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Problemas de otimização se fazem cada vez mais presentes nos mais diversos ramos da Engenharia. Encontrar configurações ótimas para um determinado problema significa, por exemplo, melhorar desempenho, reduzir custos entre outros ganhos. Existem hoje diversas maneiras de atacar um problema de otimização, cada qual com suas particularidades, vantagens e desvantagens. Dentre os métodos de otimização que utilizam gradientes de sensibilidade, o cálculo numérico dos mesmos consiste em uma importante etapa do projeto que, dependendo do problema, pode acarretar em custos computacionais muito elevados inviabilizando a abordagem escolhida. Este trabalho visa desenvolver e apresentar uma nova metodologia para o cálculo desses gradientes de sensibilidade, com base no Método Adjunto. O Método Adjunto é um método amplamente estudado e com diversas aplicações principalmente em Engenharia Aeronáutica. Nesse trabalho, todo o conhecimento prévio é utilizado para a derivação do método para aplicá-lo a escoamentos viscosos e incompressíveis. É desenvolvido também o cálculo do gradiente de sensibilidade com respeito a parâmetros geométricos e não geométricos. Para validar a metodologia proposta são feitas simulações numéricas das equações governantes do escoamento e adjuntas utilizando dois códigos computacionais distintos, SEMTEX e FreeFem++, o primeiro baseado no Método dos Elementos Espectrais e o segundo no Método dos Elementos Finitos, mostrando assim a independência do Método Adjunto na sua formulação contínua em relação a métodos computacionais. Para a validação são cujos gradientes possam ser calculados de outras formas permitindo comparações para calibrar e aperfeiçoar o cálculo do gradiente de sensibilidade. / Optimization problems are widely present in differents fields of Engineering. Finding optimal configurations in a problem means, for example, improving performance, reducing costs, among other achievements. There are several wellknown ways to tackle an optimization problem, each one has its own advantages and disadvantages. Considering the gradient-based optimization methods, the step of their numerical calculation is extremely important, as it may result in huge computational costs, thus making the chosen method impracticable. This work aims to develop and present a new methodology to compute these sensitivity gradients based on the Adjoint Method. The Adjoint Method is a widely studied method with several applications chiefly in A eronautical Engineering. In the present work, all the previous knowledge will be used to derive the equations of the method in order to apply them to viscous incompressible flows. The calculation of the sensitivity gradient, with respect to both geometric and non-geometric paramatersm will be developed as well. To validate the proposed methodology, numerical simulations of the governing and adjoint equations are carried out, using two computational codes called SEMTEX and FreeFem++, the former is based on the Spectral Element Method and the later, on the Finite Element Method, thus showing that the Adjoint Method, in its continuous formulation, is independent of the particular numerical method that is used. In order to validate the algorithm, simple problems are chosen, for which the gradients can be computed by other methods. This choice admits comparison between numerical values of gradients in order to calibrate and improve the methodology proposed.

In the thesis we study incompressible turbulent flow using spectral element method (SEM). We present fundamentals of SEM which can be seen as a combination of spectral method and finite element method. Turbulent flow is described with the help of two-equation k-ω turbulence models. We give reasons for the choice of Kolmogorov's and Wilcox model and implement them in 2D into an existing SEM C++ framework, Nektar++. Analytical solutions of Navier- Stokes equation were found and used as test cases for the implementation of the models. We simulated turbulent flow in channel and compared the results with direct numerical simulation. 1

Theoretical predictions of behaviour of various parameters governing the free flow of a single, very long, denser-than-liquid carrier, cylindrical capsule in a horizontal pipeline are reported in this dissertation. The study was carried out for average flow velocities of approximately 1 to 10 ft/sec in pipes of diameters 4, 6, 12 and 24 in. with diameter ratios varying from 0.9 to 0.99. While two liquid carriers -- water and oil (µ = 10 cP and sp. gr. = 0.85) were used, the eccentricity of the capsule-pipe system was fixed at 0.999. The theoretical solution was found to be in good agreement with the experimental results. / Thesis / Master of Engineering (ME)

