Sub-harmonic resonant wave interactions in the presence of a linear interfacial instability

Abstract

This work considers the role of nonlinear sub-harmonic resonant wave interactions in the development of interfacial waves, which may be under the influence of a linear interfacial instability, in an inviscid two-fluid stratified flow through a horizontal channel. We begin by examining the case of resonant interactions between one linearly unstable mode and its sub-harmonic that may be linearly stable or unstable. Using the method of multiple scales, we derive the nonlinear interaction equations governing the time evolution of interacting wave amplitudes. These nonlinear equations account for the combined effects of both the nonlinear resonant interaction and the linear instability. We show that through this nonlinear coupling, the linearly stable sub-harmonic mode can achieve faster than exponential growth. It is found that such a mechanism is capable of generating large-amplitude long waves that are stable by linear stability analysis. The analytical predictions are cross validated by comparisons with direct nonlinear numerical simulations based on a more general perturbation based spectral method. Good agreement between the analytical and numerical solutions is observed. The more complicated case where a single mode is simultaneously involved in multiple (sub-harmonic and triad) resonances is also investigated numerically. The results demonstrate that chains of resonances can permit the energy generated by the linear instability, among high wavenumber components, to be passed across the spectrum to the longest wave components creating an efficient mechanism for the generation of large-amplitude long waves from unstable short waves.

Experimental Characterization of Linear Interfacial Waves in a Stratified Turbulent Gas-Liquid Pipe Flow Using Particle Image Velocimetry

“…Within this regime, resonant interactions can occur and lead to a change of regime. For two phase flows, the resonance of disturbances was described recently in the work of Campbell and Liu (2014).…”

“…It is important to mention that due to signal aliasing it is not possible to distinguish between harmonic and subharmonic frequencies because both appear around the frequency of 2 Hz. The resonant interaction between waves of different frequency is a wellstudied phenomenon (Jurman et al 1992, Valluri et al 2008, Campbell and Liu 2014, Ayati 2018. This mechanism is characterized by fast energy transfer from (b) (a) the fundamental to the sub-harmonic mode.…”

“…This mechanism is characterized by fast energy transfer from (b) (a) the fundamental to the sub-harmonic mode. Campbell and Liu (2014) derived weakly nonlinear formulation to account for nonlinear resonant interaction. They have shown that resonant coupling can induce wave growth faster than exponential.…”