This (Diplom-) thesis deals with the particle trajectories of an incompressible and ideal fluid flow in 𝑛 ≥ 2 dimensions. It presents a complete and detailed proof of the surprising fact that the trajectories of a smooth solution of the incompressible Euler equations are locally analytic in time. In following the approach of P. Serfati, a complex ordinary differential equation (ODE) is investigated which can be seen as a complex extension of a partial differential equation, which is solved by the trajectories. The right hand side of this ODE is in fact given by a singular integral operator which coincides with the pressure gradient along the trajectories. Eventually, we may apply the Cauchy-Lipschitz existence theorem involving holomorphic maps between complex Banach spaces in order to get a unique solution for the above mentioned ODE. This solution is real-analytic in time and coincides with the particle trajectories.

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This study seeks to validate a recently introduced global preconditioning technique for the Euler equations. Energy and enthalpy equations are nondimensionalized by means of a reference enthalpy, resulting in increased numerical accuracy for low-speed flows. A cellbased, finite volume formulation is used, with Roe flux difference splitting and both explicit and implicit time integration schemes. A Newton-linearized iterative implicit algorithm is implemented, with Symmetric Gauss-Seidel (LU/SGS) nested sub-iterations. This choice allows one to retain time accuracy, and eliminates approximate factorization errors, which become dominant at low speed flows. The linearized flux Jacobians are evaluated by numerical differentiation. Higher-order discretization is constructed by means of the MUSCL approach. Locally one-dimensional characteristic variable boundary conditions are implemented at the farfield boundary. The preconditioned scheme is successfully applied to the following traditional test cases used as benchmarks for local preconditioning techniques: point disturbance, flow angle disturbance, and stagnation point arising from the impingement of two identical jets. The flow over a symmetric airfoil and a convergentdivergent nozzle are then simulated for arbitrary Mach numbers. The preconditioned scheme greatly enhances accuracy and convergence rate for low-speed flows (all the way down to M ≈ 10E − 4). Some preliminary tests of fully unsteady flows are also conducted.

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The successful application of Computational Aeroacoustics (CAA) requires high accuracy numerical schemes with good dissipation and dispersion characteristics. Unstructured meshes have a greater geometrical flexibility than existing high order structured mesh methods. This work investigates the suitability of unstructured mesh techniques by computing a two-dimensionallinearised Euler problem with various discretisation schemes and different mesh types. The goal of the present work is the development of an unstructured numerical method with the high accuracy, low dissipation and low dispersion required to be an effective tool in the study of aeroacoustics. The suitability of the unstructured method is investigated using aeroacoustic test cases taken from CAA Benchmark Workshop proceedings. Comparisons are made with exact solutions and a high order structured method. The higher order structured method was based upon a standard central differencing spatial discretisation. For the unstructured method a vertex-based data structure is employed. A median-dual control volume is used for the finite volume approximation with the option of using a Green-Gauss gradient approximation technique or a Least Squares approximation. The temporal discretisation used for both the structured and unstructured numerical methods is an explicit Runge-Kutta method with local timestepping. For the unstructured method, the gradient approximation technique is used to compute gradients at each vertex, these are then used to reconstruct the fluxes at the control volume faces. The unstructured mesh types used to evaluate the numerical method include semi-structured and purely unstructured triangular meshes. The semi-structured meshes were created directly from the associated structured mesh. The purely unstructured meshes were created using a commercial paving algorithm. The Least Squares method has the potential to allow high order reconstruction. Results show that a Weighted Least gradient approximation gives better solutions than unweighted and Green-Gauss gradient computation. The solutions are of acceptable accuracy on these problems with the absolute error of the unstructured method approaching that of a high order structured solution on an equivalent mesh for specific aeroacoustic scenarios.

L'évolution des turbines à gaz vers des régimes de combustion en mélange pauvre augmente la sensibilité de la flamme aux perturbations de l'écoulement. Plus particulièrement, cela augmente le risque que des instabilités de combustion apparaissent. Comme ces oscillations peuvent affecter le processus de combustion, il est très important d'être capable de prédire ce comportement au niveau de la conception.L'objectif du travail présenté est de développer un solveur numérique qui permet de décrire ces instabilités, et d'évaluer les effets du nombre de Mach de l'écoulement moyen sur ce phénomène. L'approche choisie consiste à résoudre les équations d'Euler linéarisées, qui sont écrites dans le domaine fréquentiel sous la forme d'un problème aux valeurs propres. Ce système d'équations permets de prendre en compte la vitesse moyenne de l'écoulement, et donc d'évaluer les effets causés par la convection et leur impact sur la stabilité des modes. Parmi les mécanismes qui peuvent être étudiés se trouve notamment l'effet des ondes d'entropie convectées, ce qui est particulièrement intéressant dans le contexte des chambres de combustions. Afin de déterminer l'effet des termes liés à la vitesse de l'écoulement moyen sur la stabilité des modes, une analyse de l'énergie contenue dans les perturbations est effectuée. Finalement, l'aspect de la non-orthogonalité des modes propres, qui permet une croissance d'énergie transitoire dans un système linéairement stable, est abordé. / The development of gas turbines towards lean combustion increases the susceptibility of the flame to flow perturbations, and leads more particularly to a higher risk of combustion instability. As these self-sustained oscillations may affect the performance of the combustion device, it is very important to be able to predict them at the design level. At present, several methods are used to describe combustion instabilities, ranging from complex LES and DNS calculations to low-order network models. An intermediate method consists in solving a set of equations describing the acoustic field using a finite volume technique, which is the approach used in the present study.This thesis discusses the impact of a non zero Mach number mean flow field on thermoacoustic instability. The study is based on the linearized Euler equations, which are stated in the frequency domain in the form of an eigenvalue problem. Using the linearized Euler equations rather than the Helmholtz equation avoids making the commonly used assumption of the mean flow being at rest, and allows to take into account convection effects and their impact on the stability of the system. Among the mechanisms that can be studied using the present approach is namely the impact of convected entropy waves, which is especially interesting in combustion applications.For this study, a 1D and a 2D numerical solver have been developed and are presented in this thesis. In order to asses the effect of the mean flow terms on the modes' stability, an analysis of the disturbance energy budget is performed. Finally, the aspect of the eigenmodes being non-orthogonal and thus allowing for transient growth in linearly stable systems is adressed.

