An optimal shape control problem for the steady-state Navier-Stokes equations is considered from an analytical point of view. We examine a rather specific model problem dealing with 2-dimensional channel flow of incompressible viscous fluid: we wish to determine the shape of a bump on a part of the boundary in order to minimize the energy dissipation. To formulate the problem in a comprehensive manner, we study some properties of the Navier-Stokes equations. The penalty method is applied to relax the difficulty of dealing with incompressibility in conjunction with domain perturbations and regularity requirements for the solutions. The existence of optimal solutions for the penalized problem is presented. The computation of the shape gradient and its treatment plays central role in the shape sensitivity analysis. To describe the domain perturbation and to derive the shape gradient, we study the material derivative method and related shape calculus. The shape sensitivity analysis using the material derivative method and Lagrange multiplier technique is presented. The use of Lagrange multiplier techniques,from which an optimality system is derived, is justified by applying a method from functional analysis. Finite element discretizations for the domain and discretized description of the problem are given. We study finite element approximations for the weak penalized optimality system. To deal with inhomogeneous essential boundary condition, the framework of a Lagrange multiplier technique is applied. The split formulation decoupling the traction force from the velocity is proposed in conjunction with the penalized optimality system and optimal error estimates are derived. / Ph. D.

Accurate and efficient numerical algorithms for solving the three dimensional Navier Stokes equations with a generalized thermodynamic and chemistry model and a one equation turbulence model on structured and unstructured mesh topologies are presented. In the thermo-chemical modeling, particular attention is paid to the modeling of the chemical source terms, modeling of equilibrium thermodynamics, and the modeling of the non-equilibrium vibrational energy source terms. In this work, nonequilibrium thermo-chemical models are applied in the unstructured environment for the first time. A three-dimensional, second-order accurate k-exact reconstruction algorithm for the inviscid and viscous fluxes is presented. Several new methods for determining the stencil required for the inviscid and viscous k-exact reconstruction are discussed. A new simplified method for the computation of the viscous fluxes is also presented. Implementation of the one equation Spalart and Allmaras turbulence model is discussed. In particular, an new integral formulation is developed for this model which allows flux splitting to be applied to the resulting convective flux. Solutions for several test cases are presented to verify the solution algorithms discussed. For the thermo-chemical modeling, inviscid solutions to the three dimensional Aeroassist Flight Experiment vehicle and viscous solutions for the axi-symmetric Ram-II C are presented and compared to experimental data and/or published results. For the hypersonic AFE and Ram-II C solutions, focus is placed on the effects of the chemistry model in flows where ionization and dissociation are dominant characteristics of the flow field. Laminar and turbulent solutions over a flat plate are presented and compared to exact solutions and experimental data. Three dimensional higher order solutions using the k-exact reconstruction technique are presented for an analytic forebody. / Ph. D.

The process of enhancing an upwind finite volume, two-dimensional, thin layer Navier Stokes solver to achieve complete Navier Stokes solutions is described. The shear stress and heat flux contributions are identified and transformed to a generalized coordinate system. The metrics which result from the transformation have a geometrical interpretation in the finite volume formulation and are presented as supporting material. The additional terms which are neglected in the thin-layer approximations, are evaluated and discretized consistently with the finite volume method. Implicit linearizations are applied to the second derivatives tangent to the body surface; however, the cross derivatives are not linearized and are treated conservatively. Validation of the Navier Stokes solver is acquired by comparison to existing computational solutions for a double throat nozzle. Additional viscous solutions for the thin layer and the complete forms of the NS equations are provided for a flat plate shock boundary layer interaction. / Master of Science

The development of the patched grid solution methodology for both the Euler and the Navier-Stokes equations in two dimensions is presented. The governing equations are written in the integral form and the basic numerical algorithm is finite volume. The method is capable of first through third order accuracy in space. The flux vectors associated with the Euler equations are split into two sub-vectors (based on the signs of the characteristic speeds) and discretized separately. The viscous and heat flux contributions are treated with central differences. Patched grid results are demonstrated on shock reflection, subsonic boundary layer, and shock-boundary layer interaction flow problems. The results are compared with non-patched or single zone grids. The patched grid approach shows an improvement in resolution while minimizing storage and computer time. / Master of Science

Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations. / text

La première partie de cette thèse traite du développement du code de simulation numérique directe YWC, principalement du développement des conditions aux limites. En effet, une forte contribution scientifique a été apportée aux conditions aux limites appelées "Three dimensional Navier-Stokes characteristic boundary condtions" (3D-NSCBC). Premièrement, la formulation de ces conditions aux arêtes et coins a été complétée, ensuite une extension de la formulation a été proposée pour supprimer les déformations observées en sortie dans le cas d'écoulements non-perpendiculaires à la frontière. <p>De plus, ces conditions ont été étendues au cas des gaz réels et une nouvelle définition du facteur de relaxation pour la pression a été proposée. Ce nouveau facteur de relaxation permet de supprimer les déformations observées en sortie pour des écoulements transcritiques. <p>Les résultats obtenus avec le code YWC ont ensuite été utilisés dans la seconde partie de la thèse pour développer une nouvelle méthode de tabulation basée sur l'analyse en composantes principales. Par rapport aux méthodes existante telles que FPI ou SLFM, la technique proposée, permet une identification automatique des variables à transporter et n'est, de plus, pas lié à un régime de combustion spécifique. Cette technique a permis d'effectuer des calculs d'interaction flamme-vortex en ne transportant que 5 espèces à la place des 9 requises pour le calcul en chimie détaillée complète, sans pour autant perdre en précision. <p>Finalement, dans le but de réduire encore le nombre d'espèces transportées, les techniques T-BAKED et HT-BAKED PCA ont été introduites. En utilisant une pondération des points sous-représentés, ces deux techniques permettent d'augmenter la précision de l'analyse par composantes principales dans le cadre des phénomènes de combustion.<p> / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

Les équations fondamentales décrivant la dynamique de l'océan sont en théorie les équations de Navier-Stokes sur une sphère en rotation, auxquelles il faut a jouter une équation d'état pour la densité, et des équations de transport-diffusion pour les traceurs. Toutefois, un certain nombre de considérations physiques et de limitations pratiques ont nécessité le développement de modèles plus simples. En effet, un certain nombre d'hypothèses simplificatrices sont pleinement justifiées du point de vue de la physique des mouvements océaniques, dont les principales sont les approximations de couche mince et de Boussinesq. D'autre part, étant donné les dimensions des bassins océaniques (plusieurs centaines à plusieurs milliers de kilomètres), les coûts de calculs sont un facteur pratique extrêmement limitant. On est, à l'heure actuelle, capable de simuler l'océan mondial avec une résolution de l'ordre de dix kilomètres, en utilisant des modèles dits aux équations primitives, dont le coût de calcul est bien inférieur à celui des équations de Navier-Stokes. On est donc bien loin d'une modélisation complète des phénomènes décrits par ces équations, qui nécessiterait en théorie de considérer des échelles de l'ordre du millimètre. Les équations primitives sont issues des équations complètes de la mécanique des fluides en effectuant l'approximation hydrostatique, justifiée par la faible profondeur des domaines considérés au regard de leur dimension horizontale. Dans cette thèse, nous considérerons les équations de Navier-Stokes (NS) qui sont le coeur du modèle complet évoqué ci-dessus, sans prendre en compte les équations de la densité et des traceurs (salinité, température, etc.). Nous utiliserons l'approximation hydrostatique dans le chapitre 10, et le modèle sera naturellement appelé Navier-Stokes hydrostatique (NSH). Il correspond aux équations primitives dans lesquelles on ne prendrait pas en compte la densité et les traceurs. C'est dans ce cadre que se situe le travail présenté dans cette thèse, avec l'objectif à moyen terme de pouvoir coupler de façon rigoureuse et efficace les équations de Navier-Stokes avec les équations primitives. Dans une première partie, on présentera quelques rappels sur les équations de Navier-Stokes, leur discrétisation, ainsi que le cas-test de la cavité entraînée qui sera utilisé dans tout ce document. Dans une deuxième partie, on met en œuvre les méthodes de Schwarz sur les équations de Stokes et Navier-Stokes, en dérivant notamment des conditions absorbantes exactes et approchées pour ces systèmes. Enfin, dans une troisième partie, on proposera des pistes vers le couplage Navier-Stokes/Navier-Stokes hydrostatique décrit ci-dessus. / Fundamental equations describing the ocean dynamic are theoretically Navier-Stokes equations over a rotating sphere, whom need to add a state equation for the fluid density, and advection-diffusion equations for tracers. However, some physical considerations and practical limitations required to developped more simple models. Indeed, some simplifying hypotheses are well justified from a ocean dynamic point of view, whose principal ones are thin layer and Boussinesq approximations. On the other hand, considering the dimensions of oceans (from serveral hundreds to serveral thousands kilometers), computations costs are a very practical limitating factor. We are, by now, able to simulate the global ocean with about ten kilometers large grid mesh. This is very far from a complete modelisation of all phenomenes decribed by the Navier-Stokes equations, which require to consider scales of milimeters order. Primitives equations derive from complete equations describing fluid mecanics, by doing the hydrostatic approximations, which is justified by the low deepness of considered domains with regard to their horizontal dimension. In this thesis, we are considering Navier-Stokes equations (NS) which are the heart of the complete modele mentionned previously, without holding in account density and tracers equations. We will use the hydrostatic approximations, and the resulting equations will be named as hydrostatic Navier-Stokes equations (NSH).The mid term objective is to couple carefully Navier-Stokes equations with primitive equation. In a first part, we will remind few results for Navier-Stokes equations, their discretization, and the lid-driven cavity which wil be used as a test-case. In a second part, we will use Schwarz method with Stokes and Navier-Stokes equations, deriving in particular exact and approched absorbing interface conditions for these systems. Finally, in a third part, we shall propose first results towards coupling Navier-Stokes and hydrostatic Navier-Stokes equations.

