(116d) Growth and Nonlinear Resonant Interactions of Interfacial Waves in Stratified Channel Flows

This work considers nonlinear wave-wave interactions and interfacial wave resonances in a two-fluid stratified flow through a horizontal channel for the purpose of understanding mechanisms capable of generating slug flow. An efficient high-order spectral method is developed for the simulation of the generation and nonlinear evolution of interfacial waves. The method is based on a potential flow formulation which includes normal viscous stresses and a pressure forcing term at the interface respectively for modeling of the damping effect and surface shear effect by the upper fluid. The method is capable of accounting for the nonlinear interactions of a large number of wave components in a broadband spectrum, and obtains an exponential convergence of the solution with the number of spectral modes and interaction order.

It is known that depending on the relative velocity between the two phases, two instability mechanisms can create initial wave disturbances on the interface: the classic Kelvin-Helmholtz instability or shear instability due to turbulent fluctuations in the upper phase. These instability mechanisms on the initial growth of the disturbances on the interface are well understood; however, the subsequent nonlinear evolution and the effects of wave resonances on the interface remain a subject of interest and will be the focus of this study.

The Kelvin-Helmholtz (KH) model is the traditional model for predicting interfacial instability in stratified flows. This linear instability occurs when the lower phase inertia and the pressure from the upper phase overcome the stabilizing effects of gravity and surface tension. This mechanism is most unstable to short waves and does not allow for modal interactions. When nonlinearity is included, additional second-order sum- and difference-wavenumber modes as well as higher harmonics must be accounted for. Given an initial spectrum of stable modes, sum-wavenumber modes and higher harmonics are generated due to nonlinear wave-wave interactions, which can be unstable to KH. The generated second-order difference-wavenumber modes are stable to KH. For certain combinations of wave modes, we found that there exist simultaneous coupled second-harmonic (overtone) and triad resonances. Due to the overtone resonance, energy from the KH unstable second harmonic is transferred to the (stable) first harmonic. Meanwhile, this first harmonic transfers its energy to other two (stable) wave modes in the resonant triad. These coupled resonant wave-wave interactions result in rapid growth of long wave components on the interface.

For flows which are KH stable; the shear instability can cause the growth of relatively short waves on the interface. As in the case of KH instability, the overtone and triad resonances also occur. The overtone resonance transfers energy from the dominant unstable mode to its subharmonic (creating a period doubling phenomenon). Similarly the triad resonance enables the transfer of energy from unstable mode(s) to stable mode(s). These resonant wave-wave interactions provide a mechanism for the growth of long waves by taking energy from (relatively short) unstable waves.

Direct comparisons between the numerical simulations and existing laboratory experiments are made. Good agreements between them are obtained. The findings in this study improve the understanding of the underlying mechanisms which cause short waves to evolve into large amplitude liquid slugs through nonlinear wave-wave interactions.