Nonlinear Effects on Interfacial Wave Growth into Slug Flow

It is known that when two fluids flow through a horizontal channel, depending on the relative velocity between the two fluids, two different instability mechanisms can create initial wave disturbances on the interface: the classic Kelvin-Helmholtz instability or shear induced instability. These instability mechanisms, which affect 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) instability 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 secondorder sumand 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 second harmonic resonance, energy from the KH unstable second harmonic is transferred to the first harmonic. Meanwhile, this first harmonic, which is involved in a triad resonance, transfers its energy to the other two (stable) wave modes. 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 can 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. 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. Stable resonances are considered and a nonlinear solution is obtained through the method of multiple scales. In addition to this, an efficient high-order spectral method is developed for the simulation of the generation and nonlinear evolution of interfacial waves. This numerical 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. Direct comparisons between the numerical simulations and the multiple scale analytic solutions are made. Excellent agreement is found between the two methods. The numerical simulations are also compared against existing laboratory experiments. Good agreement between them is observed. The findings in this study improve the understanding of the underlying mechanisms which can cause short waves to evolve into large amplitude liquid slugs through nonlinear wave-wave interactions. Thesis Supervisor: Yuming Liu Title: Principal Research Scientist