THE STOCHASTIC NAVIER STOKES EQUATIONS FOR HEAT CONDUCTING, COMPRESSIBLE FLUIDS

This dissertation is devoted to the equations of motion governing the evolution of a fluid or gas at the macroscopic scale. The classical model is a PDE description known as the Navier-Stokes equations. The behavior of solutions is notoriously complex, leading many in the scientific community to describe fluid mechanics using a statistical language. In the physics literature, this is often done in an ad-hoc manner with limited precision about the sense in which the randomness enters the evolution equation. The stochastic PDE community has begun proposing precise models, where a random perturbation appears explicitly in the evolution equation. Although this has been an active area of study in recent years, the existing literature is almost entirely devoted to incompressible fluids. The purpose of this thesis is to take a step forward in addressing this statistical perspective in the setting of compressible fluids. In particular, we study the well posedness for the corresponding system of Stochastic Navier Stokes equations, satisfied by the density, velocity, and temperature. The evolution of the momentum involves a random forcing which is Brownian in time and colored in space. We allow for multiplicative noise, meaning that spatial correlations may depend locally on the fluid variables. Our main result is a proof of global existence of weak martingale solutions to the Cauchy problem set within a bounded domain, emanating from large initial datum. The proof involves a mix of deterministic and stochastic analysis tools. Fundamentally, the approach is based on weak compactness techniques from the deterministic theory combined with martingale methods. Four layers of approximate stochastic PDE's are built and analyzed. A careful study of the probability laws of our approximating sequences is required. We prove appropriate tightness results and appeal to a recent generalization of the Skorohod theorem. This ultimately allows us to deduce analogues of the weak compactness tools of Lions and Feireisl, appropriately interpreted in the stochastic setting.

Low cost 3D global instability analysis and flow sensitivity based on dynamic mode decomposition and high-order numerical tools

We explore the recently developed snapshot-based dynamic mode decomposition (DMD) technique, a matrix-free Arnoldi type method, to predict 3D linear global flow instabilities. We apply the DMD technique to flows confined in an L-shaped cavity and compare the resulting modes to their counterparts issued from classic, matrix forming, linear instability analysis (i.e. BiGlobal approach) and direct numerical simulations. Results show that the DMD technique, which uses snapshots generated by a 3D non-linear incompressible discontinuous Galerkin Navier?Stokes solver, provides very similar results to classical linear instability analysis techniques. In addition, we compare DMD results issued from non-linear and linearised Navier?Stokes solvers, showing that linearisation is not necessary (i.e. base flow not required) to obtain linear modes, as long as the analysis is restricted to the exponential growth regime, that is, flow regime governed by the linearised Navier?Stokes equations, and showing the potential of this type of analysis based on snapshots to general purpose CFD codes, without need of modifications. Finally, this work shows that the DMD technique can provide three-dimensional direct and adjoint modes through snapshots provided by the linearised and adjoint linearised Navier?Stokes equations advanced in time. Subsequently, these modes are used to provide structural sensitivity maps and sensitivity to base flow modification information for 3D flows and complex geometries, at an affordable computational cost. The information provided by the sensitivity study is used to modify the L-shaped geometry and control the most unstable 3D mode.

Vortex Shedding Dynamics in Long Aspect-Ratio Aerodynamics Bodies

The focus of the current dissertation is to study qualitatively the underlying physics of vortex-shedding and wake dynamics in long aspect-ratio aerodynamics in incompressible viscous flow through the use of the KLE method. We carried out a long series of numerical experiments in the cases of flow around the cylinder at low Reynolds numbers. The study of flow at low Reynolds numbers provides an insight in the fluid physics and also plays a critical role when applying to stalled turbine rotors. Many of the conclusions about the qualitative nature of the physical mechanisms characterizing vortex formation, shedding and further interaction analyzed here at low Re could be extended to other Re regimes and help to understand the separation of the boundary layers in airfoils and other aerodynamic surfaces. In the long run, it aims to provide a better understanding of the complex multi-physics problems involving fluid-structure-control interaction through improved mathematical computational models of the multi-physics process. Besides the scientific conclusions produced, the research work on streamlined and bluff-body condition will also serve as a valuable guide for the future design of blade aerodynamics and the placement of wind turbines and hydrakinetic turbines, increasing the efficiency in the use of expensive workforce, supplies, and infrastructure. After the introductory section describing the main fields of application of wind power and hydrokinetic turbines, we describe the main features and theoretical background of the numerical method used here. Then, we present the analysis of the numerical experimentation results for the oscillatory regime right before the onset of vortex shedding for circular cylinders. We verified the wake length of the closed near-wake behind the cylinder and analysed the decay of the wake at the wake formation region, and then studied the St-Re relationship at the Reynolds numbers before the wake sheds compared to the experimental data. We found a theoretical model that describes the time evolution of the amplitude of fluctuations in the vorticity field on the twin vortex wake, which accurately matches the numerical results in terms of the frequency of the oscillation and rate of decay. We also proposed a model based on an analog circuit that is able to interpret the concerning flow by reducing the number of degrees of freedom. It follows the idea of the non-linear oscillator and resembles the dynamics mechanism of the closed near-wake with a common configured sine wave oscillator. This low-dimensional circuital model may also help to understand the underlying physical mechanisms, related to vorticity transport, that give origin to those oscillations.

