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Simplified equations for the interaction of nearly parallel vortex filaments

Published online by Cambridge University Press:  26 April 2006

Rupert Klein ,
Andrew J. Majda  and
Kumaran Damodaran
Rupert Klein
Affiliation:
Institut fur Technische Mechanik, RWTH, Templergraben 64, 52056, Aachen, Germany
Andrew J. Majda
Affiliation:
Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
Kumaran Damodaran
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

New simplified asymptotic equations for the interaction of nearly parallel vortex filaments are derived and analysed here. The simplified equations retain the important physical effects of linearized local self-induction and nonlinear potential vortex interaction among different vortices but neglect other non-local effects of self-stretching and mutual induction. These equations are derived systematically in a suitable distinguished asymptotic limit from the Navier–Stokes equations. The general Hamiltonian formalism and conserved quantities for the simplified equations are developed here. Properties of these asymptotic equations for the important special case involving nearly parallel pairs of interacting filaments are developed in detail. In particular, strong evidence is presented that for any filament pair with a negative circulation ratio, there is finite-time collapse in a self-similar fashion independent of the perturbation but with a structure depending on the circulation ratio. On the other hand, strong evidence is presented that no finite-time collapse is possible for perturbations of a filament pair with a positive circulation ratio. The present theory is also compared and contrasted with earlier linear and nonlinear stability analyses for pairs of filaments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 288 , 10 April 1995 , pp. 201 - 248
Copyright
© 1995 Cambridge University Press

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References

Ablowitz, M. & Segur, H. 1981 Solitons and the inverse scattering transform. SIAM Stud. Appl. Maths 4.Google Scholar
Anderson, C. & Greengard, C. 1989 The vortex ring merger problem at infinite Reynolds number. Commun. Pure Appl. Maths 42, 1123.Google Scholar
Aref, H. 1983 Integrable, chaotic and turbulent vortex motion in two-dimensional flows. Ann. Rev. Fluid Mech. 15, 345–389.Google Scholar
Arms, R. J. & Hama, F. R. 1965 Localized-induction concept on a curved vortex and motion of an elliptic vortex ring. Phys. Fluids 8, 553–559.Google Scholar
Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics, 2nd Edn. Springer.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
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
Chorin, A. J. 1982 Evolution of a turbulent vortex. Commun. Math. Phys. 83, 517.Google Scholar
Chorin, A. J. 1988 Spectrum, dimension and polymer analogies in fluid turbulence. Phys. Rev. Lett. 60, 1947–1949.Google Scholar
Chorin, A. J. 1994 Vorticity and Turbulence. Springer.
Chorin, A. J. & Akao, J. 1991 Vortex equilibria in turbulence theory and quantum analogues. Physica D 52, 403–414.Google Scholar
Chorin, A. J. & Marsden, J. E. 1990 A Mathematical Introduction to Fluid Mechanics. Springer.
Corcos, G. & Lin, S. 1984 The mixing layer: deterministic models of a turbulent flow. Part 2. The origin of three-dimensional motion. J. Fluid Mech. 139, 67–95.Google Scholar
Crow, S. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Damodaran, K. 1994 Simplified equations for the nonlinear interaction of vortex filaments in 3-D. Undergraduate Senior Thesis supervised by A. Majda, Princeton University.
Hasimoto, H. 1972 A soliton on a vortex filament. J. Fluid Mech. 51, 477–485.Google Scholar
Jimenez, J. 1975 Stability of a pair of co-rotating vortices. Phys. Fluids 18, 1580.Google Scholar
Kerr, R. M. & Hussain, A. K. M. F. 1989 Simulation of vortex reconnection. Physica D 37, 474.Google Scholar
Kida, S. & Takaoka, M. 1991 Breakdown of frozen motion fields and vorticity reconnection. J. Phys. Soc. Japan 60, 2184.Google Scholar
Klein, R. 1994 Zur Dynamik schlanker Wirbel. Habilitationsschrift, RWTH Aachen, submitted May 2, 1994.
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. 1991a Self-stretching of a perturbed vortex filament I. The asymptotic equations for deviations from a straight line. Physica D 49, 323–352.Google Scholar
Klein, R. & Majda, A. 1991b Self-stretching of perturbed vortex filaments II. Structure of solutions. Physica D 53, 267–294.Google Scholar
Klein, R. & Majda, A. 1993 An asymptotic theory for the nonlinear instability of anti-parallel pairs of vortex filaments. Phys. Fluids A 5, 369–387.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, 2271–2281.Google Scholar
Lamb, H. 1932 Hydrodynamis, 6th Edn. Cambridge University Press.
Logan, J. D. 1987 Applied Mathematics – A Contemporary Approach. John Wiley & Sons.
Melander, M. V. & Zabusky, N. 1987 Interaction and Reconnection of Vortex Tubes via Direct Numerical Simulations. Proceedings IUTAM Symposium on Fundamental Aspects of Vortex Motion, Tokyo.
Meiron, D., Shelley, M., Ashurst, W. & Orszag, S. 1989 Numerical studies of vortex, reconnection. In Mathematical Aspects of Vortex Dynamics (ed. R. Caflisch), pp. 183–194. SIAM, Philadelphia, PA.
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. Lond. A 272, 403.Google Scholar
Ting, L. & Klein, R. 1991 Viscous Vortical Flows. Springer.
Van Dyke, M., 1982 An Album of Fluid Motion, Parabolic Press, Stanford, CA.
Widnall, S. 1975 The structure and dynamics of vortex filaments. Ann. Rev. Fluid Mech. 7, 141–165.Google Scholar
Zakharov, V. E. 1988 Wave collapse. Usp. Fiz. Nauk 155, 529–533.Google Scholar