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

Both experimental and numerical investigations of converging shock waves have been performed. In the experiments, a shock tube was used to create and study converging shock waves of various geometrical shapes. Two methods were used to create polygonally shaped shocks. In the first method, the geometry of the outer boundary of the test section of the shock tube was varied. Four different exchangeable shapes of the outer boundary were considered: a circle, a smooth pentagon, a heptagon, and an octagon. In the second method, an initially cylindrical shock wave was perturbed by metal cylinders placed in various patterns and positions inside the test section. For three or more regularly spaced cylinders, the resulting diffracted shock fronts formed polygonal shaped patterns near the point of focus. Regular reflection was observed for the case with three cylinders and Mach refection was observed for cases with four or more cylinders. When the shock wave is close to the center of convergence, light emission is observed. An experimental investigation of the light emission was conducted and results show that the shape of the shock wave close to the center of convergence has a large influence on the amount of emitted light. It was found that a symmetrical polygonal shock front produced more light than an asymmetrical shape. The shock wave focusing was also studied numerically using the Euler equations for a gas obeying the ideal gas law with constant specific heats. Two problems were analyzed; an axisymmetric model of the shock tube used in the experiments and a cylindrical shock wave diffracted by cylinders in a two dimensional test section. The results showed good agreement with the experiments. The temperature field from the numerical simulations was investigated and shows that the triple points behind the shock front are hot spots that increase the temperature at the center as they arrive there. As a practical example of shock wave focusing, converging shocks in an electrohydraulic lithotripter were simulated. The maximum radius of a gas bubble subjected to the pressure field obtained from the lithotripter was calculated and compared for various geometrical shapes and materials of the reflector. Results showed that the shape had a large impact while the material did not influence the maximum radius of the gas bubble. / QC 20100706

In the thesis work, a code to solve the two dimensional compressible Euler equations for external flows around arbitrary geometries have been developed. A Cartesianmesh generator is incorporated to the solver. Hence the pre-processing can be performed together with the solution within a single code. The code is written in the C++ programming language and its object oriented capabilities have been exploited to save memory in the data structure developed. The Cartesian mesh is formed by dividing squares successively into its four quadrants. The main advantage of using this type of a mesh is the ability to generate meshes around geometries of arbitrary complexity quickly and to adapt the mesh easily based on the solution. The main disadvantage of this method is that the treatment of the cells that are cut by the geometry. For the solution procedure Roe&rsquo / s method as well as flux vector splitting methods are used for the flux evaluation. The flux vector splitting schemes used are van Leer, AUSM, AUSMD and AUSMV methods. Time discretization is performed using a multi-stage method. To increase the accuracy least squares reconstruction is employed. The code is validated by performing calculations around a NACA0012 airfoil profile. The effect of reconstruction is demonstrated by plotting the pressure coefficient on the airfoil. The distribution obtained using reconstruction is very close to the experimental one while there is a considerable deviation for the case without reconstruction. Also the shock capturing capabilities of different methods have been investigated. In addition the performance of each method is analyzed for flow around an NLR 7301 airfoil with a flap.

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the 3-D incompressible Euler equations belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve kinetic energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which do not conserve kinetic energy. The first part, relating to conservation of kinetic energy, has since been confirmed (cf. Eyink 1994, Constantin-E-Titi 1994). The second part, relating to the existence of non-conservative solutions, remains an open conjecture and is the subject of this dissertation. In groundbreaking work of De Lellis and Székelyhidi Jr. (2012), the authors constructed the first examples of non-conservative Hölder continuous weak solutions to the Euler equations. The construction was subsequently improved by Isett (2012/2013), introducing many novel ideas in order to construct 1/5− Hölder continuous weak solutions with compact support in time. Adhering more closely to the original scheme of De Lellis and Székelyhidi Jr., we present a comparatively simpler construction of 1/5− Hölder continuous non-conservative weak solutions which may in addition be made to obey a prescribed kinetic energy profile. Furthermore, we extend this scheme in order to construct weak non-conservative solutions to the Euler equations whose Hölder 1/3− norm is Lebesgue integrable in time. The dissertation will be primarily based on three papers, two of which being in collaboration with De Lellis and Székelyhidi Jr.