Vortex Dynamics in Domains Whith Boundaries

Abstract

In this thesis we consider the following problems: 1) The problem of fluid advection excited by point vortices in the presence of stationary cylinders (we also add a uniform flow to the systems). 2) The problem of motion of one vortex (or vortices) around cylinder(s). We also investigate integrable and chaotic cases of motion of two vortices around an oscillating cylinder in the presence of a uniform flow. In the fluid advection problems Milne-Thomson's Circle theorem and an analyticalnumerical solution in the form of an infinite power series are used to determine flow fields and the forces on the cylinder(s) are calculated by the Blasius theorem. In the "two vortices-one cylinder" case we generalize the problem by adding independent circulation k0 around the cylinder itself. We then write the conditions for force to be zero on the cylinder. The Hamiltonian for motion of two vortices in the case with no uniform flow and stationary cylinder is constructed and reduced. Also constant Hamiltonian (energy) curves are plotted when the system is shown to be integrable according to Liouville's definition. By adding uniform flow to the system and by allowing the cylinder to vibrate, we model the natural vibration of the cylinder in the flow field, which has applications in ocean engineering involving tethers or pipelines in a flow field. We conclude that in the chaotic case, forces on the cylinder may be considerably larger than those on the integrable case depending on the initial positions of the vortices, and that complex phenomena such as chaotic capture and escape occur when the initial positions lie in a certain region.