Vortex scattering in the point vortex model

Abstract

Here we provide both a theoretical and numerical analysis of scattering in vortex collisionsusing the 2-D point vortex model, where both the dipole-vortex collision is considered in the three vortex case, and the dipole-dipole collision is considered in the four vortex case. We begin by presenting the basics of turbulence in classical fluids, and in quantum fluids which we then use to give an exposition of the 2-D point vortex system. The point vortex model is then given completely for the general case of N point vortices in an infinite domain. We express the system as a Hamiltonian formulation as done by Kirchhoff, with the equations of motion derived using Hamilton’s canonical equations. It is then considered how to best solve the system numerically. We form a numerical simulation using the famed Dormand-Prince adaptive time step Runge-Kutta scheme, as well as an elementary controller to adjust the time- step, pseudo-code is provided to demonstrate the method. The motion of two point vortices, in both the vortex/anti-vortex and identical vortex case, is then considered both numerically and theoretically as a check of the reliability of the simulations finding precise agreement between numerical and theoretical results. We then consider the crux of the current work, the dipole-vortex collision and the dipole- dipole collision, partly inspired by the work of Aref [H. Aref, Phys. Fluids 3 (1979), 393- 400]. We solve each case numerically, with an additional theoretical treatment in the dipole- vortex case. We solve the three vortex collision for the scattering angle, minimum lengths and critical lengths. Continuing, the four vortex collision is solved for the scattering angles and the minimum lengths. We find similarities in the results of both systems, and also note an apparent mistake in Aref’s work in the three vortex case, as well as noting regions of atypical vortex behavior in the four vortex case, possibly corresponding to regions of self-similar vortex collapse.