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Vortex motion around a circular cylinder both in an unbounded domain and near a plane boundary

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Previous issue date: 2012-05-17 / Nessa disserta ̧c ̃ao estudamos a dinˆamica de v ́ortices pr ́oximos a fronteiras s ́olidas emum fluido ideal, atrav ́es do modelo de v ́ortices puntiformes. Obtivemos as configura ̧c ̃oesestacion ́arias de v ́ortices na presen ̧ca de um cilindro circular colocado em um escoamentouniforme e investigamos suas propriedades de estabilidadesob pequenas perturba ̧c ̃oes.Dois sistemas distintos foram estudados. Consideramos inicialmente o caso cl ́assico deum cilindro circular colocado em um escoamento uniforme ilimitado. Nesse caso, comose sabe, um par de v ́ortices com sentidos opostos ́e observado na esteira do cilindro, paran ́umeros de Reynolds at ́e cerca de 50, ao passo que para n ́umeros de Reynolds maiores,essa configura ̧c ̃ao torna-se inst ́avel dando lugar `a emiss ̃ao alternada de v ́ortices. Estesistema foi tratado analiticamente pela primeira vez, atrav ́es de um modelo de v ́orticespuntiformes, por F ̈oppl em 1913. Na primeira parte dessa disserta ̧c ̃ao, o modelo deF ̈oppl ́e revisto e v ́arias caracter ́ısticas novas desse sistema s ̃ao apresentadas, incluindoa existˆencia de um ponto de sela nilpotente no infinito, at ́eent ̃ao n ̃ao percebido, cujas ́orbitas homocl ́ınicas definem a regi ̃ao de estabilidade n ̃ao-linear do chamado equil ́ıbrio deF ̈oppl. Al ́em disso, estudamos tamb ́em a dinˆamica n ̃ao-linear resultante de perturba ̧c ̃oesanti-sim ́etricas do equil ́ıbrio de F ̈oppl e discutimos suarelevˆancia para a emiss ̃ao alternadade v ́ortices. Na segunda parte, consideramos o movimento de um v ́ortice em torno deum cilindro circular colocado acima de uma parede plana infinita. Em experimentos comesse arranjo, um v ́ortice estacion ́ario ́e observado na frente do cilindro, uma situa ̧c ̃ao quen ̃ao ́e encontrada no caso cl ́assico (i.e., sem o plano). Para estudar a dinˆamica de v ́orticesnessa situa ̧c ̃ao, a regi ̃ao do fluido ́e inicialmente mapeada em um anel em um planocomplexo auxiliar, e o potencial complexo correspondente ́e ent ̃ao obtido em termos dachamada fun ̧c ̃ao prima de Schottky-Klein, que neste caso pode ser escrita em termos defun ̧c ̃oes el ́ıpticas. As configura ̧c ̃oes estacion ́arias s ̃ao ent ̃ao calculadas e suas propriedadesde estabilidade s ̃ao determinadas. Discutimos tamb ́em, como as solu ̧c ̃oes do modelo dev ́ortice puntiforme podem ajudar a explicar as observa ̧c ̃oes experimentais envolvendo aforma ̧c ̃ao de v ́ortices na frente de um cilindro colocado pr ́oximo a um plano. / In this thesis the dynamics of vortices near solid boundaries in an ideal fluid is studiedusing the point vortex model. Stationary configurations of vortices in the presence of acircular cylinder placed in a uniform stream are obtained and their stability propertiesunder small disturbances are investigated. Two different systems are studied. First, theclassical case of a circular cylinder placed in a uniform stream in an otherwise unboundeddomain is considered. As is well known, in this case a pair of counter-rotating eddies isobserved downstream of the cylinder for Reynolds numbers upto about 50, whereas forlarger Reynolds number this configuration becomes unstable, leading to vortex shedding.This system was first treated analytically using point vortices by F ̈oppl in 1913. In thefirst part of the thesis, the F ̈oppl model is revisited and several novel features of this sys-tem are presented, including the existence of a hitherto unnoticed nilpotent saddle pointat infinity whose homoclinic orbits define the region of nonlinear stability of the so-calledF ̈oppl equilibrium. In addition, the nonlinear dynamics resulting from antisymmetricperturbations of the F ̈oppl equilibrium is studied and its relevance to vortex sheddingis discussed. In the second part, the motion of a vortex around a cylinder placed abovean infinite plane wall is considered. In experiments using this arrangement, a stationaryeddy is observed in front of the cylinder, a situation that isnot found in the classical case(i.e., without the plane). To study the vortex dynamics in this case, the flow domain isfirst mapped to an annulus in an auxiliary complex plane and the corresponding complexpotential is obtained in terms of the so-called Schottky-Klein prime function, which inthis case can be written in terms of elliptic functions. The stationary configurations arethen calculated and their stability properties are determined. It is also discussed how thesolutions of the point vortex model can help to explain the experimental findings for thevortex formation in front of a cylinder placed near a plane.

Identiferoai:union.ndltd.org:IBICT/oai:repositorio.ufpe.br:123456789/20014
Date17 May 2012
CreatorsMOURA, Marcel Nascimento de
Contributorshttp://lattes.cnpq.br/1091830046970956, VASCONCELOS, Giovani Lopes
PublisherUniversidade Federal de Pernambuco, Programa de Pos Graduacao em Fisica, UFPE, Brasil
Source SetsIBICT Brazilian ETDs
LanguageBreton
Detected LanguageEnglish
Typeinfo:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/masterThesis
Sourcereponame:Repositório Institucional da UFPE, instname:Universidade Federal de Pernambuco, instacron:UFPE
Rightsinfo:eu-repo/semantics/openAccess

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