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Yes / This research paper presents an incompressible smoothed particle hydrodynamics (ISPH) technique to investigate a regular wave overtopping on the coastal structure of different types. The SPH method is a mesh-free particle modeling approach that can efficiently treat the large deformation of free surface. The incompressible SPH approach employs a true hydrodynamic formulation to solve the fluid pressure that has less pressure fluctuations. The generation of flow turbulence during the wave breaking and overtopping is modeled by a subparticle scale (SPS) turbulence model. Here the ISPH model is used to investigate the wave overtopping over a coastal structure with and without the porous material. The computations disclosed the features of flow velocity, turbulence, and pressure distributions for different structure types and indicated that the existence of a layer of porous material can effectively reduce the wave impact pressure and overtopping rate. The proposed numerical model is expected to provide a promising practical tool to investigate the complicated wave-structure interactions. / Nazarbayev University Seed Grant, entitled “Environmental assessment of sediment pollution impact on hydropower plants”. S. Shao also acknowledges the Royal Society Research Grant (2008/R2 RG080561)

A three-dimensional unstructured incompressible RANS code has been developed using artificial compressibility and Spalart-Allmaras eddy viscosity model. A node-based finite volume method is used in which all flow variables are defined at the vertices of tetrahedrons in an unstructured grid. The inviscid fluxes are computed by using the Roe's flux difference splitting method, and higher order accuracy is attained by data reconstruction based on Taylor series expansion. Gauss theorem is used to formulate necessary gradients. For time integration, an implicit scheme based on linearized Euler backward method is used. A tetrahedral unstructured grid generation code has been also developed and applied to the tip clearance flow in a highly staggered cascade. Surface grids are first generated in the flow passage and blade tip by using several triangulation methods including Delaunay triangulation, advancing front method and advancing layer method. Then the whole computational domain including tip gap region is filled with prisms using the surface grids. Each prism is divided into three tetrahedrons. To accomplish this division in a consistent manner, connectivity pattern is assigned to each triangle in the surface grids. A new algorithm is devised to assign the connectivity pattern without reference to the particular method of triangulation. This technique offers great flexibility in surface grid generation. The code has been validated by comparisons with available computational and experimental results for several test cases: invisicd flow around NACA section, laminar and turbulent flow over a flat plate, turbulent flow through double-circular arc cascade and laminar flow through a square duct with 90° bend. For the laminar flat plate case, the velocity profile and skin friction coefficient are in excellent agreement with Blasius solution. For the turbulent flat plate case, velocity profiles are in full agreement with the law of the wall up to Reynolds number of 1.0E8, however, the skin friction coefficient is under-predicted by about 10% in comparison with empirical formula. Blade loading for the two-dimensional circular arc cascade is also compared with experiments. The results obtained with the experimental inflow angle (51.5° ) show some discrepancies at the trailing edge and severely under-predict the suction peak at the leading edge. These discrepancies are completely remedied if the inflow angle is increased to 53.5° . The code is also capable of predicting the secondary flow in the square duct with 90° bend, and the velocity profiles are in good agreement with measurements and published Navier-Stokes computations. Finally the code is applied to a linear cascade that has GE rotor B section with tip clearance and a high stagger angle of 56.9° . The overall structure of the tip clearance flow is well predicted. Loss of loading due to tip leakage flow and reloading due to tip leakage vortex are presented. On the end wall, separation line of the tip leakage vortex and reattachment line of passage vortex are identified. The location of the tip leakage vortex in the passage agrees very well with oil flow visualization. Separation bubble on the blade tip is also predicted. Mean streamwise velocity contours and cross sectional velocity vectors are compared with experimental results in the near wake, and good agreements are observed. It is concluded that Spalart-Allmaras turbulence model is adequate for this type of flow field except at locations where the tip leakage vortex of one blade interacts with the wake of a following blade. This situation may prevail for blades with longer span and/or in the far wake. Prediction of such an interaction presents a challenge to RANS computations. The effects of blade span on the flow structure have been also investigated. Two cascades with blades of aspect ratios of 0.5 and 1.0 are considered. By comparing pressure distributions on the blade, it is shown that the aspect ratio has strong effects on loading distribution on the blade although the tip gap height is very small (0.016 chord). Grid convergence study has been carried out with three different grids for pressure distributions and limiting streamlines on the end wall. / Ph. D.

