Skip to main content
SpringerLink
Log in

Global time evolution of viscous vortex rings

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

This article gives an overview of growing knowledge of translation speed of an axisymmetric vortex ring, with focus on the influence of viscosity. Helmholtz–Lamb’s method provides a shortcut to manipulate the translation speed at both small and large Reynolds number, for a vortex ring starting from an infinitely thin core. The resulting asymptotics significantly improve Saffman’s formula (1970) and give closer lower and upper bounds on translation speed in an early stage. At large Reynolds numbers, Kelvin–Benjamin’s kinematic variational principle achieves a further simplification. At small Reynolds numbers, the whole life of a vortex ring is available from the vorticity obeying the Stokes equations, which is closely fitted, over a long time, by Saffman’s second formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Batchelor G.K.: An Introduction to Fluid Dynamics. University Press, Cambridge (1967)

    MATH  Google Scholar 

  2. Okabe, J.: Essay on the occasion of retirement [in Japanese]. Research Institute of Applied Mechanics, Kyushu University (1984)

  3. Kosugi, T., Takamori, F.H., Takeda, T.: Design and consideration of air cannons for tactile display. In papers of the 13th annual meeting of the Japan society of virtual reality [in Japanese], pp. 401–404 (2008)

  4. von Helmholtz, H.: Über integrale der hydrodynamischen gleichungen welche den wirbelbewegungen entsprechen. Crelles, J. 55, 25 (1858). English translation by Tait, P.G.: On integrals of the hydrodynamical equations which express vortex-motion. Phil. Mag. 33(4), 485–512 (1867)

  5. Lamb H.: Hydrodynamics, Chap. 7. Cambridge University Press, Cambridge (1932)

    Google Scholar 

  6. Thomson J.J.: A Treatise on the Motion of Vortex Rings. Macmillan, London (1883)

    Google Scholar 

  7. Hicks W.M.: Researches on the theory of vortex rings—part II. Philos. Trans. R. Soc. Lond. A 176, 725–780 (1885)

    Article  Google Scholar 

  8. Dyson F.W.: The potential of an anchor ring—part II. Philos. Trans. R. Soc. Lond. A 184, 1041–1106 (1893)

    Article  Google Scholar 

  9. Fraenkel L.E.: On steady vortex rings of small cross-section in an ideal fluid. Proc. R. Soc. Lond. A 316, 29–62 (1970)

    Article  MATH  Google Scholar 

  10. Fraenkel L.E.: Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135 (1972)

    Article  MATH  Google Scholar 

  11. Norbury J.: A family of steady vortex rings. J. Fluid Mech. 57, 417–431 (1973)

    Article  MATH  Google Scholar 

  12. Fukumoto Y., Moffatt H.K.: Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity. J. Fluid Mech. 417, 1–45 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fukumoto Y.: Higher-order asymptotic theory for the velocity field induced by an inviscid vortex ring. Fluid Dyn. Res. 30, 65–92 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fukumoto Y., Moffatt H.K.: Kinematic variational principle for motion of vortex rings. Physica D 237, 2210–2217 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Saffman P.G.: The velocity of viscous vortex rings. Stud. Appl. Math. 49, 371–380 (1970)

    MATH  Google Scholar 

  16. Callegari A.J., Ting L.: Motion of a curved vortex filament with decaying vortical core and axial velocity. SIAM J. Appl. Maths 35, 148–175 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  17. Tung C., Ting L.: Motion and decay of a vortex ring. Phys. Fluids 10, 901–910 (1967)

    Article  Google Scholar 

  18. Donnelly R.J.: Quantized Vortices in Helium II, Chap. 1. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  19. Saffman P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  20. Shariff K., Leonard A.: Vortex rings. Annu. Rev. Fluid Mech. 24, 235–279 (1992)

    Article  MathSciNet  Google Scholar 

  21. Lim T., Nickels T.: Vortex rings. In: Green, S.I. (eds) Fluid Vortices, Kluwer, Dordrecht (1995)

    Google Scholar 

  22. Roberts P.H., Donnelly R.J.: Dynamics of vortex rings. Phys. Lett. A 31, 137–138 (1970)

    Article  Google Scholar 

  23. Roberts P.H.: A Hamiltonian theory for weakly interacting vortices. Mathematika 19, 169–179 (1972)

    Article  MATH  Google Scholar 

  24. Kelvin Lord.: On the stability of steady and of periodic fluid motion. Phil. Mag. 23, 459–464 (1878)

    Google Scholar 

  25. Arnol’d V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications á l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier Grenoble 16, 319–361 (1966)

