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Elliptic instability of a curved Batchelor vortex

  • Francisco J. Blanco-Rodríguez (a1) and Stéphane Le Dizès (a1)

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.

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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, 219247.
Callegari, A. J. & Ting, L. 1978 Motion of a curved vortex filament with decaying vortical core and axial velocity. SIAM J. Appl. Maths 35, 148175.
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 21722179.
Eloy, C. & Le Dizès, S. 1999 Three-dimensional instability of Burgers and Lamb–Oseen vortices in a strain field. J. Fluid Mech. 378, 145166.
Eloy, C. & Le Dizès, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13 (3), 660676.
Fabre, D., Sipp, D. & Jacquin, L. 2006 The Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235274.
Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287318.
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of a vortex ring. J. Fluid Mech. 526, 77115.
Hattori, Y. & Fukumoto, Y. 2014 Modal stability analysis of a helical vortex tube with axial flow. J. Fluid Mech. 738, 222249.
Jiménez, J., Moffatt, H. K. & Vasco, C. 1996 The structure of the vortices in freely decaying two dimensional turbulence. J. Fluid Mech. 313, 209222.
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83113.
Lacaze, L., Birbaud, A.-L. & Le Dizès, S. 2005 Elliptic instability in a Rankine vortex with axial flow. Phys. Fluids 17, 017101.
Lacaze, L., Ryan, K. & Le Dizès, S. 2007 Elliptic instability in a strained Batchelor vortex. J. Fluid Mech. 577, 341361.
Laporte, F. & Corjon, A. 2000 Direct numerical simulations of the elliptic instability of a vortex pair. Phys. Fluids 12 (5), 10161031.
Le Dizès, S. 2004 Viscous critical-layer analysis of vortex normal modes. Stud. Appl. Maths 112 (4), 315332.
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 6996.
Le Dizès, S. & Laporte, F. 2002 Theoretical predictions for the elliptic instability in a two-vortex flow. J. Fluid Mech. 471, 169201.
Leweke, T., Le Dizès, S. & Williamson, C. H. K. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48, 507541.
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.
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85119.
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91114.
Meunier, P. & Leweke, T. 2005 Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533, 125159.
Moffatt, H. K., Kida, S. & Ohkitani, K. 1994 Stretched vortices – the sinews of turbulence; large-Reynolds-number asymptotics. J. Fluid Mech. 259, 241264.
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413425.
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319342.
Quaranta, H. U., Bolnot, H. & Leweke, T. 2015 Long-wave instability of a helical vortex. J. Fluid Mech. 780, 687716.
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.
Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15, 18611874.
Ting, L. & Tung, C. 1965 Motion and decay of a vortex in a nonuniform stream. Phys. Fluids 8 (6), 10391051.
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), 721733.
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641663.
Widnall, S. E., Bliss, D. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66 (1), 3547.
Widnall, S. E. & Tsai, C.-Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Lond. A 287, 273305.
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Journal of Fluid Mechanics
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