On the motion of three point vortices in a periodic strip

Abstract

Motivated by observations of Williamson & Roshko of the wake of an oscillating cylinder with three vortices per cycle, and by the analyses of Rott and Aref of the motion of three vortices with vanishing net circulation on the unbounded plane, the integrable problem of three interacting, periodic vortex rows is solved. The problem is ‘mapped’ onto a problem of advection of a passive particle by a certain set of fixed point vortices. The results of this mapped problem are then re-interpreted in terms of the motion of the vortices in the original problem. A rather complicated structure of the solution space emerges with a surprisingly large number of regimes of motion, some of them somewhat counter-intuitive. Representative cases are analysed in detail, and a general procedure is indicated for all cases. We also trace the bifurcations of the solutions with changing linear momentum of the system. For rational ratios of the vortex circulations all motions are periodic. For irrational ratios this is no longer true. The point vortex results are compared to the aforementioned wake experiments and appear to shed light on the experimental observations. Many additional possibilities for the wake dynamics are suggested by the analysis.