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No desenho de mecanismos lubrificados, tais como Mancais hidrodinâmicos ou anéis de pistões de Motores a Combustão, atrito e desgaste são efeitos não desejados. Por exemplo, é sabido que aproximadamente 5% da energia perdida em um motor a combustão esta associada ao atrito presente no sistema de anéis/cilindro do pistão. Após vários trabalhos experimentais e teóricos, as superfícies texturizadas hão mostrado serem capazes de reduzir o atrito em algumas condições de funcionamento. O estudo da relação entre o atrito e os parâmetros de texturização é um problema difícil e de interesse tanto industrial como acadêmico. O contexto matemático e computacional destes trabalhos apresentam desafios por si mesmos, como o estudo da boa colocação dos modelos matemáticos, a consideração adequada das descontinuidades das superfícies. Este trabalho enfoca-se no contexto matemático, apresentando e estudando a equação de Reynolds junto com diferentes modelos de cavitação que podem encontrar-se na literatura. Começamos estudando a matemática da equação de Reynolds. Depois disso, modelos de cavitação são inclusos, aumentando a complexidade da matemática envolvida. Seguidamente, como aplicação da teoria apresentada, um rolamento deslizante será estudado junto com uma texturização da superfície móvel. Os resultados deste estudo revelam mecanismos básicos de redução de atrito e propriedades gerais que não haviam sido reportadas anteriormente. Possíveis trabalhos futuros são apresentados, tal como o uso de Métodos Descontínuos de Galerkin em vez dos Métodos de Volumes Finitos. O último em procura de uma melhor acomodação da formulação matemática, tentando melhorar a flexibilidade da malha e a precisão. / In the design of lubricated mechanisms, such as Journal Bearings or Piston Rings of Combustion Engines, friction and wear are undesirable effects. It is known, for instance, that about 5% of the energy loss in a Combustion Engine is associated to friction taking place in the Piston Rings/Cylinder system. Textured surfaces, after a significant number of experimental and theoretical studies, have shown to reduce friction in some operating conditions. The study of the relation between the friction and the texture parameters is a challenging problem with both industrial and academic interest. The mathematical and computational frameworks involved present challenges by themselves, such as establishing the well-posedness of the mathematical models with suitable consideration of discontinuous surfaces. In this work we focus on the mathematical framework, presenting and studying the Reynolds equation along with different state-of-the-art cavitation models. We begin by studying the Reynolds equation and then incorporate two different cavitation models of increasing mathematical complexity. Next, as an application of the theory already presented, a slider bearing is numerically studied considering a sinusoidal texture on the runner. The results of this study unveil basic mechanisms of friction reduction and global quantitative trends that had not been previously reported. In this way, the applicability of numerical tools for texture selection is established. Future research directions are also identified, such as using Discontinuous Galerkin methods instead of Finite Volume Methods, aiming at improving the mesh exibility and thus the accuracy of the discrete formulation.

La propagation d'une fissure chargée par un écoulement de fluide visqueux est un phénomène complexe où la compréhension des phénomènes mécaniques mis en jeu en pointe de fissures reste encore partielle. C'est le cas de la zone de décollement entre le solide et le fluide qui apparaît pour un certain choix de débit d'injection, de viscosité du fluide et de ténacité du matériau. Cette thèse propose une modélisation simplifiée de ce problème d'interaction fortement couplé. Dans un premier chapitre, on étudie un modèle simplifié unidimensionnel de film élastique collé sur un substrat rigide et on considère une injection de fluide visqueux entre le film et le substrat. On suppose que la propagation de la fissure est régie par la loi de Griffith. On néglige l'existence du retard possible entre le fluide et le solide et on choisit une loi de comportement non-linéaire pour le fluide visqueux. A partir d'une analyse asymptotique pour une faible viscosité, on établit une solution approchée du problème. On montre que le champ pression est singulier en pointe de fissure et on montre l'influence du débit d'injection sur la cinétique du trajet de fissuration. Dans le deuxième chapitre on propose de prendre en compte l'existence de la zone de décollement en modifiant la formulation du modèle et en le réécrivant sous la forme d'un problème d'optimisation en temps discret où les zones de décollement font partie des inconnues du problème. On valide la formulation proposée sur l'exemple analytique de l'écrasement d'une goutte par une barre rigide. On montre ensuite que cette formulation et l'algorithme lié à son implémentation sont capables de gérer l'évolution de l'écrasement de plusieurs gouttes de forme quelconque en capturant correctement les phase d'étalement des gouttes ainsi que de leur coalescence. On étend ensuite cette formulation au cas de l'écrasement d'une goutte par un film élastique. Dans le dernier chapitre, on examine la validité de l'hypothèse de lubrification utilisée en fracturation hydraulique. A l'aide de la méthode de développement asymptotique, on construit une équation de Reynolds régularisée avec des termes de gradient supérieur tenant