Four-vortex motion around a circular cylinder

TL;DR

This paper analyzes the complex dynamics of four vortices around a circular cylinder, revealing stable configurations for same-signed vortices and unstable equilibria for opposite-signed vortices, using analytical and numerical methods.

Contribution

It provides new insights into vortex equilibria around a cylinder, including stable configurations and the absence of certain equilibria, with stability analysis for symmetric and antisymmetric perturbations.

Findings

Existence of linearly stable equilibria for same-signed vortex pairs.

No equilibrium for opposite-signed vortices on opposite sides of the cylinder.

Discovery of a new family of opposite-signed equilibria on the normal line.

Abstract

The motion of two pairs of counter-rotating point vortices placed in a uniform flow past a circular cylinder is studied analytically and numerically. When the dynamics is restricted to the symmetric subspace---a case that can be realized experimentally by placing a splitter plate in the center plane---, it is found that there is a family of linearly stable equilibria for same-signed vortex pairs. The nonlinear dynamics in the symmetric subspace is investigated and several types of orbits are presented. The analysis reported here provides new insights and reveals novel features of this four-vortex system, such as the fact that there is no equilibrium for two pairs of vortices of opposite signs on the opposite sides of the cylinder. (It is argued that such equilibria might exist for vortex flows past a cylinder confined in a channel.) In addition, a new family of opposite-signed equilibria on the normal line is reported. The stability analysis for antisymmetric perturbations is also carried out and it shows that all equilibria are unstable in this case.