Journal of Fluid Mechanics



A sufficient condition for the instability of columnar vortices


S.  Leibovich a1 and K.  Stewartson a2
a1 Sibley School of Mechanical and Aerospace Engineering. Cornell University, Ithaca, NY 14853
a2 Department of Mathematics, University College London

Article author query
leibovich s   [Google Scholar] 
stewartson k   [Google Scholar] 
 

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.$

(Published Online April 20 2006)
(Received January 25 1982)
(Revised July 21 1982)



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