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Abstract(s)
A dinâmica dos vórtices e dos traçadores passivos em fluxos dominados por vórtices
forma uma vasta área de investigação que continua a atrair a atenção de numerosos
estudos, surgindo recentemente um interesse especial na utilização da teoria de
controlo aplicada à dinâmica de vórtices.
Os vórtices pontuais são soluções das equações bidimensionais incompressíveis de
Euler que correspondem ao caso limite em que a vorticidade está completamente
centralizada num número finito de pontos espaciais, cada um com uma determinada
circulação.
Nesta dissertação a grande preocupação é a dinâmica de um traçador passivo advinda
de um fluxo de vórtices de pontos bidimensionais, ou seja, pretende-se conduzir uma
partícula passiva de um ponto inicial a um ponto final, ambos dados a priori, num
determinado espaço de tempo finito sendo o fluxo provocado pelo deslocamento de
vórtices. Mais precisamente, pretende-se encontrar trajetórias ideais que minimizem a
função objetivo que corresponde à energia gasta no controlo das trajetórias, ou seja, no
movimento da partícula.
Para o deslocamento da partícula serão testados quatro casos diferentes, em que cada
caso será utilizado um dado nº de vórtices, o qual varia de um a quatro. Posteriormente
é necessário adotar uma estratégia de modo a encontrar o menor valor de energia gasto
no movimento assim como encontrar esse valor o de uma forma mais rápida,
permitindo assim otimizar o movimento da partícula.
Vortex dynamics and passive tracers in vortex-dominated flows form a vast area of research that continues to attract the attention of numerous studies, arising in recent times a special interest in the use of control theory applied to vortex dynamics. Point vortices are singular solutions of the two-dimensional incompressible Euler equations. These solutions correspond to the limiting case where the vorticity is completely concentrated on a finite number of spatial points, each with a prescribed strength/circulation. In this dissertation the great preoccupation it is the dynamics of a passive tracer advected by two-dimensional point vortex flow, that is, want to drive a passive particle from an initial starting point to a final terminal point, both given a priori, in a given finite time, being The flow is originated by the displacement of N viscous point vórtices. More precisely, we look for the optimal trajectories that minimize the objective function that correspond to the energy expended in the control of the trajectories. For the same movement of the particle, four different cases will be tested, in which each case a given number of vortexes will be used, which varies from one to four. Subsequently, it is necessary to adopt a strategy in order to find the least amount of energy spent on the movement as well as finding that value as quickly as possible, thus allowing to optimize the movement of the particle.
Vortex dynamics and passive tracers in vortex-dominated flows form a vast area of research that continues to attract the attention of numerous studies, arising in recent times a special interest in the use of control theory applied to vortex dynamics. Point vortices are singular solutions of the two-dimensional incompressible Euler equations. These solutions correspond to the limiting case where the vorticity is completely concentrated on a finite number of spatial points, each with a prescribed strength/circulation. In this dissertation the great preoccupation it is the dynamics of a passive tracer advected by two-dimensional point vortex flow, that is, want to drive a passive particle from an initial starting point to a final terminal point, both given a priori, in a given finite time, being The flow is originated by the displacement of N viscous point vórtices. More precisely, we look for the optimal trajectories that minimize the objective function that correspond to the energy expended in the control of the trajectories. For the same movement of the particle, four different cases will be tested, in which each case a given number of vortexes will be used, which varies from one to four. Subsequently, it is necessary to adopt a strategy in order to find the least amount of energy spent on the movement as well as finding that value as quickly as possible, thus allowing to optimize the movement of the particle.
Description
Keywords
Vórtice Pontos bidimensionais