Yes / A novel Euler-Lagrangian (EL) computational uid dynamics (CFD) nite volume-based model to simulate the gas mixing of sludge for anaerobic digestion is developed and described. Fluid motion is driven by momentum transfer from bubbles to liquid. Model validation is undertaken by assessing the ow eld in a labscale model with particle image velocimetry (PIV). Conclusions are drawn about the upscaling and applicability of the model to full-scale problems, and recommendations are given for optimum application.

Yes / For the first time, an Euler-Lagrange model for Computational Fluid Dynamics (CFD) is used to model a full-scale gas-mixed anaerobic digester. The design and operation parameters of a digester from a wastewater treatment works are modelled, and mixing is assessed through a novel, multi-facetted approach consisting of the simultaneous analysis of (i) velocity, shear rate and viscosity flow patterns, (ii) domain characterization following the average shear rate value, and (iii) concentration of a non-diffusive scalar tracer. The influence of sludge’s non-Newtonian behaviour on flow patterns and its consequential impact on mixing quality were discussed for the first time. Recommendations to enhance mixing effectiveness are given: (i) a lower gas mixing input power can be used in the digester modelled within this work without a significant change in mixing quality, and (ii) biogas injection should be periodically switched between different nozzle series placed at different distances from the centre. / The first author is funded via a University of Birmingham Postgraduate Teaching Assistantship award.

Yes / For the first time, a set of Lattice-Boltzmann two-way coupling pointwise Euler-Lagrange models is applied to gas mixing of sludge for anaerobic digestion. The set comprises a local model, a “first-neighbour” (viz., back-coupling occurs to the voxel where a particle sits, plus its first neighbours) and a “smoothing-kernel” (forward- and back-coupling occur through a smoothed-kernel averaging procedure). Laboratory-scale tests display grid-independence problems due to bubble diameter being larger than voxel size, thereby breaking the pointwise Euler-Lagrange assumption of negligible particle size. To tackle this problem and thereby have grid-independent results, a novel data-scaling approach to pointwise Euler-Lagrange grid independence evaluation, based on an application of the Buckingham π theorem, is proposed. Evaluation of laboratory-scale flow patterns and comparison to experimental data show only marginal differences in between the models, and between numerical modelling and experimental data. Pilot-scale simulations show that all the models produce grid-independent, coherent data if the Euler-Lagrange assumption of negligible (or at least, small) particle size is recovered. In both cases, a second-order convergence was achieved. A discussion follows on the opportunity of applying the proposed data-scaling approach rather than the smoothing-kernel model.

A numerical procedure has been developed for the computation of inviscid flows over arbitrary, complex two-dimensional geometries. The Euler equations are solved using a finite-volume method with a non-body-fitted Cartesian grid. A new numerical formulation for complicated body geometries is developed in conjunction with implicit flux-splitting schemes. A variety of numerical computations have been performed to validate the numerical methodologies developed. Computations for supersonic flow over a flat plate with an impinging shock wave are used to verify the numerical algorithm, without geometric considerations. The supersonic flow over a blunt body is utilized to show the accuracy of the non-body-fitted Cartesian grid, along with the shock resolution of flux-vector splitting scheme. Geometric complexities are illustrated with the flow through a two-dimensional supersonic inlet with and without an open bleed door. The ability of the method to deal with subsonic and transonic flows is illustrated by computations over a non-lifting NACA 0012 airfoil. The method is shown to be accurate, efficient and robust and should prove to be particularly useful in a preliminary design mode, where flows past a wide variety of complex geometries can be computed without complicated grid generation procedures. / Ph. D.

