abstract: The dynamics of a fluid flow inside 2D square and 3D cubic cavities under various configurations were simulated and analyzed using a spectral code I developed. This code was validated against known studies in the 3D lid-driven cavity. It was then used to explore the various dynamical behaviors close to the onset of instability of the steady-state flow, and explain in the process the mechanism underlying an intermittent bursting previously observed. A fairly complete bifurcation picture emerged, using a combination of computational tools such as selective frequency damping, edge-state tracking and subspace restriction. The code was then used to investigate the flow in a 2D square cavity under stable temperature stratification, an idealized version of a lake with warmer water at the surface compared to the bottom. The governing equations are the Navier-Stokes equations under the Boussinesq approximation. Simulations were done over a wide range of parameters of the problem quantifying the driving velocity at the top (e.g. wind) and the strength of the stratification. Particular attention was paid to the mechanisms associated with the onset of instability of the base steady state, and the complex nontrivial dynamics occurring beyond onset, where the presence of multiple states leads to a rich spectrum of states, including homoclinic and heteroclinic chaos. A third configuration investigates the flow dynamics of a fluid in a rapidly rotating cube subjected to small amplitude modulations. The responses were quantified by the global helicity and energy measures, and various peak responses associated to resonances with intrinsic eigenmodes of the cavity and/or internal retracing beams were clearly identified for the first time. A novel approach to compute the eigenmodes is also described, making accessible a whole catalog of these with various properties and dynamics. When the small amplitude modulation does not align with the rotation axis (precession) we show that a new set of eigenmodes are primarily excited as the angular velocity increases, while triadic resonances may occur once the nonlinear regime kicks in. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

Une étude expérimentale a été menée sur le sillage d’un cylindre circulaire stratifié dans le but de décrire l’effet de l’inclinaison et de la stratification sur les instabilités tridimensionnelles d’un sillage. Principalement 4 modes instables ont été trouvés.Ces résultats mettent ainsi en évidence le fait que la stratification et l’angle d’in- clinaison modifient fortement la transition d’un écoulement 2D à 3D du sillage d’un cylindre avec la présence de nouveaux modes instables. / An experimental study has been delineated of a stratified wake of a circular cy- linder in order to describe the effect of the tilt and of the stratification on the 3D instabilities of the wake. Principally, four unstable modes have been figured out.These results highlight the fact that the stratification strongly modifies the transition from a 2D to a 3D flow in a cylinder wake, with the presence of new unstable modes. The tilt of the cylinder with respect to the vertical plays a major role, such that a study limited to a vertical or a horizontal cylinder misses a lot of the rich dynamics of the tilted wake.

<p>Cavitation is ubiquitous in nature and scientific application where it might hinder through noise, vibration or erosion which eventually leads to reduced performance. Similarly, rising bubbles are employed in several industrial applications to homogenize the fluid. This thesis sheds light on special applications of these two phenomena. </p> <p>Once a bubble has been captured by a vortex core, the low (sometimes negative) pressure in the core causes the cavitation bubble to elongate axially while the radius of the bubble oscillates with time. Three dimensional compressible Navier-Stokes equations with surface tension are numerically solved using an all-mach solver on Basilisk software. The bubble dynamics can be categorised into separate stages: spherical growth, pinching, elongation and fragmentation. As the cylindrical bubble grows, it increases the vortex core radius. The flow and the bubble dynamics are strongly coupled. The effect of changing cavitation number and bubble to vortex size ratio has been explored. The bubble sizes and dynamics at different time steps have also been recorded. When the pressure in the core is negative, the bubble continues to grow axially forming a long tube, which is also observed in experiments. In oceans, density varies with depth due to varying salinity and temperature gradient, which prevents the vertical exchange of heat, carbon, dissolved oxygen, and nutrients as well as blooms the population of harmful bacteria such as cyanobacteria. The rising motion of a single or cluster of bubbles creates an upflow that can cause homogenization or destratification. Confined bubble columns are used for microelectronic cooling as well as in chemical reactors for mixing stratified fluids without any mechanical agitation or power. To begin realizing this complex multi-phase flow system to better understand mixing, we start with a simplified problem of a single air bubble rising in a confined Hele-Shaw channel. We performed a time-resolved stereoscopic Particle Image Velocimetry (PIV) measurement to characterize the bubble wake. Pure water and varying salt concentration were used to achieve a linear density stratification corresponding to Froude numbers (Fr) ranging from 22.1 to 40.7. Due to