O estudo de métodos de modelagem de escoamentos aerodinâmicos em regime transônico é de grande importância para a engenharia aeronáutica. O maior desafio no tratamento desses escoamentos está na sua característica não linear devido aos efeitos de compressibilidade e formação de ondas de choque. Tais efeitos não lineares influenciam no desempenho de superfícies aerodinâmicas em geral, bem como são responsáveis pelo aparecimento de fenômenos danosos para a resposta aeroelástica de aeronaves. O equacionamento para esses tipos de escoamentos pode ser obtido via as equações básicas da mecânica dos fluidos. No entanto, apenas soluções numéricas de tais equações são possíveis de ser obtidas de forma prática no presente momento. Para o caso específico do tratamento de problemas transônicos, as equações de Euler formam um conjunto de equações diferenciais a derivadas parciais capazes de capturar os efeitos não lineares de escoamentos compressíveis, porém os efeitos da viscosidade não são levados em consideração. O objetivo desse trabalho é implementar uma rotina computacional capaz de resolver numericamente escoamentos em regime transônico em torno de aerofólios. Para isso as equações de Euler não lineares são utilizadas e o campo de fluido ao redor de um perfil aerodinâmico é discretizado pelo método das diferenças finitas. Uma malha estruturada do tipo C discretizando o fluido ao redor de um aerofólio NACA0012 é considerada. A metodologia para solução numérica é baseada no método explícito de MacCormack de segunda ordem de precisão no tempo e espaço. Baseados na aproximação upwind, termos de dissipação artificial com coeficientes não lineares também são adicionados ao método. A solução do escoamento transônico estacionário ao redor do aerofólio NACA0012 é obtida e as principais propriedades do escoamento são apresentadas. Observa-se a formação de ondas de choque através de contornos de número de Mach ao redor do aerofólio. Gráficos das distribuições de pressão no intra e extradorso do aerofólio são mostrados, onde se identificam aos efeitos da brusca variação de pressão devido as ondas de choque. Os resultados são validados com valores de distribuição de pressão para o mesmo aerofólio encontradas na literatura técnica. Os resultados obtidos combinam bem com os fornecidos em códigos computacionais para solução do mesmo problema aerodinâmico / The study of aerodynamic modeling methods for the transonic flow regime is of great importance in aeronautical engineering. Major challenge on the treatment of those flows is on their nonlinear features due to compressibility effects and shock waves (appearance). Such nonlinear effects present a strong influence on aerodynamic performance, as well as they are responsible for harmful aeroelastic response phenomena in aircraft. Equations for transonic flows can be obtained from the basic fluid mechanic equations. However, only numerical methods are able to attain practical solutions for those set of differential equations at the present moment. For the specific case of treating transonic flow problems, the nonlinear Euler equations provide a set of partial differential equations with features to capture nonlinear effects of typical compressible flows, despite of not accounting for viscous flows effects. The aim of this work is to implement a computational routine for the numerical solution of transonic flows around airfoils. The Euler equations are used and the flow field around a aerodynamic profile is discretized by finite difference method. A C-type structured mesh is used to discretize the flow around a NACA0012 airfoil. The methodology for numerical solution is based on the explicit MacCormack method which has second order accuracy in time and space. Based on the upwind approximation, artificial dissipation with nonlinear coefficients is incorporated to the method. The steady transonic flow around the NACA0012 airfoil numerical solution is assessed and the main flow properties are presented. Shock wave structure can also be observed by means of the Mach number contours around the airfoil. Pressure distributions on upper and lower surfaces for different flow conditions are also shown, thereby allowing the observation of the effects of the abrupt pressure change due to shock waves. The results are validated using data presented in the technical literature. The present code solutions agree well with the solution obtained in other computational codes used for the same problem

In this thesis, we present a collection of newly obtained results concerning the existence of vanishing viscosity solutions to the one-dimensional compressible Euler equations of gas dynamics, with and without geometric structure. We demonstrate the existence of such vanishing viscosity solutions, which we show to be entropy solutions, to the transonic nozzle problem and spherically symmetric Euler equations in Chapter 4, in both cases under the simple and natural assumption of relative finite-energy. In Chapter 5, we show that the viscous solutions of the one-dimensional compressible Navier-Stokes equations converge, as the viscosity tends to zero, to an entropy solution of the Euler equations, again under the assumption of relative finite-energy. In so doing, we develop a compactness framework for the solutions and approximate solutions to the Euler equations under the assumption of a physical pressure law. Finally, in Chapter 6, we consider the Euler equations in special relativity, and show the existence of bounded entropy solutions to these equations. In the process, we also construct fundamental solutions to the entropy equations and develop a compactness framework for the solutions and approximate solutions to the relativistic Euler equations.

