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Nonlinear dynamics of the elliptic instability

Published online by Cambridge University Press:  08 March 2010

NATHANAËL SCHAEFFER  and
STÉPHANE LE DIZÈS
NATHANAËL SCHAEFFER*
Affiliation:
IRPHE, CNRS, University Aix-Marseille, 49 rue F. Joliot Curie, 13013 Marseille, France LGIT, CNRS, University Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
STÉPHANE LE DIZÈS
Affiliation:
IRPHE, CNRS, University Aix-Marseille, 49 rue F. Joliot Curie, 13013 Marseille, France
*
†Email address for correspondence: Nathanael.Schaeffer@obs.ujf-grenoble.fr
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Abstract

In this paper, we analyse by numerical simulations the nonlinear dynamics of the elliptic instability in the configurations of a single strained vortex and a system of two counter-rotating vortices. We show that although a weakly nonlinear regime associated with a limit cycle is possible, the nonlinear evolution far from the instability threshold is, in general, much more catastrophic for the vortex. In both configurations, we put forward some evidence of a universal nonlinear transition involving shear layer formation and vortex loop ejection, leading to a strong alteration and attenuation of the vortex, and a rapid growth of the vortex core size.

JFM classification

Vortex Flows: Vortex instability Vortex Flows: Vortex breakdown
Type
Papers
Information
Journal of Fluid Mechanics , Volume 646 , 10 March 2010 , pp. 471 - 480
Copyright
Copyright © Cambridge University Press 2010

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References

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Schaeffer and Le Dizes supplementary movie

Movie 1. Strongly non-linear evolution of the elliptic instability of a single strained Lamb-Oseen vortex (case 1), at Reynolds number Re=Γ/ν=10000 (where Γ is the vortex circulation, ν the kinematic viscosity of the fluid) and in an imposed external strain field of strain rate 0.01, where time scales are normalized by the vortex initial turn-over time. The movie starts with a barely noticeable perturbation, which grows to strong deformations of the vortex core. This instability does not saturate and leads to the appearance of secondary azimuthal structures and the complete breakdown of the vortex. When there is no more vortex core on which to grow, the instability dies and we can see the beginning of a relaminarisation phase at the end of the movie. The total vorticity is equal to 1.0 on the red surface, and the current time t is given in the top right corner in initial vortex turnover time units. See also fig. 5 in the paper.

Download Schaeffer and Le Dizes supplementary movie(Video)
Video 2.7 MB

Schaeffer and Le Dizes supplementary movie

Movie 2. Strongly non-linear evolution of the elliptic instability of a counter-rotating Lamb-Oseen vortex pair (case 2) of core size a separated by a distance b/a=5 and for Reynolds number Re=Γ/ν=6300 (where Γ is the vortex circulation, ν the kinematic viscosity of the fluid). On this movie, the elliptic instability follows the same evolution as in movie 1 : deformation of the vortex cores, formation of azimuthal structures and finally breakdown. By the end of the movie, one can also see some structures wrapping around both vortices. The total vorticity is equal to 1.5 on the red surface, and the current time t is given in the top right corner in initial vortex turnover time units. See also fig. 8 in the paper.

Download Schaeffer and Le Dizes supplementary movie(Video)
Video 1.1 MB