Kinematic variational principle for motion of vortex rings

Yasuhide Fukumoto, H. K. Moffatt

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

We show how the ideas of topology and variational principle, opened up by Euler, facilitate the calculation of motion of vortex rings. Kelvin-Benjamin's principle, as generalised to three dimensions, states that a steady distribution of vorticity, relative to a moving frame, is the state that maximizes the total kinetic energy, under the constraint of constant hydrodynamic impulse, on an iso-vortical sheet. By adapting this principle, combined with an asymptotic solution of the Euler equations, we make an extension of Fraenkel-Saffman's formula for the translation velocity of an axisymmetric vortex ring to third order in a small parameter, the ratio of the core radius to the ring radius. Saffman's formula for a viscous vortex ring is also extended to third order.

Original languageEnglish
Pages (from-to)2210-2217
Number of pages8
JournalPhysica D: Nonlinear Phenomena
Volume237
Issue number14-17
DOIs
Publication statusPublished - Aug 15 2008

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vortex rings
variational principles
kinematics
radii
vorticity
impulses
topology
kinetic energy
hydrodynamics
rings

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Cite this

Kinematic variational principle for motion of vortex rings. / Fukumoto, Yasuhide; Moffatt, H. K.

In: Physica D: Nonlinear Phenomena, Vol. 237, No. 14-17, 15.08.2008, p. 2210-2217.

Research output: Contribution to journalArticle

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