Les méthodes de domaines fictifs permettent de simuler numériquement des écoulements autour de structures complexes et/ou mobiles à l’aide de maillages simples. L’objet solide est alors « immergé » dans un domaine de calcul englobant le fluide et le solide. Dans un premier temps, on étudie une méthode de pénalisation, qui consiste à ajouter un terme dans l’équation de conservation de la quantité de mouvement du fluide afin d’imposer la vitesse du solide. Grâce à des développements asymptotiques, on obtient une estimation de l’erreur induite par cette approche lorsque le solide est en mouvement. Ce procédé est ensuite couplé avec un schéma de projection vectorielle permettant d’imposer la contrainte d’incompressibilité. La convergence du schéma ainsi obtenu, vers les équations de Navier-Stokes, est établie. Dans un second temps,une approche originale capable de traiter des écoulements multiphasiques est développée : la méthode de porosité variable. L’idée principale est de considérer le solide comme un milieu sans masse. La discrétisation des bilans massiques de chaque phase est alors modifiée, de sorte que le volume total occupé par l’ensemble des phases fluides soit égal au volume laissé libre parle solide. Cette méthode est validée numériquement sur un ensemble varié de cas test comprenantd es écoulements monophasiques incompressibles et compressibles ainsi que des écoulements diphasiques. / Fictitious domain methods allow to simulate flows around complex and/or moving bodies with simple meshes. The object is "immersed" in a domain that contains fluid and solid volumes. The penalization method, which consists in adding a term in the momentum balance equation, in order to impose the solid velocity, is studied in a first part. Thanks to asymptotic expansions, the order of the error induced by this method is computed for moving bodies. This approach is then coupled with a Vector Penalty Projection scheme that permits to impose the incompressibility constraint. The convergence of the penalized scheme towards the Navier-Stokes equations is established. In a second part, an original approach, able to treat multiphase flowsis presented: the Time and Space Dependent Porosity method. The key idea is to consider the solid as a medium without mass. The discretization of the mass balance equation is modified,so that the total volume occupied by all fluid phases and the solid is equal to the total volume.This method is numerically validated on a set of various test cases including incompressible or compressible single phase flows and two-phase flows.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work presents the Cauchy's theorem in its classical form, and aims to weaken their hypotheses, providing a more advantageous use in continuum mechanics. The methodology is axiomatic, that is, basic concepts are presented aiming to triggering logical statements that were made in the main theorems to achieve the objectives of this dissertation. The main result is Theorem 14, where a law of balance is folowed necessary and suficient condition for a Cauchy Flow be Weakly Balanced. / Neste trabalho é apresentado o Teorema de Cauchy em sua forma clássica e tem por objetivo enfraquecer suas hipóteses, proporcionando uma aplicação mais vantajosa na mecânica do contínuo. A metodologia empregada é axiomática, ou seja, são apresentados conceitos básicos com vistas ao desencadeamento lógico das demonstrações que foram realizadas nos teoremas principais para atingir os objetivos dessa dissertação. O resultado principal é o teorema 14, onde obedecer a uma lei de balanço, é condições necessária e suficiente para que um Fluxo de Cauchy seja Fracamente Balanceado.

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O presente trabalho de pesquisa tem por objetivo estudar a paralelização de algoritmos voltados à solução de equações diferenciais parciais. Esses algoritmos são utilizados para gerar a solução numérica das equações de Navier-Stokes em um escoamento bidimensional incompressível de um fluido newtoniano. As derivadas espaciais são calculadas através de um método de diferenças finitas compactas com a utilização de aproximações de altas ordens de precisão. Uma vez que o cálculo de derivadas espaciais com alta ordem de precisão da forma compacta adotado no presente estudo requer a solução de sistemas lineares tridiagonais, é importante realizar estudos voltados a resolução desses sistemas, para se obter uma boa performance. Ressalta-se ainda que a solução de sistemas lineares também faz-se presente na solução numérica da equação de Poisson. Os resultados obtidos decorrentes da solução das equações diferenciais parciais são comparados com os resultados onde se conhece a solução analítica, de forma a verificar a precisão dos métodos implementados. Os resultados do código voltado à resolução das equações de Navier-Stokes paralelizado para simulação de escoamentos incompressíveis são comparados com resultados da teoria de estabilidade linear, para validação do código final. Verifica-se a performance e o speedup do código em questão, comparando-se o tempo total gasto em função do número de elementos de processamento utilizados / The objective of the present work is to study the parallelization of partial differential equations. The aim is to achieve an effective parallelization to generate numerical solution of Navier-Stokes equations in a two-dimensional incompressible and isothermal flow of a Newtonian fluid. The spatial derivatives are calculated using compact finite differences approximations of higher order accuracy. Since the calculation of spatial derivatives with high order adopted in the present work requires the solution of tridiagonal systems, it is important to conduct studies to solve these systems and achieve good performance. In addiction, linear systems solution is also present in the numerical solution of a Poisson equation. The results generated by the solution of partial differential equations are compared to analytical solution, in order to verify the accuracy of the implemented methods. The numerical parallel solution of a Navier-Stokes equations is compared with linear stability theory to validate the final code. The performance and the speedup of the code in question is also checked, comparing the execution time in function of the number of processing elements

