Resonant gravity-wave interactions in a shear flow | Journal of Fluid Mechanics | Cambridge Core

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.

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