Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the downstream development and breakup of systems of trailing-line vortices


The downstream evolution of two configurations of trailing-line vortex systems is considered and analysed from the point of view of bi-global instability. The results lead to predictions for the breakup of the systems considered, based primarily on stability analysis. Throughout, the analysis is entirely rational from the theoretical (asymptotic, large Reynolds number) point of view, with a relatively long developmental lengthscale in the streamwise direction for the base flow, and a relatively short streamwise wavelength for the bi-global stability analysis. As such, the stability modes are inviscid in nature, and therefore likely to be one of the dominant instability mechanisms. The issue of adverse streamwise freestream pressure gradients is also addressed, and it is suggested that these can lead to a rapid breakdown/up of a vortex system, analogous to the axisymmetric case discussed by Hall (Ann Rev Fluid Mech 4:195–218, 1972).

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


  1. 1

    Batchelor G.K.: Axial flow in trailing-line vortices. J. Fluid Mech. 20, 645–658 (1964)

  2. 2

    Crouch J.D.: Instability and transient growth for two trailing-vortex pairs. J. Fluid Mech. 350, 311–330 (1997)

  3. 3

    Crow S.C.: Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179 (1970)

  4. 4

    Duck P.W.: The inviscid stability of swirling flows: large wavenumber disturbances. Z. Angew. Math. Phys. 37, 340–360 (1986)

  5. 5

    Duck P.W.: Transient growth in developing plane and Hagen Poiseuille flow. Proc. Roy. Soc. Lond. A 461, 1311–1333 (2005)

  6. 6

    Duck P.W.: Nonlinear growth (and breakdown) of disturbances in developing Hagen Poiseuille flow. Phys. Fluids 18, 064103 (2006)

  7. 7

    Duck P.W., Foster M.R.: The inviscid stability of a trailing line vortex. Z. Angew. Math. Phys. 31, 524–532 (1980)

  8. 8

    Duck P.W., Khorrami M.R.: A note on the effects of viscosity on the stability of a trailing-line vortex. J. Fluid Mech. 245, 175–189 (1992)

  9. 9

    Fabre D., Jacquin L.: Stability of a four-vortex aircraft wake model. Phys. Fluids 12, 2438–2443 (2000)

  10. 10

    Goldstein S.: On laminar boundary-layer flow near a position of separation. Q. J. Mech. Appl. Math. 1, 43–69 (1948)

  11. 11

    Hall M.G.: Vortex breakdown. Ann. Rev. Fluid Mech. 4, 195–218 (1972)

  12. 12

    Hall P., Horseman N.J.: The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375 (1991)

  13. 13

    Hein S., Theofilis V.: On stability characteristics of isolated vortices and models of trailing-vortex systems. Comp. Fluids 33, 741–753 (2004)

  14. 14

    Khorrami M.R.: On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197–212 (1991)

  15. 15

    Khorrami M.R.: Behavior of asymmetric unstable modes of a trailing line vortex near the upper neutral curve. Phys. Fluids A4, 1310–1313 (1992)

  16. 16

    Lacaze L., Ryan K., Le Dizès S.: Elliptic instability in a stained Batchelor vortex. J. Fluid Mech. 577, 341–361 (2007)

  17. 17

    Leibovich S., Stewartson K.: A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356 (1983)

  18. 18

    Lessen M., Paillet F.: The stability of a trailing line vortex. Part 2. J. Fluid Mech. 65, 769–779 (1974)

  19. 19

    Lessen M., Singh P.J., Paillet F.: The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763 (1974)

  20. 20

    Ludweig H.: Zur Erklärung der Instabilität der über angestellten Deltaflügeln aufteten freien Wirkbelkerne. Z. Flugwiss. 10, 242–249 (1962)

  21. 21

    Ludweig H.: Explanation of vortex breakdown by the stability theory for spiralling flows. IUTAM Symposium on Vortex Motions, Ann Arbor. Available as AVA-Bericht 64A, 14 (1964)

  22. 22

    Mayer E.W., Powell K.G.: Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114 (1992)

  23. 23

    Meunier P., Leweke T.: Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533, 125–159 (2005)

  24. 24

    Ortega J.M., Bristol R.L., Savaş Ö: Experimental study of the instability of unequal-strength counter-rotating vortex pairs. J. Fluid Mech. 474, 35–84 (2003)

  25. 25

    Otto S.R., Denier J.P.: Numerical solution of a generalized elliptic partial differential eigenvalue problem. J. Comp. Phys. 156, 352–359 (1999)

  26. 26

    Spalart P.R.: Airplane trailing vortices. Ann. Rev. Fluid Mech. 30, 107–138 (1998)

  27. 27

    Stewartson K.: The stability of swirling flows at large Reynolds number when subjected to disturbances with large azimuthal wavenumber. Phys. Fluids 25, 1953–1957 (1982)

  28. 28

    Stewartson K., Brown S.N.: Near-neutral centre-modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387–399 (1985)

  29. 29

    Stewartson K., Capell K.: On the stability of ring modes in a trailing line vortex: the upper neutral points. J. Fluid Mech. 156, 369–386 (1985)

  30. 30

    Stewartson K., Leibovich S.: On the stability of a columnar vortex to disturbances with large azimuthal wavenumber: the lower neutral points. J. Fluid Mech. 178, 549–566 (1987)

  31. 31

    Thomas P.J., Auerbach D.: The observation of the simultaneous development of a long- and a short-wave instability mode on a vortex pair. J. Fluid Mech. 265, 289–302 (1994)

Download references

Author information

Correspondence to Peter W. Duck.

Additional information

Communicated by T. Colonius

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Duck, P.W. On the downstream development and breakup of systems of trailing-line vortices. Theor. Comput. Fluid Dyn. 25, 43–52 (2011). https://doi.org/10.1007/s00162-010-0186-6

Download citation


  • Trailing vortices
  • Vortex breakdown
  • Global instabilities