Vortex motion around a circular cylinder

TL;DR

This paper analyzes the complex nonlinear dynamics of counter-rotating vortices around a circular cylinder, revealing new stability features and insights into vortex shedding phenomena.

Contribution

It presents a detailed phase portrait analysis of vortex dynamics, uncovering novel features like a nilpotent saddle point and stability regions, with implications for vortex shedding modeling.

Findings

Identification of a nilpotent saddle point at infinity.

Discovery of a stability region defined by homoclinic orbits.

Demonstration that downstream vortex pairs can move upstream against the flow.

Abstract

The motion of a pair of counter-rotating point vortices placed in a uniform flow around a circular cylinder forms a rich nonlinear system that is often used to model vortex shedding. The phase portrait of the Hamiltonian governing the dynamics of a vortex pair that moves symmetrically with respect to the centerline---a case that can be realized experimentally by placing a splitter plate in the center plane---is presented. The analysis provides new insights and reveals novel dynamical features of the system, such as a nilpotent saddle point at infinity whose homoclinic orbits define the region of nonlinear stability of the so-called F\"oppl equilibrium. It is pointed out that a vortex pair properly placed downstream can overcome the cylinder and move off to infinity upstream. In addition, the nonlinear dynamics resulting from antisymmetric perturbations of the F\"oppl equilibrium is studied and its relevance to vortex shedding discussed.