Strongly Nonlinear Long Gravity Waves In Poiseuille Background Flows

Dispersion and nonlinearity are two important characteristics for the surface and interfacial waves in fluid, and the research on them not only helps to uncover the mechanisms of their formation, evolvement and attenuation, but also can guide the ocean and costal engineering in reality.In this thesis, we just consider irrotational, inviscid and incompressible fluids. First, we focus on the long surface gravity waves in one layer fluid with Poiseuille flows, a new model is derived under the assumption that the aspect ratio between waterlength and water depth is small, using the depth-averaged velocities. The derived equations can be reduced to the case of Green & Naghi's. Since its derivation requires no assumption on wave amplitude, the model can be used to describe arbitrary amplitude waves. Then, large amplitude internal waves interacting with Poiseuille background flows in a system of two layers of different densities are derived under the long wave approximation without the smallness assumption on the amplitude, the fluid is also irrotational, inviscid and incompressible. Subsequently, we got the model equations in terms of the depth-averaged velocities can be reduced to the case of Choi W & Camassa R's. Finally, the derived internal waves equations reveal the nonlinear interactions characteristic between waves and flows and among the interfacial waves with Poiseuille background flows.