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Elliptical instability of the Moore–Saffman model for a trailing wingtip vortex

Published online by Cambridge University Press:  30 August 2016

J. Feys  and
S. A. Maslowe
J. Feys*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada
S. A. Maslowe
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec H3A 0B9, Canada
*
†Email address for correspondence: jan.feys@mail.mcgill.ca
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Abstract

In this paper, we investigate the elliptical instability exhibited by two counter-rotating trailing vortices. This type of instability can be viewed as a resonance between two normal modes of a vortex and an external strain field. Recent numerical investigations have extended earlier results that ignored axial flow to include models with a simple wake-like axial flow such as the similarity solution found by Batchelor (J. Fluid Mech., vol. 20, 1964, pp. 645–658). We present herein growth rates of elliptical instability for a family of velocity profiles found by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 333, 1973, pp. 491–508). These profiles have a parameter $n$ that depends on the wing loading. As a result, unlike the Batchelor vortex, they are capable of modelling both the jet-like and the wake-like axial flow present in a trailing vortex at short and intermediate distances behind a wingtip. Direct numerical simulations of the linearized Navier–Stokes equations are performed using an efficient spectral method in cylindrical coordinates developed by Matsushima & Marcus (J. Comput. Phys., vol. 53, 1997, pp. 321–345). We compare our results with those for the Batchelor vortex, whose velocity profiles are closely approximated as the wing loading parameter $n$ approaches 1. An important conclusion of our investigation is that the stability characteristics vary considerably with $n$ and $W_{0}$, a parameter measuring the strength of the mean axial velocity component. In the case of an elliptically loaded wing ($n=0.50$), we find that the instability growth rates are up to 50 % greater than those for the Batchelor vortex. Our results demonstrate the significant effect of the distribution and intensity of the axial flow on the elliptical instability of a trailing vortex.

JFM classification

Vortex Flows: Vortex Flows Vortex Flows: Vortex instability Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , pp. 556 - 590
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Copyright
© 2016 Cambridge University Press 

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