Nonlinear resonant interactions of interfacial waves in horizontal stratified channel flows

Abstract

AbstractWe consider the problem of nonlinear resonant interactions of interfacial waves with the presence of a linear interfacial instability in an inviscid two-fluid stratified flow through a horizontal channel. The resonant triad consists of a (linearly) unstable wave and two stable waves, one of which has a wavelength that can be much longer than that of the unstable component. Of special interest is the development of the long wave by energy transfer from the base flow due to the coupled effect of nonlinear resonance and interfacial instability. By use of the method of multiple scales, we derive the interaction equations which govern the time evolution of the amplitudes of the interacting waves including the effect of interfacial instability. The solution of the evolution equations shows that depending on the flow conditions, the (stable) long wave can achieve a bi-exponential growth rate through the resonant interaction with the unstable wave. Moreover, the unstable wave can grow unboundedly even when the nonlinear self-interaction effect is included, as do the stable waves in the associated resonant triad. For the verification of the theoretical analysis and the practical application involving a broadbanded spectrum of waves, we develop an effective direct simulation method, based on a high-order pseudo-spectral approach, which accounts for nonlinear interactions of interfacial waves up to an arbitrary high order. The direct numerical simulations compare well with the theoretical analysis for all of the characteristic flows considered, and agree qualitatively with the experimental observation of slug development near the entrance of two-phase flow into a pipe.