Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-21T17:32:42.061Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonant gravity-wave interactions in a shear flow

Published online by Cambridge University Press:  28 March 2006

Alex. D. D. Craik
Alex. D. D. Craik
Affiliation:
Department of Applied Mathematics, University of St Andrews, Fife, Scotland
Get access Rights & Permissions [Opens in a new window]

Abstract

Among a triad of gravity waves in a uniform shear flow, a remarkably powerful second-order resonant interaction may take place. This interaction is characterized by large growth rates of waves which propagate in directions oblique to that of the primary flow, and by a systematic transfer of energy from the primary flow to such waves. Most of the energy transfer takes place in the vicinity of a ‘critical layer’, where viscous forces are dominant.

Provided the resonance condition may be satisfied, a uniform shear flow which is perturbed by a two-dimensional wave of small but finite amplitude may be unstable, owing to the growth of two initially infinitesimal oblique waves which complete the resonant triad.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 34 , Issue 3 , 2 December 1968 , pp. 531 - 549
Copyright
© 1968 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. & Fier, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory J. Fluid Mech. 27, 417.Google Scholar
Benney, D. J. 1961 A non-linear theory for oscillations in a parallel flow J. Fluid Mech. 10, 209.Google Scholar
Benney, D. J. 1964 Finite amplitude effects in an unstable laminar boundary layer Phys. Fluids, 7, 319.Google Scholar
Craik, A. D. D. 1968 Wind-generated waves in contaminated liquid films J. Fluid Mech. 31, 141.Google Scholar
Davis, R. E. & Acrivos, A. 1968 The stability of oscillatory internal waves J. Fluid Mech. 30, 723.Google Scholar
Kelly, R. E. 1968 On the resonant interaction of neutral disturbances in inviscid shear flows J. Fluid Mech. 31, 789.Google Scholar
Miles, J. W. 1967 Surface-wave damping in closed basins. Proc. Roy. Soc. A 297, 459.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions J. Fluid Mech. 9, 193.Google Scholar
Raetz, G. S. 1959 A new theory of the cause of transition in fluid flows. Norair Rept. NOR-59-383. Hawthorne, California.Google Scholar
Stuart, J. T. 1962 On three-dimensional non-linear effects in the stability of parallel flows Advanc. Aero. Sci. 3, 121.Google Scholar