the large velocity dynamic range for PIV, we enhanced the signal to noise ratio of our correlation planes with pyramid correlation. We found a significant out of plane velocity component in both homogenous and stratified fluid in the vicinity of the bubble, which was assumed to be negligible in previous studies with confined fluid. The wake of the bubble carries the higher density fluid to the top, which later releases from the wake to form the reverse jet. This buoyant jet has been characterized for different Fr. Eulerian coherent structures are also considered to describe the flow. The rising bubble generates vortices that shed downstream and decay with varying timescales for different Fr. The difference in the coherent structures and decay coefficient leads to a different level of mixing with Fr. The scope of this research is in applications homogenizing the stratified flow using rising bubbles. </p>

The effects of propagation of a short internal gravity wave through an inertia wave on internal wave stability is analyzed and parameterized. The interactions are specifically between a short wave packet and a large inertia wave packet. The short wave packet is a wave bounded with a Gaussian envelope with high frequencies and scales in the hundreds of meters horizontally and tens of meters vertically. The inertia wave packet is also an enveloped wave but with frequencies close to the rotation of the earth and scales in the thousands of meters in the horizontal and hundreds of meters in the vertical. The wave-wave interactions are modeled using ray theory and 2d non-linear numerical models. Ray tracing is used because it is less computationally expensive, however it fails at regions of strong refraction also known as caustics. To measure stability the steepness is calculated from the 2d non-linear methods and it is compared with estimates found in the linear theory. It is determined that the estimates of the short wave steepness from linear theory are qualitatively comparable. A quantifiable comparison, although more difficult, resulted in adjustment factors to the ray tracing results. It is also found that for the particular cases modeled, convective instabilities are predominant and the influence of the shear exerted by the large inertia wave is insignificant. Instability time scales are included in the stability analysis and estimates of overturning and wave-breaking are developed for different wave-wave interactions. From the stability analysis it is found that in general the faster the short wave propagates the more likely it is to conform to both of the conditions required for wave breaking (i.e presence of instabilities and instability time scales longer than the timescale of the short wave).

Les ondes internes de gravité jouent un rôle essentiel dans la dynamique océanique. La relation de dispersion anisotrope de ces ondes conduit à des lois de réflexion qui sont différentes de celles dont nous avons l'habitude avec les ondes acoustiques ou les rayons lumineux. Dans cette thèse de doctorat, nous nous intéressons aux structures créées par ces ondes en deux dimensions puis en trois dimensions. Dans la plupart des géométries 2D, le parcours des ondes va converger vers un attracteur. Nous étudions d'abord expérimentalement, dans une géométrie trapézoïdale, l'aspect énergétique d'un de ces attracteurs d'ondes. Nous examinons ensuite expérimentalement la transformation de ces attracteurs dans des géométries tridimensionnelles. Dans certaines géométries, la réflexion des ondes conduit à un phénomène de piégeage dans un plan 2D. Ce phénomène, d'abord étudié à l'aide de tracés de rayons, a été reproduit dans une géométrie trapézoïdale ainsi que dans une géométrie de canal. Cette mise en évidence expérimentale du piégeage pourrait expliquer certaines mesures in-situ réalisées dans l'estuaire du Saint Laurent où la propagation des ondes internes est encore mal comprise. Cette thèse est enrichie par deux études expérimentales portant sur la propagation et la réflexion d'un faisceau d'ondes interne : d'une part, l'instabilité créant un courant moyen dans le cas d'un faisceau se propageant dans une géométrie tridimensionnelle et d'autre part la génération d'ondes rétro-réfléchies lors de la réflexion sur des surfaces courbes. / Internal waves play a critical rôle in the ocean dynamics. The anisotropic dispersion relation of these waves leads to reflexion law which are different from what we are used to with acoustic waves or light rays. In the PhD thesis, we are interested in structures generated by these waves, in two dimensions then in three dimensions. In most of the 2D geometries, wave path will converge onto an attractor. We firstly study experimentally, in a trapezoidal geometry, the energy aspect of one of these attractors. Then, we survey experimentally the future of these attractors in tridimensional geometries. In some of them, reflexion leads to a trapping event in a 2D plan. This phenomenon was firstly studied by means of ray tracing, and was reproduced in both a trapezoidal and a canal geometry. The experimental obtainment of trapping could explain some in-situ measurements done in the Saint Laurent estuary, where internal wave propagation is still under scrutiny. This thesis is enhanced by two experimental studies on propagation and reflexion of an internal wave beam. Firstly, the instability generating a mean flow from a beam propagating in a three-dimensional geometry. Secondly, the generation of back-reflected waves from beam reflexion on a curved surface.