compte de la variation spatiale de la hauteur des parois. On compare alors le comportement des champs de pression donnés par les équations de Reynolds classique et régularisée sur des exemples d'écoulement entre des conduits de formes multiples. / The crack evolution under a viscous fluid action is a complex phenomenon where the understanding of the mechanical phenomena near the crack tip is still largely limited. This is the case for the lag between the solid and the fluid front propagation which appears for some configurations of injection rate, fluid viscosity and material toughness. This thesis proposes a simplified model for this strongly coupled interaction problem.The first chapter studies a simplified one-dimensional model of a elastic film bonded to a rigid substrate. We consider a viscous fluid injection between the film and the substrate. The crack propagation is assumed to follow the Griffith's law. The existence of the lag is neglected and a non-linear behavior law is chosen for the viscous fluid. Using an asymptotic analysis, an approximate solution is established for the low viscosity case. It is shown that the pressure field diverges at the crack tip and that the kinetics of the crack is influenced by the injection rate. The second chapter proposes to take into account the existence of the lag by modifying the model formulation and rewriting it as a discrete time optimisation problem where the delamination zones are part of the unknowns of the problem. This formulation is validated for the analytical example of a drop crushed by a rigid bar. It is shown that this formulation and its implementation can manage the evolution of several drops of any shape and correctly captures the drops spreading and coalescence. This formulation is then extended to the case of a drop crushed by an elastic film. In the last chapter, the validity of the lubrication hypothesis is examinated. Using an asymptotic analysis, a regularized Reynolds equation is constructed with higher gradient terms taking into account the spatial variation of the walls height. A comparison between the pressure fields behaviour given by the classical and the regularized Reynolds equation is shown for different conducts.

No desenho de mecanismos lubrificados, tais como Mancais hidrodinâmicos ou anéis de pistões de Motores a Combustão, atrito e desgaste são efeitos não desejados. Por exemplo, é sabido que aproximadamente 5% da energia perdida em um motor a combustão esta associada ao atrito presente no sistema de anéis/cilindro do pistão. Após vários trabalhos experimentais e teóricos, as superfícies texturizadas hão mostrado serem capazes de reduzir o atrito em algumas condições de funcionamento. O estudo da relação entre o atrito e os parâmetros de texturização é um problema difícil e de interesse tanto industrial como acadêmico. O contexto matemático e computacional destes trabalhos apresentam desafios por si mesmos, como o estudo da boa colocação dos modelos matemáticos, a consideração adequada das descontinuidades das superfícies. Este trabalho enfoca-se no contexto matemático, apresentando e estudando a equação de Reynolds junto com diferentes modelos de cavitação que podem encontrar-se na literatura. Começamos estudando a matemática da equação de Reynolds. Depois disso, modelos de cavitação são inclusos, aumentando a complexidade da matemática envolvida. Seguidamente, como aplicação da teoria apresentada, um rolamento deslizante será estudado junto com uma texturização da superfície móvel. Os resultados deste estudo revelam mecanismos básicos de redução de atrito e propriedades gerais que não haviam sido reportadas anteriormente. Possíveis trabalhos futuros são apresentados, tal como o uso de Métodos Descontínuos de Galerkin em vez dos Métodos de Volumes Finitos. O último em procura de uma melhor acomodação da formulação matemática, tentando melhorar a flexibilidade da malha e a precisão. / In the design of lubricated mechanisms, such as Journal Bearings or Piston Rings of Combustion Engines, friction and wear are undesirable effects. It is known, for instance, that about 5% of the energy loss in a Combustion Engine is associated to friction taking place in the Piston Rings/Cylinder system. Textured surfaces, after a significant number of experimental and theoretical studies, have shown to reduce friction in some operating conditions. The study of the relation between the friction and the texture parameters is a challenging problem with both industrial and academic interest. The mathematical and computational frameworks involved present challenges by themselves, such as establishing the well-posedness of the mathematical models with suitable consideration of discontinuous surfaces. In this work we focus on the mathematical framework, presenting and studying the Reynolds equation along with different state-of-the-art cavitation models. We begin by studying the Reynolds equation and then incorporate two different cavitation models of increasing mathematical complexity. Next, as an application of the theory already presented, a slider bearing is numerically studied considering a sinusoidal texture on the runner. The results of this study unveil basic mechanisms of friction reduction and global quantitative trends that had not been previously reported. In this way, the applicability of numerical tools for texture selection is established. Future research directions are also identified, such as using Discontinuous Galerkin methods instead of Finite Volume Methods, aiming at improving the mesh exibility and thus the accuracy of the discrete formulation.

