ABSTRACT SIMULATION OF SURFACE WAVES GENERATED BY A RAPID RISE OF A BLOCK AT THE SEA BOTTOM SENOL, Nalan M.Sc., Department of Civil Engineering, Supervisor: Assoc. Prof. Dr. ismail AYDIN July 2005, 74 Pages A mathematical model is developed for investigating time dependent surface deformations of a hydrostatic water volume, when it is subjected to a sudden partial rise of the sea bottom. In the model, 2-dimensional, compressible, and viscous Navier-Stokes equations are solved by Marker and Cell (MAC) method. Variable mesh size in both horizontal and vertical directions with a staggered grid arrangement is used. Limited compressibility model is utilized for pressure. Various computational tests are done for the selection of computational parameters of the model. It is found that the amplitude of surface waves generated by vertical displacements of the sea bottom depends on size and speed of bottom displacements.

The aim of this study is to develop a code that is capable of solving three-dimensional compressible flows which are viscous and turbulent, and parallelization of this code. Purpose of parallelization is to obtain a computational efficiency in time respect which enables the solution of complex flow problems in reasonable computational times. In the first part of the study, which is the development of a three-dimensional Navier-Stokes solver for turbulent flows, first step is to develop a two-dimensional Euler code using Roe flux difference splitting method. This is followed by addition of sub programs involving calculation of viscous fluxes. Third step involves implementation of Baldwin-Lomax turbulence model to the code. Finally, the Euler code is generalized to three-dimensions. At every step, code validation is done by comparing numerical results with theoretical, experimental or other numerical results, and adequate consistency between these results is obtained. In the second part, which is the parallelization of the developed code, two-dimensional code is parallelized by using Message Passing Interface (MPI), and important improvements in computational times are obtained.

Thesis (Doctora).

Mémoire (M.SC.A.)--Université du Québec à Chicoutimi, 1986. / Document électronique également accessible en format PDF. CaQCU

Mémoire (M.Eng.)--Université du Québec à Chicoutimi, 1996. / Document électronique également accessible en format PDF. CaQCU

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The present work shows Cauchy's Theorem on its classical form, and it has the objective of weakening their statements, and provides aplications in continuum mechanics. The metodology used here is the axiomatic, that is, a presentation of basic concepts with fundamental results and theirs proofs. The main result here is Theorem 5, since it shows that for a Cauchy ux weakly balanced with density f there is a field such that f is linear in almost every points of R. The conclusion obtained is we can substitute the statement that f is a continuum function of position and have similar conclusions. / O presente trabalho apresenta o Teorema de Cauchy em sua forma clássica e tem por finalidade enfraquecer as suas hipóteses, proporcionando sua aplicação na mecânica do contínuo. A metodologia empregada é a axiomática, ou seja, é apresentada uma listagem de definições básicas com vistas ao desencadeamento lógico das demonstrações que foram realizadas para atingir os objetivos dessa dissertação. O resultado principal é teorema 15, pois mostra que para um Fluxo de Cauchy Fracamente Balanceado com densidade f existe um campo T:R−→L(R3) tal que f é linear em quase todos os pontos de R. A conclusão obtida é que podemos substituir a hipótese de que a referida função e contínua na variável espacial e obter conclusão semelhante.