La thèse porte sur l'étude de la réduction de traînée par injection de bulles. La réduction de traînée présente un intérêt pour les applications navales puisqu'elle est issue d'une modification des structures cohérentes qui contribuent le plus à la résistance à l'avancement. Le but de cette étude est d'analyser les mécanismes à l'origine de cette diminution du frottement pariétal. L'approche utilisée dans le cadre de cette étude est numérique, elle emploie le code JADIM par une approche Euler-Lagrange : la phase continue est simulée par Simulation Numérique Directe et la phase dispersée est simulée en suivant individuellement chaque bulle. La configuration retenue dans le cadre de cette étude est celle de l'écoulement de Taylor-Couette (écoulement compris entre deux cylindres en rotation). La première partie de la thèse vise à adapter l'outil numérique employé, afin de pouvoir prendre en compte le retour de la phase dispersée via des termes de forçage dans les équations bilan de matière et de quantité de mouvement. La deuxième partie de la thèse vise à étudier l'écoulement porteur en configuration monophasique, afin de disposer d'une référence sur l'écoulement non perturbé. La troisième partie de la thèse a pour objectif d'étudier la dispersion passive des bulles dans le système, afin d'analyser les mécanismes de migrations mis en jeu. Enfin la dernière partie de la thèse vise à étudier l'influence des bulles sur l'écoulement porteur en analysant l'effet de certains paramètres, notamment le taux de vide et la flottabilité. / The study deals with drag reduction induced by bubble injection, its application concerns naval transport. The aim of the study is to shed more light on mechanisms that are involved in this wall friction reduction. The study is based on a numerical approach, and use the JADIM code with an Euler-Lagrange approach: the continuous phase is solved by Direct Numerical Simulation, and the disperse phase by a tracking of each bubble. Within the framework of this study we consider the Taylor-Couette flow configuration (flow between two concentric cylinders in rotation). The first part of the study deals with the modification of the numerical tool, in order to take into account the influence of the disperse phase on the continuous one through forcing terms in the mass and momentum balance equations. The aim of the second part is to study de Taylor-Couette flow in its monophasic configuration, for the purpose of providing a reference of the undisturbed flow. The third part deals with the passive dispersion of bubble in Taylor-Couette flow, in order to analyze the migration mechanisms involved. And the aim of the last part is to study the effects of the disperse phase on the continuous one, by analyzing the influence of bubbly phase parameters (like void fraction and buoyancy).

Desde o importante trabalho de Barth e Frederickson (1990), um certo número de pesquisadores têm estudado o método de Volumes Finitos de alta-ordem k-exato, por exemplo o grupo do Prof. Ollivier-Gooch: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Outras discretizações espaciais de alta-ordem bastante populares são o método Galerkin Descontínuo e o método de Diferença Espectral; processos iterativos que involucram estes esquemas tem sido acelerados, nos últimos anos, por métodos p-multigrid. Porém, esta aceleração não tem sido aplicada no contexto do método de Volumes Finitos de alta-ordem, pelo menos para conhecimento do autor desta tese. Por isso, o objetivo desta pesquisa é adaptar o p-multigrid desenvolvido por Liang et al. (2009b) no contexto da Diferença Espectral, para o ambiente dos Volumes Finitos estudado pelo Prof. Ollivier-Gooch. A pesquisa começa implementando o solver VF-RK, de Volumes Finitos com avanço Runge-Kutta, para resolver as equações de advecção-difusão e de Euler aplicados a problemas estacionários, por exemplo, o escoamento transônico ao redor do NACA 0012. Depois, estuda-se o método p-multigrid no contexto da Diferença Espectral; o p-multigrid acelera o processo iterativo comutando níveis polinomiais de alta e de baixa-ordem. Após esse estudo, a adaptação ao âmbito dos Volumes Finitos é realizada resultando num p-multigrid relativamente mais simples porque, em contraposição com o p-multigrid para Diferença Espectral, não precisa de operadores de restrição e prolongação para a comunicação entre diferentes níveis polinomiais. A pesquisa conclui com uma comparação com o método de Volumes Finitos de 4a ordem sem p-multigrid (solver VF-RK). Nesse sentido, implementa-se o solver pMG, baseado no p-multigrid proposto, para resolver os problemas estacionários considerados na primeira parte do