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Mathematical Study of Free-Surface Fluids in Incompressible Dynamics Dena Kazerani

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Mathematical Study of Free-Surface Fluids in Incompressible Dynamics Dena Kazerani Mathematical study of free-surface fluids in incompressible dynamics Dena Kazerani To cite this version: Dena Kazerani. Mathematical study of free-surface fluids in incompressible dynamics. Mathematics [math]. Université Pierre & Marie Curie - Paris 6, 2016. English. tel-01422881v2 HAL Id: tel-01422881 https://tel.archives-ouvertes.fr/tel-01422881v2 Submitted on 6 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Université Pierre et Marie Curie Laboratoire Jacques-Louis Lions Paris 6 UMR 7598 THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Présentée et soutenue publiquement le 29 Novembre 2016 pour l’obtention du grade de DOCTEUR DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Spécialité : Mathématiques Appliquées par Dena KAZERANI Études mathématiques de fluides à frontières libres en dynamique incompressible après avis des rapporteurs Roberto Natalini Consiglio Nazionale delle Ricerche (CNR) Fabio Nobile École Polytechnique Fédérale de Lausanne devant un jury composé de Vincent Duchêne Centre National de la Recherche Scientifique (CNRS) Examinateur Pascal Frey Université Pierre et Marie Curie-Paris 6 Directeur de thèse Edwige Godlewski Université Pierre et Marie Curie-Paris 6 Examinatrice Frédéric Legoll École des Ponts ParisTech Examinateur Fabio Nobile École Polytechnique Fédérale de Lausanne Rapporteur Jacques Sainte-Marie Centre d’études et d’expertise sur les risques, l’environnement, Examinateur la mobilité et l’aménagement (CEREMA) Nicolas Seguin Université de Rennes 1 Directeur de thèse École Doctorale de Sciences Mathématiques Faculté de Mathématiques de Paris Centre ED 386 UFR 929 Dena KAZERANI : Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F- 75005, Paris, France. CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France. Adresse électronique: [email protected] 3 4 The test of a good teacher is not how many questions he can ask his pupils that they will answer readily, but how many questions he inspires them to ask him which he finds it hard to answer. Alice Wellington Rollins We are what we repeatedly do. Excellence then, is not an act, but a habit. Aristotle Remerciements Mes premiers remerciements s’adressent à mes directeurs de thèse, Pascal Frey et Nicolas Seguin. Je remercie Pascal pour m’avoir confié son code et m’avoir bénéficié de son expertise. Je lui suis particulièrement reconnaissante pour son soutien, sa compréhension et ses conseils. Je remercie de même, Nicolas Seguin pour m’avoir accordé sa confiance. J’ai été réellement impressionnée par son intuition, sa vision globale et sa vision stratège. Le contact simultané avec ces deux spécialistes m’a permis de bénéficier d’une formation pluridisciplinaire et enrichissante. Je suis très honorée que des personnalités scientifiques comme Roberto Natalini et Fabio No- bile acceptent de reporter cette thèse. Je remercie vivement Vincent Duchêne, Edwige Godlewski, Frédéric Legoll et Jacques Sainte-Marie pour avoir accepté de participer au jury. Le laboratoire Jacques-Louis Lions est un grand centre de recherche. Il est aussi riche sur le plan humain que sur le plan scientifique. Je remercie particulièrement Edwige Godlewski et Benoît Perthame de nous avoir donné l’occasion d’établir le concept du thé du LJLL. J’ai beaucoup appris pendant la période que j’ai passé dans ce laboratoire et ça c’est en grande partie, dû aux rencontres exceptionnelles que j’ai pu y faire. Je remercie Alain Haraux et Jean-Paul Penot pour les partages qu’ils m’ont accordés. J’estime avoir eu beaucoup de chance pour avoir été encadrée par Bruno Després pendant le stage de pré-M2. Il m’a généreusement montré la beauté et la richesse de la recherche en analyse numérique. La liste des gens qui ont indirectement contribué à la réalisation de cette thèse est exhaustive. Je remercie la gestion administrative et informatique du laboratoire. J’ai une pensée particulière envers les doctorants et anciens doctotrants du bureau 15-16-301 avec qui j’ai partagé des restaurants, des expositions et des cours de zumba. Je remercie aussi tous les doctorants/post-doctorants qui m’ont accompagné pour les déjeuners à la cantine, pour des pauses café et pour la gestion des bureaux des doctorants. De même, je remercie les doctorants que j’ai rencontrés pendant les conférences, pour les échanges, les balades et les courses à pied. Enfin, j’arrive à ceux dont la présence dépasse grandement le contexte de la thèse, ceux dont la présence chauffe mon coeur. Je pense à la personne à avoir cru le plus en mon intelligence: ma soeur Parna. Je pense aussi à mon père pour m’avoir fait comprendre entre autres que mes capacités étaient bien plus que ce que j’imaginais. Ma mère de son coté, n’a jamais cessé de me soutenir. C’est aussi grâce à eux que j’ai compris que le savoir n’est pas donné mais ça s’acquiert lentement, bout par bout. Mes derniers remerciements sont pour Alex, Archibald, Chiara, Dalila, Donya, Dounia, Flavien, François, Gazal, Géraldine, Jasmine, Hoel, Lara, Luiz, Maxime, Pato et Yiting. 