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Elliptic instability in a strained Batchelor vortex

Published online by Cambridge University Press:  19 April 2007

LAURENT LACAZE ,
KRIS RYAN  and
STÉPHANE LE DIZÈS
LAURENT LACAZE
Affiliation:
Institut de Recherche sur les Phénoménes Hors Équilibre (IRPHE), CNRS/Universités Aix-Marseille I & II, 49 rue F. Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
KRIS RYAN
Affiliation:
Fluids Laboratory for Aeronautical and Industrial Research (FLAIR), Department of Mechanical Engineering, Monash University, Melbourne, Victoria 3800, Australia
STÉPHANE LE DIZÈS
Affiliation:
Institut de Recherche sur les Phénoménes Hors Équilibre (IRPHE), CNRS/Universités Aix-Marseille I & II, 49 rue F. Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
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Abstract

The elliptic instability of a Batchelor vortex subject to a stationary strain field is considered by theoretical and numerical means in the regime of large Reynolds number and small axial flow. In the theory, the elliptic instability is described as a resonant coupling of two quasi-neutral normal modes (Kelvin modes) of the Batchelor vortex of azimuthal wavenumbers m and m + 2 with the underlying strain field. The growth rate associated with these resonances is computed for different values of the azimuthal wavenumbers as the axial flow parameter is varied. We demonstrate that the resonant Kelvin modes m = 1 and m = −1 which are the most unstable in the absence of axial flow become damped as the axial flow is increased. This is shown to be due to the appearance of a critical layer which damps one of the resonant Kelvin modes. However, the elliptic instability does not disappear. Other combinations of Kelvin modes m = −2 and m = 0, then m = −3 and m = −1 are shown to become progressively unstable for increasing axial flow. A complete instability diagram is obtained as a function of the axial flow parameter for several values of the strain rate and Reynolds number.

The numerical study considers a system of two counter-rotating Batchelor vortices in which the strain field felt by each vortex is due to the other vortex. The characteristics of the most unstable linear modes developing on the frozen base flow are computed by direct numerical simulations for two axial flow parameters and compared to the theory. In both cases, a very good agreement is obtained for the most unstable modes. Less unstable modes are also identified in the numerics and shown to correspond to peculiar resonances involving Kelvin modes from branches of different labels.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 577 , 25 April 2007 , pp. 341 - 361
Copyright
Copyright © Cambridge University Press 2007

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