TEIXEIRA, Rivania Maria do Nascimento. Leis de escala em mapeamentos discretos. 2016. 85 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2016. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T18:20:56Z No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) / Approved for entry into archive by Edvander Pires (edvanderpires@gmail.com) on 2016-07-18T18:22:35Z (GMT) No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) / Made available in DSpace on 2016-07-18T18:22:35Z (GMT). No. of bitstreams: 1 2016_tese_rmnteixeira.pdf: 5826571 bytes, checksum: 6cde4ae78436e469a71b8b9608331776 (MD5) Previous issue date: 2016 / In this work we are going to investigate the scale formalism in discret mappings. In 1D mappings, we explore the asymptotic decays to the steady state with focus in three types of bifurcation: transcriptical, pitchfork and period-doubling. We identify this behavior through a well defined generalized homogeneous function with critical exponents. Next to the bifurcation point, the decay to the fix point occurs by an exponential function, which is given by a power law that is independent of the non-linearity mapping. The numerical results obtained agree with the analytical results. We also apply the scale formalism in conservatives and dissipatives bidimensional mappings. In the conservative case, our goal was analyze the behavior of the chaotics orbits next to the phase transition from the integrable to the non-integrable. Next to that transition, we describe the dynamical system using a generalized homogeneous function for which we found a power law that describe the behavior of the criticality. Through a phenomenological discussion, we found critical exponents in agree with the analytical description. In the dissipative case, our main goal was to investigate the influence of a dissipative term in the dynamics, causing a phase transition - suppression of unlimited difusion of the action variable. Following a phenomenological approach with an analytical description, we were able to determine the critical exponents using a generalized homogeneous function. / Neste trabalho investigamos algumas aplicações do formalismo de escala em mapeamentos discretos. Exploramos os decaimentos assintóticos ao estado estacionário com foco em três tipos de bifurcações em mapeamentos unidimensionais: bifurcação transcrítica, bifurcação supercrítica de forquilha e bifurcação de duplicação de período. Caracterizamos este comportamento através de uma função homogênea generalizada com expoentes críticos bem definidos. Próximo ao ponto de bifurcação o decaimento ao ponto fixo ocorre através de uma função exponencial cujo o tempo de relaxação é caracterizado por uma lei de potência que independe da não linearidade do mapa. Os resultados obtidos numericamente harmonizam com os resultados analíticos. Aplicamos também o formalismo de escala em mapeamentos bidimensionais conservativos e dissipativos. No caso conservativo, nosso objetivo foi analisar o comportamento de órbitas caóticas próximas à transição de fase de integrável para não integrável. Próximo à esta transição, descrevemos o sistema dinâmico utilizando uma função homogênea generalizada para a qual encontramos um lei de escala que descreve o comportamento da ação quadrática média próximo à transição. Através de uma discussão fenomenológica, encontramos expoentes críticos que corroboram com a descrição analítica. No caso dissipativo, nosso principal objetivo foi investigar a influência na dinâmica ao ser introduzido um termo dissipativo, causando a supressão da difusão ilimitada da variável ação quadrática média. Seguimos uma descrição fenomenológica acompanhada de uma descrição analítica e assim, determinamos os expoentes críticos usando uma função homogênea generalizada.

