In Hamiltonian systems with symmetry, many previous studies have centred their attention on compact symmetry groups, but relatively little is known about the effects of noncompact groups. This thesis investigates the properties of the system of N point vortices on the hyperbolic plane H2, which has noncompact symmetry SL (2, R).The Poisson Hamiltonian structure of this dynamical system is presented and the relative equilibria conditions are found. We also describe the trajectories of relative equilibria with momentum value not equal to zero. Finally, stability criteria are found for a number of cases, focusing on N = 2, 3. These results are placed in context with the study of point vortices on the sphere, which has compact symmetry.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20154号 / 理博第4239号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 上田 哲生, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The thesis develops singly-periodic mathematical models for complex laminar wakes which are formed behind vortex-shedding bluff bodies. These wake structures exhibit a variety of patterns as the bodies oscillate or are in close proximity of one another. The most well-known formation comprises two counter-rotating vortices in each shedding cycle and is popularly known as the vk vortex street. Of the more complex configurations, as a specific example, this thesis investigates one of the most commonly occurring wake arrangements, which consists of two pairs of vortices in each shedding period. The paired vortices are, in general, counter-rotating and belong to a more general definition of the 2P mode, which involves periodic release of four vortices into the flow. The 2P arrangement can, primarily, be sub-classed into two types: one with a symmetric orientation of the two vortex pairs about the streamwise direction in a periodic domain and the other in which the two vortex pairs per period are placed in a staggered geometry about the wake centerline. The thesis explores the governing dynamics of such wakes and characterizes the corresponding relative vortex motion. In general, for both the symmetric as well as the staggered four vortex periodic arrangements, the thesis develops two-dimensional potential flow models (consisting of an integrable Hamiltonian system of point vortices) that consider spatially periodic arrays of four vortices with their strengths being +/-1 and +/-2. Vortex formations observed in the experiments inspire the assumed spatial symmetry. The models demonstrate a number of dynamic modes that are classified using a bifurcation analysis of the phase space topology, consisting of level curves of the Hamiltonian. Despite the vortex strengths in each pair being unequal in magnitude, some initial conditions lead to relative equilibrium when the vortex configuration moves with invariant size and shape. The scaled comparisons of the model results with experiments conducted in a flowing soap film with an airfoil, which was imparted with forced oscillations, are satisfactory and validate the reduced order modeling framework. The experiments have been performed by a collaborator group at the Department of Physics and Fluid Dynamics at the Technical University of Denmark (DTU), led by Dr. Anders Andersen. Similar experiments have also been run at Virginia Tech as part of this dissertation and the preliminary results are included in this treatise. The thesis also employs the same dynamical systems techniques, which have been applied to study the 2P regime dynamics, to develop a mathematical model for the P+S mode vortex wakes, with three vortices present in each shedding cycle. The model results have also been compared favorably with an experiment and the predictions regarding the vortex circulation data match well with the previous results from literature. Finally, the thesis introduces a novel concept of clean and renewable energy extraction from vortex-induced vibrations of bluff bodies. The slow-moving currents in the off-shore marine environments and riverine flows are beyond the operational capabilities of the more established hydrokinetic energy converters and the discussed technology promises to be a significant tool to generate useful power from these copiously available but previously untapped sources. / Ph. D.

A dynamical system approach is utilized to reduce the representation order of unsteady fluid flows and fluid-structure interaction systems. This approach allows for significant reduction in the computational cost of their numerical simulations, implementation of optimization and control methodologies and assessment of their dynamic stability. In the first chapter, I present a new Lagrangian function to derive the equations of motion of unsteady point vortices. This representation is a reconciliation between Newtonian and Lagrangian mechanics yielding a new approach to model the dynamics of these vortices. In the second chapter, I investigate the flutter of a helicopter rotor blade using finite-state time approximation of the unsteady aerodynamics. The analysis showed a new stability region that could not be determined under the assumption of a quasi-steady flow. In the third chapter, I implement the unsteady vortex lattice method to quantify the effects of tail flexibility on the propulsive efficiency of a fish. I determine that flexibility enhances the propulsion. In the fourth chapter, I consider the stability of a flapping micro air vehicle and use different approaches to design the transition from hovering to forward flight. I determine that first order averaging is not suitable and that time periodic dynamics are required for the controller to achieve this transition. In the fifth chapter, I derive a mathematical model for the free motion of a two-body planar system representing a fish under the action of coupled dynamics and hydrodynamics loads. I conclude that the psicform fish family are inherently stable under certain conditions that depend on the location of the center of mass. / Ph. D. / We present modeling approaches of the interaction between flying or swimming bodies and the surrounding fluids. We consider their stability as they perform special maneuvers. The approaches are applied to rotating blades of helicopters, fish-like robots, and micro-air vehicles. We develop and validate a new mathematical representation for the flow generated by moving or deforming elements. We also assess the effects of fast variations in the flow on the stability of a rotating helicopter blade. The results point to a new stable regime for their operation. In other words, the fast flow variations could stabilize the rotating blades. These results can also be applied to the analysis of stability of rotating blades of wind turbines. We consider the effects of flexing a tail on the propulsive force of fish-like robots. The results show that adding flexibility enhances the efficiency of the fish propulsion. Inspired by the ability of some birds and insects to transition from hovering to forward motion, we thoroughly investigate different approaches to model and realize this transition. We determine that no simplification should be applied to the rigorous model representing the flapping flight in order to model transition phenomena correctly. Finally, we model the forward-swim dynamics of psciform and determine the condition on the center of mass for which a robotic fish can maintain its stability. This condition could help in designing fish-like robots that perform stable underwater maneuvers.