Optimisation thermodynamique d’un procédé solaire utilisant un système de réfrigération à éjecto-compression pour la production du froid

L’objectif essentiel de cette thèse est de développer un système industriel de réfrigération ou de climatisation qui permet la conversion du potentiel de l’énergie solaire en production du froid. Ce système de réfrigération est basé sur la technologie de l’éjecto-compression qui propose la compression thermique comme alternative économique à la compression mécanique coûteuse. Le sous-système de réfrigération utilise un appareil statique fiable appelé éjecteur actionné seulement par la chaleur utile qui provient de l’énergie solaire. Il est combiné à une boucle solaire composée entre autres de capteurs solaires cylindro-paraboliques à concentration. Cette combinaison a pour objectif d’atteindre des efficacités énergétiques et exergétiques globales importantes. Le stockage thermique n’est pas considéré dans ce travail de thèse mais sera intégré au système dans des perspectives futures. En première étape, un nouveau modèle numérique et thermodynamique d’un éjecteur monophasique a été développé. Ce modèle de design applique les conditions d’entrée des fluides (pression, température et vitesse) et leur débit. Il suppose que le mélange se fait à pression constante et que l’écoulement est subsonique à l’entrée du diffuseur. Il utilise un fluide réel (R141b) et la pression de sortie est imposée. D’autre part, il intègre deux innovations importantes : il utilise l'efficacité polytropique constante (plutôt que des efficacités isentropiques constantes utilisées souvent dans la littérature) et n’impose pas une valeur fixe de l'efficacité du mélange, mais la détermine à partir des conditions d'écoulement calculées. L’efficacité polytropique constante est utilisée afin de quantifier les irréversibilités au cours des procédés d’accélérations et de décélération comme dans les turbomachines. La validation du modèle numérique de design a été effectuée à l’aide d’une étude expérimentale présente dans la littérature. La seconde étape a pour but de proposer un modèle numérique basé sur des données expérimentales de la littérature et compatible à TRNSYS et un autre modèle numérique EES destinés respectivement au capteur solaire cylindro-parabolique et au sous-système de réfrigération à éjecteur. En définitive et après avoir développé les modèles numériques et thermodynamiques, une autre étude a proposé un modèle pour le système de réfrigération solaire à éjecteur intégrant ceux de ses composantes. Plusieurs études paramétriques ont été entreprises afin d’évaluer les effets de certains paramètres (surchauffe du réfrigérant, débit calorifique du caloporteur et rayonnement solaire) sur sa performance. La méthodologie proposée est basée sur les lois de la thermodynamique classique et sur les relations de la thermodynamique aux dimensions finies. De nouvelles analyses exergétiques basées sur le concept de l’exergie de transit ont permis l'évaluation de deux indicateurs thermodynamiquement importants : l’exergie produite et l’exergie consommée dont le rapport exprime l’efficacité exergétique intrinsèque. Les résultats obtenus à partir des études appliquées à l’éjecteur et au système global montrent que le calcul traditionnel de l’efficacité exergétique selon Grassmann n’est désormais pas un critère pertinent pour l'évaluation de la performance thermodynamique des éjecteurs pour les systèmes de réfrigération.

One-dimensional Model for Fluids of Third-grade in Tubes with Constant Radius

Our goal in this paper is to extend previous results obtained for Newtonian and secondgrade fluids to third-grade fluids in the case of an axisymmetric, straight, rigid and impermeable tube with constant cross-section using a one-dimensional hierarchical model based on the Cosserat theory related to fluid dynamics. In this way we can reduce the full threedimensional system of equations for the axisymmetric unsteady motion of a non-Newtonian incompressible third-grade fluid to a system of equations depending on time and on a single spatial variable. Some numerical simulations for the volume flow rate and the the wall shear stress are presented.