Theoretical and Experimental Study of the Stability of a Vortex Pair

  • S. E. Widnall
  • D. Bliss
  • A. Zalay


The linear stability of the trailing vortex pair from an aircraft is discussed. The method of matched asymptotic expansions is used to obtain a general solution for the flow field within and near a curved vortex filament with an arbitrary distribution of swirl and axial velocities. The velocity field induced in the neighborhood of the vortex core by distant portions of the vortex line is calculated for a sinusoidally perturbed vortex filament and for a vortex ring. General expressions for the self-induced motion are given for these two cases. It is shown that the details of the vorticity and axial velocity distributions affect the self-induced motion only through the kinetic energy of the swirl and the axial momentum flux. The presence of axial velocity in the core reduces both the angular velocity of the sinusoidal vortex filament and the speed of the ring. The vortex pair instability is then considered in terms of the more general model for self-induced motion of the sinusoidal vortex. The presence of axial velocity within the core slightly decreases the amplification rate of the instability. Experimental results for the distortion and breakup of a perturbed vortex pair are presented.


Axial Velocity Vortex Ring Vortex Pair Vortex Breakdown Vorticity Distribution 
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  1. 1.
    Crow, W. C., “Stability Theory for a Pair of Trailing Vortices”, AIAA Paper No. 70–53 [January, 1970 ].Google Scholar
  2. 2.
    Landahl, M. and S. Widnall, “Vortex Control”, Symposium on Aircraft Wake Turbulence, September 1970.Google Scholar
  3. 3.
    Van Dyke, M., Perturbation Methods in Fluid Mechanics, Academic Press, New York, [ 1964 ].MATHGoogle Scholar
  4. 4.
    Thomson, Sir W. (Lord Kelvin), “Vibrations of a Columnar Vortex”, Mathematical and Physical Papers, Volume IV, Cambridge University Press, Cambridge, England, [ 1910 ].Google Scholar
  5. 5.
    Lamb, Sir. H., Hydrodynamics, Dover Publications, Inc., New York, [ 1945 ].Google Scholar
  6. 6.
    Tung, C. and Ting, L., “The Motion and Decay of a Vortex Ring”, Physics of Fluids, Vol. 10, No. 5, pp. 901–10, [May, 1967 ].CrossRefGoogle Scholar
  7. 7.
    Saffman, P. G., “The Velocity of Viscous Vortex Rings”, Symposium on Aircraft Wake Turbulence, September 1970.Google Scholar
  8. 8.
    Bliss, D. B., “The Dynamics of Curved Rotational Vortex Lines”, Massachusetts Institute of Technology Masters Thesis, September 1970.Google Scholar
  9. 9.
    Hildebrand, F. B., Advanced Calculus for Applications, Fifth Printing, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, pp. 29–31, [ 1965 ].Google Scholar
  10. 10.
    Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, England, p. 529, [ 1967 ].Google Scholar
  11. 11.
    Zalay, A. D., “Experimental Investigation of the Decay of a Vortex Pair”, S.M. Thesis, Massachusetts Institute of Technology, June, 1970.Google Scholar
  12. 12.
    Withycombe, E. II, “Wingtip Vortex Decay: An Experimental Investigation”, S.B. Thesis, Massachusetts Institute of Technology, August 1970.Google Scholar

Copyright information

© Plenum Press, New York 1971

Authors and Affiliations

  • S. E. Widnall
    • 1
  • D. Bliss
    • 1
  • A. Zalay
    • 1
  1. 1.Massachusetts Institute of TechnologyUSA

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