Cílem této diplomové práce je najít optimální profil mezery mazání pro turbodmychadlo. Cílem je minimalizovat tření, udržovat nosnost a nezvyšovat průtok maziva. Tato multiobjektivní optimalizace se provádí pomocí genetického algoritmu (GA) v MATLABu. Minimalizace třecí síly snižuje ztráty třecího výkonu turbodmychadla. Řešení Reynoldsovy rovnice je počítáno numericky pomocí MATLABu. Je zjištěna minimální tloušťka mazací mezery pro počáteční problém. Funkce spline se používá ke generování obecného profilu mazací mezery. Tento profil je poté optimalizován pomocí GA v MATLABu.

This thesis is devoted to the study of some homogenization problems with applications in lubrication theory. It consists of an introduction, five research papers (I–V) and a complementary appendix.Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficients. Many important problems in physics with one or several microscopic scales give rise to this kind of equations, whence the need for methods that enable an efficient treatment of such problems. To this end several mathematical techniques have been devised. The main homogenization method used in this thesis is called multiscale convergence. It is a notion of weak convergence in  Lp spaces which is designed to take oscillations into account. In paper II we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with a finite number of microscopic scales. The main idea in the proof is closely related to a decomposition of vector fields due to Hermann Weyl. The Weyl decomposition is further explored in paper III.Lubrication theory is devoted to the study of fluid flows in thin domains. More generally, tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. The mathematical foundations of lubrication theory are given by the Navier–Stokes equation which describes the motion of a viscous fluid. In thin domains several simplifications are possible, as shown in the introduction of this thesis. The resulting equation is named after Osborne Reynolds and is much simpler to analyze than the Navier--Stokes equation.The Reynolds equation is widely used by engineers today. For extremely thin films, it is well-known that the surface micro-topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelengths upon the solution of the Reynolds equation. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are “averaged out”. One problem consists of finding an algorithm for computing the solution of the homogenized equation. Another problem consists of showing, on introducing the appropriate mathematical definitions, that the homogenized equation is the correct method of averaging. Papers I, II, IV and V investigate the effects of surface roughness by homogenization techniques in various situations of hydrodynamic lubrication. To compare the homogenized solution with the solution of the deterministic Reynolds equation, some numerical examples are also included. / Godkänd; 2011; 20110408 (johfab); DISPUTATION Ämnesområde: Matematik/Mathematics Opponent: Professor Guy Bayada, Institut National des Sciences Appliquées de Lyon (INSA-LYON), Lyon, France, Ordförande: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Tid: Tisdag den 7 juni 2011, kl 10.00 Plats: D2214/15, Luleå tekniska universitet

Les travaux de recherche constituent une contribution au développement de la théorie de la fiabilité en mécanique des fluides. Pour la conception de machines et de systèmes mécatroniques complexes, de nombreux composants fluides, difficiles à dimensionner, sont utilisés. Ces derniers ont des caractéristiques intrinsèques statiques et dynamiques sensibles et ont donc une grande importance sur la fiabilité et la durée de vie de la plupart des machines et des systèmes.Le développement effectué se concentre spécialement sur l'évaluation de la fiabilité d’un palier fluide grâce à un couplage « mécanique des fluides - fiabilité ». Ce couplage exige une définition propre de la fonction d’état limite permettant d’estimer la probabilité de défaillance d’un palier fluide. La modélisation par l'équation de Reynolds modifiée permet de déterminer la capacité de charge d’un palier fluide en fonction des conditions de fonctionnement. Plusieurs formes simples de paliers fluides ont été modélisées analytiquement et leurs probabilités de défaillance ont été estimées grâce à des méthodes d'approximation FORM/SORM (First Order Reliability, Second Order Reliability) et de simulation Monte Carlo. / These research is a contribution to the development of reliability theory in fluid mechanics. For the machines design and complex mechatronic systems, many fluid components are used. These components have static and dynamic sensitive characteristics and thus have a great significance on the reliabilityand lifetime of the machines and systems. Development performed focuses specifically on the reliability evaluation of a fluid bearing using a"fluid mechanics - reliability" interaction approach. This coupling requires a specific definition of the limit state function for estimating the failure probability of a fluid bearing. The Reynolds equation permits to determine the fluid bearing load capacity according to the operating conditions. Several simple geometries of fluid bearings were modeled analytically and their failure probabilities were estimated using the approximation methods FORM / SORM (First Order Reliability Method,Second Order Reliability Method) and Monte Carlo simulation.

