Global bifurcation of hollow vortex streets

TL;DR

This work rigorously connects discrete periodic point-vortex configurations with full stationary hollow-vortex streets in the plane. Using an analytic global bifurcation approach and a periodic desingularization framework, the authors prove that any nondegenerate periodic point-vortex configuration yields a large family of hollow-vortex solutions via a near-identity conformal mapping, and they describe the two possible breakdown mechanisms along the global curve. The paper develops a periodic layer-potential formulation, reduces the problem to a nonlocal operator equation, and establishes local existence before extending to a global curve, with explicit applications to von Kármán streets, translating vortex arrays, and 2P configurations, including a wave-speed bound and symmetry-driven refinements. The results bridge point-vortex models and full free-boundary Euler hollow-vortex solutions, offering non-perturbative constructions and detailed geometric/analytic characterizations of the vortex streets.

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 Kármán 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.

Figures (3)

• Figure 1: An illustration of the physical domain $\mathscr{D}$ in a case that $M=2$. In this case, we have two hollow vortices per period
• Figure 2: Left: fundamental periodic strip for a starting configuration of four point vortices. Center: fundamental strip for the conformal domain $\mathcal{D}_\rho$ with $0 < |\rho|$. Right: one period of the corresponding physical domain $\mathscr{D} = f(\mathcal{D}_\rho)$. Each vortex boundary $\Gamma_k = f(\partial B_\rho(\zeta_k))$.
• Figure 3: A schematic illustration of the global bifurcation curve ${\mathscr C}$ given by Theorem \ref{['global desingularization theorem']}. It bifurcates from an initial (non-degenerate) point vortex figuration and contains the local curve ${\mathscr C}_{\mathrm{loc}}$ given by Theorem \ref{['local desingularization theorem']}. While ${\mathscr C}_{\mathrm{loc}}$ is parameterized by the conformal radius $\rho$, the global curve may have multiple turning points. It is also possible that secondary bifurcations occur, which are not pictured. As we follow ${\mathscr C}$ to its extreme, either $|\lambda(s)|$ is unbounded or else conformal degeneracy or velocity degeneracy occurs.

Theorems & Definitions (51)

• Theorem 1.1: Desingularization
• Theorem 1.2: Global continuation
• Corollary 1.3: von Kármán vortex street
• Remark 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof
• Proposition 2.5