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Mathematical Analysis of the Interaction of an Inviscid Fluid with Immersed Structures Krisztian Benyo

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Mathematical Analysis of the Interaction of an Inviscid Fluid with Immersed Structures Krisztian Benyo Mathematical analysis of the interaction of an inviscid fluid with immersed structures Krisztian Benyo To cite this version: Krisztian Benyo. Mathematical analysis of the interaction of an inviscid fluid with immersed structures. Fluid mechanics [physics.class-ph]. Université de Bordeaux, 2018. English. NNT : 2018BORD0156. tel-01927910 HAL Id: tel-01927910 https://tel.archives-ouvertes.fr/tel-01927910 Submitted on 20 Nov 2018 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. Thèse de doctorat de l’Université de Bordeaux École doctorale de mathématiques et d’informatique Spécialité : Mathématiques Appliquées Présentée par Krisztián Benyó pour obtenir le grade de docteur de l’Université de Bordeaux Analyse mathématique de l’interaction d’un fluide non-visqueux avec des structures immergées Date de soutenance : 25 septembre 2018 Directeurs de thèse : M. David Lannes DR CNRS (Université de Bordeaux) M. Franck Sueur Professeur (Université de Bordeaux) Rapporteurs : M. Pascal Noble Professeur (Université Paul Sabatier) M. Takéo Takahashi CR Inria (Université de Lorraine) Devant la commission d’examen composée de : Emmanuel Audusse MdC Université Paris 13 Examinateur Afaf Bouharguane MdC Université de Bordeaux Examinatrice Catherine Choquet Professeur Université de La Rochelle Examinatrice David Lannes DR CNRS Université de Bordeaux Co-directeur Pascal Noble Professeur Université Paul Sabatier Rapporteur Franck Sueur Professeur Université de Bordeaux Directeur Takéo Takahashi CR Inria Université de Lorraine Rapporteur 2018 Doctoral dissertation University of Bordeaux Doctoral School of Mathematics and Computer Science Discipline: Applied Mathematics Presented by Krisztián Benyó for the degree of Doctor of Philosophy Mathematical analysis of the interaction of an inviscid fluid with immersed structures PhD supervisors: Prof. David Lannes DR CNRS (Université de Bordeaux) Prof. Franck Sueur Professeur (Université de Bordeaux) Reviewers: Prof. Pascal Noble Professeur (Université Paul Sabatier) Prof. Takéo Takahashi CR Inria (Université de Lorraine) Examiners: Prof. Emmanuel Audusse MdC (Université Paris 13) Prof. Afaf Bouharguane MdC (Université de Bordeaux) Prof. Catherine Choquet Professeur (Université de La Rochelle) 2018 Résumé : Cette thèse porte sur l’analyse mathématique de l’interaction d’un fluide non-visqueux avec des structures immergées. Plus précisément, elle est structurée autour de deux axes principaux. L’un d’eux est l’analyse asymptotique du mouvement d’une particule infinitésimale en milieu liquide. L’autre concerne l’interaction entre des vagues et une structure immergée. La première partie de la thèse repose sur l’analyse mathématique d’un système d’équa- tions différentielles ordinaires non-linéaires d’ordre 2 modélisant le mouvement d’un solide infiniment petit dans un fluide incompressible en 2D. Les inconnues du modèle décrivent la position du solide, c’est-à-dire la position du centre de masse et son angle de rotation. Les équations proviennent de la deuxième loi de Newton avec un prototype de force de type Kutta-Joukowski. Plus précisément, nous étudions la dynamique de ce système lorsque l’inertie du solide tend vers 0. Les principaux outils utilisés sont des développements asymptotiques multi- échelles en temps. Pour la dynamique de la position du centre de masse, l’étude met en évidence des analogies avec le mouvement d’une particule chargée dans un champ électromagnétique et la théorie du centre-guide. En l’occurrence, le mouvement du centre- guide est donné par une équation de point-vortex. La dynamique de l’angle est quant à elle donnée par une équation de pendule non-linéaire lentement modulée. Des régimes très différents se distinguent selon les données initiales. Pour de petites vitesses angulaires initiales la méthode de Poincaré-Lindstedt fait apparaitre une modulation des oscillations rapides, alors que pour de grandes vitesses angulaires initiales, un movement giratoire bien plus irrégulier est observé. C’est une conséquence particulière et assez spectaculaire de l’enchevêtrement des trajectoires homocliniques. La deuxième partie de la thèse porte sur le problème des vagues dans le cas où le domaine occupé par le fluide est à surface libre et avec un fond plat sur lequel un objet solide se translate horizontalement sous l’effet des forces de pression du fluide. Nous avons étudié deux systèmes asymptotiques qui décrivent le cas d’un