Dans cette thèse, nous étudions les propriétés des écoulements de fluides qui interagissent avec un corps rigide ou avec une source et un puits. Dans le cas d'un fluide visqueux incompressible qui satisfait les équations de Navier Stokes dans un domaine borné 2D, les solutions faibles de Leray-Hopf sont bien comprises. L'existence et l'unicité sont prouvées. De plus, les solutions sont continues en temps `a valeurs dans L 2 (Omega) et satisfont l’égalité d'énergie classique. Plus récemment, le problème d'un corps rigide en mouvement dans un fluide visqueux incompressible modélisé par les équations de Navier-Stokes couplées aux lois de Newton qui décrivent le mouvement du solide a également été abordé dans le cas où des conditions aux limites sans glissement ont été prescrites. Des résultats analogues concernant les solutions de Leray-Hopf ont également été démontrés dans ce contexte. Dans ce manuscrit, nous étudions le cas de conditions aux limites de Navier-Slip. Dans ce cadre, le résultat d'existence pour le système couplé a été prouvée par G'erard-Varet et Hillairet en 2014. Ici, nous montrons que les solutions sont continues en temps, qu'elles satisfont l’égalité d'énergie et qu’elles sont uniques. De plus, nous montrons un résultat d'existence des solutions faibles dans le cas d'un fluide incompressible visqueux auquel s'ajoute un corps rigide dans le cas où la vitesse du fluide a une partie orthonormale d'énergie infinie.Pour un fluide incompressible non visqueux modélisé par les équations d'Euler dans un domaine borné 2D, le cas où le fluide est autorisé à entrer et à sortir de la frontière a été traité par Judovic qui a introduit certaines conditions limites consistant à prescrire la composante normale de la vitesse et de la vorticité entrante. Dans ce manuscrit, nous considérons un domaine borné qui possède deux trous. L'un d'eux est une source, ce qui signifie que le fluide est autorisé à entrer dans le domaine et l'autre est un puits où le fluide peut sortir. En particulier, nous établissons les équations limites vérifiées par le fluide lorsque la source et le puits se contractent en deux points différents. Le système limite est caractérisé par un point source/puits et un point vortex en chacun des deux points où les trous se sont contractés. / In this thesis, we investigate properties of incompressible flows that interact with a rigid body or a source and a sink. In the case of an incompressible viscous fluid that satisfies the Navier Stokes equations in a 2D bounded domain well-posedness of Leray-Hopf weak solutions is well-understood. Existence and uniqueness are proved. Moreover solutions are continuous in time with values in L 2 (Omega) and they satisfy the energy equality. Recently the problem of a rigid body moving in a viscous incompressible fluid modeled by the Navier-Stokes equations coupled with the Newton laws that prescribe the motion of the solid, was also tackled in the case where the no-slip boundary conditions were imposed. And the correspondent well-posedness result for Leray-Hopf type weak solutions was proved. In this manuscript we consider the case of the Navier-slip boundary conditions. In this setting, the existence result for the coupled system was proved by G'erard-Varet and Hillairet in 2014. Here, we prove that solutions are continuous in time, that they satisfy the energy equality and that they are unique. Moreover we show an existence result for weak solutions of a viscous incompressible fluid plus rigid body system in the case where the fluid velocity has an orthoradial part of infinite energy.For an inviscid incompressible fluid modelled by the Euler equations in a 2D bounded domain, the case where the fluid is allowed to enter and to exit from the boundary was tackled by Judovic who introduced some conditions which consist in prescribing the normal component of the velocity and the entering vorticity. In this manuscript we consider a bounded domain with two holes, one of them is a source which means that the fluid is allowed to enter in the domain and the other is a sink from where the fluid can exit. In particular we find the limiting equations satisfied by the fluid when the source and the sink shrink to two different points. The limiting system is characterized by a point source/sink and a point vortex in each of the two points where the holes shrunk.

