Equilibrium And Stability In Vortex And Wave Flows

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This dissertation focuses on the development of theoretical and numerical methodologies to study equilibrium and stability in conservative fluid flows. These techniques include: a bifurcation-diagram approach to obtain the stability properties of families of steady flows; a theory of Hamiltonian resonance for vortex arrays; an efficient numerical method for computing vortices with arbitrary symmetry; and a variational principle for compressible, barotropic or baroclinic flows. We employ these theoretical and numerical approaches to obtain new results regarding the structure and stability of several fundamental vortex and wave flows. The applications that we examine involve simple representations of fundamental fluid problems, which may be regarded as prototypical of flows associated with transport and mixing in the ocean and in the atmosphere, with aquatic animal propulsion, and with the dynamics of vortices in quantum condensates. We address two issues affecting the use of a variational argument to determine stability of families of steady flows. By building on ideas from bifurcation theory, we link turning points in a velocity-impulse diagram to gains or losses of stability. We introduce concepts from imperfection theory into these problems, enabling us to reveal hidden solution branches. The resulting methodology detects exchanges of stability through an "imperfect velocity-impulse" (IVI) diagram. We apply the IVI diagram approach to wide variety of vortex and wave flows. These examples include elliptical vortices, translating and ro- tating vortex pairs, single and double vortex rows, distributed vortices, as well as steep gravity waves. For a few of the flows considered, our work yields the first available stability boundaries. In addition, the IVI diagram methodology leads us to the discovery of several new families of steady flows, which exhibit lower symmetry. We next examine conditions for the development of an oscillatory instability in two-dimensional vortex arrays. By building on the theory of Krein signatures for Hamiltonian systems, we show that a resonant instability cannot occur for one or two vortices. To predict the onset of resonance for three or more vortices, we develop a simple approximate technique, which compares favorably with full analyses. In addition, we propose a simple technique to immediately check the accuracy of a detailed linear stability analysis. All of the uniform-vorticity equilibria analyzed in this dissertation were computed using a newly developed numerical approach. This methodology, which is based on Newton iteration, employs a new discretization to radically increases the efficiency of the calculation. In addition, we introduce a procedure to remove the degeneracies in the steady vorticity equation, thus ensuring convergence for general vortex configurations. Our method enables the computation, for the first time, of steady vortices that do not exhibit any geometric symmetry, in an unbounded flow. Finally, we re-examine the variational principle that underpins the IVI diagram stability approach. We show that this principle may be obtained, in a conceptually straightforward manner, by first considering the classical principle of virtual work. This link enables us to readily formulate generalizations to compressible, barotropic and baroclinic flows.

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Vortex Dynamics; Fluid Stability; Variational Principles


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Williamson, Charles Harvey Kaye

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Leibovich, Sidney
Collins, Lance

Degree Discipline

Aerospace Engineering

Degree Name

Ph. D., Aerospace Engineering

Degree Level

Doctor of Philosophy

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dissertation or thesis

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