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Eroding dipoles and vorticity growth for Euler flows in $\mathbb{R}^{3}$: axisymmetric flow without swirl

Published online by Cambridge University Press:  16 September 2016

Stephen Childress ,
Andrew D. Gilbert  and
Paul Valiant
Stephen Childress*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
Andrew D. Gilbert
Affiliation:
Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK
Paul Valiant
Affiliation:
Department of Computer Science, Brown University, Providence, RI 02912, USA
*
†Email address for correspondence: [email protected]
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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 $\mathbb{R}^{3}$. We examine here the possible formation of such a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity as $t^{4/3}$. Our study suggests that the optimizing flow giving the $t^{4/3}$ 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.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 805 , 25 October 2016 , pp. 1 - 30
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Copyright
© 2016 Cambridge University Press 

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