5 6 Table of contents Introduction 9 1 Some results on a shallow water model called the Green-Naghdi equations . 10 2 Numerical simulation of the free surface incompressible Navier–Stokes equations . 29 I Some results on a shallow water model called the Green-Naghdi equations 35 1 The symmetric structure of Green–Naghdi type equations 37 1.1 Introduction . 37 1.2 Weak symmetric structure . 42 1.3 Application to Green–Naghdi type equations . 49 1.4 Conclusion . 56 1.A Computation of the second variation . 57 2 Global existence for small data of the viscous Green–Naghdi type equations 59 2.1 Introduction . 59 2.2 Main results . 64 2.3 A priori estimates . 68 2.4 Conclusion and Perspectives . 84 2.A Special Symmetric Structure . 85 2.B Local well-posedness . 87 2.C Linear stability of equilibriums of the Green-Naghdi equation . 88 II Numerical simulation of the free surface incompressible Navier–Stokes equa- 7 8 Table of contents tions 91 3 Numerical tools and global algorithm 93 3.1 Introduction . 93 3.2 Mathematical model . 95 3.3 Numerical tools . 97 3.4 Global numerical scheme . 112 3.5 Numerical results . 114 3.6 Conclusion and perspectives . 121 Bibliography 130 Introduction This thesis is about the mathematical study of free surface incompressible fluids. An incompressible flow is a fluid with a constant density. This implies a free divergence velocity. Obviously, water like most liquids is incompressible. In practice, if the density change of a fluid is small, it can be considered incompressible. This happens when the Mach number associated to the flow is low i.e. when the ratio of flow velocity to the speed of sound is small. Particularly, in our framework, air fits this regime and is considered incompressible. Incompressible fluids satisfy the incompressible Navier–Stokes equations: 8 ρ @u + (u · r)u − µ∆u + rp = ρf <> @t div u = 0 (0.1) > :u(0; x) = u0(x): Here ρ is the density whereas µ is the viscosity of the fluid. The unknowns of the problem are u the velocity flow and p the pressure. On the other hand, f is the density of force exerted on the fluid while u0 is the initial velocity flow. This equation is endowed with boundary conditions on the boundary of fluid domain. An incompressible fluid satisfying the Navier–Stokes equation (0.1) with a viscosity µ > 0, is called a Newtonian fluid. Nevertheless, if the Reynolds number associated to the flow i.e. the ratio of inertial forces to viscous forces is large, we can suppose that the fluid is perfect. In this case, it 9 10 1. Some results on a shallow water model called the Green-Naghdi equations satisfies the incompressible Euler equation (Navier–Stokes equation without the Laplace operator): 8 ρ @u + (u · r)u + rp = ρf <> @t div u = 0 (0.2) > :u(0; x) = u0(x): As an example, let us consider water. The viscosity of water is 0:001 P a:s while its density is 1000 kg=m3. Therefore, it is modeled by incompressible Euler equations in most applications specially in oceanography. For this reason, system (0.2) is the equation used in this latter domain, for the derivation of approximative models. These models are called shallow water and concern the regime where the fluid height is small compared to the characteristic wavelength. This regime coincides with the frame of some geophysical phenomena like tsunamis and earthquakes. Let us mention that in this work, our concern is the fluid’s free surface. Indeed, free surface flows satisfy the fluid equation on the fluid domain which moves following the velocity flow. For this reason, the treatment of the free surface raises some challenges which have led to the derivation of shallow water models and to creation of some specific numerical tools. The first part of the thesis is on the theoretical study of one of these approximative models called the Green–Naghdi equation whereas the second part is on the numerical study of System (0.1). 1 Some results on a shallow water model called the Green-Naghdi equations Shallow water models are essentially considered in Oceanography where the water height is small compared to the characteristic wavelength and the rotational effects are negligible.
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