On the dynamics of a free vortex pair in the presence of a fixed vortex

Abstract

The dynamics of a constrained three-vortex problem, a free point vortex pair in the velocity field of a fixed point vortex, is investigated. The underlying dynamical system is simplified using a coordinate transformation and categorized into two cases based on the zero and non-zero values of the constant of angular impulse. For each case, dynamical features of the vortex motion are studied analytically in the transformed plane, in order to completely classify the vortex motions and to understand the boundedness and periodicity of the inter-vortex distances. The theoretical predictions are also verified numerically and illustrated for various sets of initial conditions and circulations.