This work is concerned with the Almost Axisymmetric Flows with Forcing Terms which are derived from the inviscid Boussinesq equations. It is our hope that these flows will be useful in Meteorology to describe tropical cyclones. We show that these flows give rise to a collection of Monge-Ampere equations for which we prove an existence and uniqueness result. What makes these equations unusual is the boundary conditions they are expected to satisfy and the fact that the boundary is part of the unknown. Our study allows us to make inferences in a toy Almost Axisymmetric Flows with a forcing term model.

El presente trabajo estudia la convección Rayleigh-Bénard en cavidades cúbicas sin inclinación con respecto a la horizontal y calentadas por debajo para números de Rayleigh tanto dentro del estado estacionario como dentro del régimen turbulento.Inicialmente se estudian los efectos que tiene la variación de las propiedades físicas con la temperatura sobre la estructura de flujo, el mecanismo de transporte y la transferencia de calor para dos números de Rayleigh bajos (Ra=104 y Ra=5×104) en los que el flujo es laminar y estacionario y para un Rayleigh dentro del régimen turbulento (Ra=107). Para este estudio numérico se utilizó agua como fluido convectivo (Pr=5.9) y se supuso que las paredes laterales de la cavidad eran perfectamente conductoras. De acuerdo con Gray y Giorgini (1976) la aproximación de Boussinesq debe aplicarse para diferencias menores al 10% en las propiedades físicas del fluido. De acuerdo con este criterio, para agua a T0=26ºC las variaciones de temperatura máximas permitidas para poder aplicar la aproximación de Boussinesq serían ΔT ≤ 2.9ºC para el coeficiente de expansión térmica, ΔT ≤ 4.5ºC para la viscosidad. Los cálculos sin la aproximación de Boussinesq llevados a cabo en el presente trabajo fueron calculados con una diferencia de temperaturas de 17.5ºC entre placas fría y caliente (6 veces mayor que el limite para la propiedad física más crítica) y por tanto los presentes resultados pueden ser considerados fuera de la aproximación de Boussinesq. En estas condiciones las variaciones del coeficiente de expansión térmica y de la viscosidad entre las paredes caliente y fría respecto al valor evaluado a la temperatura media, son del 62% y 40%, respectivamente, siendo el fluido cerca a la pared fría más viscoso y con menor coeficiente de expansión.Posteriormente se han identificado numéricamente las estructuras de flujo promedio temporal para cinco números de Rayleigh dentro del régimen turbulento (para el rango 107≤Ra≤108). Debido a la similitud de cuatro de estas estructuras (para Ra>3×107) sólo dos de ellas fueron verificadas experimentalmente (Ra=7×107 y Ra=108) además de la estructura encontrada a Ra=107. Para los cálculos se asumió la aproximación de Boussinesq debido a que las diferencias de temperatura en la experimentación eran suficientemente bajas como para considerar propiedades físicas constantes de acuerdo con los resultados obtenidos en el análisis anterior. Con el fin de reproducir al máximo las condiciones experimentales, los resultados numéricos fueron obtenidos teniendo presente la conductividad térmica del vidrio de las paredes laterales. La visualización de las estructuras de flujo y la medición de los campos de velocidad en el plano vertical medio de la cavidad se realizó con el método PIV (de Particle Image Velocimetry). Estos resultados nos permitieron validar los resultados obtenidos con las simulaciones, comparándose tanto las topologías del flujo como los valores de velocidad puntuales en perfiles dentro del plano analizado.De los resultados numéricos se encontró que aun con porcentajes de variación de las propiedades físicas del fluido entre las paredes fría y caliente muy por encima del criterio normalmente utilizado (10%) tanto en régimen laminar como turbulento, las estructuras de flujo y las condiciones de transporte de calor, no se ven afectadas considerablemente por esta variación. Esto es debido principalmente a que los gradientes de temperatura mas altos están localizados dentro de la delgada límite térmica cerca de las paredes y por tanto la variación de las propiedades físicas con la temperatura solo es significativa en esta zona. En los resultados numéricos a régimen turbulento para números de Rayleigh en el rango 3×107≤Ra≤108 se observa que la estructura cambia de dirección a través del cambio de orientación del eje de rotación dela estructura permaneciendo períodos de tiempo indefinidos en una posición determinada. Los resultados experimentales para Ra=7×107 y Ra=108, no presentan este cambio de orientación en la estructura. Se observa que la estructura de flujo no cambia de posición en el tiempo conservando la misma orientación durante todo el tiempo de muestreo. La diferencia entre resultados numéricos y experimentales puede atribuirse a las inevitables imperfecciones en las condiciones de contorno de los experimentos, especialmente en la distribución de temperaturas en las paredes de la cavidad, o bien a la pequeña pero también inevitable desviación de la cavidad respecto la perfecta horizontalidad. Estos efectos pueden ser los causantes de la