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Hollow vortices in shear

Published online by Cambridge University Press:  16 November 2016

Luca Zannetti ,
Michele Ferlauto  and
Stefan G. Llewellyn Smith
Luca Zannetti*
Affiliation:
Accademia delle Scienze di Torino, Via Accademia delle Scienze 6, 10123 Torino, Italy
Michele Ferlauto
Affiliation:
DIMEAS, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Stefan G. Llewellyn Smith
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
*
†Email address for correspondence: luca.zannetti@polito.it
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Abstract

An analytical method for determining the shape of hollow vortices in shear flows is presented in detail. In a non-dimensional formulation, it is shown that the problem has one degree of freedom represented by the free choice of the non-dimensionalized speed $\unicode[STIX]{x1D705}$ at the boundary of the vortex. The solutions form two families of shapes corresponding to vortex circulation and shear-flow vorticity having the opposite or same sign. When the signs are opposite, the shape family resembles that described by Llewellyn Smith & Crowdy (J. Fluid Mech., vol. 691, 2012, pp. 178–200) for hollow vortices in a potential flow with strain. As for that flow, there is a minimum value of $\unicode[STIX]{x1D705}$ below which there is no solution as the boundary of the vortex self-intersects, while, when the signs are the same, solutions exist for $0<\unicode[STIX]{x1D705}$.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 809 , 25 December 2016 , pp. 705 - 715
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Copyright
© 2016 Cambridge University Press 

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