Vortex polygons: Dynamics and associated particle advection

Abstract

This paper presents a numerical study of the advection of passive and active particles by three and four equal vortices located on the vertices of a polygon. The vortices, which have either singular or uniform vorticity distribution, are immersed in an incompressible, unbounded, and inviscid fluid. Under these conditions, a regular polygon rotates steadily if it consists of point vortices and unsteadily if it consists of Rankine vortices. When the point-vortex polygon is perturbed by making it slightly irregular, the flow becomes time periodic. In this case, dynamical-system methods, such as lobe dynamics and Poincaré maps, serve to compute the fluid exchanged between different regions and the area of the chaotic sea. Both quantities are found to grow with the amplitude of the perturbation: the former does it in a weakly nonlinear way, the latter in a piecewise linear manner. The Rankine-vortex polygons always produce a time-aperiodic flow, which depends on their relative size. Small vortices deform slightly and produce particle advection, which is analogous to the perturbed point-vortex case; large vortices deform strongly and merge to form a single one. The critical distance for merger is found to be δ/a≈3.6 and δ/a≈3.2 for three and four vortices, respectively, where a is the vortex radius, and δ is the side length of the polygon. In both cases, the vortices expel the largest amount of vortical fluid at their critical distance, thus producing the least efficient merger.