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Elliptic instability of a curved Batchelor vortex

Published online by Cambridge University Press:  09 September 2016

Francisco J. Blanco-Rodríguez  and
Stéphane Le Dizès
Francisco J. Blanco-Rodríguez
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, F-13013 Marseille, France
Stéphane Le Dizès*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE, F-13013 Marseille, France
*
†Email address for correspondence: ledizes@irphe.univ-mrs.fr
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Abstract

The occurrence of the elliptic instability in rings and helical vortices is analysed theoretically. The framework developed by Moore & Saffman (Proc. R. Soc. Lond. A, vol. 346, 1975, pp. 413–425), where the elliptic instability is interpreted as a resonance of two Kelvin modes with a strained induced correction, is used to obtain the general stability properties of a curved and strained Batchelor vortex. Explicit expressions for the characteristics of the three main unstable modes are obtained as a function of the axial flow parameter of the Batchelor vortex. We show that vortex curvature adds a contribution to the elliptic instability growth rate. The results are applied to a single vortex ring, an array of alternate vortex rings and a double helical vortex.

JFM classification

Vortex Flows: Vortex Flows Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 804 , 10 October 2016 , pp. 224 - 247
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Copyright
© 2016 Cambridge University Press 

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References

Blanco-Rodriguez, F. J., Le Dizès, S., Selçuk, C., Delbende, I. & Rossi, M. 2015 Internal structure of vortex rings and helical vortices. J. Fluid Mech. 785, 219–247.CrossRefGoogle Scholar
Callegari, A. J. & Ting, L. 1978 Motion of a curved vortex filament with decaying vortical core and axial velocity. SIAM J. Appl. Maths 35, 148–175.Google Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 2172–2179.Google Scholar
Eloy, C. & Le Dizès, S. 1999 Three-dimensional instability of Burgers and Lamb–Oseen vortices in a strain field. J. Fluid Mech. 378, 145–166.Google Scholar
Eloy, C. & Le Dizès, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13 (3), 660–676.Google Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 The Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235–274.CrossRefGoogle Scholar
Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287–318.Google Scholar
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of a vortex ring. J. Fluid Mech. 526, 77–115.Google Scholar
Hattori, Y. & Fukumoto, Y. 2014 Modal stability analysis of a helical vortex tube with axial flow. J. Fluid Mech. 738, 222–249.Google Scholar
Jiménez, J., Moffatt, H. K. & Vasco, C. 1996 The structure of the vortices in freely decaying two dimensional turbulence. J. Fluid Mech. 313, 209–222.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.Google Scholar
Lacaze, L., Birbaud, A.-L. & Le Dizès, S. 2005 Elliptic instability in a Rankine vortex with axial flow. Phys. Fluids 17, 017101.CrossRefGoogle Scholar
Lacaze, L., Ryan, K. & Le Dizès, S. 2007 Elliptic instability in a strained Batchelor vortex. J. Fluid Mech. 577, 341–361.CrossRefGoogle Scholar
Laporte, F. & Corjon, A. 2000 Direct numerical simulations of the elliptic instability of a vortex pair. Phys. Fluids 12 (5), 1016–1031.CrossRefGoogle Scholar
Le Dizès, S. 2004 Viscous critical-layer analysis of vortex normal modes. Stud. Appl. Maths 112 (4), 315–332.CrossRefGoogle Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96.CrossRefGoogle Scholar
Le Dizès, S. & Laporte, F. 2002 Theoretical predictions for the elliptic instability in a two-vortex flow. J. Fluid Mech. 471, 169–201.Google Scholar
Leweke, T., Le Dizès, S. & Williamson, C. H. K. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48, 507–541.CrossRefGoogle Scholar
Leweke, T., Quaranta, H. U., Bolnot, H., Blanco-Rodriguez, F. J. & Le Dizès, S. 2014 Long- and short-wave instabilities in helical vortices. J. Phys.: Conf. Ser. 524, 012154.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.Google Scholar
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114.Google Scholar
Meunier, P. & Leweke, T. 2005 Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533, 125–159.CrossRefGoogle Scholar
Moffatt, H. K., Kida, S. & Ohkitani, K. 1994 Stretched vortices – the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241–264.Google Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.Google Scholar
Quaranta, H. U., Bolnot, H. & Leweke, T. 2015 Long-wave instability of a helical vortex. J. Fluid Mech. 780, 687–716.Google Scholar
Roy, C., Schaeffer, N., Le Dizès, S. & Thompson, M. 2008 Stability of a pair of co-rotating vortices with axial flow. Phys. Fluids 20, 094101.Google Scholar
Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15, 1861–1874.CrossRefGoogle Scholar
Ting, L. & Tung, C. 1965 Motion and decay of a vortex in a nonuniform stream. Phys. Fluids 8 (6), 1039–1051.CrossRefGoogle Scholar
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73 (4), 721–733.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.CrossRefGoogle Scholar
Widnall, S. E., Bliss, D. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66 (1), 35–47.Google Scholar
Widnall, S. E. & Tsai, C.-Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Lond. A 287, 273–305.Google Scholar