Chaotic capture of vortices by a moving body. I. The single point vortex case

The study of the dynamical properties of vortex systems is an important and topical research area, and is becoming of ever increasing usefulness to a variety of physical applications. In this paper, we present a study of a model of a rotational singularity which obeys a logarithmic potential interacting with a bluff body in a uniform inviscid laminar flow, e.g., a line vortex interacting with a cylinder in three dimensions or a point vortex with a circular boundary in two dimensions. We show that this system is Hamiltonian and simple enough to be solved analytically for the stagnation points and separatrices of the flow, and a bifurcation diagram for the relevant parameters and classification of the various types of motion is given. We also show that, by introducing a periodic perturbation to the body, chaotic motion of the vortex can be readily generated, and we present analytic criteria for the generation of chaos using the Poincare-Melnikov-Arnold method. This leads to an important dynamical effect for the model, i.e., that the possibility exists for the vortex to be chaotically captured around the body for periods of time which are extremely sensitive to initial conditions. The basic mechanism for this capture is due to the chaotic dynamics and is similar to that of other chaotic scattering phenomena. We show numerically that cases exist where the vortex can be captured around an elliptic point external to (and possibly far from) the body, and the existence of other very complicated motions are also demonstrated. Finally, generalizations of the problem of the vortex-body interaction are indicated, and some possible applications are postulated such as the interaction of line vortices with aircraft wings.