trabalho; o smoother do p-multigrid é o esquema Runge-Kutta do código VF-RK, e cada problema estacionário é resolvido utilizando diferentes Vciclos procurando sempre soluções de 4a ordem. Os resultados indicam que o método p-multigrid proposto é mais eficiente que o método de Volumes Finitos de 4a ordem sem p-multigrid, isto é, os dois métodos oferecem a mesma precisão mas o primeiro pode levar menos de 50% do tempo de CPU do segundo. / Since Barth and Frederickson\'s important work (Barth e Frederickson, 1990), a number of researchers have studied high-order k-exact Finite Volume method, for example Prof. Ollivier-Gooch\'s group: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Other quite popular high-order spatial discretizations are the Discontinuous Galerkin methods and the Spectral Difference methods; the iterative processes involving these schemes have been accelerated in recent years by p-multigrid methods. However, this acceleration has not been applied in the context of the high-order Finite Volume method, at least for the knowledge of the author of this thesis. Therefore, the objective of this research is to adapt the p-multigrid developed by Liang et al. (2009b) in the context of Spectral Difference methods, to the environment of Finite Volume studied by Prof. Ollivier-Gooch. This research begins by implementing the solver VF-RK, Finite Volume solver with Runge-Kutta advance, to compute the advection-diffusion equation and Euler equations applied to steady state problems, for example, the transonic flow around NACA 0012. Then, it is studied the p-multigrid method in the context of Spectral Difference schemes; p-multigrid accelerates the iterative process by switching polynomial levels of high- and low-order. After this study, the adaptation to the context of the Finite Volume scheme is performed resulting in a relatively simple p-multigrid because, in contrast to the p-multigrid for Spectral Difference schemes, it doesn\'t need restriction and prolongation operators for communication between different polynomial levels. The research concludes with a comparison with 4th order Finite Volume method without p-multigrid (solver VF-RK). Accordingly, the solver pMG, based on the proposed p-multigrid, is implemented to resolve the steady state problems considered in the first part of the work; the p-multigrid smoother is the Runge-Kutta scheme from VF-RK code, and each steady state problem is solved using different Vcycles, looking for 4th order solutions ever. The results indicate that the proposed p-multigrid method is more efficient than the 4th order Finite Volume method without p-multigrid: the two methods give the same accuracy but the first one can take less than 50% of second one\'s CPU time.

O presente trabalho tem por objetivo o estudo da otimização multiobjetivo aplicada ao projeto de perfis aerodinâmicos em regime transônico, analisando comparativamente diferentes formas de definir as funções objetivo. A otimização é efetuada pelo algoritmo genético NSGA-II. Os resultados são avaliados utilizando métricas de diversidade da população e otimalidade das soluções, das quais duas são propostas. As funções objetivo são constituidas de diferentes parametrizações da geometria e diferentes técnicas de simulação numérica. A parametrização da geometria é feita utilizando a paramentrização Parsec ou a parametrização baseada em pontos de controle. A discretização do domínio espacial é feita utilizando malha estruturada conformada ao perfil e suavização por EDP elíptica. As duas técnicas de volumes finitos com diferentes modelos para o cálculo do fluxo na face do volume implementadas foram o método de Jameson (esquema centrado) e o método de Roe (esquema upwind). As comparações feitas são as seguintes: utilização de modelo viscoso e invíscido, com o uso do código Mses com a parametrização por ponto de controle; a utilização da parametrização por ponto de controle e parametrização Parsec usando o método de Jameson; e a comparação entre o método centrado e o upwind, utilizando a parametrização Parsec. Conclui-se dos resultados obtidos que a utilização da parametrização por pontos de controle é melhor. Entretanto, ainda é necessária a utilização de uma parametrização que garanta maior suavidade ou a imposição de restrições sobre a suavidade da solução. A utilização do modelo viscoso torna os resultados da otimização melhores do ponto de vista da otimalidade. Na utilização de modelos de correção viscosa, como no caso do Mses, é necessária a utilização de métodos invíscidos que forneçam resultados com maior representatividade física / The objective of present study is analyze the multi-objective