Sistema de leis de balanço em problemas de dinâmicas de fluidos : modelagem matemática e aproximação numérica

Orientador: Eduardo Cardoso de AbreuTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação CientíficaResumo: Nesta tese, estamos preocupados com o comportamento limite de sistemas hiperbólicos de leis de conservação com termos de relaxamento {\it stiff} para os sistemas locais de leis de conservação, com particular interesse na questão da estabilidade e limites singulares dessas soluções no tempo zero de relaxão. O relaxamento é importante em muitas situações físicas, tais como, em teoria cinética, dinâmica de gases fora do equilíbrio termodinâmico local, em elasticidade com memória (histerese), transição de fase em fluxo multifásico e problemas lineares e não lineares de progação de ondas. Embora a teoria matemática para modelos não lineares de leis equilíbrio com relaxamento tem apresentado algum significativo progresso na boa colocação no contexto de modelos em termodinâmica e teoria cinética, uma compreensão completa sobre o comportamento assintótico para sistemas maiores que $2\times2$, sobre os quais soluções evoluem a partir de um determinado dado inicial com regularidade, permanece indefinida, notadamente para soluções fracas de sistemas hiperbólicos. Assim, devido à complexidade inerente a esta classe de modelos, existem poucas soluções para tais leis de equilíbrio de relaxamento por meio de métodos analíticos. Então, uma análise abstrata, bem como a computação numérica prática por meio de algoritmos de aproximação, constituem ferramentas importantes para estudar tal classe de modelos, bem como para obter novas perspectivas para ampliar o conhecimento geral de sistemas de leis de balanço, ou de leis de equilíbrio. Portanto, foi também desenvolvido um novo método de volumes finitos de tipo {\it unsplitting}, localmente conservativo, via construção formal. Este método foi capaz de computar para sistemas de Euler tanto novas soluções não monótonas como também de reproduzir soluções qualitativamente corretas em regime de fricção alta e gravidade, recentemente publicados na literatura. De fato, os novos algoritmos de apro\-xima\-ção {\it unsplitting} também foram usados para ajudar a compreender um problema de injeção de nitrogênio e de vapor em meios poroso. Outro ponto de vista fundamental perseguido nesta tese é a comparação entre duas metodologias para abordar a questão da resolução de leis equilíbrio com termos fonte de relaxamento: uma metodologia baseia-se do pressuposto que o fenômeno físico está sob equilíbrio termodinâmico (equilíbrio instantâneo), que é modelado por sistemas de leis de conservação hiperbólicas, e a outra metodologia é baseada no relaxamento de tal equilíbrio, que por sua vez dá origem à utilização dos sistemas de leis de equilíbrio na modelagem do processo de relaxamento, como por exemplo, em modelos de transição de fase. Neste momento, uma série de perguntas naturais surgem: quão diferentes são essas soluções de ambas as soluções obtidas por meio destas duas abordagens? A este respeito, uma pergunta mais rigorosa - e mais fundamental - é: como é o comportamento de tais soluções durante o processo de relaxamento e qual é o seu limite? A fim de entender melhor essas metodologias, vamos considerar dois formalismos matemáticos distintos. Nesta tese, nós damos um exemplo de modelagem utilizando esta nova metodologia para a injeção de nitrogênio e de vapor de água em meios porosos. Nós não fomos capazes de dar uma resposta assertiva a todas as perguntas acima, mas um sólido ponto de partida é um estudo aprofundado do caso unidimensional para um problema concreto, que é feito nesta tese. Acreditamos que temos um campo muito interessante (e promissor) de trabalho pela frente, que temos a intenção de continuar a estudar, a fim de entender melhor, via análises abstrata e numérica, tais perguntas importantes e que permanecem indefinidas. Esta tese é uma pequena tentativa de obter uma nova compreensão sobre tais modelos de leis de balançoAbstract: In this thesis, we are concerned with the limit behaviour of hyperbolic systems of conservation laws with stiff relaxation terms to the local systems of conservation laws, particularly the question of stability and singular limits of such solutions to the zero relaxation time. Relaxation is important in many physical situations, as such, in kinetic theory, gases not in local thermodynamic equilibrium, elasticity with memory (hysteresis), multiphase and phase transition and linear and nonlinear waves. Although the mathematical theory of nonlinear balance law with relaxation has presented significant progress on well-posedness linked to extended thermodynamics and kinetic theory, a complete understanding for systems larger than $2\times2$ about how solutions evolve from a given initial data and their regularity and asymptotic behaviour remains elusive, mainly for weak solutions of hyperbolic systems. Thus, due to the complexity inherent to this class of models, there are few solutions for such relaxation balance laws by means of analytical methods. Then, abstract analysis as well as practical computing via approximation algorithms are both significant mathematical tools to tackle as well as to get further insights to enlarge the knowledge for systems of balance laws. Therefore, it was also developed a new unsplitting finite volume methods, which in turn is locally conservative by formal construction. This method was able to corroborate the new solutions for Euler systems with a non-monotonic character as well as to reproduce correct qualitatively solutions of the Euler models with high friction regime and gravity, recently published in the literature. Indeed, the novel unsplitting approximation algorithms were also used to address injection problems of nitrogen and steam in porous media. Another crucial viewpoint pursued in this thesis is the comparison between two methodologies to tackle the issue of solving balance laws with relaxation source terms: one methodology is based by assuming that the physical phenomenon is under thermodynamic equilibrium (instantaneous equilibrium), which is modelled by systems of conservation laws, and the other methodology is based in the relaxation of such equi\-li\-brium, which in turn gives rise to the use of systems of balance laws in the modelling of the relaxation process, for instance, in the modelling of phase transition. At this moment a natural questions is: how different are these both solutions obtained by means of two approaches? In this regard, a more stringent -- and more fundamental -- question is: how is the behaviour of such solutions during the relaxation process and how is its limit? In order to better understand these methodologies we will consider two distinct mathematical formalisms. In thesis, we give an example of modelling using this novel methodology for the injection of nitrogen and steam in porous media. We were not able to give assertive answers to the above questions, but a solid starting point is a thorough study of the one-dimensional case for a concrete problem, which is done in this thesis. We believe we have a very interesting (and promising) field of work ahead of us, which we intend to continue studying in order to better understand abstract and numerical analysis for these important questions that remains elusive. This thesis is a small attempt to get new insights in this directionDoutoradoMatematica AplicadaDoutor em Matemática Aplicada2011/23628-0FAPES