Ce travail de thèse est consacré à l'étude numérique de la forme instationnaire des équations de Navier--Stokes incompressibles en dimension trois d'espace. Ces équations sont étudiées dans leur formulation classique mais aussi dans leur formulation hydrostatique. Le schéma en temps est résolu à l'aide d'une méthode de projection, la méthode de projection incrémentale standard qui sert de référence à l'étude numérique des nouvelles méthodes proposées: la méthode de projection hydrostatique et la méthode de projection dynamique étendue. Ces méthodes sont combinées à la méthode de pénalisation de volume qui permet de tenir compte de la présence d'obstacles de forme quelconque dans le domaine de simulation en utilisant un maillage cartésien. Une nouvelle expression des équations de Navier--Stokes hydrostatiques issue de l'application de la méthode de pénalisation de volume est obtenue et étudiée. La discrétisation en espace s'effectue en utilisant la méthode des volumes finis sur une maillage décalé de type MAC. Un nouveau code de calcul baptisé VivAn'O a été conçu et validé durant ce travail. Il a permis l'étude numérique des différentes méthodes proposées et a également permis de réaliser des simulations dans la zone portuaire de Pointe-à-Pitre à partir de données bathymétriques réelles. / The present work is about the numerical study of the three dimensional form of the unstationnary Navier--Stokesequations. These equations are studied under their classical form but also under their hydrostatic one. Thetime--dependent problem is solved using a projection method called the standard incremental projection method which isused as the reference to the numerical study of the new projection methods we proposed : the hydrostatic projectionmethod and the extended dynamical projection method. These methods are combined with a volume penalization methodto take into account obstacles of any shape in the simulation domain using cartesian meshes. A new formulation of the hydrostatic Navier--Stokes equations is obtained from the use of the penalization method and studied. The space isdiscretized using the finite volume method on a staggered MAC mesh.A new computational fluid dynamic code nammed VivAn'O have been developped during this work. It had permitted thenumerical study of the proposed methods. It was also used to carry out simulations in the harbour area of Pointe--à--Pitre usine real bathvmetric data.

Cette thèse présente une famille de schémas explicites pour la résolution d'un problème couplé d'interaction entre un fluide visqueux incompressible et une structure élastique (avec possiblement un comportement visco-élastique et/ou non linéaire). La principale propriété de ces schémas est une condition de Robin consistante à l'interface, qui représente une caractéristique fondamentale du problème continu dans le cas où la structure est mince. Si le couplage s'effectue avec une structure épaisse, une condition de Robin généralisée peut être formulée pour le problème semi-discret en espace, à l'aide d'une condensation de la matrice de masse de la structure. Une deuxième caractéristique majeure de ces schémas est la capacité d'obtenir une condition de Robin qui intègre à la fois des extrapolations de la vitesse et des efforts du solide (donnant lieu à un schéma de couplage explicite), mais également un traitement implicite de l'inertie de la structure, qui rend le schéma stable quelle que soit l'intensité de l'effet de masse ajoutée. Un résultat général de stabilité et de convergence est présenté pour tous les ordres d'extrapolations dans un cadre linéaire représentatif. On montre, en particulier, que les propriétés de stabilité se conservent lorsque le couplage s'effectue avec une structure mince ou épaisse. En revanche, la précision optimale obtenue dans le cas d'une structure mince n'est pas retrouvée avec une structure épaisse. L'erreur introduite par le schéma de couplage comporte en effet une non-uniformité en espace, qui provient de la non-uniformité des reconstructions discrètes des opérateurs viscoélastiques. L'approximation induite par la condensation de la matrice de masse solide n'est pas responsable de cette non-uniformité. À partir de ce schéma,on propose également des méthodes itératives pour la résolution du schéma fortement couplé.La convergence de cette méthode est démontrée dans un cadre linéaire et ne montre pas de sensibilité à l'effet de masse ajoutée. Finalement, les résultats théoriques obtenus sont illustrés par des exemples numériques