    MATH  Google Scholar 

  26. Benjamin, T.B.: The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. Lecture notes in math no. 503, pp. 8–29, Springer, Berlin (1976)

  27. Moffatt H.K.: Structure and stability of solutions of the Euler equations: a lagrangian approach. Philos. Trans. R. Soc. Lond. A 333, 321–342 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  28. Jones C.A., Roberts P.H.: Motions in a Bose condensate: IV. Axisymmetric solitary waves. J. Phys. A: Math. Gen. 15, 2599–2619 (1982)

    Article  Google Scholar 

  29. Maxworthy T.: The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32 (1972)

    Article  Google Scholar 

  30. Dabiri J., Gharib M.: Fluid entrainment by isolated vortex rings. J. Fluid Mech. 511, 311–331 (2004)

    Article  MATH  Google Scholar 

  31. Danaila I., Hélie J.: Numerical simulation of the postformation evolution of a laminar vortex ring. Phys. Fluids 20, 073602 (2008)

    Article  Google Scholar 

  32. Kaplanski F., Rudi U.: A model for the formation of ‘optimal’ vortex rings taking into account viscosity. Phys. Fluids 17, 087101 (2005)

    Article  MathSciNet  Google Scholar 

  33. Fukumoto Y., Kaplanski F.: Global time evolution of an axisymmetric vortex ring at low Reynolds numbers. Phys. Fluids 20, 053103 (2008)

    Article  Google Scholar 

  34. Kambe T., Oshima Y.: Generation and decay of viscous vortex rings. J. Phys. Soc. Japan 38, 271–280 (1975)

    Article  Google Scholar 

  35. Rott N., Cantwell B.: The decays of a viscous vortex pair. Phys. Fluids 31, 3213–3224 (1988)

    Article  MATH  Google Scholar 

  36. Rott N., Cantwell B.: Vortex drift. I: Dynamic interpretation. Phys. Fluids A 5, 1443–1450 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  37. Fukumoto Y., Okulov V.L.: The velocity field induced by a helical vortex tube. Phys. Fluids 17, 107101 (2005)

    Article  MathSciNet  Google Scholar 

  38. Fukumoto Y.: Three-dimensional motion of a vortex filament and its relation to the localized induction hierarchy. Euro. Phys. J. B 29, 167–171 (2002)

    Article  Google Scholar 

  39. Fukumoto, Y., Moffatt, H.K.: In preparation (2009)

  40. Burton G.R.: Vortex-rings of prescribed impulse. Math. Proc. Camb. Philos. Soc. 134, 515–528 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  41. Stanaway, S.K., Cantwell, B.J., Spalart, P.R.: A numerical study of viscous vortex rings using a spectral method. NASA Technical Memorandum 101041 (1988)

  42. Weigand A., Gharib M.: On the evolution of laminar vortex rings. Exps. Fluids 22, 447–457 (1997)

    Article  Google Scholar 

  43. Fukumoto Y.: A unified view of topological invariants of fluid flows. Topologica 1, 003 (2008)

    Article  Google Scholar 

  44. Mohseni K.: A formulation for calculating the translational velocity of a vortex rings or pair. Bioinsp. Biomim. 1, S57–S64 (2006)

    Article  Google Scholar 

  45. Phillips O.M.: The final period of decay of non-homogeneous turbulence. Proc. Camb. Phil. Soc. 252(Pt. 1), 135–151 (1956)

    Article  Google Scholar 

  46. Kaltaev, A.: Investigation of dynamic characteristics of motion of a vortex ring of viscous fluid, vol. 63 [In Russian]. In: Continuum Dynamics. Kazah State University, Alma-Ata (1982)

  47. Moffatt H.K.: Generalised vortex rings with and without swirl. Fluid Dyn. Res. 3, 22–30 (1988)

    Article  Google Scholar 

  48. Kaplanski F., Sazhin S.S., Fukumoto Y., Begg S., Heikal M.: A generalised vortex ring model. J. Fluid Mech. 622, 233–258 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Y. Fukumoto

Authors
  1. Y. Fukumoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. Fukumoto.

Additional information

Communicated by H. Aref

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fukumoto, Y. Global time evolution of viscous vortex rings. Theor. Comput. Fluid Dyn. 24, 335–347 (2010). https://doi.org/10.1007/s00162-009-0155-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-009-0155-0

Keywords

PACS

Search

Navigation