fluide parfait incompressible en faible profondeur. Ceux-ci correspondent respectivement aux équations de Saint-Venant et de Boussinesq. Grâce à leur caractère bien-posé en temps long, les modèles traités permettent de prendre en compte certains effets de la mécanique du solide, comme les forces de friction, ainsi que les effets non-hydrostatiques. Notre analyse théorique a été complétée par des études numériques. Nous avons développé un schéma de différences finies d’ordre élevé et nous l’avons adapté à ce prob- lème couplé afin de mettre en évidence les effets d’un solide (dont le mouvement est limité à des translations sur le fond) sur les vagues qui passent au dessus de lui. A la suite de ces travaux, nous avons souligné l’influence des forces de friction sur ce genre de systèmes couplés ainsi que sur le déferlement des vagues. Quant à l’amortissement dû aux effets hydrodynamiques, une vague ressemblance avec le phénomène de l’eau morte est mise en évidence. Fluid-structure interaction v Mots-clés : interaction fluide-structure, analyse asymptotique, dynamique des fluides, équations d’Euler, particules immergées, équation de Newton vi Krisztián Benyó Abstract: This PhD thesis concerns the mathematical analysis of the interaction of an inviscid fluid with immersed structures. More precisely it revolves around two main problems: one of them is the asymptotic analysis of an infinitesimal immersed particle, the other one being the interaction of water waves with a submerged solid object. Concerning the first problem, we studied a system of second order non-linear ODEs, serving as a toy model for the motion of a rigid body immersed in a two-dimensional perfect fluid. The unknowns of the model describe the position of the object, that is the position of its center of mass and the angle of rotation; the equations arise from Newton’s second law with the consideration of a Kutta-Joukowski type lift force. It concerns the detailed analysis of the dynamic of this system when the solid inertia tends to 0. For the evolution of the position of the solid’s center of mass, the study highlights similarities with the motion of a charged particle in an electromagnetic field and the well- known “guiding center approximation”; it turns out that the motion of the corresponding guiding center is given by a point-vortex equation. As for the angular equation, its evo- lution is given by a slowly-in-time modulated non-linear pendulum equation. Based on the initial values of the system one can distinguish qualitatively different regimes: for small angular velocities, by the Poincaré-Lindstedt method one observes a modulation in the fast time-scale oscillatory terms, for larger angular velocities however erratic rota- tional motion is observed, a consequence of Melnikov’s observations on the presence of a homoclinic tangle. About the other problem, the Cauchy problem for the water waves equations is consid- ered in a fluid domain which has a free surface on the upper vertical limit and a flat bottom on which a solid object moves horizontally, its motion determined by the pressure forces exerted by the fluid. Two shallow water asymptotic regimes are detailed, well-posedness results are obtained for both the Saint-Venant and the Boussinesq system coupled with Newton’s equation characterizing the solid motion. Using the particular structure of the coupling terms one is able to go beyond the standard scale for the existence time of solutions to the Boussinesq system with a moving bottom. An extended numerical study has also been carried out for the latter system. A high order finite difference scheme is developed, extending the convergence ratio of previous, staggered grid based models. The discretized solid mechanics are adapted to represent important features of the original model, such as the dissipation due to the friction term. We observed qualitative differences for the transformation of a passing wave over a moving solid object as compared to an immobile one. The movement of the solid not only influ- ences wave attenuation but it affects the shoaling process as well as the wave breaking. The importance of the coefficient of friction is also highlighted, influencing qualitative and quantitative properties of the coupled system. Furthermore, we showed the hydrodynamic damping effects of the waves on the solid motion, reminiscent of the so-called dead water phenomenon. Fluid-structure interaction vii Keywords: fluid-structure interaction, asymptotic analysis, fluid dynamics, Euler equa- tion, immersed particule, Newton equation viii Krisztián Benyó Laboratoire d’accueil : Institut de Mathématiques de Bordeaux, Université de Bordeaux Bât. A33,
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