Propomos o estudo do meio desordenado onde a caminhada determinista parcialmente autorrepulsiva (CDPA) é desenvolvida e o estudo da caminhada aleatória autorrepulsiva (SAW) em rede regular. O meio desordenado na CDPA, gerado por um processo Poissônico espacial, é caracterizado pela estatística de vizinhança e de distâncias. A estatística de vizinhança mede a probabilidade de um ponto ser $m$-ésimo vizinho mais próximo de seu $n$-ésimo vizinho mais próximo. A estatística de distâncias mede a distribuição de distância de um ponto ao seu $k$-ésimo vizinho mais próximo. No problema da estatística de distâncias, calculamos a função densidade de probabilidade (pdf) e estudamos os casos limites de alta ordem de vizinhança e alta dimensionalidade. Um caso particular dessa pdf pode verificar se um conjunto de pontos foi gerado por um processo Poissônico. Na SAW em rede regular, um caminhante escolhe aleatoriamente um sítio adjacente para ser visitado no próximo passo, mas é proibido visitar um sítio duas ou mais vezes. Desenvolvemos uma nova abordagem para estudar grandezas conformacionais por meio do produto escalar entre o vetor posição e vetor deslocamento no $j$-ésimo passo: $\\langle\\vec{R}_{j}\\cdot\\vec{u}_{j}angle_{N}$. Mostramos que para $j=N$ o produto escalar é igual ao comprimento de persistência (projeção do vetor posição na direção do primeiro passo) e que converge para uma constante. Calculamos a distância quadrática média ponta-a-ponta, $\\langle \\vec{R}_{N}^{2}angle_{N}\\sim N^{2 u_{0}}$, como o somatório de $1\\leq j \\leq N$ do produto escalar. Os dados gerados pelo algoritmo de simulação Monte Carlo, codificado em linguagem C e paralelizado em MPI, fornecem o expoente $ u_{0}$ da regra de escala $\\langle \\vec{R}_{j}\\cdot\\vec{u}_{j}angle_{N}\\sim j^{2 u_{0}-1}$, para $1\\leq j \\leq \\Theta(N)$, próximo ao valor esperado. A partir de $\\Theta(N)\\approx N/2$ para rede quadrada e $\\Theta(N)\\approx N/3$ para rede cúbica, a caminhada torna-se mais flexível devido ao maior número de graus de liberdade disponível nos últimos passos. / We propose the study of disordered media where the deterministic partially self-avoiding walk (DPSW) is developed and the study of self-avoiding random walk (SAW) in regular lattices. The disordered media in the DPSW, generated by a spatial Poissonian process, is characterized by neighborhood and distance statistics. Neighborhood statistics quantifies the probability of a point to be the $m$th nearest neighbor of its $n$th nearest neighbor. Distance statistics quantifies the distance distribution of a given point to its $k$th nearest neighbor. For the distance statistics problem, we obtain the probability density function (pdf) and study the high dimensionality and high neighborhood order limits. A particular case of this pdf can verify if a points set is generated by a Poissonian process. In a SAW in regular lattice, the walker randomly chooses an adjacent site to be visited in the next step, but is forbidden to visit a site two or more times. We developed a new approach to study conformational quantities of SAW by means of the scalar product between the position vector and the displacement vector in the $j$th step: $\\langle\\vec{R}_{j}\\cdot\\vec{u}_{j}angle_{N}$. We show that for $j=N$ the scalar product is equal to the persistence length (projection of position vector in the direction of the first step) and that converges to a constant. We compute the square end-to-end distance, $\\langle \\vec{R}_{N}^{2}angle_{N}\\sim N^{2 u_{0}}$, as the summation $1\\leq j \\leq N$ of scalar product. The data generated by Monte Carlo simulation algorithm, coded in C language and parallelized in MPI, provides the exponent $ u_{0}$ of the scaling law $\\langle \\vec{R}_{j}\\cdot\\vec{u}_{j}angle_{N}\\sim j^{2 u_{0}-1}$, for $1\\leq j \\leq \\Theta(N)$, close to the expected value. Starting from $\\Theta(N)\\approx N/2$ for square lattice and $\\Theta(N)\\approx N/3$ for cubic lattice, the walk becomes more flexible due to the large number of degrees of freedom available in the last steps.