Aquesta tesi proposa una millora del clàssic mètode dels elements finits (finite element method, FEM) per a un tractament eficient de dominis amb contorns corbs: el denominat NURBS-enhanced finite element method (NEFEM). Aquesta millora permet descriure de manera exacta la geometría mitjançant la seva representació del contorn CAD amb non-uniform rational B-splines (NURBS), mentre que la solució s'aproxima amb la interpolació polinòmica estàndard. Per tant, en la major part del domini, la interpolació i la integració numèrica són estàndard, retenint les propietats de convergència clàssiques del FEM i facilitant l'acoblament amb els elements interiors. Només es requereixen estratègies específiques per realitzar la interpolació i la integració numèrica en elements afectats per la descripció del contorn mitjançant NURBS.La implementació i aplicació de NEFEM a problemes que requereixen una descripció acurada del contorn són, també, objectius prioritaris d'aquesta tesi. Per exemple, la solució numèrica de les equacions de Maxwell és molt sensible a la descripció geomètrica. Es presenta l'aplicació de NEFEM a problemes d'scattering d'ones electromagnètiques amb una formulació de Galerkin discontinu. S'investiga l'habilitat de NEFEM per obtenir solucions precises amb malles grolleres i aproximacions d'alt ordre, i s'exploren les possibilitats de les anomenades malles NEFEM, amb elements que contenen singularitats dintre d'una cara o aresta d'un element. Utilitzant NEFEM, la mida de la malla no està controlada per la complexitat de la geometria. Això implica una dràstica diferència en la mida dels elements i, per tant, suposa un gran estalvi tant des del punt de vista de requeriments de memòria com de cost computacional. Per tant, NEFEM és una eina poderosa per la simulació de problemes tridimensionals a gran escala amb geometries complexes. D'altra banda, la simulació de problemes d'scattering d'ones electromagnètiques requereix mecanismes per aconseguir una absorció eficient de les ones scattered. En aquesta tesi es discuteixen, optimitzen i comparen dues tècniques en el context de mètodes de Galerkin discontinu amb aproximacions d'alt ordre.La resolució numèrica de les equacions d'Euler de la dinàmica de gasos és també molt sensible a la representació geomètrica. Quan es considera una formulació de Galerkin discontinu i elements isoparamètrics lineals, una producció espúria d'entropia pot evitar la convergència cap a la solució correcta. Amb NEFEM, l'acurada imposició de la condició de contorn en contorns impenetrables proporciona resultats precisos inclús amb una aproximació lineal de la solució. A més, la representació exacta del contorn permet una imposició adequada de les condicions de contorn amb malles grolleres i graus d'interpolació alts. Una propietat atractiva de la implementació proposada és que moltes de les rutines usuals en un codi d'elements finits poden ser aprofitades, per exemple rutines per realitzar el càlcul de les matrius elementals, assemblatge, etc. Només és necessari implementar noves rutines per calcular les quadratures numèriques en elements corbs i emmagatzemar el valor de les funciones de forma en els punts d'integració. S'han proposat vàries tècniques d'elements finits corbs a la literatura. En aquesta tesi, es compara NEFEM amb altres tècniques populars d'elements finits corbs (isoparamètics, cartesians i p-FEM), des de tres punts de vista diferents: aspectes teòrics, implementació i eficiència numèrica. En els exemples numèrics, NEFEM és, com a mínim, un ordre de magnitud més precís comparat amb altres tècniques. A més, per una precisió desitjada NEFEM és també més eficient: necessita un 50% dels graus de llibertat que fan servir els elements isoparamètrics o p-FEM per aconseguir la mateixa precisió. Per tant, l'ús de NEFEM és altament recomanable en presència de contorns corbs i/o quan el contorn té detalls geomètrics complexes. / This thesis proposes an improvement of the classical finite element method (FEM) for an efficient treatment of curved boundaries: the NURBSenhanced FEM (NEFEM). It is able to exactly represent the geometry by means of the usual CAD boundary representation with non-uniform rational Bsplines (NURBS), while the solution is approximated with a standard piecewise polynomial interpolation. Therefore, in the vast majority of the domain, interpolation and numerical integration are standard, preserving the classical finite element (FE) convergence properties, and allowing a seamless coupling with standard FEs on the domain interior. Specifically designed polynomial interpolation and numerical integration are designed only for those elements affected by the NURBS boundary representation.The implementation and application of NEFEM to problems demanding an accurate boundary representation are also primary goals of this thesis. For instance, the numerical solution of Maxwell's equations is highly sensitive to geometry description. The application of NEFEM to electromagnetic scattering problems using a discontinuous Galerkin formulation is presented. The ability of NEFEM to compute an accurate solution with coarse meshes and high-order approximations is investigated, and the possibilities of NEFEM meshes, with elements containing edge or corner singularities, are explored. With NEFEM, the mesh size is no longer subsidiary to geometry complexity, and depends only on the accuracy requirements on the solution, whereas standard FEs require mesh refinement to properly capture the geometry. This implies a drastic difference in mesh size that results in drastic memory savings, and also important savings in computational cost. Thus, NEFEM is a powerful tool for large-scale scattering simulations with complex geometries in three dimensions. Another key issue in the numerical solution of electromagnetic scattering problems is using a mechanism to perform the absorption of outgoing waves. Two perfectly matched layers are discussed, optimized and compared in a high-order discontinuous Galerkin framework.The numerical solution of Euler equations of gas dynamics is also very sensitive to geometry description. Using a discontinuous Galerkin formulation and linear isoparametric elements, a spurious entropy production may prevent convergence to the correct solution. With NEFEM, the exact imposition of the solid wall boundary condition provides accurate results even with a linear approximation of the solution. Furthermore, the exact boundary representation allows using coarse meshes, but ensuring the proper implementation of the solid wall boundary condition. An attractive feature of the proposed implementation is that the usual routines of a standard FE code can be directly used, namely routines for the computation of elemental matrices and vectors, assembly, etc. It is only necessary to implement new routines for the computation of numerical quadratures in curved elements and to store the value of shape functions at integration points. Several curved FE techniques have been proposed in the literature. In this thesis, NEFEM is compared with some popular curved FE techniques (namely isoparametric FEs, cartesian FEs and p-FEM), from three different perspectives: theoretical aspects, implementation and performance. In every example shown, NEFEM is at least one order of magnitude more accurate compared to other techniques. Moreover, for a desired accuracy NEFEM is also computationally more efficient. In some examples, NEFEM needs only 50% of the number of degrees of freedom required by isoparametric FEs or p-FEM. Thus, the use of NEFEM is strongly recommended in the presence of curved boundaries and/or when the boundary of the domain has complex geometric details.