La marée interne générée sur une topographie est un élément clé des transferts énergétiques des échelles de forçage de l'océan vers les échelles de mélange turbulent. Elle contribuerait à près de la moitié du mélange turbulent nécessaire au maintien de la stratification océanique. Une compréhension plus approfondie des processus mis en jeu est nécessaire pour décrire plus précisément son rôle dans le maintien de la circulation océanique. Cette thèse s'inscrit dans la continuité des travaux d'Y. Dossmann (2012) portant sur les ondes internes solitaires générées au dessus d'une dorsale océanique. Ces travaux reposent sur une utilisation complémentaire d'expériences physiques menées dans le grand canal du CNRM-GAME et d'expériences numériques à l'échelle du laboratoire effectuées avec le modèle d'océanographie côtière SNH. L'utilisation simultanée de ces deux outils a notamment permis d'évaluer la validité des hypothèses sous-jacentes de ce modèle et le développement de nouveaux schémas numériques. Dans cette thèse, des simulations numériques utilisant la version non-hydrostatique et non-Boussinesq du modèle SNH sont utilisées pour décrire les différents régimes d'ondes internes dans des régions " supercritiques ". Le terme supercritique désigne à la fois des courants de marée intenses dont la vitesse U est supérieure à la vitesse de propagation des ondes internes cn, et des topographies très abruptes dont l'angle de la pente est supérieur à l'angle du rayon d'onde interne dans la pycnocline . De telles conditions environnementales correspondent à des régions de mélange intense, jusqu'à 10 000 fois supérieur au mélange turbulent observé dans l'océan ouvert. Les processus physiques ayant lieu dans ces régions restent encore mal compris et mal représentés par les paramétrisations d'ondes internes existantes. De plus, ces régions sont également des zones propices à la génération d'ondes internes non-linéaires pouvant se propager pendant plusieurs jours et entraînant ainsi des transferts d'énergie significatifs loin de leur zone de génération. La description des processus turbulents en jeu dans ces régions " extrêmes " constitue le cœur de ma thèse. Dans une première partie, des configurations académiques à l'échelle du laboratoire sont mises en place pour étudier les processus en jeu dans différents régimes " supercritiques " de génération d'onde internes. Des simulations numériques directes sont réalisées et permettent d'identifier un nombre limité de paramètres physiques adimensionnés contrôlant la dynamique des ondes internes dans ces régions. Une attention particulière est portée sur le rôle joué par la topographie sur la génération des modes verticaux d'ondes internes et sur la formation de modes " hauts " d'ondes internes solitaires. Le second objectif de cette thèse est de faire le lien entre les précédentes études académiques à l'échelle du laboratoire et l'échelle océanique. Pour cela, un principe de similitude permettant de conserver la dynamique des ondes internes tout en modifiant l'échelle de l'écoulement est mise en place. Par le biais de ce principe de similitude, nous partons de cas idéalisés à l'échelle du laboratoire, que nous transposons à l'échelle océanique, pour nous rapprocher de cas océaniques plus réalistes et de plus en plus complexes. Puis notre étude de régime est étendue à deux régions océaniques " supercritiques " bien connues : le détroit de Gibraltar et le plateau situé à l'entrée du golfe du Maine (nommé " Georges Bank " en anglais). L'applicabilité de nos paramètres clés est étudiée dans le cas de ces deux environnements complexes par le biais de simulations haute résolution de grande échelle (LES). / Internal tides are involved in the Meridional Overturning Circulation energy balance. The issue about the relative importance of the mechanical and thermodynamical energy sources induces a need for a quantitative evaluation of the energy transfers and for a clear understanding of the physical processes involved in these energy transfers. In supercritical regions such as the strait of Gibraltar or the Hawaiian Ridge, large topography variations and strong currents lead to more complex generation mechanisms of internal waves and environmental interactions. They can be subject locally to spectacular breaking, with turbulent structures observed hundreds of meters above the seafloor, and driving turbulence orders of magnitude higher than open-ocean levels. These regions are also effective at generating nonlinear internal waves (ISWs) which persist for days after their generation and are suspected to be responsible for important remote energy transfers. In these "extreme" regions, ISWs dynamics is also more difficult