optimization applied to transonic airfoils project comparing different ways to define the objective functions. The optimization is evaluated by the genetic algorithm NSGA-II. The results is analyzed using metrics of diversity and optimality for multi-objective problems, which two are proposed. The objective functions are defined by different parametrizations of geometry and different techniques of numerical simulation. The geometry parametrization was made by two distinct forms: using Parsec parametrization; and the control points based parametrization. The space domain discretization was made using structured body-fitted mesh with elliptical PDE smooth. A finite volume code with two different techniques for calculations of flux interface had been implemented: the Jamesons method (centered); and the Roes method ( it upwind). For viscous model usage analysis was used the Mses code that has implemented a finite volumes technique with viscous model correction. The following comparisons has been made: viscous and inviscid model using the Mses code with the control points parametrization; the control points and Parsec parametrizations using the Jamesons method; and the comparison among the centered method and upwind using the parametrization Parsec. From the results, it is concluded that the used of control points parametrization is interesting. Although, is still needed the used of a parametrization that guarantees a better smoothness or the imposes of a geometrical or property distribution restriction. The uses of viscous model gives better optimizations results in optimality requirement. It is needed the uses of inviscid method that forces better physical representation when using viscous correction model

Made available in DSpace on 2017-07-21T20:56:28Z (GMT). No. of bitstreams: 1 Jose Robyson Aggio Molinari.pdf: 837254 bytes, checksum: a577b2742ab4df347179d61529e63767 (MD5) Previous issue date: 2016-02-03 / This study presents some of the encryption methods used in antiquity as well as the advance in the way of encrypting. The main objective of this work is the study of RSA Method: its Historical context, the importance of prime numbers, the inefficiency of factorization algorithms, coding, decoding, its security and a study of the Euler function. Some activities with mathematical content related to encryption have been developed. Thus, it is expected that this research can present an auxiliary methodology for teaching certain math content, linked to the utilization of cryptography. / Este trabalho apresenta alguns métodos de criptografia utilizados na antiguidade e também o avanço na maneira de criptografar. O objetivo principal é o estudo do Método RSA: contextualização histórica, a importância dos números primos, a ineficiência dos algoritmos de fatoração, codificação, decodificação, a segurança e um estudo sobre a função de Euler. Desenvolveu-se algumas atividades com conteúdos matemáticos relacionadas à criptografia. Desta maneira, espera-se que esta pesquisa possa apresentar uma metodologia auxiliar para o ensino de certos conteúdos da matemática, articulados com a utilização da criptografia.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-08-29T20:47:14Z No. of bitstreams: 2 Poliedros_E_Teorema_de_Euler.pdf: 4810573 bytes, checksum: f1f57ad45cfd7dc575fe3c3e963b24c6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-08-29T20:47:14Z (GMT). No. of bitstreams: 2 Poliedros_E_Teorema_de_Euler.pdf: 4810573 bytes, checksum: f1f57ad45cfd7dc575fe3c3e963b24c6 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2014-03-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work aims is to demonstrate the Euler's Theorem for polyhedra, given by the equation V 􀀀 A + F = 2, where V; A and F are the numbers of vertices, edges and faces, respectively, the polyhedron. A historical survey of the main characters who contributed to the theme was elaborated. De nitions and properties of polygons and polyhedra were given. The statements were constructed in three distinct ways. The rst by Cauchy, commented by Professor Elon Lages Lima. This statement is valid for any polyhedron homeomorphic to a sphere and has the path planning of the polyhedron withdrawing one of its faces. The second statement was prepared by the professor Zoroastro Azambuja Filho, valid for any convex polyhedron, and its path projection of the polyhedron on a plane and comparison of the internal angles of polygons with projection angles of the polygon faces. The third statements was presented by Legendre, also valid for any convex polyhedron, and its path in the projection of a spherical polyhedron surface. We use the Girard's Formula, the sum