la hidrodinámica subelemental. Los resultados presentados se basan en la combinación de la cinética molecular linealizada y las teorías hidrodinámicas. Para representación del comportamiento realista de los líquidos en contacto con sustratos sólidos, el fenómeno de histéresis del ángulo de contacto se tiene en cuenta imponiendo un mecanismo de anclado / desanclado consistente desarrollado en el marco del método de level-set. Aparte de los desarrollos principales, también se propone una técnica novedosa para la convección de la función ”level-set”. Ésta permite mejorar significativamente la precisión, minimizando a su vez la pérdida en las características geométricas de la interfaz asociadas al transporte. Esta nueva metodología está basada en el algoritmo de corrección de compensación de errores (BFECC). La herramienta numérica desarrollada en esta tesis es validada para varios casos que involucran burbujas de gas en un líquido y pequeñas gotas de líquido en un gas. Para el último escenario, además de las gotas libres, se evalúa la precisión de la herramienta propuesta para capturar la dinámica de las gotas sobre sustratos sólidos. A continuación, se analiza el rendimiento del modelo para capturar la configuración de una gota de agua sobre un sustrato inclinado en presencia de la histéresis del ángulo de contacto. El método propuesto finalmente se aplicaPostprint (published version

Numerical methods for drift-diffusion models

The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume discretization scheme and recent efforts to generalize this approach to general statistical distribution functions

Institute for Computational Mechanics in Propulsion (ICOMP)

The Institute for Computational Mechanics in Propulsion (ICOMP) is operated by the Ohio Aerospace Institute (OAI) and the NASA Lewis Research Center in Cleveland, Ohio. The purpose of ICOMP is to develop techniques to improve problem-solving capabilities in all aspects of computational mechanics related to propulsion. This report describes the accomplishments and activities at ICOMP during 1993

Numerical methods for drift-diffusion models

The van Roosbroeck system describes the semi-classical transport of free electrons and holes in a self-consistent electric field using a drift-diffusion approximation. It became the standard model to describe the current flow in semiconductor devices at macroscopic scale. Typical devices modeled by these equations range from diodes, transistors, LEDs, solar cells and lasers to quantum nanostructures and organic semiconductors. The report provides an introduction into numerical methods for the van Roosbroeck system. The main focus lies on the Scharfetter-Gummel finite volume disretization scheme and recent efforts to generalize this approach to general statistical distribution functions

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

Non-Standard Discretization of the Advection-Diffusion-Reaction Equation with Logistic Growth Reaction

The goal of this work is to make a comparative analysis between the standard finite difference method and the non-standard finite difference method, then to make a non-standard discretization of the advection-diffusion-reaction equation with a reaction modelling a logistic growth which can be the evolution of the concentration of a microbial population in a medium, the equation will thus model transport and diffusion of this population in the aforementioned medium in one dimension of space and one makes numerical simulations to compare the non-standard scheme and the Euler’s scheme, explicit in time, implicit for the first order derivative in  and centered for the second order derivative in . One arrives by constructing a scheme of the advection-reaction equation, then adds the term of diffusion to obtain the non-standard scheme

Formulações numéricas conservativas para aproximação de modelos hiperbólicos com termos de fonte e problemas de transporte relacionados

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: O objetivo desta tese é desenvolver, pelo menos no aspecto formal, algoritmos construtivos e bem-balanceados para a aproximação de classes específicas de modelos diferenciais. Nossas principais aplicações consistem em equações de água rasa e problemas de convecção-difusão no contexto de fenômenos de transporte, relacionados a problemas de pressão capilar descontínua em meios porosos. O foco principal é desenvolver sob o framework Lagrangian-Euleriano um esquema simples e eficiente para, em nível discreto, levar em conta o delicado equilíbrio entre as aproximações numéricas não lineares do fluxo hiperbólico e o termo fonte, e entre o fluxo hiperbólico e o operador difusivo. Os esquemas numéricos são propostos para ser independentes de estruturas particulares das funções de fluxo. Apresentamos diferentes abordagens que selecionam a solução entrópica qualitativamente correta, amparados por um grande conjunto de experimentos numéricos representativosAbstract: The purpose of this thesis is to develop, at least formally by construction, conservative methods for approximating specific classes of differential models. Our major applications consist in shallow water equations and nonstandard convection-diffusion problems in the context of transport phenomena, related to discontinuous capillary pressure problems in porous media. The main focus in this work is to develop under the Lagrangian-Eulerian framework a simple and efficient scheme to, on the discrete level, account for the delicate nonlinear balance between the numerical approximations of the hyperbolic flux and source term, and between the hyperbolic flux and the diffusion operator. The proposed numerical schemes are aimed to be independent of particular structures of the flux functions. We present different approaches that select the qualitatively correct entropy solution, supported by a large set of representative numerical experimentsDoutoradoMatematica AplicadaDoutor em Matemática Aplicada165564/2014-8CNPQCAPE

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching