Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-05T08:19:51.282Z Has data issue: false hasContentIssue false
  • English
  • Français

The evolution of an elliptic vortex ring

Published online by Cambridge University Press:  20 April 2006

M. R. Dhanak  and
B. DE Bernardinis
M. R. Dhanak
Affiliation:
Department of Mathematics, Imperial College, London
B. DE Bernardinis
Affiliation:
Department of Mathematics, Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

The evolution of a vortex ring in an ideal fluid under self-induction from a flat and elliptic configuration is followed numerically using the cut-off approximation (Crow 1970) for the velocity at the vortex. Calculations are presented for four different axes ratios of the initial ellipse. A particular choice is made for the core size and vorticity distribution in the core of the vortex ring. When the initial axes ratio is close to 1, the vortex ring oscillates periodically. The periodicity is lost as more eccentric cases are considered. For initial axes ratio 0·2, the calculations suggest a break-up of the ring through the core at one portion of the ring touching that at another, initially distant, portion of the ring.

Results from quantitative experiments, conducted at moderate Reynolds number with the vortex rings produced by puffing air through elliptic orifices, are compared with the calculations. The agreement is fairly good and it is found that a vortex ring produced from an orifice of axes ratio 0·2 breaks up into two smaller rings. The relevance of the results to the vortex trail of an aircraft is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 109 , August 1981 , pp. 189 - 216
Copyright
© 1981 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

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.Google Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. A.I.A.A. J. 8, 2172.Google Scholar
Didden, N. 1979 On the formation of vortex rings. Z. angew. Math. Phys. 30, 101.Google Scholar
Fohl, T. & Turner, J. S. 1975 Colliding vortex rings. Phys. Fluids 18, 433.Google Scholar
Fraenkel, L. E. 1970 On steady vortex rings of small cross-section in an ideal fluid. Proc. Roy. Soc. A 316, 29.Google Scholar
Kambe, T. & Takao, T. 1971 Motion of distorted vortex rings. J. Phys. Soc. Japan 31, 591.Google Scholar
Kokshaysky, N. V. 1979 Tracing the wake of a flying bird. Nature 279, 176.Google Scholar
Lamb, H. 1932 Hydrodynamics. Dover.
Leonard, A. 1974 Numerical simulation of interacting three-dimensional vortex filaments. Proc. 4th Int. Conf. Numerical Methods in Fluid Dynamics, p. 245. Springer.
Moore, D. W. 1972 Finite amplitude waves on aircraft trailing vortices. Aeronaut. Quart. 23, 307.Google Scholar
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. R. Soc. A 272, 403.Google Scholar
Norbury, J. 1978 A family of steady vortex rings. J. Fluid Mech. 57, 417.Google Scholar
Oshima, Y. 1972 Motion of vortex rings in water. J. Phys. Soc. Japan 32, 1125.Google Scholar
Oshima, Y. 1978 Head-on collision of two vortex rings. J. Phys. Soc. Japan 44, 328.Google Scholar
Oshima, Y. & Asaka, S. 1977a Interaction of two vortex rings along parallel axes in air, J. Phys. Soc. Japan 42, 708.Google Scholar
Oshima, Y. & Asaka, S. 1977b Interaction of multi-vortex rings, J. Phys. Soc. Japan 42, 1391.Google Scholar
Pullin, D. I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88, 401.Google Scholar
Rayner, J. M. V. 1979 A vortex theory of animal flight. Part 2. The forward flight of birds. J. Fluid Mech. 91, 731.Google Scholar
Saffman, P. G. 1970 The velocity of viscous vortex rings. Stud. Appl. Math. 49, 371.Google Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625.Google Scholar
Sallet, D. W. 1975 Impulsive motion of a circular disk which causes a vortex ring. Phys. Fluids 18, 109.Google Scholar
Sallet, D. W. & Widemayer, R. S. 1974 An experimental investigation of laminar and turbulent vortex rings in air. Z. Flugwiss. 22, 207.Google Scholar
Sheffield, J. S. 1977 Trajectories of an ideal vortex pair near an orifice. Phys. Fluids 20, 543.Google Scholar
Taylor, G. I. 1953 Formation of a vortex ring by giving an impulse to a circular disc and then dissolving it away. J. Appl. Phys. 24, 104.Google Scholar
Viets, H. & Sforza, P. M. 1975 Dynamics of bilaterally summetric vortex rings. Phys. Fluids 15, 230.Google Scholar
Widnall, S. E., Bliss, D. B. & Zalay, A. 1971 Theoretical and experimental study of the stability of a vortex pair. In Aircraft Wake Turbulence and its Detection (ed. J. H. Olsen, A. Goldburg & M. Rogers), pp. 305–338. Plenum.
Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex rings. Proc. R. Soc. A 322, 335.Google Scholar
Widnall, S. E. & Tsai, C. Y. 1977 The instability of a thin vortex ring of constant vorticity. Phil. Trans. R. Soc. A 287, 273.Google Scholar