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Effects of an axial flow on the centrifugal, elliptic and hyperbolic instabilities in Stuart vortices

Published online by Cambridge University Press:  10 October 2014

Manikandan Mathur ,
Sabine Ortiz ,
Thomas Dubos  and
Jean-Marc Chomaz
Manikandan Mathur*
Affiliation:
Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai 600036, India
Sabine Ortiz
Affiliation:
LadHyX, CNRS-École Polytechnique, F-91128 Palaiseau CEDEX, France UME/DFA, ENSTA, Chemin de la Hunière, 91761 Palaiseau CEDEX, France
Thomas Dubos
Affiliation:
Laboratoire de Météorologie Dynamique/IPSL, École Polytechnique, Palaiseau, France
Jean-Marc Chomaz
Affiliation:
LadHyX, CNRS-École Polytechnique, F-91128 Palaiseau CEDEX, France
*
†Email address for correspondence: [email protected]
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Abstract

Linear stability of the Stuart vortices in the presence of an axial flow is studied. The local stability equations derived by Lifschitz & Hameiri (Phys. Fluids A, vol. 3 (11), 1991, pp. 2644–2651) are rewritten for a three-component (3C) two-dimensional (2D) base flow represented by a 2D streamfunction and an axial velocity that is a function of the streamfunction. We show that the local perturbations that describe an eigenmode of the flow should have wavevectors that are periodic upon their evolution around helical flow trajectories that are themselves periodic once projected on a plane perpendicular to the axial direction. Integrating the amplitude equations around periodic trajectories for wavevectors that are also periodic, it is found that the elliptic and hyperbolic instabilities, which are present without the axial velocity, disappear beyond a threshold value for the axial velocity strength. Furthermore, a threshold axial velocity strength, above which a new centrifugal instability branch is present, is identified. A heuristic criterion, which reduces to the Leibovich & Stewartson criterion in the limit of an axisymmetric vortex, for centrifugal instability in a non-axisymmetric vortex with an axial flow is then proposed. The new criterion, upon comparison with the numerical solutions of the local stability equations, is shown to describe the onset of centrifugal instability (and the corresponding growth rate) very accurately.

JFM classification

Vortex Flows: Vortex Flows Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 565 - 585
Copyright
© 2014 Cambridge University Press 

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