Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-27T00:07:21.942Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of nearly parallel interacting vortex filaments

Published online by Cambridge University Press:  27 November 2017

James A. Kwiecinski  and
Robert A. Van Gorder Open the ORCID record for Robert A. Van Gorder [Opens in a new window]
James A. Kwiecinski
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Robert A. Van Gorder*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The dynamics of interacting vortex filaments in an incompressible fluid, which are nearly parallel, have been approximated in the Klein–Majda–Damodaran model. The regime considers the deflection of each filament from a central axis; that is to say, the vortex filaments are assumed to be roughly parallel and centred along parallel lines. While this model has attracted a fair amount of mathematical interest in the recent literature, particularly concerning the existence of certain specific vortex filament structures, our aim is to generalise several known interesting filament solutions, found in the self-induced motion of a single vortex filament, to the case of pairwise interactions between multiple vortex filaments under the Klein–Majda–Damodaran model by means of asymptotic and numerical methods. In particular, we obtain asymptotic solutions for counter-rotating and co-rotating vortex filament pairs that are separated by a distance, so that the vortex filaments always remain sufficiently far apart, as well as intertwined vortex filaments that are in close proximity, exhibiting overlapping orbits. For each scenario, we consider both co- and counter-rotating pairwise interactions, and the specific kinds of solutions obtained for each case consist of planar filaments, for which motion is purely rotational, as well as travelling wave and self-similar solutions, both of which change their form as they evolve in time. We choose travelling waves, planar filaments and self-similar solutions for the initial filament configurations, as these are common vortex filament structures in the literature, and we use the dynamics under the Klein–Majda–Damodaran model to see how these structures are modified in time under pairwise interaction dynamics. Numerical simulations for each case demonstrate the validity of the asymptotic solutions. Furthermore, we develop equations to study a co-rotating hierarchy of many satellite vortices orbiting around a central filament. We numerically show that such configurations are unstable for plane-wave solutions, which lead to collapse of the hierarchy. We also consider more general travelling wave and self-similar solutions for co-rotating hierarchies, and these give what appears to be chaotic dynamics.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex Flows Vortex Flows: Vortex interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 575 - 623
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

del Álamo, J., Jimenez, J., Zandonade, P. & Moser, R. D. 2006 Self-similar vortex clusters in the turbulent logarithmic region. J. Fluid Mech. 561, 329–358.Google Scholar
Banica, V., Faou, E. & Miot, E. 2014 Collisions of vortex filament pairs. J. Nonlinear Sci. 24 (6), 1263–1284.Google Scholar
Banica, V., Faou, E. & Miot, E. 2016 Collision of almost parallel vortex filaments. Commun. Pure Appl. Maths 70 (2), 378–405.Google Scholar
Banica, V. & Miot, E. 2011 Global existence and collisions for symmetric configurations of nearly parallel vortex filaments. Ann. Inst. Henri Poincaré 29 (5), 813–832.CrossRefGoogle Scholar
Banica, V. & Miot, E. 2013 Evolution, interaction and collisions of vortex filaments. Differ. Integral Equ. 26 (3/4), 355–388.Google Scholar
Bewley, G. P., Paoletti, M. S., Sreenivasan, K. R. & Lathrop, D. P. 2008 Characterization of reconnecting vortices in superfluid helium. Proc. Natl Acad. Sci. USA 105, 13707–13710.Google Scholar
Bleckmann, H. & Zelick, R. 2009 Lateral line system of fish. Integrative Zoology 4 (1), 13–25.Google Scholar
Boersma, J. & Wood, D. H. 1999 On the self-induced motion of a helical vortex. J. Fluid Mech. 384, 263–280.Google Scholar
