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The decay of Hill's vortex in a rotating flow

Published online by Cambridge University Press:  20 May 2021

Matthew N. Crowe*
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Cameron J.D. Kemp
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
Edward R. Johnson
Affiliation:
Department of Mathematics, University College London, LondonWC1E 6BT, UK
*
Email address for correspondence: m.crowe@ucl.ac.uk

Abstract

Hill's vortex is a classical solution of the incompressible Euler equations which consists of an axisymmetric spherical region of constant vorticity matched to an irrotational external flow. This solution has been shown to be a member of a one-parameter family of steady vortex rings and as such is commonly used as a simple analytic model for a vortex ring. Here, we model the decay of a Hill's vortex in a weakly rotating flow due to the radiation of inertial waves. We derive analytic results for the modification of the vortex structure by rotational effects and the generated wave field using an asymptotic approach where the rotation rate, or inverse Rossby number, is taken to be small. Using this model, we predict the decay of the vortex speed and radius by combining the flux of vortex energy to the wave field with the conservation of peak vorticity. We test our results against numerical simulations of the full axisymmetric Navier–Stokes equations.

Type
JFM Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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Crowe et al. supplementary movie

The temporal evolution of the velocity fields for ε = 0.4 for a vortex with initial speed and radius of 1. We plot ρuρ, ρuz and w as functions of ρ and z from t = 0 until t = 50. We observe that the wave wake is formed behind the travelling vortex and the periodicity in the z direction does not significantly affect the evolution.
Download Crowe et al. supplementary movie(Video)
Video 732 KB
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