Air velocities induced by underground vehicles in complex metro systems are obtained using artificial neural networks. Complex tunnel shaft-systems with any number of tunnels and shafts and with most of the practically possible geometries encountered in underground structures can be simulated with the proposed method. A single neural network, of type feed-forward back propagation, with a single hidden layer is trained for modelling a single tunnel segment. Train and tunnel parameters that have influence on the vehicle induced flow characteristics are used together to obtain non-dimensional input and target parameters. First input parameter is the major head loss coefficient of tunnel, (L/D)_Tunnel. Blockage ratio A_Train/A_Tunnel and train aspect ratio (D/L)_Train are selected to be non-dimensional input parameters to represent the system geometry. As the final input parameter, skin friction coefficient of the train, f_Train drag coefficient of the train, C_D / frontal area of the train, A_Train and lateral area of the train, A_Lateral are combined into a single overall drag coefficient based on the train frontal area. Non-dimensional V_Air/V_Train speed ratio is selected to be the target parameter. Using maximum air velocity predicted by the trained neural network together with non dimensional system parameters and time, an additional neural network is trained for predicting the deceleration of air in case of train stoppage within the tunnel system and departure of the train from the system. A simulation tool for predicting time dependent velocity profile of air in metro systems is developed with the trained neural networks.

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint.

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O objetivo desta dissertação de mestrado é estudar técnicas numéricas para simular escoamentos incompressíveis multifásicos e implementar uma ferramenta computacional utilizando malhas hierárquicas e discretizações por diferenças finitas. São apresentados a formulação matemática e o desenvolvimento do método numérico, levando em consideração o caráter multifásico do escoamento. Foi adotado o modelo de força superficial contínua e a representação da interface foi feita pelo método de acompanhamento de fronteira. São expostos todos os testes realizados durante o desenvolvimento da ferramenta para checar cada etapa do método. Finalmente, testes visando verificar o código foram feitos e os resultados obtidos foram considerados satisfatórios para a verificação da ferramenta aqui desenvolvida. / The objective of this masters degree essay is to study numerical techniques to simulate incompressible multiphase flows and to implement a computational tool using hierachical meshes and discretizations by finite diferences. We introduce the mathematical formulation and the development of the numerical method, for the multiphase flow problem. A continuum surface force model is employed with the interface representation by the front tracking method. We show all tests performed to verify each stage of the methods development. Finally, results obtained in classical benchmark flow tests show good agreement with previous published results, corroborating the validity of this newly developed numerical tool.

Foram feitas simulações de um jato plano livre e incompressível usando o modelo de uma equação Spalart-Allmaras padrão e um modelo Spalart-Allmaras modificado através da alteração da escala de comprimento turbulenta. Sabe-se da literatura que no caso de jatos livres o modelo Spalart-Allmaras não consegue predizer adequadamente os resultados observados experimentalmente. Os resultados das simulações foram comparados com experimentos da literatura através de perfis de velocidade e da taxa de expansão do jato. Como esperado, os resultados obtidos das simulações utilizando o modelo Spalart-Allmaras padrão foram considerados insatisfatórios, porém o modelo Spalart-Allmaras modificado teve uma melhor concordância com os resultados experimentais. Além disso, o modelo Spalart-Allmaras modificado foi usado para simular os casos do escoamento sobre uma placa plana sem gradiente de pressão e o escoamento em um degrau com separação e gradiente adverso de pressão. Quando comparados com resultados experimentais da literatura e com resultados obtidos usando o modelo padrão, os resultados do modelo modificado obtidos para ambos os casos foram muito satisfatórios, concluindo-se que a modificação da escala de comprimento permite obter uma maior generalidade para o modelo Spalart-Allmaras. / Simulations of a plane and incompressible free jet using the standard Spalart-Allmaras model and a Spalart-Allmaras model modified by changing the turbulent length scale were carried out. It is known from literature that, in the case of the free jet, the Spalart-Allmaras model fails to adequately predict the experimentally observed results. The results of our simulations were compared with published experiments using the velocity profiles and the jet spreading rate. As expected, the results of simulations using the standard Spalart-Allmaras model were considered unsatisfactory while the modified Spalart-Allmaras model had a better agreement with the experimental results. Furthermore, the modified Spalart-Allmaras model was used to simulate the cases of flow over a flat plate with no pressure gradient and flow through a backward facing step with separation and adverse pressure gradient. When compared with experimental results from the literature and with results obtained using the standard model, the results of the modified model for both cases were very satisfactory, allowing the conclusion that the change in the length scale provided a greater generality for the Spalart-Allmaras model.