Resumo: A compreensão do escoamento em válvulas de compressores herméticos alternativos é de fundamental importância para introduzir modificações no projeto delas de maneira a aumentar a performance dos compressores, e por fim, dos ciclos de refrigeração. A válvula do compressor é um dispositivo ímpar, umas vez que seu funcionamento se dá pela ação da pressão exercida pelo escoamento, caracterizando um problema de forte interação fluido- estrutura. O uso da modelagem numérica através das ferramentas da mecânica dos fluidos computacional (CFD) tem se destacado como a alternativa mais dinâmica para o estudo do fenômeno. O trabalho desenvolvido foi voltado para o estudo numérico do escoamento através do difusor radial, o qual é um modelo simplificado da válvula, com o emprego do Método da Fronteira Imersa com Modelo Físico Virtual para a modelagem do disco superior do difusor (palheta). O ponto forte desta metodologia é que a representação de regiões sólidas é feita pelo cálculo de um campo de força, o qual é introduzido nas equações das células na vizinhança do sólido. Este procedimento dispensa o uso de malhas que se adaptam ao corpo, possibilitando o uso de malhas cartesianas convencionas para modelar geometrias complexas e móveis. A metodologia foi acoplada com a solução das equações governantes do escoamento em coordenadas cilíndricas através do Método dos Volumes Finitos. Inicialmente, a metodologia foi validada, utilizando como dados de referência resultados provenientes de estudos numéricos e experimentais, e foi avaliada a influência dos parâmetros do procedimento na qualidade final dos resultados. Na segunda etapa foram desenvolvidos estudos preliminares referentes ao movimento do disco superior, com a imposição artificial dos processos de abertura e fechamento da válvula. Os resultados obtidos mostraram que a metodologia adotada ... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The fully understanding of the flow through automatic valves of alternative hermetic compressors is essentiall to introduce modifications in its project aiming the improvement of the compressor performance and, also, the refrigeration cycle. The compressor valve is a singular device, once its operation is ruled by the flow pressure, characterizing a case with a strong fluid-structure interaction. The using of numerical tools trough the methods of computational fluid dynamics (CFD) has gained especial attention due to its flexibility to study the phenomenon. The present work was developed to study numerically the flow through the radial diffuser, which is a simplified model of the valve, with the employment of the Immersed Boundary Method with Virtual Physical Model to modeling the superior disk (valve reed). The main advantage of this methodology is that the modeling of solid boundaries is performed with the calculus of a force field, which is introduced in the cells equations nearby the solid. This procedure dispenses the using of body-fitted meshes, enabling the adoption of conventional Cartesian meshes to model complex and moving geometries. The methodology was coupled with the solution of the governing equations in cylindrical coordinates though the Finite Volume Method. Firstly, the methodology was validated, confronting the results obtained with data from numerical and experimental studies, where the influence of the main parameters in the quality of the final results was evaluated. In the second step were developed preliminary studies concerning the movement of the superior disk, whose opening and closing movements were artificially imposed. The results obtained showed that the adopted methodology is quit promising and flexible, and can be employed in more refined studies to the comprehension of the flow through the valve regarding the fluid-structure interaction that rules the problem / Orientador: José Luiz Gasche / Coorientador: Júlio Militzer / Banca: Cassio Roberto Macedo Maia / Banca: Elie Luis Martinez Padilla / Mestre

The thesis deals with the construction of an adaptive 1D and 2D mesh in the framework of the cell- centered finite volume scheme. The adaptive strategy is applied to the numerical solution of problems governed by the Euler equations, which is a hyperbolic system of PDE's. The used algorithm is applicable to nonstationary problems and consists of three independent parts, which are cyclically repeated. These steps are PDE evolution, then mesh adaptation and recovery of numerical solution from the old mesh to the newly adapted mesh. Owing to this the algorithm can be used also for other hyperbolic systems. The thesis is focused on the development of our mesh adaptation strategy, based on the anisotropic mesh adaptation, which preserves the geometric mass conservation law in each computational step. The proposed method is suitable to solve problems with moving discontinuities. Several test problems with moving discontinuity are computed to compare our algorithm with Moving Mesh algorithms.