Yes / This study uses an improved k –ε coupled shallow water equations (SWE) model that equipped with the numerical computation of the velocity fluctuation terms to investigate the turbulence structures of the open channel flows. We adapted the Kolmogorov K41 scaling model into the k –ε equations to calculate the turbulence intensities and Reynolds stresses of the SWE model. The presented model was also numerically improved by a recently proposed surface gradient upwind method (SGUM) to allow better accuracy in simulating the combined source terms from both the SWE and k –ε equations as proven in the recent studies. The proposed model was first tested using the flows induced by multiple obstructions to investigate the utilised k –ε and SGUM approaches in the model. The laboratory experiments were also conducted under the non-uniform flow conditions, where the simulated velocities, total kinetic energies (TKE) and turbulence intensities by the proposed model were used to compare with the measurements under different flow non-uniformity conditions. Lastly, the proposed numerical simulation was compared with a standard Boussinesq model to investigate its capability to simulate the measured Reynolds stress. The comparison outcomes showed that the proposed Kolmogorov k –ε SWE model can capture the flow turbulence characteristics reasonably well in all the investigated flows. / The Major State Basic Research Development Program (973 program) of China (No. 2013CB036402)

En ingénierie mécanique, la rupture des matériaux est un risque qu'il convient d'anticiper et qui reste aujourd'hui une menace pour les structures. La rupture des systèmes pré-fissurés a lieu quand l'énergie libérée par la propagation de la fissure préexistante excède un seuil critique (taux de restitution d'énergie) qui représente une propriété du matériau. Au contraire, la rupture de systèmes sans défauts préexistants survient lorsque la contrainte appliquée atteint la résistance, également propriété du matériau. L'existence de deux critères pour la rupture semble indiquer des mécanismes d'amorçage différents, ce qui soulève la question des cas réels intermédiaires qui présentent des concentrations de contrainte modérées. Différentes approches existantes sont cohérentes avec les deux situations limites mais il n'y a pas de consensus clair dans la communauté scientifique. Dans cette thèse, nous étudions les mécanismes de la rupture fragile à l'échelle atomique afin d'en comprendre l'origine physique pour différentes concentrations de contraintes. La rupture provient de la rupture des liaisons à l'échelle atomique. Nous utilisons donc des techniques de simulation moléculaire pour étudier la physique élémentaire de l'initiation de la rupture fragile. Dans ce but, on étudie deux types de structure atomique. Le premier est un matériau modèle à maille triangulaire, dont le potentiel permet d'interpréter analytiquement, et avec précision, les résultats des simulations moléculaires. L'étude est étendue à un système plus réel : le graphène. Ce matériau, qui présente une résistance élevée au regard de sa faible ténacité, a l'une des plus petites tailles de zone d'élaboration par rapport aux autres matériaux fragiles, ce qui permet d'appliquer numériquement les concepts de la rupture fragile jusqu'à l'échelle nanométrique de la simulation moléculaire. On s'intéresse dans un premier temps à la rupture des matériaux à 0K. À cette température, un système atomique est en équilibre statique. La rupture peut donc être traitée comme une instabilité. L'analyse du profil énergétique du système atomique fournit un moyen d'identifier les mécanismes de rupture. Nous montrons qu'on peut identifier la rupture en cherchant les valeurs propres nulles ou négative de la matrice hessienne. Les vecteurs propres correspondants indiquent les modes de rupture et montrent l'apparition de bandes de transition entre mouvements de groupes d'atomes pour des systèmes intacts, dont la largeur rappelle la longueur d'élaboration, généralement introduite dans des théories macroscopiques d'initiation de la rupture. On étudie aussi l'effet de la présence de défauts sur les modes d'instabilité et leur dégénérescence. Cette étude fournit une technique générale pour identifier les mécanismes d'initiation de rupture quelle que soit la concentration de contrainte dans la structure. On s'intéresse ensuite aux températures non nulles. On étudie les effets combinés de la température, de la taille du système et du taux de chargement. En partant de la théorie cinétique, nous montrons qu'il existe des lois d'échelle générales fournissant une équivalence taille-vitesse de chargement-température et permettant de relier résistance et ténacité à la limite à 0K. La différence entre la loi d'échelle en résistance et celle en ténacité réside dans le fait que la rupture ne soit pas sensible à la taille du système pré-fissuré mais au nombre de pointes de fissure. Cela indique une différence statistique essentielle entre la rupture en résistance et la fracture ce qui permet de mieux comprendre la transition de l'une à l'autre. Dans l'esprit de mieux comprendre la transition entre les deux types de rupture, on traite le cas de trous elliptiques à différents rapports d'aspects et on analyse en même temps l'effet de cette transition sur les modes d'instabilité. On étudie en dernière partie, l'effet des surfaces libres et les différents paramètres caractérisant cette situation / In mechanical engineering, failure is a risk that must be anticipated and is still a threat for structures. The failure of pre-cracked systems occurs when the energy released by the propagation of the pre-existing crack exceeds a critical threshold (Griffith's energy release rate) which represents a property of the material. On the contrary, the failure of systems without pre-existing defects occurs