本文研究的目的是建立一個數值模式用以探討非線性、頻散長波互制作用的波浪與流場特性，因此包含兩個主題：『兩孤立波碰撞』及『橢圓餘弦波與波狀地形互制』。此模式特點是以流函數-渦度方程式為基礎，發展具有虛擬造波機的二維黏性數值波浪水槽，結合了瞬時貼壁格網系統來處理隨時變動的自由水面、移動的造波板邊界與曲面底床邊界，進而可應用於模擬具自由水面的波流場現象。當考慮兩孤立波碰撞或橢圓餘弦波與波狀地形互制問題時需解析近底床邊界層流場，無論是平底床或曲面底床邊界，貼壁格網系統亦可藉由適當的代數式決定計算格網，有效率的分配計算格點來解析波動過程中驅動流場引起的細部渦流現象。
在探討應用性問題前，先對模式進行基本測試以確認數值造波水槽的可靠性。測試的兩案例分別是孤立波與橢圓餘弦波在平底床傳遞問題，藉由波傳後的水面波形、相位與速度剖面分布和現有的理論或實驗結果比較，計算結果皆可得到合理的一致性。另方面因橢圓餘弦波的波長甚長，在下游端使用開放邊界條件來節省計算量與耗時，下游邊界能否適當的處理透射波，亦透過量化的分析來謹慎評估。
本文研究的第一個應用案例為孤立波碰撞問題，包含了反向碰撞與同向追撞。過去已有相當多研究著墨於自由水面現象討論，本文除了有關於水面的波形演變、溯升高度、相位延遲等方面的討論，並進一步以數值模式解析兩波互制過程的邊界層內流場現象及質點軌跡運動，而相關的流場演變渦流現象恰可與近期學者以PIV量測的實驗影像比較。在同向追撞問題中，另一有趣的現象是，當兩波追撞時，其大波與小波的波高比值會影響交互作用時形成的波峰型態有所不同，在此也一併討論並與過去學者的理論、數值解相比較。
第二個應用案例為橢圓餘弦波與波狀底床互制作用。當所考慮的波狀地形尺度甚小，橢圓餘弦波具有強烈的非線性特性(波峰、波谷不對稱性)故產生極強列的非對稱的振盪流場，會在沙漣地形間產生週期性的渦流生成與消散，此種流場特性影響底床沙粒移動與堆積的機制，本文以不同的入射波、地形尖銳度條件來討論對流場的影響；當考慮的波狀地形尺度較大，即地形波長為入射波長的一半，根據理論會在地形前方產生類似駐波現象，即所謂布拉格共振現象，此處討論著重於線性波(微小振幅波)與非線性波(橢圓餘弦波)的比較，計算結果顯示，對非線性波來說，形成的包絡波形具上下對稱特性，而非線性波的包絡波形則為上下非對稱，這樣的結果也導致非線性波在估算反射率會被低估；對線性波而言，最大共振條件發生於入射波長為地形波長的兩倍，而非線性波則會偏離此條件，且當考慮較大的入射波高或地形高地，偏離的程度會越顯著；另以FFT分析共振現象時的水面發現，線性波只有單頻成分，故共振現象單純由此成分波所影響，而非線性波具有高頻成分，當共振發生時，主頻成分波的振幅反而衰減，但高頻成分波的振幅增加，顯示共振發生時造成振幅增大的主要貢獻來自於高頻成分波。

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.

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