This manuscript demonstrates the well-posedness (existence, uniqueness, and regularity of solutions) of the Cauchy problem for simplified equations of nematic liquid crystal hydrodynamic flow in three dimensions for initial data that is uniformly locally L3(R3) integrable (L3U(R3)). The equations examined are a simplified version of the equations derived by Ericksen and Leslie. Background on the continuum theory of nematic liquid crystals and their flow is provided as are explanations of the related mathematical literature for nematic liquid crystals and the Navier–Stokes equations.

Two-dimensional flow in a viscous incompressible fluid, generated by a circular cylinder executing large-amplitude rectilinear oscillations in a plane perpendicular to its axis and parallel to one of the sides of a surrounding rectangular box filled with incompressible fluid is studied numerically. The circular cylinder moves back and forth through its own wake, resulting in an extremely complex flow field. For ease of implementing boundary conditions, a numerically generated body-fitted coordinate system is used. At each time step, the physical domain is doubly-connected, and a cut is introduced in order to map it into a rectangular computational domain. A body-fitted grid is generated by solving a pair of Laplace equations with a simple grid spacing control method which preserves the essential one-to-one property of the mapping. A finite difference/pseudo-spectral technique is used in this work to solve the Navier-Stokes equations in velocity-vorticity formulation. The time integration of the vorticity transport equation is handled by a fully explicit three-level Adams-Bashforth method. The two Poisson equations for the velocity components are 11-banded and block-diagonal in form, and are solved by a preconditioned biconjugate gradient routine. An integral constraint on the vorticity field is used to determine the boundary vorticity that simultaneously satisfies the no-slip and no-penetration conditions. The surface vorticity is uniquely determined by a general solution procedure developed in this study which is valid for flows over multiple solid bodies. With this approach, the physical process of vorticity generation on the solid boundary is properly simulated and the principle of vorticity conservation is satisfied. Results for various test cases and the complex vortex shedding phenomena generated by an oscillating circular cylinder are presented and discussed.

For simulations of highly complex geometries, frequently encountered in many fields of science and engineering, the process of generating a high-quality, body-fitted grid is very complicated and time-intensive. Thus, one of the principal goals of contemporary CFD is the development of numerical algorithms, which are able to deliver computationally efficient, and highly accurate solutions for a wide range of applications involving multi-physics problems, e.g. Fluid Structure Interaction (FSI). Immersed interface/boundary methods provide considerable advantages over conventional approaches, especially for flow problems containing moving boundaries.In the present work, a novel, robust, highly-accurate, Immersed Interface Method (IIM) is developed, which is based on a local Taylor-series expansion at irregular grid points enforcing numerical stability through a local stability condition. Various immersed methods have been developed in the past; however, these methods only considered the order of the local truncation error. The numerical stability of these schemes was demonstrated (in a global sense) by considering a number of different test-problems. None of these schemes used a concrete local stability condition to derive the irregular stencil coefficients. This work will demonstrate that the local stability constraint is valid as long as the DFL-number does not reach a limiting value. The IIM integrated into a newly developed Incompressible Navier-Stokes (INS) solver is used herein to simulate fully coupled FSI problems. The extension of the novel IIM to a higher-order method, the compressible Navier-Stokes equations and the Maxwell's equations demonstrate the great potential of the novel IIM.In the second part of this dissertation, the newly developed INS solver is employed to study the flow of a stalled airfoil and steady/unsteady stenotic flows. In this context, a new biglobal stability analysis approach based on solving an Initial Value Problem (IVP), instead of the traditionally used EigenValue Problem (EVP), is presented. It is demonstrated that this approach based on an IVP is computationally less expensive compared to EVP approaches while still capturing the relevant physics.