variés, dans les cas linéaire et non linéaire. / This thesis introduces a class of explicit coupling schemes for the numerical solution of fluid-structure interaction problems involving a viscous incompressible fluid and a general elastic structure (thin-walled or thick-walled, viscoelastic and non-linear).The first fundamental ingredient of these methods is the notion of interface Robin consist encyon the interface. This is an intrinsic (parameter free) feature of the continuous problem, in the case of the coupling with thin-walled solids. For thick-walled structures, we show that an intrinsic interface Robin consistency can also be recovered at the space semi-discrete level, using a lumped-mass approximation in the structure.The second key ingredient of the methods proposed consists in deriving an explicit Robin interface condition for the fluid, which combines extrapolations of the solid velocity and stresses with an implicit treatment of the solid inertia. The former enables explicit coupling,while the latter guarantees added-mass free stability. Stability and error estimates are provided for all the variants (depending on the extrapolations), using energy arguments within a representative linear setting. We show, in particular, that the stability properties do not depend on the thin- or thick-walled nature of the structure. The optimal first-order accuracy obtained in the case of the coupling with thin-walled structuresis, however, not preserved when the structure is thick-walled, due to the spatial non uniformityof the splitting error. The genesis of this problem is the non-uniformity of the discrete viscoelastic operators, related to the thick-walled character of the structure,and not to the mass-lumping approximation. Based on these splitting schemes, new, parameter-free, Robin-Neumann iterative procedures for the partitioned solution of strong coupling are also proposed and analyzed. A comprehensive numerical study, involving linear and non linear models, confims the theoretical findings reported in this thesis.

In this thesis, evaluation of H(div)-conforming finite elements is implemented in a high-performance setting and used to solve the incompressible Navier-Stokes equation, obtaining an exactly point-wise divergence-free velocity field. In particular, the anisotropic Raviart-Thomas tensor-product polynomial space is considered, where the finite element operators are evaluated with quadrature in a matrix-free fashion using sum-factorization on tensor-product meshes. The implementation includes evaluation over elements and faces in two- and three-dimensional space, supporting non-conforming meshes with hanging nodes, and using the contravariant Piola transformation to preserve normal components on element boundaries. In terms of throughput, the implementation achieves up to an order of magnitude faster evaluation of finite element operators compared to a matrix-based evaluation. Correctness is demonstrated with optimal convergence rates for various polynomial degrees, as well as exactly divergence-free solutions for the velocity field.

Cette thèse est consacrée à l'étude des opérateurs strictement hyperboliques à coefficients peu réguliers, aussi bien qu'à l'étude du système d'Euler incompressible à densité variable. Dans la première partie, on montre des estimations a priori pour des opérateurs strictement hyperboliques dont les coefficients d'ordre le plus grand satisfont une condition de continuité log-Zygmund par rapport au temps et une condition de continuité log-Lipschitz par rapport à la variable d'espace. Ces estimations comportent une perte de dérivées qui croît en temps. Toutefois, elles sont suffisantes pour avoir encore le caractère bien posé du problème de Cauchy associé dans l'espace H^inf (pour des coefficients du deuxième ordre ayant assez de régularité).Dans un premier temps, on considère un opérateur complet en dimension d'espace égale à 1, dont les coefficients du premier ordre étaient supposés hölderiens et celui d'ordre 0 seulement borné. Après, on traite le cas général en dimension d'espace quelconque, en se restreignant à un opérateur de deuxième ordre homogène: le passage à la dimension plus grande exige une approche vraiment différente. Dans la deuxième partie de la thèse, on considère le système d'Euler incompressible à densité variable. On montre son caractère bien posé dans des espaces de Besov limites, qui s'injectent dans la classe des fonctions globalement lipschitziennes, et on établit aussi des bornes