The nonlinear evolution of high-frequency resonant-triad waves in an oscillatory Stokes layer at high Reynolds number

Abstract

The nonlinear evolution of high-frequency disturbances in high-Reynolds-number Stokes layers is studied. The disturbances are composed of a two-dimensional wave (2α, 0) of magnitude δ, and a pair of oblique waves (α, ± β) of magnitude ε, where α, β are the streamwise and spanwise wavenumbers respectively. We assume that β = √3α so that the waves form a resonant triad when they are nearly neutral. It is shown that the growth rate of the disturbance is controlled by nonlinear interactions inside ‘critical layers’. In order for there to be a nonlinear feedback mechanism between the two-dimensional and the three-dimensional waves, the former is required to have a smaller magnitude than the latter, namely .As in Goldstein & Lee (1992), the amplitude equations turn out to be significantly different from those of Raetz (1959), Craik (1971) and Smith & Stewart (1987) in two respects. Firstly, they are integro-differential equations, i.e. the local growth rate depends on the whole history of the evolution. Secondly the back reaction of the oblique waves on the two-dimensional wave is represented by two cubic terms and one quartic term, rather than by one quadratic term. Our numerical investigations show that the amplitudes of the two- and three-dimensional waves can develop a finite-time singularity, a result of some importance. The structure of the finite-time singularity is identified, and it is found that the two-dimensional wave has a ‘more singular’ structure than the three-dimensional waves. The finite-time singularity implies that explosive growth is induced by nonlinear effects. We suggest that this nonlinear blow-up of high-frequency disturbances is related to the bursting phenomena observed in oscillatory Stokes layers and can lead to transition to turbulence.