inhibición del cambio de orientación de la estructura observado en las simulaciones numéricas con condiciones de contorno ideales y llevadas a cabo suponiendo una perfecta horizontalidad de la cavidad. Sin embargo, es importante resaltar que las estructuras de flujo medias y los contornos de la desviación estándar de la velocidad vertical obtenidos experimentalmente son cualitativamente similares a aquellos obtenidos numéricamente para un periodo de tiempo en el cual la estructura media permanece en la misma posición, indicando que la estructura de flujo experimental corresponde a una determinada orientación de la estructura obtenida numéricamente. Por otro lado las estructuras de flujo promedio temporal y los valores de velocidad obtenidos numéricamente concuerdan significativamente con las correspondientes medidas experimentales para los números de Rayleigh analizados. / In the first part of this study, the effects of a non-Boussinesq fluid are numerically studied and discussed for Rayleigh-Bénard convection in a cubical cavity with perfectly conducting sidewalls at low and high Rayleigh numbers using water as a convecting fluid (Pr=5.9). Numerical simulations at all Rayleigh numbers considered were carried out for two different cases. In the first case a Boussinesq fluid was considered (Boussinesq Fluid Simulation-BFS) and in the second case, the dependence of viscosity and thermal conductivity of water on temperature was adopted in the simulations (Non-Boussinesq Fluid Simulation-NBFS). At the low Rayleigh numbers used in this study (Ra=104 and Ra=5x104) the flow is laminar and steady and at the high Rayleigh number considered (Ra=107) the flow is turbulent. At Ra=104 and Ra=5x104 we focus our analysis on the effect of variation of the fluid viscosity with temperature on the more stable flow structures from the set of seven different topologies reported in previous studies. At Ra=107 and Pr=5.9 the non-Boussinesq effects on the turbulent flow are analysed in detail and the flow structures and heat transfer rates compared with those available in the literature at Pr=0.71. Previous works recommend that temperature difference should be less than 4.5ºC in order to obtain less than 10% of variation in the viscosity. Non-Boussinesq simulations in the present work were calculated with a viscosity variation of 40% between cold and hot plates. The numerical simulations at high and low Rayleigh numbers were conducted with a second order finite volume code without any turbulence model because the time-steps and grid sizes used are adequate for the time and spatial resolution requirements reported in previous direct numerical simulations of Rayleigh-Bénard flows. The structure of the flow topologies at Ra=104 and Ra=5x104 are not significantly affected by the effects of the variation of viscosity and thermal conductivity with temperature. However, results obtained with a NBFS show an increase of the ascending flow velocities compared with those obtained with the Boussinesq approximation according to the decrease of viscosity with increasing temperature. At Ra=107 the instantaneous flow shows large deviations with respect to the time-average flow field that consists in two counter rotating vortex rings located near the horizontal plates. The temperature gradients and, thus the viscosity variation are located close to the walls within the thermal boundary layers. This causes that the time-averaged flow field topologies corresponding to BFS and NBFS are not greatly affected by the effects of the variation of viscosity and thermal conductivity with temperature.In the second part, experimental measurements and numerical simulations of natural convection in a cubical cavity heated from below and cooled from above are reported at turbulent Rayleigh numbers using water as a convective fluid (Pr=6.0). The numerical simulations were carried out, in the range 107≤Ra≤108, with a second order finite volume code without any turbulence model because the time-steps and grid sizes used are adequate for the time and spatial resolution requirements reported in previous direct numerical simulations of Rayleigh-Bénard flows. The Boussinesq approximation was considered in the simulations according to the thermal conditions and the dimensions of the cavities used in the experiments. The Particle Image Velocimetry technique was used to measure the two velocity components parallel to a vertical mid-plane of the cavity at Ra=107, Ra=7×107 and Ra=108. Both experiments and simulations show that at Ra=107 the time averaged flow structure consists in two horizontal counter-rotating vortex rings located near the horizontal walls of the cavity. At the higher Rayleigh numbers considered, the simulations predict an unsteady single roll motion in which the direction of the axis of rotation rotates in the horizontal plane with very low frequencies. This rotation produces a time averaged flow structure similar to that found at Ra=107. There is a general agreement between the predicted time averaged localvelocities and those experimentally measured if the heat conduction through the sidewalls occurring in the experiments is considered in the simulations.