when the applied stress reaches the strength, also property of the material. The existence of two criteria for failure suggests different driving mechanisms, which raises the question of intermediate cases with moderate stress concentrations. Different existing approaches are consistent with the two limit cases but there is no clear consensus in the scientific community.In this work, we study the mechanisms of brittle failure on the atomic scale in order to understand the underlying physical mechanisms. Macroscopic failure comes from the breaking of bonds at the atomic scale. We therefore use molecular simulation techniques to study the elementary physics of brittle failure initiation. Two types of atomic structure are studied. The first one is a triangular lattice toy model whose simplicity allows precise analytical interpretation of the molecular simulation results. The study is extended to a more realistic system: graphene. This material, which has a high strength and a rather low toughness in comparison, has one of the smallest process zones compared to other brittle materials, which makes it possible to apply the concepts of brittle failure up to the nanometric scale of molecular simulation. We first investigate the failure of materials at 0K. At this temperature, an atomic system is in static equilibrium. The breaking of bonds can be treated as instability. The analysis of the energy profile of the atomic system provides a means of identifying the mechanisms of failure. We show that we can identify failure initiation by looking for negative or zero eigenvalues of the Hessian matrix. The corresponding eigenvectors indicate the modes of failure and show the appearance of transition bands between motions of groups of atoms for intact systems, whose width recalls the size of the process zone, generally introduced in macroscopic theories of failure initiation. We also study the effect of defects on the instability modes and their degeneracy. This study provides a general technique to capture fracture initiation mechanisms irrespective of the stress concentration in the structure. We focus afterwards on finite temperatures. We study the combined effects of temperature, system size and loading rate. Starting from the kinetic theory, we identify general scaling laws providing a size-loading rate-temperature equivalence and relating the strength and toughness to the limit at 0K. The difference between the scaling law of strength and that of toughness lies in the fact that failure is not sensitive to the size of the pre-cracked system but to the number of crack tips. This indicates an essential statistical difference between strength and fracture failures which makes it possible to better understand the transition from one to the other.In order to better understand the transition between the two types of failure, we treat the case of elliptic holes with different aspect ratios and we focus at the same time on the effect of this transition on instability modes. We study in the last part the case of non-periodic structures with free surfaces. We determine the various parameters characterizing this situation and the effect of the presence of surface phenomena on instability modes

Simultaneous operation of different wireless applications in the same geographical region and the same frequency band gives rise to undesired interference issues. Since licensed (primary) applications have been granted priority access to the frequency spectrum, unlicensed (secondary) services should avoid imposing interference on the primary system. In other words, secondary system’s activity in the same bands should be in a controlled fashion so that the primary system maintains its quality of service (QoS) requirements. In this thesis, we consider collocated point-to-multipoint primary and secondary networks that have simultaneous access to the same frequency band. Particularly, we examine three different levels at which the two networks may coexist: pure interference, asymmetric co-existence, and symmetric co-existence levels. At the pure interference level, both networks operate simultaneously regardless of their interference to each other. At the other two levels, at least one of the networks attempts to mitigate its interference to the other network by deactivating some of its users. Specifically, at the asymmetric co-existence level, the secondary network selectively deactivates its users based on knowledge of the interference and channel gains, whereas at the symmetric level, the primary network also schedules its users in the same way. Our aim is to derive optimal sum-rates (i.e., throughputs) of both networks at each co-existence level as the number of users grows asymptotically and evaluate how the sum-rates scale with the network size. In order to find the asymptotic throughput results, we derive two propositions; one on the asymptotic behaviour of the largest order statistic and one on the asymptotic behaviour of the sum of lower order statistics. As a baseline comparison, we calculate primary and secondary sum-rates for the time division (TD) channel sharing. Then, we compare the asymptotic secondary sum-rate in TD to that under simultaneous channel sharing, while ensuring the primary network maintains the same sum-rate in both cases. Our results indicate that simultaneous channel sharing at both asymmetric and symmetric co-existence levels can outperform TD. Furthermore, this enhancement is achievable when user scheduling in uplink mode is based only on the interference gains to the opposite network and not on a network’s own channel gains. In other words, the optimal secondary sum-rate is achievable by applying a scheduling strategy, referred to as the least interference strategy, for which only the knowledge of interference gains is required and can be performed in a distributed way.

