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Internal structure of vortex rings and helical vortices

  • Francisco J. Blanco-Rodríguez (a1), Stéphane Le Dizès (a1), Can Selçuk (a2) (a3), Ivan Delbende (a2) (a3) and Maurice Rossi (a4) (a5)...


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.


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Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Alekseenko, S. V., Kuibin, P. A. & Okulov, V. L. 2007 Theory of Concentrated Vortices: An Introduction. Springer.
Boersma, J. & Wood, D. H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263280.
Bolnot, H.2012 Instabilités des tourbillons hélicoïdaux: application au sillage des rotors. PhD thesis, Aix Marseille University.
Bolnot, H., Le Dizès, S. & Leweke, T. 2014 Spatio-temporal development of the pairing instability in an infinite array of vortex rings. Fluid Dyn. Res. 46, 061405.
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.
Delbende, I., Piton, B. & Rossi, M. 2015 Merging of two helical vortices. Eur. J. Mech. (B/Fluids) 49, 363372.
Delbende, I., Rossi, M. & Daube, O. 2012a DNS of flows with helical symmetry. J. Theor. Comput. Fluid Dyn. 26, 141160.
Delbende, I., Rossi, M. & Piton, B. 2012b Direct numerical simulation of helical vortices. Intl J. Engng Systems Model. Simul. 4, 94101.
Eloy, C. & Le Dizès, S. 1999 Three-dimensional instability of Burgers and Lamb-Oseen vortices in a strain field. J. Fluid Mech. 378, 145166.
Fukumoto, Y. 2002 Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring. Fluid Dyn. Res. 30, 6795.
Fukumoto, Y. & Hattori, Y. 2005 Curvature instability of a vortex ring. J. Fluid Mech. 526, 77115.
Fukumoto, Y. & Miyazaki, T. 1991 Three-dimensional distorsions of a vortex filament with axial velocity. J. Fluid Mech. 222, 369416.
Fukumoto, Y. & Moffatt, H. K. 2000 Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity. J. Fluid Mech. 417, 145.
Fukumoto, Y. & Okulov, V. L. 2005 The velocity induced by a helical vortex tube. Phys. Fluids 17, 107101.
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25 (11), 19491952.
Hattori, Y. & Fukumoto, Y. 2003 Short-wavelength stability analysis of thin vortex rings. Phys. Fluids 15, 31513163.
Hattori, Y. & Fukumoto, Y. 2009 Short-wavelength stability analysis of a helical vortex tube. Phys. Fluids 21, 014104.
Hattori, Y. & Fukumoto, Y. 2014 Modal stability analysis of a helical vortex tube with axial flow. J. Fluid Mech. 738, 222249.
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83113.
Kuibin, P. A. & Okulov, V. L. 1998 Self-induced motion and asymptotic expansion of the velocity field in the vicinity of a helical vortex filament. Phys. Fluids 10, 607614.
Lacaze, L., Ryan, K. & Le Dizès, S. 2007 Elliptic instability in a strained batchelor vortex. J. Fluid Mech. 577, 341361.
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Le Dizès, S. & Verga, A. 2002 Viscous interaction of two co-rotating vortices before merging. J. Fluid Mech. 467, 389410.
Levy, H. & Forsdyke, A. G. 1927 The stability of an infinite system of circular vortices. Proc. R. Soc. Lond. A 114, 594604.
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.
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. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. Lond. A 272, 403429.
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319342.
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241259.
Roy, C., Leweke, T., Thompson, M. C. & Hourigan, K. 2011 Experiments on the elliptic instability in vortex pairs with axial core flow. J. Fluid Mech. 677, 383416.
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
Ting, L. & Tung, C. 1965 Motion and decay of a vortex in a nonuniform stream. Phys. Fluids 8, 1039.
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, 3547.
Widnall, S. E., Bliss, D. B. & Zalay, Y. 1971 Theoretical and experimental study of the stability of a vortex pair. In Aircraft Wake Turbulence and Its Detection (ed. Olsen, J. H., Goldburg, A. & Rogers, M.), pp. 305338. Springer.
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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