Global bifurcation of hollow vortex streets

TL;DR

This paper develops a mathematical framework to construct and analyze hollow vortex street solutions in fluid dynamics, revealing how vortex configurations can be smoothly transformed and identifying potential singular behaviors.

Contribution

It introduces a global bifurcation approach to desingularize point vortex configurations into hollow vortex streets, extending previous techniques to periodic flows.

Findings

Constructed global solution curves for hollow vortex streets.

Characterized singular behaviors at the extremes of solution curves.

Applied the theory to von Kármán, translating, and 2P vortex configurations.

Abstract

Vortex streets are periodic configurations of vortices propagating through an irrotational flow. In this paper, we study streets of hollow vortices, which are solutions to the free boundary $2$-d irrotational incompressible Euler equations. Each vortex core is a region of constant pressure in the complement of the fluid domain with a nonzero circulation around it. We prove that any non-degenerate singly-periodic point vortex configuration can be ``desingularized'' to create a global curve of solutions to the steady hollow vortex street problem, and we further characterize the types of singular behavior that can develop as one transverses the curve to its extreme. As specific examples, we study von K\'arm\'an vortex streets, translating vortex arrays, and a two-pair (2P) configuration. Our method is based on analytic global bifurcation theory and adapts the desingularization technique of Chen, Walsh, and Wheeler to the periodic setting.