Breitsamter, C. 2011 Wake vortex characteristics of transport aircraft. Prog. Aerosp. Sci. 47 (2), 89–134.Google Scholar
Chagnaud, B. P., Bleckmann, H. & Engelmann, J. 2006 Neural responses of goldfish lateral line afferents to vortex motions. J. Expl Biol. 209 (2), 327.Google Scholar
Craig, W., Garcia-Azpeitia, C. & Yang, C.-R. 2017 Standing waves in near-parallel vortex filaments. Commun. Math. Phys. 350 (1), 175–203.Google Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8 (12), 2172–2179.CrossRefGoogle Scholar
Da Rios, L. S. 1906 Sul moto d’un liquido indefinite con un filetto vorticoso di forma qualunque. Rend. Circ. Mat. Palermo 22, 117.CrossRefGoogle Scholar
Das, C., Kida, S. & Goto, S. 2001 Overall self-similar decay of two-dimensional turbulence. J. Phys. Soc. Japan 70, 966–976.Google Scholar
Felli, M., Camussi, R. & Di Felice, F. 2011 Mechanisms of evolution of the propeller wake in the transition and far fields. J. Fluid Mech. 682, 5–53.CrossRefGoogle Scholar
Fernandez, V. M., Zabusky, N. J. & Gryanik, V. M. 1995 Vortex intensification and collapse of the Lissajous-elliptic ring: single-and multi-filament Biot–Savart simulations and visiometrics. J. Fluid Mech. 299, 289–331.Google Scholar
Fonda, E., Meichle, D. P., Ouellette, N. T., Hormoz, S. & Lathrop, D. P. 2014 Direct observation of Kelvin waves excited by quantized vortex reconnection. Proc. Natl Acad. Sci. USA 111, 4707–4710.Google Scholar
Franosch, J. P., Hagedorn, H. J. A., Goulet, J., Engelmann, J. & van Hemmen, J. L. 2009 Wake tracking and the detection of vortex rings by the canal lateral line of fish. Phys. Rev. Lett. 103, 078102.CrossRefGoogle ScholarPubMed
Fukumoto, Y. 1997 Stationary configurations of a vortex filament in background flows. Proc. R. Soc. Lond. A 453, 1205.CrossRefGoogle Scholar
Gutiérrez, S., Rivas, J. & Vega, L. 2003 Formation of singularities and self-similar vortex motion under the localized induction approximation. Commun. Part. Diff. Equ. 28, 927–968.Google Scholar
Hardin, J. C. 1982 The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952.Google Scholar
Hasimoto, H. 1971 Motion of a vortex filament and its relation to elastica. J. Phys. Soc. Japan 31, 293.Google Scholar
Hasimoto, H. 1972 A soliton on a vortex filament. J. Fluid Mech. 51 (3), 477–485.Google Scholar
Jimenez, J. 1975 Stability of a pair of co-rotating vortices. Phys. Fluids 18 (11), 1580–1581.Google Scholar
Kida, S. 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397.CrossRefGoogle Scholar
Kida, S. 1982 Stability of a steady vortex filament. J. Phys. Soc. Japan 51, 1655.Google Scholar
Kimura, Y. 1987 Similarity solutions of two-dimensional point vortices. J. Phys. Soc. Japan 56, 2024–2030.Google Scholar
Kimura, Y. 2009 Self-similar collapse of a 3d straight vortex filament model. Geophys. Astrophys. Fluid Dyn. 103, 135–142.Google Scholar
Kimura, Y. 2010 Self-similar collapse of 2d and 3d vortex filament models. Theor. Comput. Fluid Dyn. 24, 389–394.CrossRefGoogle Scholar
Klein, R. & Knio, O. M. 1995 Asymptotic vorticity structure and numerical simulation of slender vortex filaments. J. Fluid Mech. 284, 275–321.Google Scholar
Klein, R. & Majda, A. J. 1991 Self-stretching of a perturbed vortex filament I. The asymptotic equation for deviations from a straight line. Physica D 49 (3), 323–352.Google Scholar
Klein, R., Majda, A. J. & Damodaran, K. 1995 Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248.Google Scholar
Klein, R., Majda, A. J. & McLaughlin, R. M. 1992 Asymptotic equations for the stretching of vortex filaments in a background flow field. Phys. Fluids A 4 (10), 2271–2281.CrossRefGoogle Scholar
Lauder, G. V., Nauen, J. C. & Drucker, E. G. 2002 Experimental hydrodynamics and evolution: function of median fins in ray-finned fishes. Integr. Compar. Biol. 42 (5), 1009–1017.Google Scholar
Lazer, A. C. & McKenna, P. J. 1990 Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 32 (4), 537–578.Google Scholar
Leweke, T., Le Dizés, S. & Williamson, C. H. K. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48 (1), 507–541.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.Google Scholar
Leweke, T. & Williamson, C. H. K. 2011 Experiments on long-wavelength instability and reconnection of a vortex pair. Phys. Fluids 23 (2), 024101.Google Scholar