inférieures pour le temps de vie de la solution ne dépendant que des données initiales. Cela fait, on prouve la persistance des structures géométriques, comme la régularité stratifiée et conormale, pour les solutions de ce système. À la différence du cas classique de densité constante, même en dimension 2 le tourbillon n'est pas transporté par le champ de vitesses. Donc, a priori on peut s'attendre à obtenir seulement des résultats locaux en temps. Pour la même raison, il faut aussi laisser tomber la structure des poches de tourbillon. La théorie de Littlewood-Paley et le calcul paradifférentiel nous permettent d'aborder ces deux différents problèmes. En plus, on a besoin aussi d'une nouvelle version du calcul paradifférentiel, qui dépend d'un paramètre plus grand que ou égal à 1, pour traiter les opérateurs à coefficients peu réguliers. Le cadre fonctionnel adopté est celui des espaces de Besov, qui comprend en particulier les ensembles de Sobolev et de Hölder. Des classes intermédiaires de fonctions, de type logarithmique, entrent, elles aussi, en jeu / The present thesis is devoted both to the study of strictly hyperbolic operators with low regularity coefficients and of the density-dependent incompressible Euler system. On the one hand, we show a priori estimates for a second order strictly hyperbolic operator whose highest order coefficients satisfy a log-Zygmund continuity condition in time and a log-Lipschitz continuity condition with respect to space. Such an estimate involves a time increasing loss of derivatives. Nevertheless, this is enough to recover well-posedness for the associated Cauchy problem in the space $H^infty$ (for suitably smooth second order coefficients).In a first time, we consider acomplete operator in space dimension $1$, whose first order coefficients were assumed Hölder continuous and that of order $0$only bounded. Then, we deal with the general case of any space dimension, focusing on a homogeneous second order operator: the step to higher dimension requires a really different approach. On the other hand, we consider the density-dependent incompressible Euler system. We show its well-posedness in endpoint Besov spaces embedded in the class of globally Lipschitz functions, producing also lower bounds for the lifespan of the solution in terms of initial data only. This having been done, we prove persistence of geometric structures, such as striated and conormal regularity, for solutions to this system. In contrast with the classical case of constant density, even in dimension $2$ the vorticity is not transported by the velocity field. Hence, a priori one can expect to get only local in time results. For the same reason, we also have to dismiss the vortex patch structure. Littlewood-Paley theory and paradifferential calculus allow us to handle these two different problems .A new version of paradifferential calculus, depending on a parameter $ggeq1$, is also needed in dealing with hyperbolic operators with nonregular coefficients. The general framework is that of Besov spaces, which includes in particular Sobolev and Hölder sets. Intermediate classes of functions, of logaritmic type, come into play as well

La méthode IIM (Immersed Interface Method) permet d'étendre certaines méthodes numériques à des problèmes présentant des discontinuités. Elle est utilisée ici pour étudier un fluide incompressible régi par les équations de Navier-Stokes, dans lequel est immergée une membrane exerçant une force singulière. Nous utilisons une méthode de projection dans une grille de différences finies de type MAC. Une dérivation très complète des conditions de saut dans le cas où la viscosité est continue est présentée en annexe. Deux exemples numériques sont présentés : l'un sans membrane, et l'un où la membrane est immobile. Le cas général d'une membrane mobile est aussi étudié en profondeur. / The Immersed Interface Method allows us to extend the scope of some numerical methods to discontinuous problems. Here we use it in the case of an incompressible fluid governed by the Navier-Stokes equations, in which a membrane is immersed, inducing a singular force. We use a projection method and staggered (MAC-type) finite difference approximations. A very complete derivation for the jump conditions is presented in the Appendix, for the case where the viscosity is continuous. Two numerical examples are shown : one without a membrane, and the other where the membrane is motionless. The general case of a moving membrane is also thoroughly studied.