Simultaneous operation of different wireless applications in the same geographical region and the same frequency band gives rise to undesired interference issues. Since licensed (primary) applications have been granted priority access to the frequency spectrum, unlicensed (secondary) services should avoid imposing interference on the primary system. In other words, secondary system’s activity in the same bands should be in a controlled fashion so that the primary system maintains its quality of service (QoS) requirements. In this thesis, we consider collocated point-to-multipoint primary and secondary networks that have simultaneous access to the same frequency band. Particularly, we examine three different levels at which the two networks may coexist: pure interference, asymmetric co-existence, and symmetric co-existence levels. At the pure interference level, both networks operate simultaneously regardless of their interference to each other. At the other two levels, at least one of the networks attempts to mitigate its interference to the other network by deactivating some of its users. Specifically, at the asymmetric co-existence level, the secondary network selectively deactivates its users based on knowledge of the interference and channel gains, whereas at the symmetric level, the primary network also schedules its users in the same way. Our aim is to derive optimal sum-rates (i.e., throughputs) of both networks at each co-existence level as the number of users grows asymptotically and evaluate how the sum-rates scale with the network size. In order to find the asymptotic throughput results, we derive two propositions; one on the asymptotic behaviour of the largest order statistic and one on the asymptotic behaviour of the sum of lower order statistics. As a baseline comparison, we calculate primary and secondary sum-rates for the time division (TD) channel sharing. Then, we compare the asymptotic secondary sum-rate in TD to that under simultaneous channel sharing, while ensuring the primary network maintains the same sum-rate in both cases. Our results indicate that simultaneous channel sharing at both asymmetric and symmetric co-existence levels can outperform TD. Furthermore, this enhancement is achievable when user scheduling in uplink mode is based only on the interference gains to the opposite network and not on a network’s own channel gains. In other words, the optimal secondary sum-rate is achievable by applying a scheduling strategy, referred to as the least interference strategy, for which only the knowledge of interference gains is required and can be performed in a distributed way.

FundaÃÃo de Amparo à Pesquisa do Estado do Cearà / Neste trabalho investigamos algumas aplicaÃÃes do formalismo de escala em mapeamentos discretos. Exploramos os decaimentos assintÃticos ao estado estacionÃrio com foco em trÃs tipos de bifurcaÃÃes em mapeamentos unidimensionais: bifurcaÃÃo transcrÃtica, bifurcaÃÃo supercrÃtica de forquilha e bifurcaÃÃo de duplicaÃÃo de perÃodo. Caracterizamos este comportamento atravÃs de uma funÃÃo homogÃnea generalizada com expoentes crÃticos bem definidos. PrÃximo ao ponto de bifurcaÃÃo o decaimento ao ponto fixo ocorre atravÃs de uma funÃÃo exponencial cujo o tempo de relaxaÃÃo à caracterizado por uma lei de potÃncia que independe da nÃo linearidade do mapa. Os resultados obtidos numericamente harmonizam com os resultados analÃticos. Aplicamos tambÃm o formalismo de escala em mapeamentos bidimensionais conservativos e dissipativos. No caso conservativo, nosso objetivo foi analisar o comportamento de Ãrbitas caÃticas prÃximas à transiÃÃo de fase de integrÃvel para nÃo integrÃvel. PrÃximo à esta transiÃÃo, descrevemos o sistema dinÃmico utilizando uma funÃÃo homogÃnea generalizada para a qual encontramos um lei de escala que descreve o comportamento da aÃÃo quadrÃtica mÃdia prÃximo à transiÃÃo. AtravÃs de uma discussÃo fenomenolÃgica, encontramos expoentes crÃticos que corroboram com a descriÃÃo analÃtica. No caso dissipativo, nosso principal objetivo foi investigar a influÃncia na dinÃmica ao ser introduzido um termo dissipativo, causando a supressÃo da difusÃo ilimitada da variÃvel aÃÃo quadrÃtica mÃdia. Seguimos uma descriÃÃo fenomenolÃgica acompanhada de uma descriÃÃo analÃtica e assim, determinamos os expoentes crÃticos usando uma funÃÃo homogÃnea generalizada. / In this work we are going to investigate the scale formalism in discret mappings. In 1D mappings, we explore the asymptotic decays to the steady state with focus in three types of bifurcation: transcriptical, pitchfork and period-doubling. We identify this behavior through a well defined generalized homogeneous function with critical exponents. Next to the bifurcation point, the decay to the fix point occurs by an exponential function, which is given by a power law that is independent of the non-linearity mapping. The numerical results obtained agree with the analytical results. We also apply the scale formalism in conservatives and dissipatives bidimensional mappings. In the conservative case, our goal was analyze the behavior of the chaotics orbits next to the phase transition from the integrable to the non-integrable. Next to that transition, we describe the dynamical system using a generalized homogeneous function for which we found a power law that describe the behavior of the criticality. Through a phenomenological discussion, we found critical exponents in agree with the analytical description. In the dissipative case, our main goal was to investigate the influence of a dissipative term in the dynamics, causing a phase transition - suppression of unlimited difusion of the action variable. Following a phenomenological approach with an analytical description, we were able to determine the critical exponents using a generalized homogeneous function.