L’amélioration des turbomachines passe par l’augmentation de leurs vitesses de rotation et peut conduire leurs composants à se trouver en présence d’un régime d’écoulement supersonique, en particulier leurs systèmes de pivoterie à air. L’étude d’une butée aérodynamique en régime supersonique est traitée dans ce manuscrit et s’inscrit dans la continuité de la recherche sur les butées hautes vitesses mais qui n’a que très peu été abordée dans la littérature. Ce problème se trouve à la frontière entre deux domaines scientifiques : la lubrification et l’aérodynamique. L’enjeu ici est développer un modèle réaliste traduit par un code de calcul écrit en FORTRAN, capable de capturer les phénomènes liés au régime supersonique (choc, détente) et d’être adapté à la géométrie des films minces. Pour cela, deux modèles ont été développés et codés à l’aide de la méthode numérique des différences finies : les équations de Reynolds Modifiées et les équations de Navier-Stokes adaptées aux films minces. Ce premier modèle est une extension de l’équation de Reynolds généralisée, prenant en compte l’inertie et déjà utilisé dans des études de la lubrification. Le second modèle a été établi à partir des équations de Navier-Stokes et conserve leur forme. Ce système possède l’avantage de pouvoir utiliser les outils numériques adaptés à la capture de choc (WENO). La comparaison des deux modèles montre que les équations de Reynolds Modifiées ne sont pas suffisantes pour l’étude d’un écoulement film mince en régime supersonique. Les résultats des simulations menées montrent la présence d’une détente sur le changement d’inclinaison du double profil qui dépend de la vitesse, de la température et de l’angle de la géométrie. Cette détente, même dans des conditions sévères de fonctionnement (grandes vitesses ou fort convergent), n’a pas beaucoup d’influence sur le comportement global statique de la butée. Les résultats montrent également que contrairement à la théorie des écoulements supersoniques, aucun choc n’est observé en film mince supersonique. Une transition géométrique obtenue par homothétie, entre un écoulement contenant un choc et un autre n’en contenant pas, a été observée à une certaine valeur du rapport des longueurs d’adimensionnement pour une vitesse et une géométrie donnée. L’extrapolation de ces résultats pour un cas réaliste montre qu’un choc ne peut se produire qu’à partir de 5500 m/s pour une épaisseur de film de 40 μm avec epsilon = 0.001. Il est donc peu probable qu’un choc ne se produise dans une butée en conditions supersoniques dans le cadre industriel. / The improvement of turbomachines requires to increase their rotational speeds and can leads components to be in presence of a supersonic regime, particularly their air bearing systems. This manuscript deals with a study of an aerodynamic thrust bearing in a supersonic regime. This work belongs to the research field on the high-speed thrust bearings, but very few studies are focused on this subject. This problem is at the boundary between two scientific fields: lubrication and aerodynamic. The aim of this study is to develop models transcribed as a FORTRAN code, able to capture phenomenon related to the supersonic regime (shock, expansion wave) and adapted to the thin film geometry. For this, two models have been developed as well as computer codes related to these models using the finite difference method: the Modified Reynolds equations and the Navier-Stokes equations adapted to thin films. The first model is an extension of the generalized Reynolds equation, taking into account inertia effects. It is a model already used in lubrication studies. The second model was developed from the Navier-Stokes and has their shape. This system has the advantage of using numerical schemes for shock capturing (WENO). The comparison of the two models shows that the Modified Reynolds equations are not appropriate to the study of a supersonic air thin film. Numerical results show the presence of an expansion wave at the end of the sloping region of the tapper flat geometry which depends on speed, on temperature and on the angle of the geometry. This expansion wave, under severe conditions, does not have much influence on the overall static behavior of the thrust bearing. The results also show that, contrary to the supersonic flow theory, a shock is not observed in a supersonic thin film. A geometrical transition obtained by homothetic, between a flow containing a shock and another with no shock, is observed at a given value of the characteristic length ratio for a given speed and geometry. Extrapolation of these results for a realistic case shows that a shock occurs from 5500 m/s for a film thickness of 40 μm with epsilon = 0.001. It is therefore unlikely that a shock occurs in a supersonic thrust bearing in industrial settings.