This research extends the capabilities of Non-oscillatory Forward-in-Time (NFT) solvers operating on unstructured meshes to allow for accurate simulation of incompressible turbulent flows. This is achieved by the development of Large Eddy Simulation (LES) and Detached Eddy Simulation (DES) turbulent flow methodologies and the development of parallel option of the flow solver. The effective use of LES and DES requires a development of a subgrid-scale model. Several subgrid-scale models are implemented and studied, and their efficacy is assessed. The NFT solvers employed in this work are based on the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) that facilitates novel implicit Large Eddy Simulation (ILES) approach to treating turbulence. The flexibility and robustness of the new NFT MPDATA solver are studied and successfully validated using well established benchmarks and concentrate on a flow past a sphere. The flow statistics from the solutions are compared against the existing experimental and numerical data and fully confirm the validity of the approach. The parallel implementation of the flow solver is also documented and verified showing a substantial speedup of computations. The proposed method lays foundations for further studies and developments, especially for exploring the potential of MPDATA in the context of ILES and associated treatments of boundary conditions at solid boundaries.

Les travaux de cette thèse, réalisés au sein de l’équipe CLEF/DMFN de l’ONERA (Office National d’ Etudes et de Recherches Aérospatiales) en partenariat avec le laboratoire DynFluid et le CIRT (Consortium Industrie-Recherche en Turbomachines) s’inscrivent dans une demarche d’amélioration des outils de simulations pour les turbomachines. Compte tenu de ce contexte, l’objectif de cette étude est de développer de nouvelles méthodes pour le traitement des raccords non coincidents dans les turbomachines qui soit à la fois d’ordre elevé et conservatifs. Les développements proposés sont validés et composés de configurations de difficulté croissante. / A high-order and conservative method is developed for the numerical treatment of interface conditions in patched grids, based on the use of a ctitious grid methodology. The proposed approach is compared with a non-conservative interpolation of the state variables from the neighbouring domain for selected internal fow problems.

Un modèle numérique décrivant le comportement de fibres souples en suspension dans un écoulement de fluide en régime inertiel a été développé au moyen d'un couplage entre la méthode des éléments discrets et la méthode des volumes finis. Chaque fibre est discrétisée en plusieurs éléments de type poutre permettant de prendre en compte une déformation (flexion, torsion, allongement) et un mouvement de corps rigide. Les équations du mouvement des fibres sont résolues au moyen d'un schéma explicite du second ordre (temps et espace). Le mouvement de la phase fluide est décrit par les équations de Navier-Stokes, qui sont discrétisées et résolues au moyen d'un schéma aux volumes finis non structurés, d'ordre 4 (temps et espace). Le couplage entre la phase solide (discrète) et la phase fluide (continue) est obtenue par une pseudo méthode IBM (Immersed Boundary Method) dans laquelle l'effort hydrodynamique est calculé analytiquement. Plusieurs modèles de force hydrodynamique issus de la littérature sont analysés et leur validité ainsi que leurs limites sont identifiées. Pour des nombres de Reynolds (Re) correspondant au régime inertiel (0.01 < Re < 100, Re défini à l'échelle de la fibre), des formulations non-linéaires de la force hydrodynamique exercée par un écoulement uniforme sur un cylindre infini sont utilisées. Le couplage a aussi été utilisé pour des fibres rigides en écoulement de Stokes, en utilisant l'expression de la force de traînée issue de la théorie des corps élancés (`slender body theory'). Une expression du moment hydrodynamique par unité de longueur est obtenu à partir de simulations numériques par volumes finis de l'écoulement autour d'un cylindre élancé.Le modèle développé a été validé par comparaison avec plusieurs résultats expérimentaux et analytiques, du régime de Stokes (pour des fibres rigides) jusqu'aux régimes inertiels. Dans le cas du régime de Stokes, des simulations numériques du cisaillement de suspensions de fibres semi-diluées ont été réalisées. Le modèle développé permet de capturer les interactions hydrodynamiques et non-hydrodynamiques entre les fibres. Les interactions élasto-hydrodynamiques pour $Re$ fini ont été validées dans deux cas. Dans le premier cas, la flèche d'une fibre encastrée-libre dans un écoulement uniforme a été obtenu par calcul numérique et le résultat validé par comparaison aux résultats expérimentaux de la littérature. Dans le second cas, la conformation de fibres élancées et très déformables dans un écoulement turbulent homogène et isotrope a été obtenu par calcul numérique et le résultat validé par comparaison aux résultats expérimentaux de la littérature. Deux études numériques ont été réalisées pour étudier l'effet de la présence de fibres en suspension sur la turbulence au sein du fluide suspensif. Le modèle numérique a permis de reproduire le phénomène de réduction/amplification de la turbulence dans un écoulement en canal ou en conduite, dû à l'évolution microstructurale de la phase fibreuse. / A numerical model describing the behavior of flexible fibers under inertial flows was developed by coupling a discrete element solver with a finite volume solver.Each fiber is discretized into several beam segments, such that the fiber can bend, twist and rotate. The equations of the fiber motion were solved usinga second order accurate explicit scheme (space and time). The three dimensional Navier-Stokes equations describing the motion of the fluid phase was discretizedusing a fourth th order accurate (space and time) unstructured finite volume scheme. The coupling between the discrete fiber phase and the continuous fluid phasewas obtained by a pseudo immersed boundary method as the hydrodynamic force on the fiber segments were calculated based on analytical expressions.Several hydrodynamic force models were analyzed and their validity and short-comings were identified. For Reynolds numbers (Re) at the inertial regime(0.01 < Re < 100, Re defined at the fiber scale), non linear drag force formulations based on the flow past an infinite cylinder was used. For rigid fibers in creeping flow, the drag force formulation from the slender body theory was used. A per unit length hydrodynamic torque model for the fibers was derived from explicit numerical simulations of shear flow past a high aspect ratio cylinder. The developed model was validated against several experimental studies and analytical theories ranging from the creeping flow regime (for rigid fibers) to inertial regimes. In the creeping flow regime, numerical simulations of semi dilute rigid fiber suspensions in shear were performed.The developed model wasable to capture the fiber-fiber hydrodynamic and non-hydrodynamic interactions. The elasto-hydrodynamic interactions at finite Reynolds was validated with against two test cases. In the first test case, the deflection of the free end of a fiber in an uniform flow field was obtained numerically and the results were validated. In the second test case the conformation of long flexible fibers in homogeneous isotropic turbulence was obtained numerically and the results were compared with previous experiments. Two numerical studies were performed to verify the effects of the suspended fibers on carrier phase turbulence and the numerical model was able to reproduce the damping/enhancement phenomena of turbulence in channel and pipe flows as a consequence of the micro-structural evolution of the fibers.

