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Transient growth in vortices with axial flow

Published online by Cambridge University Press:  31 August 2007

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

We investigate transient growth in high-Reynolds-number vortices with axial flow. Manycases of vortex instability are not fully explained by strong exponential instability modes, and transient growth could offer an alternative route to breakdown in such cases. Strong transient growth is found, in agreement with previous studies. We first discuss the problem by reference to ducted vortices which model aeroengine flow. The transient growth is inviscid in character, and in this paper we specifically interpret it as an effect of the inviscid continuous spectrum. The relevant inviscid theory explains new scalings which we find for the transient growth, which are generalizations of the quadratic scaling seen previously in two-dimensional flows and non-swirling pipe flows. We then turn to a second case, of interest for vortex breakdown, the Batchelor vortex, and present calculations of the transient growth. Large growth is possible, especially for the helical modes (with azimuthal wavenumber |m| = 1). The general trends are complicated by a number ofissues, including a long-wavelength effect and a resonance effect, both of which were recently discovered for a vortex without axial flow and are found here to be present in the Batchelor vortex also. Overall, the results suggest that strong transient effects are present in the moderate- to high-swirl regime of practical interest (swirl number q ≳ 2). Foraxisymmetric (m = 0) and higher (|m| > 1) modes, however, transient effects are not found to be significant.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 587 , 25 September 2007 , pp. 271 - 301
Copyright
Copyright © Cambridge University Press 2007

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