Nous considérons dans cette thèse la discrétisation par la méthode spectrale et la simulation numérique de l'écoulement d'un fluide visqueux incompressible occupant le domaine ? modélisé par les équations de Navier-Stokes. Nous avons choisi de les coupler avec l'équation de la chaleur dans le cas ou la viscosité dépend de la température avec des conditions aux limites portant sur la vitesse et la température.La méthode s'avère optimale en ce sens que l'erreur obtenue n'est limitée que par la régularité de la solution. Elle est de type spectrale. Nous donnons des estimations d'erreur a priori optimales et nous confirmons l'étude théorique par des résultats numériques. Nous considérons aussi les équations de Navier-Stokes/chaleur instationnaires dont nous proposons une discrétisation en temps et en espace en utilisant le schéma d'Euler implicite et les méthodes spectrale. Quelques expériences numériques confirment l'intérêt de la discrétisation. / In this thesis we consider the discretization by spectral method and the numerical simulation of a viscous incompressible fluid in the domain ?, the model being the Navier-Stokes equations. We have chosen to couple them with the heat equation where the viscosity of the fluid depends on the temperature, with boundary conditions which involve the velocity and the temperature. The method is proved to be optimal in the sense that the order of convergence is only limited by the regularity of the solution. The numerical analysis of the discrete problem is performed and numerical experiments are presented, they turn out to be in good coherence with the theoretical results. Finally, we consider the unsteady Navier-Stokes/heat equations which models the time-dependent flow. We propose a discretization of this problem that relies on a backward Euler's scheme in time and spectral methods in space and present some numerical experiments which confirm the interest of the discretization.

Nous nous intéressons dans cette thèse aux équations de Navier-Stokes pour un fluide visqueux incompressible. Dans la première partie, nous étudions le cas d’un fluide homogène. Rappelons que la grande question de la régularité globale en dimension 3 est plus ouverte que jamais : on ne sait pas si la solution de l’équation correspondant à un état initial suffisamment régulier mais arbitrairement loin du repos, va perdurer indéfiniment dans cet état (régularité globale) ou exploser en temps fini(singularité). Une façon d’aborder le problème est de supposer cette éventuelle rupture de régularité et d’envisager les différents scenarii possibles. Après un rapide survol de la structure propre aux équations de Navier-Stokes et des résultats connus à ce jour (chapitre 1), nous nous intéressons(chapitre 2) à l’existence locale (en temps) de solutions dans des espaces de Sobolev qui ne sont pas invariants d’échelle. Partant d’une donnée initiale qui produit une singularité, on prouve l’existence d’une constante optimale qui minore le temps de vie de la solution. Cette constante, donnée parla méthode rudimentaire du point fixe, fournit ainsi un bon ordre de grandeur sur le temps de vie maximal de la solution. Au chapitre 3, nous poursuivons les investigations sur le comportement de telles solutions explosives à la lumière de la méthode des éléments critiques.Dans le seconde partie de la thèse, nous sommes intéressés à un modèle plus réaliste du point de vue de la physique, celui d’un fluide incompressible à densité variable. Ceci est modélisé par les équations de Navier-Stokes incompressible et inhomogènes. Nous avons étudié le caractère globalement bien posé de ces équations dans la situation d’un fluide évoluant dans un tore de dimension 3, avec des données initiales appartenant à des espaces critiques et sans hypothèse de petitesse sur la densité. / This thesis is concerned with incompressible Navier-Stokes equations for a viscous fluid. In the first part, we study the case of an homogeneous fluid. Let us recall that the big question of the global regularity in dimension 3 is still open : we do not know if the solution associated with a data smooth enough and far from the immobile stage will last over time (global regularity) or on the contrary will stop living in finite time and blow up (singularity). The goal of this thesis is to study this regularity break. One way to deal witht his question is to assume that such a phenomen on occurs and to study differents scenarii. The chapter 1 is devoted to a recollection of well-known results. In chapter 2, we are interesting in the local (in time) existence of a solution in some Sobolev spaces which are not invariant under the natural sclaing of Navier-Stokes. Starting with a data generating a singularity, we can prove there exists an optimal lower boundary of the lifes pan of such a solution. In this way, the lower boundary provided by the elementary procedure of fixed-point, gives the correctorder of magnitude. Then, we keep on investigations about the behaviour of regular solution near the blow up, thanks to the method of critical elements (chapter 3).In the second part, we are concerned with a more relevant model, from a physics point of view : the inhomogeneous Navier-Stokes system. We deal with the global well poseness of such a model for a inhomogeneous fluid, evolving on a tor us in dimension 3, with critical data and without smallnes sassumption on the density.

