2003
DOI: 10.1016/j.fluiddyn.2003.05.001
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Leapfrogging vortex rings: Hamiltonian structure, geometric phases and discrete reduction

Abstract: We present two interesting features of vortex rings in incompressible, Newtonian uids that involve their Hamiltonian structure.The ÿrst feature is for the Hamiltonian model of dynamically interacting thin-cored, coaxial, circular vortex rings described, for example, in the works of Dyson (Philos. Trans. Roy. Soc. London Ser. A 184 (1893) 1041) and Hicks (Proc. Roy. Soc. London Ser. A 102 (1922) 111). For this model, the symplectic reduced space associated with the translational symmetry is constructed. Using t… Show more

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Cited by 23 publications
(28 citation statements)
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“…In 9(c) is a plot of the streamlines at t = 106.65 at which the ellipse is moving with velocity V 1 = −1.1824 in the same direction as the point vortices. It is worth this behavior reminds of the leapfrogging of vortex rings considered in Shashikanth and Marsden (2003). Indeed, the vortex pair and its image vorticity may be thought of as a pair of vortex rings subject to the constraint that the boundary of the solid is a streamline -this analogy remains to be investigated.…”
Section: Application To Swimmingmentioning
confidence: 99%
“…In 9(c) is a plot of the streamlines at t = 106.65 at which the ellipse is moving with velocity V 1 = −1.1824 in the same direction as the point vortices. It is worth this behavior reminds of the leapfrogging of vortex rings considered in Shashikanth and Marsden (2003). Indeed, the vortex pair and its image vorticity may be thought of as a pair of vortex rings subject to the constraint that the boundary of the solid is a streamline -this analogy remains to be investigated.…”
Section: Application To Swimmingmentioning
confidence: 99%
“…Note that the self-induced kinetic energy term obtained by applying the local induction approximation, H LI , satisfies the consistency approximation mentioned earlier i.e u n LI = δH LI δC i × t i everywhere on the ith ring (see [10]). …”
Section: Regularization Of the Self-induced Fieldmentioning
confidence: 70%
“…A simple though somewhat crude fix to regularize the self-induced velocity field is to use the local induction approximation (see references cited above for the history of this method). Using this approximation the self-induced velocity field of the N rings is obtained as [10]:…”
Section: Regularization Of the Self-induced Fieldmentioning
confidence: 99%
“…Using the ideas of discrete symmetries and fixed point subspaces, [35], in their study of leapfrogging vortex rings, showed that for closed filaments modeled as curves-which we term vortex rings throughout the present paper-a special Hamiltonian structure is derivable in the case of circular coaxial rings (using discrete reduction) from that for rings of arbitrary shape. A Poisson formulation for the dynamics of a passive particle advected by the flow due to a single vortex filament was given in [8] and for a circular vortex ring interacting with a stationary rigid sphere was given in [9].…”
Section: Introductionmentioning
confidence: 90%
“…A simple though somewhat crude fix to regularize this infinite self-induced velocity field and to obtain u S I is to use the local induction approximation (see references cited above for the history of this method). Using this approximation (see, for example, [35]), one gets…”
Section: Equations Of Motion For the Body-rings Systemmentioning
confidence: 99%