Recent technological advances are based on effectively using complex multiphysics concepts. Therefore, there is an ever increasing need for accurate numerical al-gorithms of reduced complexity for solving multiphysics problems. Traditional mesh-based simulation methods depend on a neighbor connectivity information for formulation of operators like derivatives. In large deformation problems, de-pendence on a mesh could prove a limitation in terms of accuracy and cost of preprocessing. Meshless methods obviate the need to construct meshes thus al-lowing simulations involving severe geometric deformations such as breakup of a contiguous domain into multiple fragments. Smoothed Particle Hydrodynamics (SPH) is a meshless particle based Lagrangian numerical method that has the longest continuous history of development ever since it was introduced in 1977. Commensurate with the significant growth in computational power, SPH has been increasingly applied to solve problems of greater complexity in fluid mechanics, solid mechanics, interfacial flows and astrophysics to name a few. The SPH approximation of the continuity and momentum equations govern-ing fluid flow traditionally involves a stiff equation of state relating pressure and density, when applied to incompressible flow problems. Incompressible Smoothed Particle Hydrodynamics (ISPH) is a variant of SPH that replaces this weak com-pressibility approach with a pressure equation that gives a hydrostatic pressure field which ensures a divergence-free (or density invariant) velocity field. The present study explains the development of an ISPH algorithm and its implementa-tion with focus on application to free surface flows, interaction of fluid with rigid bodies and coupling of incompressible fluids with a compressible second phase. Several improvements to the exiting ISPH algorithm are proposed in this study. A semi-analytic free surface model which is more accurate and robust compared to existing algorithms used in ISPH methods is introduced, validated against experi-ments and grid based CFD results. A surface tension model with specific applica-bility to free surfaces is presented and tested using 2D and 3D simulations. Using theoretical arguments, a volume conservation error in existing particle methods in general is demonstrated. A deformation gradient based approach is used to derive a new pressure equation which reduces these errors. The method is ap-plied to both free surface and internal flow problems and is shown to have better volume conservation and therefore reduced density fluctuations. Also, comments on instabilities arising from particle distributions are made and the role of the smoothing functions in such instabilities is discussed. The challenges in imple-menting the ISPH algorithm in a computer code are discussed and the experience of developing an in-house ISPH code is described. A parametric study on water entry of cylinders of different shapes, angular velocity and density is performed and aspects such as surface profiles, impact pressures and penetration velocities are compared. An analysis on the energy transfer between the solid and the fluid is also performed. Low Froude number water entry of a sphere is studied and the impact pressure is compared with the theoretical estimates. The Incompressible SPH formulation, employing the proposed improvements from the study is then coupled with a compressible SPH formulation to perform two phase flow simulations interacting compressible and incompressible fluids. To gain confidence in its applicability, the simulations are compared against the theoretical predication given by the Rayleigh-Plesset equation for the problem of compressible drop in an incompressible fluid.

L’objectif de cette thèse est de répondre au besoin d’accélérer la restitution des résultats de calcul d’un code CFD pour la simulation d’écoulements hydrodynamiques quasi-incompressibles. Ce code présente l’originalité de résoudre explicitement les équations de Navier-Stokes sous l’hypothèse de faible compressibilité avec des schémas numériques d’ordre élevé. Les développements effectués visent à réduire les temps de calcul à précision équivalente.Une première partie est consacrée à l’implémentation d’une formulation purement incompressible avec une résolution implicite de la pression par un schéma de projection. La formulation incompressible autorise des pas de temps plus grand en s’affranchissant de la vitesse du son, mais au prix d’une algorithmique plus complexe et de la nécessité de résoudre un système linéaire. La comparaison des deux formulations,faiblement-compressible et incompressible, tend à montrer la pertinence du schéma de projection pour les écoulements laminaires instationnaires.Un deuxième axe de développement a consisté en la proposition d’une amélioration de la méthode de frontière immergée initialement présente dans le code.Si les résultats obtenus ne sont pas encore pleinement satisfaisants, ils montrent que la montée en ordre d’une méthode de frontière immergée peut être moins contraignante en formulation incompressible.Enfin la dernière partie traite de l’immersion rapide et robuste de géométries complexes telles qu’elles peuvent être rencontrées dans l’industrie. La localisation géométrique par arbre octal permet d’évaluer rapidement une fonction de distance signée indispensable pour la méthode de frontière immergée. / An original strategy to address hydrodynamic flow was recently proposed through a high-order weakly-compressible Cartesian grid approach. The method is based on a fully-explicit temporal scheme for solving the Navier-Stokes equations. The present thesis aims to reduce the computational time required to obtain the results without deteriorating the accuracy.A first part is dedicated to the implementation of a truly incompressible formulation with an implicit solution for the pressure field through a projection scheme. The incompressible solver allows larger time step size for time integration since the speed of sound tends to infinity. In return the algorithms are no longer straight forward and a linear system has to be solved through the Pressure Poisson Equation. Comparisons carried out between both formulations show that the projection scheme can be better adapted to efficiently simulate unsteady viscous flows. Then an improvement of the immersed boundary method has been proposed. Results are not fully satisfactory for now. However, it seems easier to develop a numerical scheme for the incompressible approach rather than the weakly-compressible one.Finally, the last part addresses the setup up of complex triangulations in immersed boundary simulations. A fast and robust procedure is developed for distance computation with an octree data structure.