Eroding dipoles and vorticity growth for Euler flows in ℝ3: Axisymmetric flow without swirl

Stephen Childress, Andrew D. Gilbert, Paul Valiant

Research output: Contribution to journalReview articlepeer-review

2 Scopus citations

Abstract

A review of analyses based upon anti-parallel vortex structures suggests that structurally stable dipoles with eroding circulation may offer a path to the study of vorticity growth in solutions of Euler's equations in ℝ3. We examine here the possible formation of such a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity as t4/3 Our study suggests that the optimizing flow giving the t4/3 growth mimics an exact solution of Euler's equations representing an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is analysed asymptotically at large times, and its predicted properties are verified numerically. We also show numerically that if mirror symmetry of the dipole is not imposed but axial symmetry maintained, an instability leads to breakup into smaller vortical structures.

Original languageEnglish
Pages (from-to)1-30
Number of pages30
JournalJournal of Fluid Mechanics
Volume805
DOIs
StatePublished - 25 Oct 2016
Externally publishedYes

Keywords

  • vortex dynamics
  • vortex flows
  • vortex instability

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