Les écoulements gaz-liquide constituent un axe de recherche très actif en microfluidique. Le rapport des débits entre les deux phases, la formation de bulles et les champs de vitesse des microcanaux ont été étudiés dans cette thèse, en utilisant une caméra numérique rapide et un microsystème de Particule Image Velocimetry (micro-PIV). En particulier, le diagramme des phases gaz-liquide ont été établi dans des microcanaux carrés ; la formation des bulles en fluides tant newtoniens que non newtoniens a été étudiée en détail dans plusieurs configurations géométriques telles que T-injonction et flow-focusing. Les mécanismes régissant la formation d'une bulle ont été modélisés pour chaque étape : expansion, amincissement et rupture. L'étape amincissement de la traînée d'une bulle est notamment contrôlée par une pression orthogonale qui dépend du débit du liquide. Dans le cadre de flow-focusing, le mécanisme de la rupture du film gazeux peut être décrit par une loi d'échelle reliant l'épaisseur minimale du film au temps restant juste avant la rupture avec un exposant 1/3. Le caractère non newtonien de fluides PAAm allonge la traînée d'une bulle par rapport aux fluides newtoniens. Enfin, l'étude de la coalescence entre bulles a été entreprise à l'échelle microscopique ainsi que le comportement complexe des trains de bulles dans des réseaux de microcannaux / Gas-liquid two-phase flow is an important research project in microfluidics. The gas-liquid two-phase flow, the bubble formation and moving behaviours in microchannels were investigated, by using a high speed digital camera and a micro Particle Image Velocimetry (micro-PIV). The gas-liquid two-phase flow in vertical rectangular microchannels was investigated and a flow pattern map was constructed; the bubble formation in both Newtonian and non-Newtonian fluids in cross-flowing microfluidic T-junctions and flow-focusing devices was investigated; the bubble formation process could be divided into expansion, collapse and pinch-off stages; the collapse speed of the gaseous thread in the second stage is controlled by the squeezing pressure, and is proportional to the liquid flow rates; while the minimum width of the neck of the gaseous thread in the third stage for bubble formation in flow-focusing devices could be scaled with the remaining time to the ultimate pinch-off as a power law relationship with an exponent of 1/3; the PAAm solutions prolong the gaseous thread in the tangential direction of the neck; bubble coalescence in a microchannel with an expansion section was studied; the bubble behavior in a microchannel with a loop was also investigated

Les moteurs de fusées à ergols solides (SRMs) sont sensibles aux instabilités hydrodynamiques qui peuvent déclencher des oscillations auto-entretenues de pression de grandes amplitudes lorsqu’elles se couplent à l’un des modes acoustiques du système. Le moteur de ces instabilités est la formation de structures tourbillonnaires cohérentes synchronisées par des ondes acoustiques longitudinales. Pour certaines conditions de fonctionnement, les ondes acoustiques générées par l’interaction de ces tourbillons avec la tuyère amorcée du moteur renforcent l’oscillation acoustique. L’objectif des travaux menés dans cette thèse est de déterminer l’amplitude et la fréquence des oscillations de pression au cycle limite des instabilités. Celui-ci est atteint par saturation non linéaire des sources, qui est la conséquence de la formation de grosses structures cohérentes. Dans ce cas l’interaction tourbillon tuyère devient insensible à l’amplitude de l’onde du mode acoustique établi dans le foyer. Dans ces conditions, on peut se concentrer sur l’interaction d’un tourbillon avec la tuyère dans le mécanisme de production sonore. En considérant un écoulement incompressible et l’absence de frottement, un premier modèle analytique est développé permettant de déterminer la production sonore d’un tourbillon ingéré par une tuyère bidimensionnelle plane, lorsque le tourbillon est traité comme une ligne vorticité. Des expériences précédentes indiquent que le volume de la cavité autour de l’entrée d’une tuyère intégrée a une grande influence sur l’amplitude des oscillations de pression dans les grands SRMs. On montre que ceci est dû au champ de vitesse acoustique induit par la compressibilité du gaz dans la cavité qui produit une fluctuation de vitesse transverse à la trajectoire du tourbillon. Une seconde alternative au modèle analytique incompressible est développée en considérant toujours l’absence de frottement, mais un modèle compressible de l’interaction tourbillon-tuyère. Celui-ci repose sur un code aéroacoustique pour les écoulements internes basé sur les équations d’Euler (EIA) qui est utilisé ici pour la simulation de l’interaction