Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-22T22:21:10.386Z Has data issue: false hasContentIssue false
  • English
  • Français

Some developments in the theory of vortex breakdown

Published online by Cambridge University Press:  28 March 2006

T. Brooke Benjamin
T. Brooke Benjamin
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

The primary aim of the analysis presented herein is to consolidate the ideas of the ‘conjugate-flow’ theory, which proposes that vortex breakdown is fundamentally a transition from a uniform state of swirling flow to one featuring stationary waves of finite amplitude. The original flow is assumed to be supercritical (i.e. incapable of bearing infinitesimal stationary waves), and the mechanism of the transition is explained on the basis of physical principles that are well established in relation to the analogous supercritical-flow phenomenon of the hydraulic jump or bore. In previous presentations of the theory the existence of appropriately descriptive solutions to the full equations of motion has only been inferred from these general principles, but here the solutions are demonstrated explicitly by means of a perturbation analysis. This has basically much in common with the classical theory of solitary and cnoidal waves, which is known to explain well the essential properties of weak bores.

In § 2 the basic equations of the problem are set out and the leading results of the original theoretical treatment are recalled. The new developments are mainly presented in § 3, where an analysis of finite-amplitude waves is completed by two different methods, each serving to illustrate points of interest. The effects of small energy losses and of small flow-force reductions (i.e. wave-resistance effects) are considered, and the analysis leads to a general classification of possible phenomena accompanying such changes of integral properties in either slightly supercritical or slightly subcritical vortex flows. The application to vortex breakdown remains the focus of attention, however, and § 3 includes a careful appraisal of some experimental observations on the phenomenon. In § 4 a summary is given of a variant on the previous methods which is required when the radial boundary of the flow is taken to infinity. The main analysis is developed without restriction to particular flow models, but in § 5 the results are applied to a specific example.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 28 , Issue 1 , 12 April 1967 , pp. 65 - 84
Copyright
© 1967 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. B. 1956 On the flow in channels when rigid obstacles are placed in the stream J. Fluid Mech. 1, 227.Google Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon J. Fluid Mech. 14, 593.Google Scholar
Benjamin, T. B. 1965 Significance of the vortex breakdown phenomenon Trans. Am. Soc. Mech. Engrs., J. Basic Engng. 87, 518.Google Scholar
Benjamin, T. B. 1966 Internal waves of finite amplitude and permanent form J. Fluid Mech. 25, 241.Google Scholar
Benjamin, T. B. & Barnard, B. J. S. 1964 A study of the motion of a cavity in a rotating liquid J. Fluid Mech. 19, 193.Google Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. Roy. Soc. A, 224, 448.Google Scholar
Fraenkel, L. E. 1966 On Benjamin's theory of conjugate vortex flows J. Fluid Mech. 28, 85.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 reprint 1956.)
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press. (Dover reprint 1945.)
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics. New York: McGraw-Hill.
Titchmarsh, E. C. 1962 Eigenfunction Expansions. Part 1, 2nd ed. Oxford University Press.
Whittaker, E. T. & Watson, G. N. 1927 Modern Analysis, 4th ed. Cambridge University Press.