to model. These situations are highly non-hydrostatic and non-linear with strong instabilities, strong velocity and density gradients and steep slopes. Moreover, in these regions, actual internal wave's parameterizations are often inadequate. So there is a real need to understand and represent better the ISWs dynamic in these areas. This thesis follows the line of research of Dossmann (2012), on topographically induced internal solitary waves which used a complementary approach relying on numerical and experimental configurations at laboratory scale. In this context, we continue to explore internal tide regimes but in "supercritical" regions: internal tide generation area with supercritical topography and hydraulic control. Simulations are performed using the nonhydrostatic and non-Boussinesq version of the regional oceanic circulation model SNH. In a first part, taking an idealized modeling approach at laboratory scale, we examined a range of different internal waves regimes in "supercritical" regions. Relying on quasi-direct numerical simulations (quasi-DNS), a regime analysis has been proposed using and identifying key non-dimensional parameters for ISWs dynamics. This analysis has permitted to recover a topographic control on vertical mode generation characterized by the ratio of vertical mode wavelength to topography width, even above supercritical topography. The topographic selection criterion has proven to be a useful indicator of high mode solitary wave formation in non-linear regime. The purpose of the second part is to extend the previous studies at laboratory scales towards more realistic oceanic conditions. In this regard, the regime analysis is applied to a idealized large scale oceanic strait through a similitude principle. The idealized strait configuration succeeds in representing laboratory scale strait regime at largest and realistic scales. Then our analysis is applied to two well-known realistic cases: the Strait of Gibraltar and Georges Bank through large eddy simulations. These two oceanographic "supercritical" regions are particularly interesting for their specific topography and stratification conditions.

Les ondes internes jouent un rôle essentiel dans la dynamique de l'atmosphère et de l'océan. Elles sont entres autres invoquées pour expliquer le mélange irréversible de ce dernier. Dans des géométries particulières, ces ondes sont susceptibles de converger vers des trajectoires limites appelées attracteurs. Dans cette thèse, nous étudions expérimentalement ces attracteurs dans une cuve trapézoïdale. Dans un premier temps, nous nous intéressons à leur bilan énergétique, en mesurant les différentes dissipations ainsi que la puissance injectée par le générateur d'onde. Si cette puissance est suffisante, l'attracteur est susceptible de se déstabiliser par interactions triadiques. Nous quantifions la part de dissipation portée par ces ondes secondaires ainsi générées. La mesure de la puissance injectée en régime linéaire a montré qu'elle est plus importante pour les attracteurs. Cette résonance, observée jusqu'ici sous une excitation monochromatique, est ici observée en étudiant la réponse du système à une impulsion. Un développement théorique est proposé et permet de comparer quantitativement les deux méthodes. Enfin, nous étudions le régime non linéaire des attracteurs, qui présente des spectres temporel et spatial très riches. Après avoir caractérisé le régime non linéaire, nous exhibons deux comportements non linéaires très différents et apportons des pistes d'explication à leur existence / Internal waves are very important to atmospheric and oceanic dynamics. Among others, they might explain the irreversible mixing of the oceans. In particular geometries, these waves can focus on limit cycles called attractors. Firstly, we investigate the energy budget of these attractor by measuring the different dissipative terms and the injected power coming from the waves source. If this power is sufficient, the attractor destibilise throught triadic instabilities. We quantify how much these secondary are important to the global dissipation rate.The measurement of injected power in the linear regime has shown that it is bigger for the attractors. This resonance, which had been observed under a monochromatic excitation, is here study by analysing the response of the system to an impulse. A theoretical development is proposed to compare quantitatively these two methods. Finally, we study the non linear regime of attractors, which present some complex spatial and temporal spectrum. We caracterise this non linear regime and show that two very different non linear beahviors exist. We bring up some explanation to these behaviors