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Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex

Published online by Cambridge University Press:  28 March 2007

C. J. HEATON
C. J. HEATON*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
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Abstract

We identify a family of centre-mode disturbances to inviscid swirling flows such as jets, wakes and other vortices. The centre modes form an infinite family of modes, increasingly concentrated near to the symmetry axis of the mean flow, and whose frequencies accumulate to a single point in the complex plane. This asymptotic accumulation allows analytical progress to be made, including a theoretical stability boundary, in O(1) parameter regimes. The modes are located close to the continuous spectrum of the linearized Euler equations, and the theory is closely related to that of the continuous spectrum. We illustrate our analysis with the inviscid Batchelor vortex, defined by swirl parameter q. We show that the inviscid instabilities found in previous numerical studies are in fact the first members of an infinite set of centre modes of the type we describe. We investigate the inviscid neutral curve, and find good agreement of the neutral curve predicted by the analysis with the results of numerical computations. We find that the unstable region is larger than previously reported. In particular, the value of q above which the inviscid vortex stabilizes is significantly larger than previously reported and in agreement with a long-standing theoretical prediction.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 576 , 10 April 2007 , pp. 325 - 348
Copyright
Copyright © Cambridge University Press 2007

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