Le titre de ma thèse de doctorat est "Equations de Stokes et de Navier-Stokes avec la con- dition de Navier", où j’ai considéré l’écoulement d’un fluide newtonien visqueux, incompressible dans un domaine borné de R3. L’écoulement du fluide est décrit par les équations bien connues de Navier-Stokes, données par le système suivant ∂t − ∆u + (u • ∇)u + ∇π = 0, div u = 0 dans Ω × (0, T )u • n = 0, 2[(Du)n]τ + αuτ = 0 sur Γ × (0, T )u(0) = u0 dans Ω (0.1) dans un domaine borné Ω ⊂ R3 de frontière Γ, éventuellement non simplement connexe, de classe C1,1. La vitesse initiale u0 et le coefficient de friction α, scalaire, sont des fonctions don- nées. Les vecteurs unitaires normal extérieur et tangents à Γ sont notés n et τ respectivement et Du = 1 (∇u + ∇uT ) est le tenseur des déformations. Les fonctions u et π décrivent respective- ment les champs de vitesses et de pression du fluide dans Ω satisfaisant la condition aux limites (0.1.2).Cette condition aux limites, proposée par H. Navier en 1823, a été abondamment étudiée ces dernières années, qui pour de nombreuses raisons convient parfois mieux que la condition aux limites de Dirichlet sans glissement : elle offre plus de liberté et est susceptible de fournir une solution physiquement acceptable au moins pour certains des phénomènes paradoxaux résultant de la condition de non-glissement, comme par exemple le paradoxe de D’Alembert ou le paradoxe de non-collision.Ma thèse comporte trois parties. Dans la première, je cherche à savoir si le problème (0.1) est bien posé en théorie Lp, en particulier l’existence, l’unicité de solutions faibles, fortes dans W 1,p(Ω) et W 2,p(Ω) pour tout p ∈ (1, ∞), en considérant la régularité minimale du coefficient de friction α. Ici α est une fonction, pas simplement une constante qui reflète les diverses propriétés du fluide et/ou de la frontière, ce qui nous permet d’analyser le comportement de la solution par rapport au coefficient de frottement.Utilisant le fait que les solutions sont bornées indépendamment de α, on montre que la solution des équations de Navier-Stokes avec la condition de Navier converge fortement vers une solution des équations de Navier-Stokes avec la condition de Dirichlet, correspondant à la même donnée initiale dans l’espace d’énergie lorsque α → ∞. Des résultats similaires ont été obtenus pour le cas stationnaire.Le dernier chapitre concerne les estimations pour le problème de Robin pour le laplacien : l’opérateur elliptique de second ordre suivant, sous forme divergentielle dans un domaine bornéΩ ⊂ Rn de classe C1, avec la condition aux limites de Robin a été considéré div(A∇)u = divf + F dans Ω, ∂u+ αu = f n + g sur Γ.∂n (0.2) Les coefficients de la matrice symétrique A sont supposés appartenir à l’espace V MO(R3). Aussi α est une fonction appartenant à un certain espace Lq . En plus de prouver l’existence, l’unicité de solutions faibles et fortes, nous obtenons une borne sur u, uniforme par rapport à α pour α suffisamment large, en norme Lp. Pour plus de clarté, nous avons étudié séparément les deux cas: l’estimation intérieure et l’estimation au bord. / My PhD thesis title is "Navier-Stokes equations with Navier boundary condition" where I have considered the motion of an incompressible, viscous, Newtonian fluid in a bounded do- main in R3. The fluid flow is described by the well-known Navier-Stokes equations, given by thefollowing system 1 )t − L1u + (u ⋅ ∇)u + ∇n = 0, div u = 01u ⋅ n = 0, 2[(IDu)n]r + aur = 0 in Q × (0, T )on Γ × (0, T ) (0.1) 11lu(0) = u0 in Qin a bounded domain Q ⊂ R3 with boundary Γ, possibly not connected, of class C1,1. The initialvelocity u0 and the (scalar) friction coefficient a are given functions. The unit outward normal and tangent vectors on Γ are denoted by n and r respectively and IDu = 1 (∇u + ∇uT ) is the rate of strain tensor. The functions u and n describe respectively the velocity2 and the pressure of a fluid in Q satisfying the boundary condition (0.1.2).This boundary condition, first proposed by H. Navier in 1823, has been studied extensively in recent years, among many reasons due to its contrast with the no-slip Dirichlet boundary condition: it offers more freedom and are likely to provide a physically acceptable solution at least to some of the paradoxical phenomenons, resulting from the no-slip condition, for example, D’Alembert’s paradox or no-collision paradox.My PhD work consists of three parts. primarily I have discussed the Lp -theory of well-posedness of the problem (0.1), in particular existence, uniqueness of weak and strong solutions in W 1,p (Q) and W 2,p (Q) for all p ∈ (1, ∞) considering minimal regularity on the friction coefficienta. Here a is a function, not merely a constant which reflects various properties of the fluid and/or of the boundary. Moreover, I have deduced estimates showing explicitly the dependence of u on a which enables us to analyze the behavior of the solution with respect to the friction coefficient.Using this fact that the solutions are bounded with respect to a, we have shown the solution of the Navier-Stokes equations with Navier boundary condition converges strongly to a solution of the Navier-Stokes equations with Dirichlet boundary condition corresponding to the sameinitial data in the energy space as a → ∞. The similar results have also been deduced for thestationary case.The last chapter is concerned with estimates for a Laplace-Robin problem: the following second order elliptic operator in divergence form in a bounded domain Q ⊂ Rn of class C1, withthe Robin boundary condition has been considered1div(A∇)u = divf + F in Q, 11 )u + u = f ⋅ n + g on Γ. (0.2) 2The coefficient matrix A is symmetric and belongs to V MO(R3). Also a is a function belonging to some Lq -space. Apart from proving existence, uniqueness of weak and strong solutions, we obtain the bound on u, uniform in a for a sufficiently large, in the Lp -norm. We have separately studied the two cases: the interior estimate and the boundary estimate to make the main idea clear in the simple set up.