tourbillon-tuyère. Une étude systématique de cette interaction a été menée pour une tuyère amorcée. Les résultats ont permis de proposer un modèle de sources localisées pour des ondes planes basé sur une analyse théorique des lois d’échelles de ces phénomènes. Les simulations de ces interactions tourbillons-tuyères ont été réalisées pour différents types de tuyères. En employant un bilan énergétique, un modèle avec un seul paramètre de contrôle est formulé, qui permet de reproduire qualitativement le comportement du cycle limite d’oscillations de pression observées dans des expériences réalisées avec des gaz froids décrites dans la littérature. Finalement le modèle Euler est utilisé pour comparer la production de son par interaction tourbillon-tuyère avec celle due à l’ingestion d’une onde d’entropie, appelée aussi tache d’entropie. Contrairement au cas des tourbillons, le bruit produit par ingestion de taches d’entropie n’est pas sensible au volume de la cavité d’une tuyère intégrée. Ces résultats indiquent que le bruit produit par les tourbillons est dominant dans le cas des SRMs étudiés. L’ensemble de ces travaux permet d’améliorer la compréhension des phénomènes d’interaction entre des non-homogénéités de l’écoulement et la tuyère. Elle permet surtout de déterminer quels sont les facteurs de l’écoulement et les éléments géométriques importants qui pilotent le niveau sonore produit par ces interactions. Les modèles développés dans ces travaux, avec divers degrés d’approximation et de complexité permettent d’enrichir la gamme des outils de conception des SRMs. / Solid Rocket Motors (SRMs) can display self-sustained acoustic oscillations driven by coupling between hydrodynamic instabilities of the internal flow and longitudinal acoustic standing waves. The hydrodynamic instabilities are triggered by the acoustic standing wave and results in the formation of coherent vortical structures. For nominal ranges of flow conditions the sound waves generated by the interaction between these vortices and the choked nozzle at the end of the combustion chamber reinforces the acoustic oscillation. Most available literature on this subject focuses on the threshold of instability using a linear model. The focus of this work is on the prediction of the limit-cycle amplitude. The limit-cycle is reached due to nonlinear saturation of the source, as a consequence of the formation of large coherent vortical structures. In this case the vortex-nozzle interaction becomes insensitive to the amplitude of the acoustic standing wave. Hence, one can focus on the sound generation of a vortex with the nozzle. Sound production can be predicted from an analytical two-dimensional planar incompressible frictionless model using the so-called Vortex Sound Theory. In this model the vorticity is assumed to be concentrated in a line vortex. Experiments indicate that the volume of cavities around so-called “integrated nozzles” have a large influence on the pulsation amplitude for large SRMs. This is due to the acoustical field normal to the vortex trajectory, induced by the compressibility of the gas in this cavity. As an alternative to the incompressible analytical model a compressible frictionless model with an internal Euler Aeroacoustic (EIA) flow solver is used for simulations of vortex-nozzle interaction. A dedicated numerical simulation study focusing on elementary processes such as vortex-nozzle and entropy spot-nozzle interaction allows a systematic variation of relevant parameters and yields insight which would be difficult by means of limit cycle studies of the full engine. A systematic study of the vortex-nozzle interaction in the case of a choked nozzle has been undertaken. The results are summarized by using a lumped element model for plane wave propagation, which is based on theoretical scaling laws. From EIA simulations it appears that sound due to vortex-nozzle interaction is mainly generated during the approach phase and that for the relevant parameter range there is no impingement of the vortex on the nozzle wall as has been suggested in the literature. Using an energy balance approach, a single fit-parameter model is formulated which qualitatively predicts limit-cycle observations in cold gas-scale experiments reported in the literature. Finally the Euler model is used to compare the sound production by vortex-nozzle interaction with that due to the ingestion of an entropy non-uniformity also called entropy spot. In addition to insight, this study provides a systematic procedure to develop a lumped element model for the sound source due to non-homogeneous flow-nozzle interactions in SRMs. Such lumped models based on experimental data or a limited number of flow simulations can be used to ease the design of SRMs.

