數值研究『兩孤立波互制』與『橢圓餘弦(Cn)波與波狀底床互制』之黏性效應

Abstract

The purpose of present numerical study is to enhance the advanced understanding of detailed wave characteristics and global viscous flow behaviors related to the nonlinear, dispersive long waves during interaction, especially on the interactions of two solitary waves and on the interaction of the cnoidal wave with the rippled bottom. To this end, this investigation is performed by developing a numerical wave tank equipped with a wavemaker at one end of a long water channel and using 2-D stream function-vorticity ( ) flow model to simulate the laminar flow field under the free surface. The usage of transient body-fitted grid system in this model is advantageous to treat the movement of free surface and waveplate more accurate at the actual location. This enables to work well for a large-stroke wavemaker of piston-type, and thus offers a flexible tool to generate various kinds of long waves in the simulation. Moreover, the grid system is also ready to justify the resolution and geometry shape of grid lines to adopt for irregular boundary with highly dense mesh. The different long wave interacting problems, including the interaction of two solitary waves, cnoidal waves induced transient separated/attached flow over a wavy bottom, and Bragg interactions of cnoidal wave with bottom ripples are studied in the dissertation. The model was first validated to simulate the propagation of a solitary wave and that of cnoidal wave over a constant water depth. The good agreement of numerical results for the free surface elevation and velocity profiles (including their in the boundary-layer region) with theoretical solutions or experimental data confirm the model to be proper for further application. In the first case, the boundary layer flow near the bottom induced by collisions of two solitary waves, including head-on collisions and overtaking collisions, were studied. Waveform evolution, the run-up, phase lag and particle trajectories were discussed in detail. In addition, for those less discussed results of the boundary layer flow during interaction process were presented. The evolved patterns of streamlines and equip-vorticity lines over the boundary layer gave the new comprehensive view of vorticity exchange on the bottom region. This interaction not only appears visibly on the free surface but also occurs silently along the bottom. Considering overtaking collisions, another interesting topic is that the critical ratio between the amplitudes of two overtaking solitary waves. At the center of encounter, the wave profile is fore-and-aft symmetric, but it could have a single peak or double peaks. The present model made a series of tests and compared with previous theoretical or numerical results. In the second case, the cnoidal wave inducing oscillatory boundary-layer flow over a ripple bottom was investigated. The asymmetric bottom flow pattern traversing over several ripples due to nonlinear wave motion leads to the asymmetric vorticity transport, therefore, the bottom region with separated/attached flow developed periodically. With different phases of wave motion within one period, those vortices generated on the bottom first grow and then are convected, diffused outward and dissipated finally around the ripples by the wave-induced flow. The flow velocity affected by the vortex evolution can be further applied to trace fluid particles among ripples. The trajectories of flow particles do not form a closed orbit as do those under the linear waves during each wave period. Instead, the particles move to and fro around the ripples with in variously wide range trajectories that depends on the proximity from the ripple bottom. Finally, the last flow case studied the Bragg resonance characterized as the cnoidal wave reflected from the ripple bottom. To clearly identify the effects caused by nonlinearity and viscosity, the cases of linear waves were also performed to compare with the cases of cnoidal waves. The investigations include interferential pattern of water elevation, spatial distribution of steady wave height, resonant reflection due to the wave height, ripple height, and ripple number, and spectrum analysis.