Dynamics of a Circular Foil and Two Pairs of Point Vortices: New Relative Equilibria and a Generalization of Helmholtz Leapfrogging

Abstract

In this paper, we study the plane-parallel motion of a circular foil interacting with two vortex pairs in an infinite volume of an ideal fluid. We assumed that the circulation of the velocity of the fluid around the foil was zero. We showed that the equations of motion possess an invariant submanifold such that the foil performed translational motion and the vortices were symmetric relative to the foil’s direction of motion. A qualitative analysis of the motion on this invariant submanifold was made. New relative equilibria were found, a bifurcation diagram was constructed, and a stability analysis is given. In addition, trajectories generalizing Helmholtz leapfrogging were found where the vortices passed alternately through each other, while remaining at a finite distance from the foil.

Keywords

• point vortices
• ideal fluid
• bifurcation diagram
• stability
• relative equilibria
• Poincaré map