Dynamics of Numerics & Spurious Behaviors in CFD Computations

The global nonlinear behavior of finite discretizations for constant time steps and fixed or adaptive grid spacings is studied using tools from dynamical systems theory. Detailed analysis of commonly used temporal and spatial discretizations for simple model problems is presented. The role of dynamics in the understanding of long time behavior of numerical integration and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in computational fluid dynamics (CFD) is explored. The study is complemented with examples of spurious behavior observed in steady and unsteady CFD computations. The CFD examples were chosen to illustrate non-apparent spurious behavior that was difficult to detect without extensive grid and temporal refinement studies and some knowledge from dynamical systems theory. Studies revealed the various possible dangers of misinterpreting numerical simulation of realistic complex flows that are constrained by available computing power. In large scale computations where the physics of the problem under study is not well understood and numerical simulations are the only viable means of solution, extreme care must be taken in both computation and interpretation of the numerical data. The goal of this paper is to explore the important role that dynamical systems theory can play in the understanding of the global nonlinear behavior of numerical algorithms and to aid the identification of the sources of numerical uncertainties in CFD

Upwind and symmetric shock-capturing schemes

The development of numerical methods for hyperbolic conservation laws has been a rapidly growing area for the last ten years. Many of the fundamental concepts and state-of-the-art developments can only be found in meeting proceedings or internal reports. This review paper attempts to give an overview and a unified formulation of a class of shock-capturing methods. Special emphasis is on the construction of the basic nonlinear scalar second-order schemes and the methods of extending these nonlinear scalar schemes to nonlinear systems via the extact Riemann solver, approximate Riemann solvers, and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows is discussed. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas dynamics problems

A class of high resolution explicit and implicit shock-capturing methods

An attempt is made to give a unified and generalized formulation of a class of high resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock wave computations. Included is a systematic review of the basic design principle of the various related numerical methods. Special emphasis is on the construction of the basis nonlinear, spatially second and third order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and the flux vector splitting approaches. Generalization of these methods to efficiently include equilibrium real gases and large systems of nonequilibrium flows are discussed. Some issues concerning the applicability of these methods that were designed for homogeneous hyperbolic conservation laws to problems containing stiff source terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for 1-, 2- and 3-dimensional gas dynamics problems

Progress in the Development of Compressible, Multiphase Flow Modeling Capability for Nuclear Reactor Flow Applications

In nuclear reactor safety and optimization there are key issues that rely on in-depth understanding of basic two-phase flow phenomena with heat and mass transfer. Within the context of multiphase flows, two bubble-dynamic phenomena – boiling (heterogeneous) and flashing or cavitation (homogeneous boiling), with bubble collapse, are technologically very important to nuclear reactor systems. The main difference between boiling and flashing is that bubble growth (and collapse) in boiling is inhibited by limitations on the heat transfer at the interface, whereas bubble growth (and collapse) in flashing is limited primarily by inertial effects in the surrounding liquid. The flashing process tends to be far more explosive (and implosive), and is more violent and damaging (at least in the near term) than the bubble dynamics of boiling. However, other problematic phenomena, such as crud deposition, appear to be intimately connecting with the boiling process. In reality, these two processes share many details

A simple phase transition relaxation solver for liquid-vapor flows

International audienceDetermining liquid-vapor phase equilibrium is often required in multiphase flow computations. Existing equilibrium solvers are either accurate but computationally expensive, or cheap but inaccurate. The present paper aims at building a fast and accurate specific phase equilibrium solver, specifically devoted to unsteady multiphase flow computations. Moreover, the solver is efficient at phase diagram bounds, where non-equilibrium pure liquid and pure gas are present. It is systematically validated against solutions based on an accurate (but expensive) solver. Its capability to deal with cavitating, evaporating and condensing two-phase flows is highlighted on severe test problems both 1D and 2D

Structural Properties of the Stability of Jamitons

It is known that inhomogeneous second-order macroscopic traffic models can reproduce the phantom traffic jam phenomenon: whenever the sub-characteristic condition is violated, uniform traffic flow is unstable, and small perturbations grow into nonlinear traveling waves, called jamitons. In contrast, what is essentially unstudied is the question: which jamiton solutions are dynamically stable? To understand which stop-and-go traffic waves can arise through the dynamics of the model, this question is critical. This paper first presents a computational study demonstrating which types of jamitons do arise dynamically, and which do not. Then, a procedure is presented that characterizes the stability of jamitons. The study reveals that a critical component of this analysis is the proper treatment of the perturbations to the shocks, and of the neighborhood of the sonic points.Comment: 22 page, 6 figure