Eroding dipoles and vorticity growth for Euler flows in R 3 : Axisymmetric flow without swirl
Gilbert, AD; Childress, S; Valiant, P
Date: 16 September 2016
Article
Journal
Journal of Fluid Mechanics
Publisher
Cambridge University Press (CUP)
Publisher DOI
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 ...
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 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
analyzed 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.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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