| Research on fully nonlinear water waves is important for coastal engineering and offshore engineering. Tsunamis have a high potential to cause damage and loss of life in coastal areas. Therefore, predicting the wave transformation accurately is very important.The Green-Naghdi wave theory, called G-N theory for short, is a fully nonlinear wave theory. Some researchers call G-N model a fully nonlinear Boussinesq model. The G-N approach, which is fundamentally different from the perturbation method, only introduces some simplification of the velocity variation in the vertical direction across the fluid sheets. By introducing proper velocity assumption, G-N model can be used to analyse both shallow water waves and deep water waves. In G-N models the dimension of a free surface problem is reduced by one, and nonlinear boundary conditions are satisfied on the instantaneous free surface. Therefore, the G-N theory is very suitable for fully nonlinear water waves.Different degrees of complexity of the G-N theory are distinguished by"levels". The higher the level, the more complicated the mathematical formula is. Its derivation can be done by using mathematical symbolic software (mathematica) without too much effort. A lot of work has been done on the two-dimensional G-N model. However, the numerical implementation of three-dimensional G-N model is still a difficult task. The major work is as follows:1. The governing equations of Green-Naghdi theory in general form were derived in detail. The governing equations of G-N theory for shallow water waves and deep water waves were derived, respectively.2. The linear analytical solution and dispersion relations corresponding to Level I up to Level VII G-N theory for shallow water waves has been researched. It's found that Level VII G-N theory can predict the waves with kd≤26 (k is the wave number, d is the water depth). The highest-order derivatives in G-N equations are third derivatives. The linear dispersion relations corresponding to Level I up to Level III G-N theory for deep water was also discussed.3. The numerical model of G-N theory was rewritten in another form. The two-dimensional algorithm was introduced. The three-dimensional algorithm, which combines the three-dimensional algorithm of Boussinesq model and the two-dimensional algorithm of G-N model, was presented for the first time.4. The fully nonlinear wave-maker boundary condition, which is based on the stream function wave theory, was presented. The least-squares method was used when fitting the fluid velocity along water depth. This makes the proper wave-maker boundary condition for strong nonlinear shallow water waves.5. The head-on collision and following collision between two solitary waves were simulated. The results show that G-N model can simulate collision between two solitary waves accurately. The wave transformation problems with uneven seabed were reproduced numerically. The G-N theory presented some advantages in some details compared with other fully nonlinear Bousssinesq model. The G-N model can simulate wave transformation in shallow water accurately.6. By making the bottom a function of time, G-N theory can simulate tsunami. The two-dimensional earthquake- and landslide- induced tsunamis, and the three-dimensional earthquake-induced tsunamis were modeled. G-N theory can reproduce the most of the detail of these events, from their generation to their later propagation.7. The G-N model for deep water waves was applied to simulating the unidirectional waves travelling atα= 0°, 15°, 45°. The results of G-N model matches well with the stream function wave theory. The development of nonlinear waves created by an oscillating pressure disturbance in a closed square tank was modeled. The G-N model shows high accuracy and high efficiency.
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