Cartesian grids offer a valuable option to simulate aerodynamic flows around complex geometries such as multi-element airfoils, aircrafts, and rockets. Therefore, an adaptively-refined Cartesian grid generator and Euler solver are developed. For the mesh generation part of the algorithm, dynamic data structures are used to determine connectivity information between cells and uniform mesh is created in the domain. Marching squares and cubes algorithms are used to form interfaces of cut and split cells. Geometry-based cell adaptation is applied in the mesh generation. After obtaining appropriate mesh around input geometry, the solution is obtained using either flux vector splitting method or Roe&rsquo / s approximate Riemann solver with cell-centered approach. Least squares reconstruction of flow variables within the cell is used to determine high gradient regions of flow. Solution based adaptation method is then applied to current mesh in order to refine these regions and also coarsened regions where unnecessary small cells exist. Multistage time stepping is used with local time steps to increase the convergence rate. Also FAS multigrid technique is used in order to increase the convergence rate. It is obvious that implementation of geometry and solution based adaptations are easier for Cartesian meshes than other types of meshes. Besides, presented numerical results show the accuracy and efficiency of the algorithm by especially using geometry and solution based adaptation. Finally, Euler solutions of Cartesian grids around airfoils, projectiles and wings are compared with the experimental and numerical data available in the literature and accuracy and efficiency of the solver are verified.

Least-squares finite element method (LSFEM) is employed to simulate 2-D and axisymmetric flows governed by the compressible Euler equations. Least-squares formulation brings many advantages over classical Galerkin finite element methods. For non-self-adjoint systems, LSFEM result in symmetric positive-definite matrices which can be solved efficiently by iterative methods. Additionally, with a unified formulation it can work in all flight regimes from subsonic to supersonic. Another advantage is that, the method does not require artificial viscosity since it is naturally diffusive which also appears as a difficulty for sharply resolving high gradients in the flow field such as shock waves. This problem is dealt by employing adaptive mesh refinement (AMR) on triangular meshes. LSFEM with AMR technique is numerically tested with various flow problems and good agreement with the available data in literature is seen.

