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Internal structure of vortex rings and helical vortices

Published online by Cambridge University Press:  16 November 2015

Francisco J. Blanco-Rodríguez ,
Stéphane Le Dizès ,
Can Selçuk ,
Ivan Delbende  and
Maurice Rossi
Francisco J. Blanco-Rodríguez
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, UMR 7342, 13384 Marseille, France
Stéphane Le Dizès*
Affiliation:
Aix Marseille Université, CNRS, Centrale Marseille, IRPHE, UMR 7342, 13384 Marseille, France
Can Selçuk
Affiliation:
LIMSI, CNRS, Université Paris-Saclay, 91405 Orsay CEDEX, France Sorbonne Universités, UPMC Univ Paris 06, UFR d’Ingénierie, 75252 Paris CEDEX 05, France
Ivan Delbende
Affiliation:
LIMSI, CNRS, Université Paris-Saclay, 91405 Orsay CEDEX, France Sorbonne Universités, UPMC Univ Paris 06, UFR d’Ingénierie, 75252 Paris CEDEX 05, France
Maurice Rossi
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, 75005, Paris, France CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, 75005, Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

The internal structure of vortex rings and helical vortices is studied using asymptotic analysis and numerical simulations in cases where the core size of the vortex is small compared to its radius of curvature, or to the distance to other vortices. Several configurations are considered: a single vortex ring, an array of equally-spaced rings, a single helix and a regular array of helices. For such cases, the internal structure is assumed to be at leading order an axisymmetric concentrated vortex with an internal jet. A dipolar correction arises at first order and is shown to be the same for all cases, depending only on the local vortex curvature. A quadrupolar correction arises at second order. It is composed of two contributions, one associated with local curvature and another one arising from a non-local external 2-D strain field. This strain field itself is obtained by performing an asymptotic matching of the local internal solution with the external solution obtained from the Biot–Savart law. Only the amplitude of this strain field varies from one case to another. These asymptotic results are thereafter confronted with flow solutions obtained by direct numerical simulation (DNS) of the Navier–Stokes equations. Two different codes are used: for vortex rings, the simulations are performed in the axisymmetric framework; for helices, simulations are run using a dedicated code with built-in helical symmetry. Quantitative agreement is obtained. How these results can be used to theoretically predict the occurrence of both the elliptic instability and the curvature instability is finally addressed.

JFM classification

Vortex Flows: Vortex Flows Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 785 , 25 December 2015 , pp. 219 - 247
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© 2015 Cambridge University Press 

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