Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-06-23T07:53:20.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional distortions of a vortex filament with axial velocity

Published online by Cambridge University Press:  26 April 2006

Yasuhide Fukumoto  and
Takeshi Miyazaki
Yasuhide Fukumoto
Affiliation:
Department of Physics, Faculty of Science, University of Tokyo Bunkyo-ku, Tokyo 113, Japan Present address: Department of Applied Physics, Faculty of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan.
Takeshi Miyazaki
Affiliation:
National Institute for Environmental Studies, Tsukuba, Ibaraki 305, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

Three-dimensional motion of a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated. The deflections of the core centreline are not restricted to be small compared with the core radius. We first derive the equation of the vortex motion, correct to the second order in the ratio of the core radius to that of curvature, by a matching procedure, which recovers the results obtained by Moore & Saffman (1972). An asymptotic formula for the linear dispersion relation is obtained up to the second order. Under the assumption of localized induction, the equation governing the self-induced motion of the vortex is reduced to a nonlinear evolution equation generalizing the localized induction equation. This new equation is equivalent to the Hirota equation which is integrable, including both the nonlinear Schrödinger equation and the modified KdV equation in certain limits. Therefore the new equation is also integrable and the soliton surface approach gives the N-soliton solution, which is identical to that of the localized induction equation if the pertinent dispersion relation is used. Among other exact solutions are a circular helix and a plane curve of Euler's elastica. This local model predicts that, owing to the existence of the axial flow, a certain class of helicoidal vortices become neutrally stable to any small perturbations. The non-local influence of the entire perturbed filament on the linear stability of a helicoidal vortex is explored with the help of the cutoff method valid to the second order, which extends the first-order scheme developed by Widnall (1972). The axial velocity is found to discriminate between right- and left-handed helices and the long-wave instability mode is found to disappear in a certain parameter range when the successive turns of the helix are not too close together. Comparison of the cutoff model with the local model reveals that the non-local induction and the core structure are crucial in making quantitative predictions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 222 , January 1991 , pp. 369 - 416
Copyright
© 1991 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

