Curvature instability of a vortex ring

Journal of Fluid Mechanics

Volumn 526, Issue , 2005, Pages 77-115

  Fukumoto, Yasuhide ^a   Hattori, Yuji ^b  

^a KYUSHU UNIVERSITY   (Japan)

^b KYUSHU INSTITUTE OF TECHNOLOGY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; FUNCTIONS; RESONANCE; STABILITY; VISCOSITY;

AZIMUTHAL WAVENUMBERS; CORE RADIUS; EULER EQUATIONS; QUADRUPOLE FIELDS;

VORTEX FLOW;

STABILITY; VORTEX RING;

EID: 15944362739     PISSN: 00221120     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0022112004002678     Document Type: Article

References (47)

1. Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
2. 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.
3. Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160-2163.
4. Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1-14.
5. Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172-2179.
6. Dellnitz, M., Melbourne, I. & Marsden, J. E. 1992 Generic bifurcation of Hamiltonian vector fields with symmetry. Nonlinearity 5, 979-996.
7. Dyson, F. W. 1893 The potential of an anchor ring. - part II. Phil. Trans. R. Soc. Lond. A 184, 1041-1106.
8. 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.
9. Eloy, C. & Le Dizès, S. 2001 Stability of the Rankine vortex in a multipolar strain field. Phys. Fluids 13, 660-676.
10. Fraenkel, L. E. 1970 On steady vortex rings of small cross-section in an ideal fluid. Proc. R. Soc. Lond. A 316, 29-62.
11. Fraenkel, L. E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119-135.
12. Fukumoto, Y. 2002 Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring. Fluid Dyn. Res. 30, 67-95.
13. Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287-318.
14. Fukumoto, Y. & Hattori, Y. 2002 Linear stability of a vortex ring revisited. In Proc. IUTAM Symposium on Tubes, Sheets and Singularities in Fluid Dynamics (ed. H. K. Moffatt & K. Bajer), pp. 37-48. Kluwer.
15. 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, 1-45.
16. Gledzer, E. B., Dolzhansky, F. V., Obukhov, A. M. & Ponomarev, V. M. 1975 An experimental and theoretical study of the stability of motion of a liquid in an elliptical cylinder. Izv. Atoms. Ocean. Phys. 11, 617-622.
17. Gledzer, E. B. & Ponomarev, V. M. 1992 Instability of bounded flows with elliptical streamlines. J. Fluid Mech. 240, 1-30.
18. Guckenheimer, J. & Mahalov, A. 1992 Instability induced by symmetry reduction. Phys. Rev. Lett. 68, 2257-2260.
19. Hattori, Y. & Fukumoto, Y. 2003 Short-wavelength stability analysis of thin vortex rings. Phys. Fluids 15, 315l-3163.
20. Kerswell, R. R. 1993 The instability of precessing flow. Geophys. Astrophys. Fluid Dyn. 72, 107-144.
21. Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83-113.
22. Knobloch, E., Mahalov, A. & Marsden, J. E. 1994 Normal forms for three-dimensional parametric instabilities in ideal hydrodynamics. Physica D 73, 49-81.
23. Kop'ev, V. F. & Chernyshev, S. A. 2000 Vortex ring oscillations, the development of turbulence in vortex rings and generation of sound. Phys. Uspekhi 43, 663-690.
24. Kopiev, V. F. & Chernyshev, S. A. 1997 Vortex ring eigen-oscillations as a source of sound. J. Fluid Mech. 341, 19-57.
25. Krasny, R., Lindsay, K. & Nitsche, M. 2002 Simulation of vortex sheet roll-up: chaos, azimuthal waves, ring merger. In Proc. IUTAM Symposium on Tubes, Sheets and Singularities in Fluid Dynamics (ed. H. K. Moffatt & K. Bajer), pp. 3-12. Kluwer.
26. Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85-119.
27. Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3, 2644-2651.
28. Lugomer, S. & Fukumoto, Y. 2005 Hierarchical instability of a vortex ring array in multipulse laser-matter interactions. Fluid Dyn. Res. (to appear).
29. MacKay, R. S. 1986 Stability of equilibria of Hamiltonian systems. In Nonlinear Phenomena and Chaos (ed. S. Sarkar), pp. 254-270. Adam Hilger, Bristol.
30. Mahalov, A. 1993 The instability of rotating fluid columns subjected to a weak external Coriolis force. Phys. Fluids A 5, 891-900.
31. Marsden, J. E. 1992 Lecture on Mechanics. London Mathematical Society Lecture Notes Series vol. 174, Chap. 10. Cambridge University Press.
32. Mason, R. M. & Kerswell, R. R. 2002 Chaotic dynamics in a strained rotating flow: a precessing plane fluid layer. J. Fluid Mech. 471, 71-106.
33. Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 15-32.
34. Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465-495.
35. 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.
36. Naitoh T., Fukuda, N., Gotoh, T., Yamada, H. & Nakajima, K. 2002 Experimental study of axial flow in a vortex ring. Phys. Fluids 14, 143-149.
37. Norbury, J. 1973 A family of steady vortex rings. J. Fluid Mech. 57, 417-431.
38. Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625-639.
39. Saffman, P. G. 1992 Vortex Dynamics, Chap. 12. Cambridge University Press.
40. Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24, 235-279.
41. Shariff, K., Verzicco, R. & Orlandi, P. 1994 A numerical study of three-dimensional vortex ring instabilities: viscous corrections and early nonlinear stage. J. Fluid Mech. 279, 351-375.
42. 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, 721-733.
43. Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2, 76-80.
44. Weigand, A. & Gharib, M. 1997 On the evolution of laminar vortex rings. Exps. Fluids 22, 447-457.
45. Widnall, S. E., Bliss, D. B. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35-47.
46. Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex ring. Proc. R. Soc. Lond. A 332, 335-353.
47. 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 (referred to herein as WT77).