Lipniacki, T. 2000 Evolution of quantum vortices following reconnection. Eur. J. Mech. (B/Fluids) 19, 361–378.CrossRefGoogle Scholar
Lipniacki, T. 2003a Quasi-static solutions for quantum vortex motion under the localized induction approximation. J. Fluid Mech. 477, 321–337.Google Scholar
Lipniacki, T. 2003b Shape-preserving solutions for quantum vortex motion under localized induction approximation. Phys. Fluids 15, 1381–1395.Google Scholar
Meunier, P., Le Dizés, S. & Leweke, T. 2005 Physics of vortex merging. C. R. Phys. 6 (4), 431–450.Google Scholar
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. Lond. A 272 (1226), 403–429.Google Scholar
Nemirovskii, S. K. 2008 Kinetics of a network of vortex loops in He II and a theory of superfluid turbulence. Phys. Rev. B 77, 214509.Google Scholar
Nemirovskii, S. K. 2013 Quantum turbulence: theoretical and numerical problems. Phys. Rep. 524, 85–202.Google Scholar
Nemirovskii, S. K. & Baltsevich, A. J. 2001 Stochastic dynamics of a vortex loop, large-scale stirring force. Quantized Vortex Dynamics and Superfluid Turbulence. Springer.Google Scholar
Okulov, V. L. 2004 On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.Google Scholar
Okulov, V. L. & Sørensen, J. N. 2007 Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 1–25.Google Scholar
Parslew, B. & Crowther, W. J. 2013 Theoretical modelling of wakes from retractable flapping wings in forward flight. Peer J. 1, e105.Google Scholar
Pelz, R. B. 1997 Locally self-similar, finite-time collapse in a high-symmetry vortex filament model. Phys. Rev. E 55, 1617.Google Scholar
Pohlmann, K., Grasso, F. W. & Breithaupt, T. 2001 Tracking wakes: the nocturnal predatory strategy of piscivorous catfish. Proc. Natl Acad. Sci. USA 98 (13), 7371–7374.Google Scholar
Ricca, R. L. 1993 Torus knots and polynomial invariants for a class of soliton equations. Chaos 3 (83), 83–91.Google Scholar
Ricca, R. L. 1994 The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241.Google Scholar
Ricca, R. L. 1996 The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics. Fluid Dyn. Res. 18, 245.Google Scholar
Sonin, E. B. 2012 Dynamics of helical vortices and helical-vortex rings. Europhys. Lett. 97, 46002.Google Scholar
Tytell, E. D. 2006 Median fin function in bluegill sunfish Lepomis macrochirus streamwise vortex structure during steady swimming. J. Expl Biol. 209 (8), 1516.Google Scholar
Umeki, M. 2010 A locally induced homoclinic motion of a vortex filament. Theor. Comput. Fluid Dyn. 24, 383.Google Scholar
Van Gorder, R. A. 2012a Exact solution for the self-induced motion of a vortex filament in the arclength representation of the local induction approximation. Phys. Rev. E 86, 057301.Google Scholar
Van Gorder, R. A. 2012b Integrable stationary solution for the fully nonlinear local induction equation describing the motion of a vortex filament. Theor. Comput. Fluid Dyn. 26, 591.Google Scholar
Van Gorder, R. A. 2013a Orbital stability for rotating planar vortex filaments in the Cartesian and arclength forms of the local induction approximation. J. Phys. Soc. Japan 82, 094005.Google Scholar
Van Gorder, R. A. 2013b Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation. Phys. Rev. E 87, 043203.Google Scholar
Van Gorder, R. A. 2013c Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall–Vinen model including superfluid friction. Phys. Fluids 25, 095105.Google Scholar
Van Gorder, R. A. 2015a Helical vortex filament motion under the non-local Biot–Savart model. J. Fluid Mech. 762, 141–155.Google Scholar
Van Gorder, R. A. 2015b Non-local dynamics governing the self-induced motion of a planar vortex filament. Phys. Fluids 27, 065105.Google Scholar
Van Gorder, R. A. 2016 Self-similar vortex filament motion under the non-local Biot–Savart model. J. Fluid Mech. 802, 760–774.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641.Google Scholar
Yoshimoto, H. & Goto, S. 2007 Self-similar clustering of inertial particles in homogeneous turbulence. J. Fluid Mech. 577, 275–286.Google Scholar
Zhou, H. 1997 On the motion of slender vortex filaments. Phys. Fluids 9, 970.Google Scholar