Aref, H. & Flinchem, E. P., 1984 Dynamics of a vortex filament in a shear flow. J. Fluid Mech. 148, 477–497.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
Betchov, R.: 1965 On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22, 471–479.Google Scholar
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
Crow, S. C.: 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Fukumoto, Y. & Miyazaki, T., 1986 N-solitons on a curved vortex filament. J. Phys. Soc. Japan 55, 4152–4155.Google Scholar
Fukumoto, Y. & Miyazaki, T., 1988 N-solitons propagating on a thin curved vortex filament. In Proc. 36th Japan Natl Congr. for Theoretical and Applied Mechanics, 1986; Theoretical and applied Mechanics, Vol. 36. Tokyo University Press.
Hama, F. R.: 1962 Progressive deformation of a curved vortex filament by its own induction. Phys. Fluids 5, 1156–1162.Google Scholar
Hama, F. R.: 1963 Progressive deformation of a perturbed line vortex filament. Phys. Fluids 6, 526–534.Google Scholar
Hasimoto, H.: 1971 Motion of a vortex filament and its relation to elastica. J. Phys. Soc. Japan 31, 293–294.Google Scholar
Hasimoto, H.: 1972 A soliton on a vortex filament. J. Fluid Mech. 51, 477–485.Google Scholar
Hasimoto, H. & Kambe, T., 1985 Simulation of invariant shape of a vortex filament with an elastic rod. J. Phys. Soc. Japan 54, 5–7.Google Scholar
Hirota, R.: 1973 Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14, 805–809.Google Scholar
Heirota, R.: 1982 Bilinearization of soliton equations. J. Phys. Soc. Japan 51, 323–331.Google Scholar
Hopfinger, E. J., Browand, F. K. & Gagne, Y., 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505–534.Google Scholar
Howard, L. N. & Gupta, A. S., 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Kida, S.: 1981 A vortex filament moving without change of form. J. Fluid Mech. 112, 397–409.Google Scholar
Kida, S.: 1982 Stability of a steady vortex filament. J. Phys. Soc. Japan 51, 1655–1662.Google Scholar
Kirchhoff, G.: 1859 Über das gleichgewicht und die bewegung eines unendlich dünnen elastischen stabes. J. f. Math. (Crelle), Bd. 56.Google Scholar
Krishnamoorthy, V.: 1966 Vortex breakdown and measurements of pressure fluctuation over slender wing. Ph.D. thesis, Southampton University.
Lamb, G. L.: 1977 Solitons on moving space curves. J. Math. Phys. 18, 1654–1661.Google Scholar
Leibovichr, S.: 1970 Weakly non-linear waves in rotating fluids. J. Fluid Mech. 42, 803–822.Google Scholar
Leibovich, S., Brown, S. N. & Patel, Y., 1986 Bending waves on inviscid columnar vortices. J. Fluid Mech. 173, 595–624.Google Scholar
Leibovich, S. & Ma, H.-Y. 1983 Soliton propagation on vortex cores and the Hasimoto soliton. Phys. Fluids 26, 3173–3179.Google Scholar
Lessen, M., Deshpande, N. V. & Hadji-Ohanes, B. 1973 Stability of a potential vortex with a non-rotating and rigid-rotating top-hat jet core. J. Fluid Mech. 60, 459–466.Google Scholar
Lessen, M., Singh, P. J. & Paillet, F., 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763.Google Scholar
Levi, D., Ragnisco, O. & Sym, A., 1984 Dressing method v.s. classical Darboux transformation. Nuovo Cim. 83B, 34–4l.Google Scholar
Levi, D., Sym, A. & Wojciechowski, S., 1983 N-solitons on a vortex filament. Phys. Lett. 94A, 408–411.Google Scholar
Levy, H. & Forsdyke, A. G., 1928 The steady motion and stability of a helical vortex. Proc. R. Soc. Lond. A 120, 670–690.Google Scholar
Lundgren, T. S. & Ashurst, W. T., 1989 Area-varying waves on curved vortex tubes with application to vortex breakdown. J. Fluid Mech. 200, 283–307.Google Scholar
Maxworthy, T., Hopfinger, E. J. & Redekopp, L. G., 1985 Wave motions on vortex cores. J. Fluid Mech. 151, 141–165.Google Scholar
Maxworthy, T., Mory, M. & Hopfinger, E. J., 1983 Waves on vortex cores and their relation to vortex breakdown. In Proc. AOARD Conf. on Aerodynamics of Vortical Type Flows in Three Dimensions: AOARD CPP-342, paper 29/Google Scholar
Miyazaki, T. & Fukumoto, Y., 1988 N-solitons on a curved vortex filament with axial flow. J. Phys. Soc. Japan. 57, 3365–3370.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, 403–429.Google Scholar
Moore, D. W. & Saffman, P. G., 1974 A note on the stability of a vortex ring of small cross-section. Proc. R. Soc. Lond. A 338, 535–537.Google Scholar
Moore, D. W. & Saffman, P. G., 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346. 415–425.Google Scholar
Pritchard, W. G.: 1970 Solitary waves in rotating fluids. J. Fluid Mech. 42, 61–83.Google Scholar
Saffman, P. G.: 1970 The velocity of viscous vortex rings. Stud. Appl. Maths 49, 371–380.Google Scholar
Saffman, P. G.: 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625–639.Google Scholar
Sym, A.: 1982 Soliton surfaces. Lett. Nuovo Cim. 33, 394–400.Google Scholar
Sym, A.: 1984 Soliton surfaces. VI. — Gauge invariance and final formulation of the approach. Lett. Nuovo Cim. 41, 353–360.Google Scholar
Sym, A.: 1985 Soliton surfaces and their applications. (Soliton geometry from spectral problems). In Geometric Aspects of The. Einstein Equations and Integrable Systems (ed. R. Martini), Lecture Notes in Physics, Vol. 239, pp. 154–231. Springer.
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.Google Scholar
Widnall, S. E.: 1972 The stability of a helical vortex filament. J. FluidMech. 54, 641–663.Google Scholar
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.Google Scholar
Widnall, S. E., Bliss, D. B. & Zalay, A., 1971 Theoretical and experimental study of the stability of a vortex pair. In Proc. Symp. on Aircraft Wake Turbulence. Seattle. Plenum.
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.Google Scholar