Obsahem této práce je numerická simulace dvoudimenzionálního proudění nestlačitelné vazké kapaliny. Uvažujeme rotující elipsu soustředně umístěnou v kružnici. Prostor mezi elipsou a kružnicí je vyplněn kapalinou. Cílem je popsat proudění kapaliny vyvolané otáčející se elipsou, tzn. stanovit rychlostní pole a rozložení tlaku. Dále pak chceme stanovit přídavné silové účinky kapaliny působící na elipsu. Tyto výsledky získáme řešením Navierových-Stokesových rovnic metodou konečných prvků. Důraz je kladen na odvození numerického schématu v maticové formě vhodné pro numerickou implementaci. Časově závislá výpočetní síť je popsána pomocí Arbitrary Lagrangian-Eulerian (ALE) formulace. Pro obdržení relevantních výsledků je nutná stabilizace metody konečných prvků. Uvedené výsledky naznačují, že odvozená metoda je dostatečně přesná.

MSc (Applied Mathematics) / Mathematics and Applied Mathematics Department / In this study the iteration of boundary conditions method (Chizhonkov and Kargin, 2006) is used together with the well known Finite difference numerical method to solve the Stokes problem over a rectangular domain as well as in irregular domain. The iteration of boundary conditions method has been applied to the Stokes problem in a rectangular domain, 􀀀 2 <x< 2 , 􀀀 d 2 < y < d 2 , by the above mentioned researchers. Our main task here is to validate the results of the approximate methods by this analytical method in case of the rectangular domain and extend that to the case of irregular domain.The (Chizhonkov and Kargin, 2006) algorithm is typically the best choice for validation purposes because of its high accuracy. It is known in literature that increasing the parameter d, which represents the ratio of the sides, leads to slow down in convergence of the approximate methods like the conjugate Gradients of Uzawa (Kobelkov and Olshanskii, 2000). It is therefore important that an algorithm that converges uniformly with respect to the parameter d is considered. The (Chizhonkov and Kargin, 2006) algorithm is typical of such an algorithm, and hence our choice of the method in this work. In this project the non-homogeneous Stokes problem is transformed into a homogeneous Stokes problem and the resulting problem is then decomposed into two sub problems that are solvable by the eigenfunction expansion method. Once all necessary coefficients of the generalised Fourier series are known and the functions describing the boundary conditions are prescribed and represented in terms of the Fourier series, we then proceed to formulate the iteration of boundary conditions numerical algorithm. Finally we develop a numerical scheme, using the finite difference methods, for solving the problem in both rectangular and irregular domains. Coding of the numerical algorithm is done using MATLAB 9.0,R2016a programming language, and implemented by the author. The results of the two methods in both cases of boundary conditions are then compared for validation of our purely numerical results. / NRF

The first part of the thesis focuses on developing a numerical methodology to simulate the acoustic properties of a hybrid liner consisting of a perforated plate, a porous layer and a Helmholtz cavity. Liners are always a standard way to reduce noise in today’s aeroengines, e.g. the fan noise can be reduced effectively through the installation of acoustic liners as wall treatments in the ducts. In order to optimize a liner in the design phase, an accurate and efficient prediction tool is of interests. Hence, a unified Linearized Navier-Stokes equations(LNSE) approach has been implemented in the thesis, combining the LNSE in frequency domain with the fluid equivalent model. The LNSE is applied in the vicinity of the perforated plate to simulate sound propagation including viscous damping effect, and the fluid equivalent model is used to model the sound propagation in the porous material including absorption. The second part of the thesis focuses on the prediction of thermoacoustic instabilities. Thermoacoustic instabilities arise when positive coupling occurs between the flame and the acoustics in the feedback loop, i.e. the flame acts as an amplifier of the disturbances (acoustic or fluid) at a natural frequency of the combustion system. Once the thermoacoustic instabilities occur, it will lead to extremely high noise levels within a relatively narrow frequency range, resulting in a huge damage to the structure of the combustors. Hence, a solution must be found, which breaks the link between the combustion process and the structural acoustics. The numerical prediction of thermoacoustic instabilities in the thesis is performed by two different numerical methodologies. One solves the Helmholtz equation in combination of the flame n − tau model with the low Mach number assumptions, and the other solves the Linearized Navier-Stokes equations in frequency domain with mean flow. The result show that the mean flow has a significant effect on the thermoacoustic instabilities, which is non-negligible when the Mach number reaches to 0.15. / <p>QC 20151221</p> / TANGO