Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-16T13:29:19.345Z Has data issue: false hasContentIssue false
  • English
  • Français

The decay of Hill's vortex in a rotating flow

Published online by Cambridge University Press:  20 May 2021

Matthew N. Crowe Open the ORCID record for Matthew N. Crowe [Opens in a new window] ,
Cameron J.D. Kemp  and
Edward R. Johnson Open the ORCID record for Edward R. Johnson [Opens in a new window]
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
Get access Rights & Permissions [Opens in a new window]

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.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 919 , 25 July 2021 , A6
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Batchelor, G.K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Bretherton, F.P. 1967 The time-dependent motion due to a cylinder moving in an unbounded rotating or stratified fluid. J. Fluid Mech. 28 (3), 545–570.CrossRefGoogle Scholar
Burns, K.J., Vasil, G.M., Oishi, J.S., Lecoanet, D. & Brown, B.P. 2020 Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2, 023068.CrossRefGoogle Scholar
Flierl, G.R. & Haines, K. 1994 The decay of modons due to Rossby wave radiation. Phys. Fluids 6 (10), 3487–3497.CrossRefGoogle Scholar
Ford, R., McIntyre, M.E. & Norton, W.A. 2000 Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57 (9), 1236–1254.2.0.CO;2>CrossRefGoogle Scholar
Fraenkel, L.E. 1956 On the flow of rotating fluid past bodies in a pipe. Proc. R. Soc. A 233 (1195), 506–526.Google Scholar
Fraenkel, L.E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51 (1), 119–135.CrossRefGoogle Scholar
Griffiths, R.M. 1999 The interaction between vorticity and internal gravity waves. PhD thesis, University of Cambridge.Google Scholar
Hill, M.J.M. 1894 On a spherical vortex. Phil. Trans. R. Soc. A 185, 213–245.Google Scholar
Johnson, E.R. & Crowe, M.N. 2021 The decay of a dipolar vortex in a weakly dispersive environment. J. Fluid Mech. 917, A35.CrossRefGoogle Scholar
Lighthill, M.J. 1967 On waves generated in dispersive systems by travelling forcing effects, with applications to the dynamics of rotating fluids. J. Fluid Mech. 27 (4), 725–752.CrossRefGoogle Scholar
Llewellyn Smith, S.G. & Ford, R. 2001 Three-dimensional acoustic scattering by vortical flows. II. Axisymmetric scattering by Hill's spherical vortex. Phys. Fluids 13 (10), 2890–2900.CrossRefGoogle Scholar
Long, R.R. 1953 Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. J. Met. 10, 197–203.2.0.CO;2>CrossRefGoogle Scholar
Machicoane, N., Labarre, V., Voisin, B., Moisy, F. & Cortet, P.-P. 2018 Wake of inertial waves of a horizontal cylinder in horizontal translation. Phys. Rev. Fluids 3, 034801.CrossRefGoogle Scholar
McIntyre, M.E. 1972 On Long's hypothesis of no upstream influence in uniformly stratified or rotating flow. J. Fluid Mech. 52 (2), 209–243.CrossRefGoogle Scholar
Moffatt, H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35 (1), 117–129.CrossRefGoogle Scholar
Moffatt, H.K. & Moore, D.W. 1978 The response of Hill's spherical vortex to a small axisymmetric disturbance. J. Fluid Mech. 87 (4), 749–760.CrossRefGoogle Scholar
Norbury, J. 1972 A steady vortex ring close to Hill's spherical vortex. Proc. Camb. Phil. Soc. 72 (2), 253–284.CrossRefGoogle Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57 (3), 417–431.CrossRefGoogle Scholar
Pozrikidis, C. 1986 The nonlinear instability of Hill's vortex. J. Fluid Mech. 168, 337–367.CrossRefGoogle Scholar
Protas, B. & Elcrat, A. 2016 Linear stability of Hill's vortex to axisymmetric perturbations. J. Fluid Mech. 799, 579–602.CrossRefGoogle Scholar
Scase, M.M. & Terry, H.L. 2018 Spherical vortices in rotating fluids. J. Fluid Mech. 846, R4.CrossRefGoogle Scholar
Taylor, G.I. 1922 The motion of a sphere in a rotating liquid. Proc. R. Soc. A 102 (715), 180–189.Google Scholar

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 731.9 KB