Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T04:17:08.168Z Has data issue: false hasContentIssue false
  • English
  • Français

The response of Hill's spherical vortex to a small axisymmetric disturbance

Published online by Cambridge University Press:  12 April 2006

H. K. Moffatt  and
D. W. Moore
H. K. Moffatt
Affiliation:
School of Mathematics, University of Bristol, England
D. W. Moore
Affiliation:
Department of Mathematics, Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

The response of a Hill's spherical vortex to an irrotational axisymmetric small perturbation is examined on the assumption that viscous diffusion of vorticity is negligible. The problem of determining the response is first reduced to a system of differential equations for the evolution of the Legendre coefficients of the disturbance stream function. This system is then solved approximately and it is shown that, if the initial disturbance is such as to make the vortex prolate spheroidal, the vortex detrains a fraction ⅗ε of its original volume, where ε is the fractional extension of the axis of symmetry in the imposed distortion. The detrained fluid forms a thin spike growing from the rear stagnation point. If the vortex is initially oblate, irrotational fluid is entrained at the rear stagnation point to the interior of the vortex.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 87 , Issue 4 , 29 August 1978 , pp. 749 - 760
Copyright
© 1978 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benjamin, T. B. 1976 The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. Lecture Notes in Mathematics, no. 503 (ed. P. Germain & B. Nayroles), pp. 8–29. Springer.
Bliss, D. B. 1973 The dynamics of flows with high concentrations of vorticity. Ph.D. thesis, Dept. of Aeronautics and Astronautics, MIT.
Hill, M. J. M. 1894 On a spherical vortex. Phil. Trans. Roy. Soc. A 185, 213–245.Google Scholar
Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32.Google Scholar
Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417–431.Google Scholar
Proudman, I. & Johnson, K. 1962 Boundary layer growth near a rear stagnation point. J. Fluid Mech. 12, 161–168.Google Scholar