Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T09:11:47.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave breakdown in stratified shear flows

Published online by Cambridge University Press:  11 April 2006

M. T. Landahl  and
W. O. Criminale
M. T. Landahl
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge and Department of Mechanics, The Royal Institute of Technology, Stockholm 70, Sweden
W. O. Criminale
Affiliation:
Department of Oceanography, Geophysics Program and Applied Mathematics Group, University of Washington, Seattle
Get access Rights & Permissions [Opens in a new window]

Abstract

The wave-mechanical condition (Landahl 1972) for breakdown of an unsteady laminar flow into strong small-scale secondary instabilities is applied to some simple stratified inviscid shear flows. The cases considered have one or two discrete density interfaces and simple discontinuous or continuous velocity profiles. A primary wavelike disturbance to such a flow produces a perturbation velocity that is discontinuous at a density interface. The resulting instantaneous system, defined as the sum of the mean flow and the primary oscillation, develops a local secondary shear-flow instability that has a group velocity equal to the arithmetic mean of the instantaneous velocities on the two sides of the interface. According to the breakdown criterion, the disturbed flow will become critical whenever this velocity reaches a value equal to the phase velocity of the primary wave. The calculations show that for a single density interface breakdown may occur for low to moderate wave amplitudes in a fairly narrow range of Richardson numbers on the stable side of the stability boundary. On the other hand, in the unstable regime maximum wave slopes of order unity may be reached before breakdown occurs; this conclusion is in qualitative agreement with experiments. When the system includes two density interfaces, it is found that there exists a range of high Richardson numbers far into the stable regime for which breakdown may take place even for very small and zero wave interface deflexions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 79 , Issue 3 , 9 March 1977 , pp. 481 - 497
Copyright
© 1977 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

Browand, F. K. & Wang, H. Y. 1972 Proc. Int. Symp. Stratified Flow Novosibirsk, U.S.S.R. (ed. O. F. Vasiliev), p. 491. (See also Int. Symp. Stratified Flows, Am. Soc. Engrs, 1973.)
Browand, F. K. & Winant, C. D. 1973 Boundary-Layer Met. 5, 66.
Criminale, W. O. 1972 Proc. Int. Symp. Stratified Flow Novosibirsk, U.S.S.R. (ed.O. F.Vasiliev), p. 175. (See also Int. Symp. Stratified Flows, Am. Soc. Engrs, 1973.)
Gargett, A. E. & Hughes, B. A. 1972 J. Fluid Mech. 52, 179.
Goldstein, S. 1932 Proc. Roy. Soc. A 132, 524.
Holmboe, J. 1962 Geophys. Publ. 24, 67.
Howard, L. H. 1961 J. Fluid Mech. 10, 509.
Klebanoff, P., Tidstrom, K. D. & Sargent, L. H. 1962 J. Fluid Mech. 12, 1.
Landahl, M. 1972 J. Fluid Mech. 56, 775.
Mather, G. K. 1969 In Clear-Air Turbulence and Its Detection (ed. Y. H. Pao & A. Goldburg), p. 271. Plenum.
Maxworthy, T. B. & Browand, F. K. 1975 Ann. Rev. Fluid Mech. 7, 273.
Miles, J. W. 1961 J. Fluid Mech. 10, 496.
Phillips, O. M. 1973 Isv. Akad. Nauk SSSR, 9, 954.
Phillips, O. M. 1976 The Dynamics of the Upper Ocean, 2nd edn. Cambridge University Press.
Taylor, G. I. 1932 Proc. Roy. Soc. A 132, 499.
Thorpe, S. A. 1968 J. Fluid Mech. 32, 693.
Thorpe, S. A. 1969 J. Fluid Mech. 39, 25.
Thorpe, S. A. 1971 J. Fluid Mech. 46, 299.
Thorpe, S. A. 1973 J. Fluid Mech. 61, 731.
Woods, J. D. 1968 J. Fluid Mech. 32, 791.