This work presents a model based on the known Parametric Finite-Volume Theory to analyze the geometrically nonlinear behavior of two-dimensional elastic structures made of heterogeneous materials. The formulation employed by the model is purely incremental and based on a kinematic Total Lagrangian description. The Second Piola-Kirchhoff tensor and Green-Lagrange tensor are used as stress and strain measurements, respectively. Based on this formulation a computer code is developed in C++ language. To verify the efficiency of the model, examples of structures with large displacements and involving heterogeneous materials, as well as homogeneous materials, were analyzed. The results for these examples are shown and compared with others published in the literature. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho apresenta-se o desenvolvimento de uma nova formulação baseada na teoria paramétrica de volumes finitos, para a análise do comportamento geometricamente não linear de sólidos elásticos bidimensionais heterogêneos. A formulação é do tipo incremental e fundamentada em uma descrição cinemática Lagrangeana Total. Como medidas de tensão e de deformação são empregados o segundo tensor tensão de Piola-Kirchhoff e o tensor de Green-Lagrange, respectivamente. Com base na referida formulação, desenvolve-se um código computacional em linguagem de programação C++. Para verificar e validar o modelo proposto analisam-se diferentes problemas submetidos a grandes deslocamentos envolvendo materiais homogêneos e heterogêneos. Comparam-se os resultados obtidos com outros encontrados na literatura científica.

In the present work, the Parametric Finite-Volume Theory is expanded by using the inclusion of homogenization processes on mechanical and thermal properties of periodically elastic and viscoelastic composites, with interphase effects. This work introduces models for evaluation of thermal conductivity, coefficient of thermal expansion, elastic and viscoelastic constitutive tensors with unit cell concept applied and periodical boundary conditions in state of generalized plane strain. The current interphase model between inclusion and matrix are considered thin and replaced by equivalent imperfect interfaces. The materials of the interphase are admitted to have lower thermal and mechanical properties. The formulation of the thermal and mechanic homogenization models is based on the traditional strategy of temperature and displacement field expansion in macroscopic and fluctuating components. The effective coefficients of thermal expansion are analytically evaluated using the effective elastic properties and coefficient of thermal expansion of the constitutive phases. Procedures on viscoelastic homogenization are based on Generalized Maxwell model, which allows the assessment of the effective relaxation tensors, providing the effective creep tensors using the Correspondence Principle. The viscoelastic model applied is incremental in time and uses the internal variables formulation. The created models are applied on the solution of many case studies, involving different kinds of composite materials (including size-dependent effects), volume fraction, unit cell geometry and interphase properties influence on the behavior of effective mechanical and thermal properties. The results are compared to analytical and FEM solutions to calculate the method’s efficiency. / No presente trabalho, a versão paramétrica da teoria dos volumes finitos (TVF) é expandida pela incorporação de processos de homogeneização de propriedades térmicas e mecânicas de compósitos periódicos elásticos e viscoelásticos, incluindo efeitos de interfase. O trabalho apresenta modelos para avaliação de condutividade térmica, coeficiente de dilatação térmica, tensores constitutivos elástico e viscoelástico efetivos, considerando o conceito de célula unitária, com condição de contorno periódica em estado plano de deformações generalizado. As interfases existentes entre as inclusões e a matriz são consideradas finas e substituídas por interfaces imperfeitas equivalentes. Os materiais que constituem tais interfases são admitidos com propriedades térmicas e mecânicas inferiores àquelas dos outros constituintes. Os modelos de homogeneização térmica e mecânica são formulados com base na tradicional estratégia de expansão dos campos de temperatura e deslocamentos, respectivamente, em parcelas macroscópica e flutuante. Os coeficientes de dilatação térmica efetivos são avaliados analiticamente em função das propriedades efetivas elásticas e dos coeficientes de dilatação térmica das fases constituintes. Os procedimentos de homogeneização viscoelástico tem como base o modelo de Maxwell generalizado e permitem a determinação dos tensores de relaxação efetivos, a partir dos quais são obtidos os tensores de fluência efetivos através do Princípio da Correspondência. O modelo viscoelástico utilizado é do tipo incremental e utiliza o conceito de variáveis internas. Por fim, os modelos são aplicados na solução de exemplos envolvendo diferentes tipos de materiais compósitos, os quais incluem investigação de efeitos de tamanho, fração volumétrica, geometria da célula unitária, influência das propriedades da interfase sobre as propriedades efetivas térmicas e mecânicas. Os resultados de tais exemplos são comparados com soluções analíticas e com o método dos elementos finitos, demonstrando desempenho dos modelos apresentados.

