Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-13T01:40:06.640Z Has data issue: false hasContentIssue false
  • English
  • Français

On the generation of surface waves by shear flows Part 3. Kelvin-Helmholtz instability

Published online by Cambridge University Press:  28 March 2006

John W. Miles
John W. Miles
Affiliation:
Department of Engineering, University of California, Los Angeles
Get access Rights & Permissions [Opens in a new window]

Abstract

The Kelvin-Helmholtz model for the formation of surface waves at the interface between two fluids in relative motion is generalized for parallel shear flows. It is assumed that phase changes across the flow are negligible and hence that the aerodynamic pressure on the wave is in phase with its displacement (rather than its slope). A variational formulation is established and leads to the determination of appropriately weighted means for the velocity profiles. The principal application is to flow of a light inviscid fluid over a viscous liquid; it is shown that the principle of exchange of stabilities is applicable to such a configuration, and a critical wind speed in satisfactory agreement with observation is predicted for an air-oil interface. The results also are applied to an air-water interface and lead to the conclusion that Kelvin-Helmholtz instability of such an interface is unlikely at commonly observed wind speeds. A more general formulation of the Kelvin-Helmholtz boundary-value problem and variational principle, allowing for variations in both velocity and density, is given in two appendices.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 6 , Issue 4 , November 1959 , pp. 583 - 598
Copyright
© 1959 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

Carrier, G. F. 1954 Interface stability of the Helmholtz type. Los Alamos Internal Report.
Courant, R. & Hilbert, D. 1931 Methoden der Mathematischen Physik. Berlin: Julius Springer.
Esch, R. E. 1957 J. Fluid Mech. 3, 289.
Francis, J. R. D. 1954 Phil. Mag. (7), 45, 695.
Francis, J. R. D. 1956 Phil. Mag. (8), 1, 685.
Francis, J. R. D. 1959 Private communication.
Goldstein, S. 1931 Proc. Roy. Soc. A, 132, 524.
Hay, J. S. 1955 Quart. J. Roy. Met. Soc. 81, 307.
Jeffreys, H. 1924 Proc. Roy. Soc. A, 107, 189.
Jeffreys, H. 1925 Proc. Roy. Soc. A, 110, 341.
Lord Kelvin 1871 Mathematical and Physical Papers, vol. iv, pp. 76–85. Cambridge University Press (1910).
Keulegan, G. H. 1951 J. Res. Nat. Bur. Stand. 46, 358.
Lamb, H. 1945 Hydrodynamics, 6th ed. New York: Dover.
Lessen, M. 1950 NACA Tech. Report 979, Washington, D.C.
Lighthill, M. J. 1957 J. Fluid Mech. 3, 113.
Miles, J. W. 1957 J. Fluid Mech. 3, 185.
Milne-Thomson, L. M. 1950 Theoretical Hydrodynamics. New York: Macmillan.
Munk, W. 1947 J. Mar. Res. 6, 203.
Phillips, O. M. 1957 J. Fluid Mech. 2, 417.
Prandtl, L. 1952 The Essentials of Fluid Dynamics. New York: Hafner.
Roll, H. 1948 Ann. Met., Hamburg, 1, 139.
Taylor, G. I. 1931 Proc. Roy. Soc. A, 132, 499.
Ursell, F. 1956 Article in Surveys in Mechanics. Cambridge University Press.
Van Dorn, W. 1953 J. Mar. Res. 12, 249.