Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-05T17:14:29.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Theory of the vortex breakdown phenomenon

Published online by Cambridge University Press:  28 March 2006

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Department of Engineering, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The phenomenon examined is the abrupt structural change which can occur at some station along the axis of a swirling flow, notably the leading-edge vortex formed above a delta wing at incidence. Contrary to previous attempts at an explanation, the belief demonstrated herein is that vortex breakdown is not a manifestation of instability or of any other effect indicated by study of infinitesimal disturbances alone. It is instead a finite transition between two dynamically conjugate states of axisymmetric flow, analogous to the hydraulic jump in openchannel flow. A set of properties essential to such a transition, corresponding to a set shown to provide a complete explanation for the hydraulic jump, is demonstrated with wide generality for axisymmetric vortex flows; and the interpretation covers both the case of mild transitions, where an undular structure is developed without the need arising for significant energy dissipation, and the case of strong ones where a region of vigorous turbulence is generated. An important part of the theory depends on the calculus of variations; and the comprehensiveness with which certain properties of conjugate flow pairs are demonstrable by this analytical means suggests that present ideas may be useful in various other problems.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 14 , Issue 4 , December 1962 , pp. 593 - 629
Copyright
© 1962 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

Benjamin, T. Brooke 1956 On the flow in channels when rigid obstacles are placed in the stream. J. Fluid Mech. 1, 227.Google Scholar
Benjamin, T. Brooke 1962 The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12, 97.Google Scholar
Benjamin, T. Brooke & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. Roy. Soc. A, 224, 448.Google Scholar
Bolza, O. 1961 Lectures on the Calculus of Variations. New York: Dover.
Burns, J. C. 1953 Long waves in running water. Proc. Camb. Phil. Soc. 48, 695.Google Scholar
Elle, B. J. 1960 On the breakdown at high incidences of the leading-edge vortices on delta wings. J. Roy. Aero. Soc. 64, 596.Google Scholar
Fox, C. 1950 Calculus of Variations. Oxford University Press.
Hall, M. G. 1961 A theory for the core of a leading-edge vortex. J. Fluid Mech. 11, 209.Google Scholar
Harvey, J. K. 1960 Analysis of the vortex breakdown phenomenon, Part II. Aero. Dept., Imperial Coll., Rep. no. 103.Google Scholar
Harvey, J. K. 1962 Some observations of the vortex breakdown phenomenon. J. Fluid Mech. 14, 585.Google Scholar
Ince, E. L. 1926 Ordinary Differential Equations. London: Longmans. (Dover edition 1956.)
Jones, J. P. 1960 The breakdown of vortices in separated flow. Dept. Aero. Astro., Univ. of Southampton, Rep. no. 140.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Long, R. R. 1953 Steady motion around a symmetrical obstacle moving along the axis of a rotating fluid. J. Met. 10, 197.Google Scholar
Ludwieg, H. 1961 Contribution to the explanation of the instability of vortex cores above lifting delta wings. Aero. Versuchsanstalt, Gottingen, Rep. AVA/61 A01.Google Scholar
Milne-Thomson, L. M. 1950 Theoretical Hydrodynamics, 2nd ed. New York: Macmillan.
Squire, H. B. 1956 Rotating fluids. Art. in Surveys in Mechanics (ed. Batchelor and Davies). Cambridge University Press.
Squire, H. B. 1960 Analysis of the vortex breakdown phenomenon, Part I. Aero. Dept., Imperial Coll., Rep. no. 102.Google Scholar
Werlé, H. 1960 Sur l'eclatement des tourbillons d'apex d'une aile delta aux faibles vitesses. Res. Aero., Paris, no. 74, 23.Google Scholar