La thèse est composée de deux parties indépendantes.<br />Dans la première partie, on étudie le comportement asymptotique de problèmes elliptiques et paraboliques à données $L^1+W^{-1,p'}$ (respectivement $L^1+L^p(0,T;W^{-1,p'})$ dans le cas parabolique), dans des domaines devenant infiniment grands. En utilisant le cadre des solutions renormalisées et les résultats de régularité des solutions pour de telles données, on prouve, sous certaines hypothèses structurelles sur les variables d'espace, des résultats de convergence dans les espaces de régularité des solutions.<br />Dans la seconde partie, dans le cas de la dimension $2$, on étudie des systèmes de type Boussinesq. Ces systèmes dérivent de modèles de mécanique des fluides et consistent en un couplage des équations de Navier-Stokes incompressibles et de l'équation de la chaleur. On s'intéresse essentiellement aux questions d'unicité de la solution, particulièrement délicate à prouver du fait du couplage très non linéaire entre les équations. On travaille dans le cadre des solutions faibles pour les équations de Navier-Stokes et dans le cadre des solutions renormalisées pour des problèmes paraboliques pour l'équation de la chaleur. On établit tout d'abord des résultats de régularité pour ces équations puis on montre plusieurs résultats d'existence et d'unicité de la solution du système pour de petites données.

Universiẗat, Diss., 2006--Karlsruhe.

Universiẗat, Diss., 2003--Dortmund.

Submitted by Maykon Nascimento (maykon.albani@hotmail.com) on 2015-09-03T20:43:05Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Modelo de escoamento multifasico para estudo da interacao onda sedimento.pdf: 5609374 bytes, checksum: e7b377578e62f1a8dd61ad6f5a35b06d (MD5) / Approved for entry into archive by Elizabete Silva (elizabete.silva@ufes.br) on 2015-09-11T19:14:09Z (GMT) No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Modelo de escoamento multifasico para estudo da interacao onda sedimento.pdf: 5609374 bytes, checksum: e7b377578e62f1a8dd61ad6f5a35b06d (MD5) / Made available in DSpace on 2015-09-11T19:14:09Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Modelo de escoamento multifasico para estudo da interacao onda sedimento.pdf: 5609374 bytes, checksum: e7b377578e62f1a8dd61ad6f5a35b06d (MD5) Previous issue date: 2015 / FAPES / Modelos de escoamento multifásico são amplamente usados em diversas áreas de pesquisa ambiental, como leitos fluidizados, dispersão de gás em líquidos e vários outros processos que englobam mais de uma propriedade físico-química do meio. Dessa forma, um modelo multifásico foi desenvolvido e adaptado para o estudo do transporte de sedimentos de fundo devido à ação de ondas de gravidade. Neste trabalho, foi elaborado o acoplamento multifásico de um modelo euleriano não-linear de ondas do tipo Boussinesq, baseado na formulação numérica encontrada em Wei et al. (1995), com um modelo lagrangiano de partículas, fundamentado pelo princípio Newtoniano do movimento com o esquema de colisões do tipo esferas rígidas. O modelo de ondas foi testado quanto à sua fonte geradora, representada por uma função gaussiana, pá-pistão e pá-batedor, e quanto à sua interação com a profundidade, através da não-linearidade e de propriedades dispersivas. Nos testes realizados da fonte geradora, foi observado que a fonte gaussiana, conforme Wei et al. (1999), apresentou melhor consistência e estabilidade na geração das ondas, quando comparada à teoria linear para um kh   . A não-linearidade do modelo de ondas de 2ª ordem para a dispersão apresentou resultados satisfatórios quando confrontados com o experimento de ondas sobre um obstáculo trapezoidal, onde a deformação da onda sobre a estrutura submersa está em concordância com os dados experimentais encontrados na literatura. A partir daí, o modelo granular também foi testado em dois experimentos. O primeiro simula uma quebra de barragem em um tanque contendo água e o segundo, a quebra de barragem é simulada com um obstáculo rígido adicionado ao centro do tanque. Nesses experimentos, o algoritmo de colisão foi eficaz no tratamento da interação entre partícula-partícula e partícula-parede, permitindo a evidência de processos físicos que são complicados de serem simulados por modelos de malhas regulares. Para o acoplamento do modelo de ondas e de sedimentos, o algoritmo foi testado com base de dados da literatura quanto à morfologia do leito. Os resultados foram confrontados com dados analíticos e de modelos numéricos, e se mostraram satisfatórios com relação aos pontos de erosão, de sedimentação e na alteração da forma da barra arenosa / Multiphase flow models are widely used in many fields of environmental research, such as fluidized beds, gas dispersion in liquids, dam breaks, oil spill and others physical and chemical processes that use more than one property. Thus, a multiphase model to study the dynamics of two distinct properties represented on reference Eulerian-Lagrangian was developed. In order to study the dynamics of sediment transport due to wave’s action, a non-linear wave Boussinesq model in Eulerian frame, in which the numerical formulation based on Wei et al. (1995), and a Lagrangian particle model, whose formulation is given by the Newtonian principle of motion, has been developed. In the Lagrangian particle model was added the scheme of hard-spheres particle collision. The coupling between the two models has been doing with the wave transferring momentum to sediment particles, which is carried by the flow. For the wave’s model, the wave maker, defined by a Gaussian function, piston-paddle and flap-paddle, and the effects due to the depth of the waves were tested. The Gaussian wave maker showed better consistency in the generation of waves, getting errors about 0.11% compared to the linear theory for kh   . The wave’s nonlinearity simulated by the 2nd order dispersion Boussinesq-wave model showed satisfactory results when compared with experiment of wave propagation over a trapezoidal obstacle, where the deformation of the wave on the underwater structure is in agreement with literature data. In addition to the granular model was also tested in two experiments: the first, which simulates a dam break in a tank containing water-air; and the second experiment, the dam break interact with an obstacle in the center of the tank. In both experiments, the collision model was effective in the treatment of the interaction between the particles and the wall, and allowed the evidence of physical processes that are complicated to be simulated by models that use regular grids. For the coupling of wavesediment model, the algorithm has been tested with data obtained from the literature as the morphology of the bed. The results were compared with analytical data and numerical models, and it showed a certain agreement on points of erosion, sedimentation and change in shape of the sand bar

