The dynamics of vortex rings: leapfrogging in an ideal and viscous fluid

Abstract

We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible and viscous fluid. Using the numerical simulation of the Navier-Stokes equations, we confirm the existence of leapfrogging of three equal vortex rings and suggest the possibility of detecting it experimentally. We also confirm the existence of leapfrogging of two vortex rings with opposite-signed vorticities in a viscous fluid.

Classical and quantum vortex leapfrogging in two-dimensional channels

“…The complex function χ(z, z 0 ) (and, correspondingly, χ(z, z * 0 )) can be physically interpreted as the complex velocity generated by an isolated vortex placed in z 0 (whose complex potential would be Ω(z, z 0 ) = −sgn(z 0 )iκ log(z−z 0 )/(2π)) and its infinite images with respect to the walls of the channel, m z = 0 and m z = 2D. The expression (10) for the complex potential w(z, z 0 ) can indeed be derived by considering two sets of infinite images of a vortex placed in z 0 and an anti-vortex in z * 0 [22]. If the channel is characterised by the presence of N vortices, the complex velocity w(z, z k {k=1,...,N } ) generated by the the set of N vortices is obtained via the superposition principle, i.e.…”

“…The time evolution of this vortex configuration is striking: the vortex ring (or pair) which is ahead widens and slows down, while the ring behind contracts, speeds up, catches up with the first ring and goes ahead through it; this 'leapfrogging' game is then repeated over and over again, unless instabilities disrupt it. A number of papers have been written on different aspects of this problem, ranging from the stability [1,26,36,64] to the deformation of the vortex cores and to the effects of viscosity [60] using numerical [10,13,54] as well as experimental methods [37,40,51,73]. The most recent developments concern leapfrogging of vortex bundles [68] and helical waves [27,52,57].…”

Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave

“…Здесь приведем лишь некоторые из них. Динамика точечных вихрей на плоскости и сфере рассматривалась в работах [7][8][9][10], кольцевых вихрей -в работах [11][12][13][14][15][16]. Общая теория динамики вихрей на плоскости и сфере в постановке идеальной жидкости, включая редук-цию уравнений и частные вопросы, а также обширный литературный обзор представлены в книге [17].…”

Classical and quantum vortex leapfrogging in two-dimensional channels

“…The time evolution of this vortex configuration is striking: the vortex ring (or pair) which is ahead widens and slows down, while the ring behind contracts, speeds up, catches up with the first ring and goes ahead through it; this 'leapfrogging' game is then repeated over and over again, unless instabilities disrupt it. A number of papers have been written on different aspects of this problem, ranging from the stability (Love 1894;Hicks 1922;Acheson 2000;Tophøj & Aref 2013) to the deformation of the vortex cores and to the effects of viscosity (Shariff & Leonard 1992) using numerical (Riley & Stevens 1993;Borisov 2014;Cheng & Lim 2015) as well as experimental methods (Maxworthy 1972;Yamada & Matsui 1978;Lim 1997;Qin, Liu & Xiang 2018). The most recent developments concern leapfrogging of vortex bundles (Wacks, Baggaley & Barenghi 2014) and helical waves (Hietala et al 2016;Selçuk, Delbende & Rossi 2018;Quaranta et al 2019).…”