Dynamics of helical vortices and helical-vortex rings

TL;DR

This paper investigates the dynamics of helical vortices and rings, providing analytical solutions and explaining anomalous velocities through Hamiltonian mechanics, with implications for vortex motion in fluid dynamics.

Contribution

It introduces an analytical solution for helical-vortex rings using elliptical integrals and explains anomalous velocities via Hamiltonian relations, extending local-induction approximation.

Findings

Analytical solutions for helical-vortex rings in elliptical integrals.

Explanation of velocity suppression and inversion at large amplitudes.

Proposal of an extended local-induction approximation.

Abstract

The Letter considers dynamics of helical vortices and helical-vortex rings either solving directly the equations of motions of a vortex line or using canonical relations following from the Hamiltonian equations of motion. An analytical solution in elliptical integrals was found for helical-vortex rings in the local-induction approximation. The analysis based on the canonical Hamilton relation provides a clear physical explanation for anomalous velocities of helical-vortex rings, i.e., for suppression of the velocity and even inversion of its direction at sufficiently large amplitude of the helical distortion. The extended local-induction approximation is suggested, which provides an exact solution for the equations of motion of helical vortices and helical-vortex rings in the limit when the small-pitch helical vortex reduces to a cylindric sheet of uniform vorticity.