On the existence of leapfrogging pair of circular vortex filaments

Abstract

We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. As a preliminary analysis, we first consider the case when the filaments are arranged as straight and parallel lines, and explicitly solve the system to show that the motion of the lines resemble that of point vortices moving in a plane. Then, we consider the motion of a pair of coaxial circular vortex filaments. We show that in this case, the system can be reduced to a two‐dimensional Hamiltonian system. Based on this formulation, we give a condition for the initial configuration and parameters of the filaments for leapfrogging to occur, and prove that the condition is in fact necessary and sufficient.

Collinear interaction of vortex pairs with different strengths—Criteria for leapfrogging

“…The criteria we have presented can be expected to be in only qualitative accord with those presented in Borisov, Kilin, and Mamaev [13] and in Aiki [4].…”

“…The rear ring then passes through the leading ring, with this process of 'leapfrogging' then repeating again and again [23]. We note that interactions of such vortex rings -primarily with filament cores -have been considered in Hicks [20] and more recently in Borisov, Kilin, and Mamaev [13], in Caplan, Talley, Carretero-González, and Kevrekidis [14], in Cheng, Lou, and Lim [15] and in Aiki [4].…”

Head-On Collision of a Pair of Coaxial Circular Vortex Filament

“…Since the model system is derived by assuming that the vortex filaments are nearly straight and parallel, it is not suitable for describing motions of circular vortex filaments. In [22], the author proposed a system of nonlinear partial differential equations describing the interaction of vortex filaments. The paper [22] focused on deriving a new system describing the interaction of vortex filaments with general shape and proving the existence of solutions corresponding to leapfrogging in the case of coaxial circular vortex filaments.…”

“…In [22], the author proposed a system of nonlinear partial differential equations describing the interaction of vortex filaments. The paper [22] focused on deriving a new system describing the interaction of vortex filaments with general shape and proving the existence of solutions corresponding to leapfrogging in the case of coaxial circular vortex filaments. The aim of this paper is to prove the existence of solutions to the model system proposed in [22] which correspond to head-on collision of coaxial circular vortex filaments.…”

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We consider the head-on collision of two coaxial vortex rings described as the motion of two circular vortex filaments under the localized induction approximation. We prove the existence of solutions to a system of nonlinear partial differential equations proposed by the author in [22] which exhibit head-on collision. We also give a necessary and sufficient condition for the initial configuration and parameters of the filaments for head-on collision to occur.

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