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A sufficient condition for the instability of columnar vortices

Published online by Cambridge University Press:  20 April 2006

S. Leibovich  and
K. Stewartson
S. Leibovich
Affiliation:
Sibley School of Mechanical and Aerospace Engineering. Cornell University, Ithaca, NY 14853
K. Stewartson
Affiliation:
Department of Mathematics, University College London
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Abstract

The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r), azimuthal velocity V(r), where r is distance from the axis, let Ω be the angular velocity V/r, and let Γ be the circulation rV. Then the flow is unstable if $ V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right] < 0.$

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 126 , January 1983 , pp. 335 - 356
Copyright
© 1983 Cambridge University Press

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