We suppose that the price of a firm follows a nonlinear stochastic delay differential equation. We also assume that any claim whose value depends on firm value and time follows a nonlinear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a random partial differential equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE for the debt value and loan guarantees within a single period and homogeneous class of debt. We then analyze the risk structure of a levered firm. We also evaluate loan guarantees in the presence of more than one debt. Furthermore, we perform numerical simulations for specific companies and compare our results with existing models.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Uma análise utilizando a série de Taylor é apresentada para se estimar a priori os erros envolvidos na solução numérica da equação de advecção unidimensional com termo fonte, através do Método dos Volumes Finitos em uma malha do tipo uniforme e uma malha não uniforme. Também faz-se um estudo a posteriori para verificar a magnitude do erro de discretização e corroborar os resultados obtidos através da análise a priori. Por meio da técnica de solução manufaturada tem-se uma solução analítica para o problema, a qual facilita a análise dos resultados numéricos encontrados, e estuda-se ainda a influência das funções de interpolação UDS e CDS e do parâmetro u na solução numérica. / An analysis based on Taylor series is presented for estimating a priori the errors involved in the numerical solution of advection equation one-dimensional with source term, using the Finite Volume Method in a mesh uniform and a nonuniform mesh. Also is accomplished a study to determine the magnitude of discretization error and corroborate the results obtained on analyzing a priori. By using the technique of solution manufactured is produced an analytical solution for the problem, which facilitates analysis of the numeric results, and was also studied the influence functions of interpolation UDS and CDS and of parameter u in the numerical solution.

Control of the transition of laminar flow to turbulence would result in lower drag and reduced energy consumption in many engineering applications. A spectral state-space model of linearised plane Poiseuille flow with wall transpiration ac¬tuation and wall shear measurements is developed from the Navier-Stokes and continuity equations, and optimal controllers are synthesized and assessed in sim¬ulations of the flow. The polynomial-form collocation model with control by rate of change of wall-normal velocity is shown to be consistent with previous interpo¬lating models with control by wall-normal velocity. Previous methods of applying the Dirichlet and Neumann boundary conditions to Chebyshev series are shown to be not strictly valid. A partly novel method provides the best numerical behaviour after preconditioning. Two test cases representing the earliest stages of the transition are consid¬ered, and linear quadratic regulators (LQR) and estimators (LQE) are synthesized. Finer discretisation is required for convergence of estimators. A novel estimator covariance weighting improves estimator transient convergence. Initial conditions which generate the highest subsequent transient energy are calculated. Non-linear open- and closed-loop simulations, using an independently derived finite-volume Navier-Stokes solver modified to work in terms of perturbations, agree with linear simulations for small perturbations. Although the transpiration considered is zero net mass flow, large amounts of fluid are required locally. At larger perturbations the flow saturates. State feedback controllers continue to stabilise the flow, but estimators may overshoot and occasionally output feedback destabilises the flow. Actuation by simultaneous wall-normal and tangential transpiration is derived. There are indications that control via tangential actuation produces lower highest transient energy, although requiring larger control effort. State feedback controllers are also synthesized which minimise upper bounds on the highest transient energy and control effort. The performance of these controllers is similar to that of the optimal controllers.