of the interior angles of a spherical triangle, to complete the demonstration. This work also suggests methods of applying the proof of Euler's Theorem in the classroom for high school students, and resolution of vestibular exercises involving the subject. / Este trabalho tem por objetivo a demonstra c~ao do Teorema de Euler para poliedros, dado pela equa ção V 􀀀 A + F = 2, onde V; A e F são os n úmeros de v értices, arestas e faces, respectivamente, do poliedro. Foi elaborada uma pesquisa hist orica dos principais personagens que contribuiram para o tema. Foram dadas de ni ções e propriedades de pol ígonos e poliedros. As demonstra ções foram constru ídas em três caminhos distintos. A primeira por Cauchy, comentada pelo professor Elon Lages Lima. Esta demonstra ção é v álida para qualquer poliedro homeomorfo a uma esfera e tem como caminho a plani fica ção do poliedro retirando-se uma de suas faces. A segunda demonstra c~ao foi elaborada pelo professor Zoroastro Azambuja Filho, v álida para qualquer poliedro convexo e tem como caminho a proje ção do poliedro num plano e a compara c~ao dos ângulos internos dos pol ígonos da proje ção com os ângulos dos pol gonos das faces. A terceira demonstra c~ao foi apresentada por Legendre, tamb ém v álida para qualquer poliedro convexo e tem como caminho a projeção do poliedro em uma superf ície esf érica. Utiliza-se a F ormula de Girard, da soma dos ângulos internos de um tri^angulo esf érico, para concluir a demonstra ção. Este trabalho tamb ém sugere metodologias de aplica ção da demonstração do Teorema de Euler em sala de aula, para alunos do Ensino M édio, e resolu ção de exercí cios de vestibulares envolvendo o tema.

Neste trabalho, apresentamos um método de simulação Monte Carlo para o cálculo do hedging dinâmico de opções do tipo europeia em mercados multidimensionais do tipo Browniano e livres de arbitragem. Baseado em aproximações martingales de variação limitada para as decomposições de Galtchouk-Kunita-Watanabe, propomos uma metodologia factível e construtiva que nos permite calcular estratégias de hedging puras com respeito a qualquer opção quadrado integrável em mercados completos e incompletos. Uma vantagem da abordagem apresentada aqui é a flexibilidade de aplicação do método para os critérios quadráticos de minimização do risco local e de variância média de forma geral, sem a necessidade de se considerar hipóteses de suavidade para a função payoff. Em particular, a metodologia pode ser aplicada para calcular estratégias de hedging quadráticas multidimensionais para opções que dependem de toda a trajetória dos ativos subjacentes em modelos de volatilidade estocástica e com funções payoff descontínuas. Ilustramos nossa metodologia, fornecendo exemplos numéricos dos cálculos das estratégias de hedging para opções vanilla e opções exóticas que dependem de toda a trajetória dos ativos subjacentes escritas sobre modelos de volatilidade local e modelos de volatilidade estocástica. Ressaltamos que as simulações são baseadas em aproximações para os processos de preços descontados e, para estas aproximações, utilizamos o método numérico de Euler-Maruyama aplicado em uma discretização aleatória simples. Além disso, fornecemos alguns resultados teóricos acerca da convergência desta aproximação para modelos simples em que podemos considerar a condição de Lipschitz e para o modelo de volatilidade estocástica de Heston. / In this work, we present a Monte Carlo simulation method to compute de dynamic hedging of european-type contingent claims in a multidimensional Brownian-type and arbitrage-free market. Based on bounded variation martingale approximations for the Galtchouk-Kunita- Watanabe decomposition, we propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to any square-integrable contingent claim in complete and incomplete markets. An advantage of our approach is the exibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff function. In particular, the methodology can be applied to compute multidimensional quadratic hedgingtype strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. We illustrate our methodology, providing some numerical examples of the hedging strategies to vanilla and exotic contingent claims written on local volatility and stochastic volatility models. The simulations are based in approximations to the discounted price processes and, for these approximations, we use an Euler-Maruyama-type method applied to a simple random discretization. We also provide some theoretical results about the convergence of this approximation in simple models where the Lipschitz condition is satisfied and the Heston\'s stochastic volatility model.