Dans la présente thèse de doctorat je présente des résultats concernant des modèles désordonnés et frustrés venant de la physique statistique et de l'optimisation combinatoire. Comme application de la théorie des verres de spins, j'étudie le modèle de Blume, Emery et Griffiths désordonné et frustré. Ce modèle est traité dans l'approximation de champ moyen dans le cadre de la méthode des répliques A l'aide de l'Ansatz symétrique dans les répliques je présente une solution numérique complète puis je discute des effets de brisure de cette symétrie La stabilité de la solution symétrique a été Rudik et les régions instables identifiées Le diagramme de phase exhibe des transitions de premier et de second ordre. Le point tricritique persiste dans le modèle frustré, Ce qui est en accord avec des travaux antérieurs une version du modèle BEG avec un potentiel chimique désordonné a également été étudiée. les calculs confirment que le point tricritique apparaît à plus basse température quand il y a du désordre. Ensuite je considère le problème de la bipartition d'un graphe. Ce problème correspond du point de vue de la physique statistique h un verre de spins soumis h une contrainte d'aimantation totale nulle. je considère les propriétés statistiques des solutions de faible énergie engendrées par des algorithmes heuristiques. de tels algorithme sont en général conçus pour résoudre des problèmes d'optimisation combinatoire qui sont NP- difficiles. Plusieurs heuristiques ont 60 implémentées pour le problème de la bipartition de graphe. des lois d'échelle ont été obtenues : en particulier la moyenne et la variance du coût obéissent A une loi linéaire en N. Par conséquent le coût obtenu par des heuristiques est une quantité auto-moyennante. je suggère que cette propriété est générale valable aussi pour les solutions aléatoires pour les solutions quasi-optimales et pour les solutions optimales. En outre je propose une procédure pour comparer des algorithmes heuristiques. Cette procédure tient compte de la qualité de la solution aussi bien que du temps de calcul utilisé. Dans la troisième partie de ma thèse j'ai étudié en détail les propriétés h température nulle des verres de spins sur des graphes aléatoires lacunaires avec une coordination fixe. les verres de spins sur de tels graphes peuvent être considérés comme une approximation aux vrais verres de spins qui est plus réaliste que le modèle de Sherrington et Kirkpatrick. J'ai conçu un nouvel algorithme pour trouver les états fondamentaux. Aussi je teste numériquement une conjecture de Banavar, Sherrington et Sourlas qui donne la densité d'énergie du fondamental dans la limite de grande taille en fonction de la coordination. La distribution du paramètre d'ordre se révèle être non triviale et les données présentent une forte indication de la présence d'ultramétricité pour toutes les valeur de la coordination. Ces résultats confirment que les propriétés particulières des verres de spin, déduites an niveau de l'approximation de champ moyen dans le cadre du modèle de Sherrington et Kirkpatrick, sont aussi présentes pour des modèles plus réalistes comme les verres de spins sur des graphes aléatoires lacunaires avec une coordination fixe.