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Long's vortex revisited

Published online by Cambridge University Press:  26 August 2009

RICHARD E. HEWITT  and
PETER W. DUCK
RICHARD E. HEWITT*
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
PETER W. DUCK
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: richard.e.hewitt@manchester.ac.uk
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Abstract

We reconsider exact solutions to the Navier–Stokes equations that describe a vortex in a viscous, incompressible fluid. This type of solution was first introduced by Long (J. Atmos. Sci., vol. 15 (1), 1958, p. 108) and is parameterized by an inverse Reynolds number ϵ. Long's attention (and that of many subsequent investigators) was centred upon the ‘quasi-cylindrical’ (QC) case corresponding to ϵ = 0. We show that the limit ϵ → 0 is not straightforward, and that it reveals other solutions to this fundamental exact reduction of the Navier–Stokes system (which are not of QC form). Through careful numerical investigation, supported by asymptotic descriptions, we identify new solutions and describe the full parameter space that is spanned by ϵ and the pressure at the vortex core. Some erroneous results that exist in the literature are corrected.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 634 , 10 September 2009 , pp. 91 - 111
Copyright
Copyright © Cambridge University Press 2009

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References

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