This thesis studies the dynamics and stability of some bicycles. The dynamics of idealized bicycles is of interest due to complexities associated with the behaviour of this seemingly simple machine. It is also useful as it can be a starting point for analysis of more complicated systems, such as motorcycles with suspensions, frame flexibility and thick tyres. Finally, accurate and reliable analyses of bicycles can provide benchmarks for checking the correctness of general multibody dynamics codes. The first part of the thesis deals with the derivation of fully nonlinear differential equations of motion for a bicycle. Lagrange’s equations are derived along with the constraint equations in an algorithmic way using computer algebra.Then equivalent equations are obtained numerically using a Newton-Euler formulation. The Newton-Euler formulation is less straightforward than the Lagrangian one and it requires the solution of a bigger system of linear equations in the unknowns. However, it is computationally faster because it has been implemented numerically, unlike Lagrange’s equations which involve long analytical expressions that need to be transferred to a numerical computing environment before being integrated. The two sets of equations are validated against each other using consistent initial conditions. The match obtained is, expectedly, very accurate. The second part of the thesis discusses the linearization of the full nonlinear equations of motion. Lagrange’s equations have been used.The equations are linearized and the corresponding eigenvalue problem studied. The eigenvalues are plotted as functions of the forward speed ν of the bicycle. Several eigenmodes, like weave, capsize, and a stable mode called caster, have been identified along with the speed intervals where they are dominant. The results obtained, for certain parameter values, are in complete numerical agreement with those obtained by other independent researchers, and further validate the equations of motion. The bicycle with these parameters is called the benchmark bicycle. The third part of the thesis makes a detailed and comprehensive study of hands-free circular motions of the benchmark bicycle. Various one-parameter families of circular motions have been identified. Three distinct families exist: (1)A handlebar-forward family, starting from capsize bifurcation off straight-line motion, and ending in an unstable static equilibrium with the frame perfectly upright, and the front wheel almost perpendicular. (2) A handlebar-reversed family, starting again from capsize bifurcation, but ending with the front wheel again steered straight, the bicycle spinning infinitely fast in small circles while lying flat in the ground plane. (3) Lastly, a family joining a similar flat spinning motion (with handlebar forward), to a handlebar-reversed limit, circling in dynamic balance at infinite speed, with the frame near upright and the front wheel almost perpendicular; the transition between handlebar forward and reversed is through moderate-speed circular pivoting with the rear wheel not rotating, and the bicycle virtually upright. In the fourth part of this thesis, some of the parameters (both geometrical and inertial) for the benchmark bicycle have been changed and the resulting different bicycles and their circular motions studied showing other families of circular motions. Finally, some of the circular motions have been examined, numerically and analytically, for stability.

A three-dimensional, parallel, anisotropic, block-based, adaptive mesh refinement (AMR) algorithm is proposed and described for the solution of fluid flows on body-fitted, multi-block, hexahedral meshes. Refinement and de-refinement in any grid block computational direction, or combination of directions, allows the mesh to rapidly adapt to anisotropic flow features such as shocks, boundary layers, or flame fronts, common to complex flow physics. Anisotropic refinements and an efficient and highly scalable parallel implementation lead to a potential for significant reduction in computational cost as compared to a more typical isotropic approach. Unstructured root-block topology allows for greater flexibility in the treatment of complex geometries. The AMR algorithm is coupled with an upwind finite-volume scheme for the solution of the Euler equations governing inviscid, compressible, gaseous flow. Steady-state and time varying, three-dimensional, flow problems are investigated for various geometries, including the cubed-sphere mesh.

A three-dimensional, parallel, anisotropic, block-based, adaptive mesh refinement (AMR) algorithm is proposed and described for the solution of fluid flows on body-fitted, multi-block, hexahedral meshes. Refinement and de-refinement in any grid block computational direction, or combination of directions, allows the mesh to rapidly adapt to anisotropic flow features such as shocks, boundary layers, or flame fronts, common to complex flow physics. Anisotropic refinements and an efficient and highly scalable parallel implementation lead to a potential for significant reduction in computational cost as compared to a more typical isotropic approach. Unstructured root-block topology allows for greater flexibility in the treatment of complex geometries. The AMR algorithm is coupled with an upwind finite-volume scheme for the solution of the Euler equations governing inviscid, compressible, gaseous flow. Steady-state and time varying, three-dimensional, flow problems are investigated for various geometries, including the cubed-sphere mesh.

In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L^p continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H^r+1 (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H^r, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.