Made available in DSpace on 2014-06-12T18:31:35Z (GMT). No. of bitstreams: 2 arquivo950_1.pdf: 1082241 bytes, checksum: 372078eb11d1777102148afdc22036de (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2011 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta tese, estudamos dois modelos de mecânica dos fluidos em espaços de Morrey, os quais contêm funções fortemente singulares. O primeiro modelo é o sistema de Boussinesq (SB) em Rn, e o segundo, as equações de Navier-Stokes (ENS) no semi-espaço Rn +: Provamos novos resultados de existência global de soluções, de simetria auto-similar, de regularidade e de comportamento assintótico das soluções. Nossos resultados nos permitem considerar novas condições iniciais e campos gravitacionais singulares

Coastal flooding from wave overtopping causes considerable damage. Presently, to model wave overtopping one can either make use of physical model tests or empirical tools such as those described in the EurOtop manual. Both these methods have limitations; therefore, a quick and reliable numerical model for wave overtopping would be a very useful tool for a coastal engineer.This research aims to test and develop a numerical model (in one horizontal dimension) for nearshore waves, runup and overtopping. The Shallow Water And Boussinesq (SWAB) model solves the Boussinesq-type equations of Madsen and Sorensen (1992) for non breaking waves and the nonlinear shallow water equations for breaking waves. Through testing against a range of physical model data using regular and random waves, the SWAB model's transfer from non-breaking to breaking waves was optimised. It was found that a wave height to water depth ratio worked consistently well as a breaking criterion.A set of physical model tests were carried out, based on previous field testing of wave overtopping that had previously taken place at Anchorsholme, Blackpool. The SWAB model was used to simulate some of these physical model tests, giving good results for mean overtopping rates. SWAB models the force imposed by steep walls and recurve walls on the incident flow; this force was found to have a significant effect on overtopping rates. A comparison was made between mean overtopping rates from the SWAB model, the physical model tests, empirically-based software (PC-Overtopping) and the field data. The physical model and SWAB results compared well with the field data, though the empirical software gave large overestimates.The SWAB model was applied to the analysis of overtopping at Walcott, Norfolk. It was found that beach levels affected overtopping rates, but not as much as different randomly phased wave trains. A simulation of a recent storm event was performed, with overtopping rates being slightly lower than those reported by local residents. A joint probability analysis showed that the predicted frequency of such an event was in line with these reports.An alternative modelling technique was also tested, where a spectral energy model was coupled with a nonlinear shallow water solver. Results for wave runup parameters were very accurate, when the coupling location is at the seaward edge of the surf zone. Extension of this modelling technique into two horizontal dimensions would be more straightforward than with the SWAB model.

No description available.

In this thesis we present theoretical and numerical results for a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and controlling energy efficient building systems. In particular, we show that it is possible to locally exponentially stabilize the nonlinear Boussinesq Equations by applying Neumann/Robin type boundary control on a bounded and connected domain. The feedback controller is obtained by solving a Linear Quadratic Regulator problem for the linearized Boussinesq equations. Applying classical results for semilinear equations where the linear term generates an analytic semigroup, we establish that this Riccati-based optimal boundary feedback control provides a local stabilizing controller for the full nonlinear Boussinesq equations. In addition, we present a finite element Galerkin approximation. Finally, we provide numerical results based on standard Taylor-Hood elements to illustrate the theory. / Ph. D.