<div>Positive displacement machines are used in a wide variety of applications, ranging from fluid power where they act as a transmission of power, to lubrication and fluid transport. As the core of the fluid system responsible for mechanical--hydraulic energy conversion, the efficiencies of these units are a major driver of the total efficiency of the system. Furthermore, the durability of these units is a strong decider in the useful life of the system in which they operate.</div><div><br></div><div>The key challenge in designing these units comes from understanding their working principles and designing their lubricating interfaces, which must simultaneously perform a load carrying and sealing function as the unit operates. While most of the physical phenomena relevant to these machines have been studied previously in some capacity, the significance of their mutual interactions has not. For this reason, the importance of these mutual interactions is a fundamental question in these machines that this thesis answers for the first time. In analysis of two different machine types, it is confirmed that mutual interactions of both physical phenomena and neighboring fluid domains of the unit contribute significantly to the overall performance of the machine. Namely, these analyses demonstrate load sharing owing to mutual interactions on average of 20% and as high as 50%, and mutual flow interactions of at least 10%.</div><div><br></div><div>In this thesis, the behavior of the thin films of fluid in the lubricating interfaces of the units, the bodies that make up these films, and the volumes which interface with them will be considered. The resulting coupled problem requires a model that can consider the effects of motion of all floating bodies on all films and volumes, and collect the resulting loads applied by the fluid as it responds. This will require a novel 6 degree of freedom dynamics model including the inertia of the bodies and the transient pressure and shear loads of all interfaces of the body and the fluid domain.</div><div><br></div><div>During operation, fluid cavitation and aeration can occur in both the displacement chambers of the machine and its lubricating interfaces. To capture this, a novel cavitation algorithm is developed in this thesis, which considers the release of bubbles due to both gas trapped within the fluid and vaporization of the operating fluid in localized low pressure regions of the films. In the absence of cavitation, this model will also be used to find the pressures and flows over the film, communicating this information with the remainder of the fluid domain.</div><div><br></div><div>Due to the high pressures that form in these units, the bodies deform. The resulting deformation changes the shape of the films and therefore its pressure distribution. This coupled effect will be captured in one of two ways, the first relying on existing geometric information of the unit, and the other using a novel analytical approach that is developed to avoid this necessity. In either case, the added damping due to the shear of the materials will be considered for the first time. Additionally in regions of low gap height, mixed lubrication occurs and the effects of the surface asperities of the floating bodies cannot be neglected. Accurate modeling of this condition is necessary to predict wear that leads to failure in these units. This work will then develop a novel implementation for mixed lubrication modeling that is directly integrated into the cavitation modeling approach.</div><div><br></div><div>Finally, effort is made to maintain a generic tools, such that the model can be applied to any positive displacement machine. This thesis will present the first toolbox of its kind, which accounts for all the mentioned aspects in such a way that they can be captured for any machine. Using both multithreaded and sequential implementations, the tool will be capable of fully utilizing a machine on which it is run for both low latency (design) and high throughput (optimization) applications respectively. In order to make these applications feasible, the various modules of the tool will be strongly coupled using asynchronous time stepping. This approach is made possible with the development of a novel impedance tensor of the mixed universal Reynolds equation, and shows marked improvements in simulation time by requiring at most 50% of the simulation time of existing approaches.</div><div><br></div><div>In the present thesis, the developed tool will be validated using experimental data collected from 3 fundamentally different machines. Individual advancements of the tool will also be verified in isolation with comparison to the state of the art and commercial software in the relevant fields. As a demonstration of the use of the tool for design, detailed analysis of the displacing actions and lubricating interfaces of these same units will be performed. These validations demonstrate the ability of the tool to predict machine efficiencies with error averaging around 1% over all operating conditions for multiple machine types, and capture transient behavior of the units. To demonstrate the utility as a virtual optimization tool, design of a complete external gear machine design will be performed. This demonstration will start from only analytical parameters, and will track a route to a complete prototype.</div>

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