Neste trabalho, utilizando o método espectral de Galerkin, provamos a existência de soluções fracas (quando a dimensão n é maior que 2) e existência e unicidade de soluções fracas (quando a dimensão é 2) para um sistema de equações diferenciais parciais que descrevem o movimento de um fluido quimicamente ativo em um domínio limitado em Rn, n 2≥2. / In this work, by using the spectral Galerkin method, we prove the existence of weak solutions (when the dimension n is great than 2) and existence and uniqueness of weak solutions (when the dimension is 2) for a system of partial differential equations that describes the motion of a chemical active fluid in a bounded domain in Rn, n≥2.

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.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T16:40:40Z No. of bitstreams: 1 DÉSIO RAMIREZ DA ROCHA SILVA - DISSERTAÇÃO PPGMAT 2010..pdf: 641903 bytes, checksum: 9b0b6f008468c08a7a9d581e14ef0d13 (MD5) / Made available in DSpace on 2018-07-25T16:40:41Z (GMT). No. of bitstreams: 1 DÉSIO RAMIREZ DA ROCHA SILVA - DISSERTAÇÃO PPGMAT 2010..pdf: 641903 bytes, checksum: 9b0b6f008468c08a7a9d581e14ef0d13 (MD5) Previous issue date: 2010-09 / CNPq / Capes / Neste trabalho, baseado numa seqüência de artigos de David Ho , é provado um teorema sobre a existência de uma solução fraca para um problema de valor inicial envolvendo as equações de Navier-Stokes para o caso de um escoamento unidimensional de um fluido compressível. São consideradas como hipóteses básicas a ausência de forças externas e que a pressão seja uma função contínua positiva crescente da densidade, cuja derivada também seja contínua. Quanto aos dados iniciais, estes podem possuir descontinuidades do tipo salto, não necessariamente pequenos, podendo se comportar inclusive como funções constantes por partes, em particular dados de Riemann. Tal teorema é provado baseado numa seqüência de lemas e proposições que fornecem estimativas para soluções aproximadas suaves obtidas a partir de dados regularizados. A solução nal é obtida por um processo de passagem ao limite das soluções aproximadas / In this work, based on a serie of papers by David Ho , it is proved a theorem on the existence of a weak solution to the initial value problem for the Navier-Stokes equations for a one space dimension ow of a compressible uid. It is assumed the absence of external forces and that the pressure is a continuous positive increasing function of density with the derivative also continuous. Concerning the initial data, they are allowed to have large jump discontinuities, such as piecewise constant functions, in particular Riemann data. The proof of the theorem is based on a sequence of lemmas and propositions which give estimates on the approximate smooth solutions obtained under regularized data. The nal solution is obtained by a limit process on the approximate solutions.