【作者】 赵彬彬；

【导师】 段文洋；

【作者基本信息】 哈尔滨工程大学 ， 流体力学， 2010， 博士

【摘要】 关于全非线性水波的研究对于海岸工程和海洋工程都具有重要的意义,另外,海啸对于近海岸居民的生命安全也存在潜在的威胁。因此,准确地预报波浪的传播变形是非常重要的。Green-Naghdi理论,简称G-N理论,是一种全非线性的水波理论。一些学者称G-N模型为一种全非线性的Boussinesq模型。G-N理论的近似方法完全不同于其他水波理论所使用的摄动法,它只是引入了流体质点沿水深方向的速度变化假设。通过引入适当的速度场假设,G-N理论不仅能用来研究浅水波,也能对深水波进行分析。G-N模型中,通过沿水深积分的方法使得二维水波问题变为一维问题,三维水波问题变为二维问题,从而提高了计算效率。另外,非线性的自由面边界条件是在瞬时位置准确满足的,因此,G-N理论非常适合全非线性水波的研究。G-N理论根据速度场假设的复杂程度不同,分为不同的级别,用“Level”表示。G-N理论的级别越高,数值模型越复杂,本文使用Mathematica数学符号软件进行公式推导。对于二维G-N理论,人们进行了较多的研究。然而,三维G-N模型的数值求解,仍然是一项困难的工作。本文的主要工作如下:1.完整推导了一般形式的G-N理论控制方程。并分别给出了浅水G-N理论和深水G-N理论的控制方程。2.研究了Level I至Level VII七种不同级别的浅水G-N理论的线性解析解和色散关系式。发现Level VII浅水G-N理论可以模拟kd≤26(k是波数,d是水深)的波浪,而且求解G-N模型时出现的最高阶导数只是3阶导数。对Level I至Level III三种不同级别的深水G-N理论的线性色散关系式也进行了讨论。3.对G-N理论的数值模型进行了重新表述。对G-N模型的二维算法进行了研究,并对三维G-N模型的求解方程进行分析。结合Boussinesq模型的三维算法和G-N模型的二维算法,提出了G-N模型的三维算法。4.利用流函数波浪理论的计算结果,使用最小二乘法对水深方向上流体质点运动速度进行拟合,提出了全非线性造波边界条件。解决了数值模拟强非线性浅水波时造波边界条件的给定问题。5.对两个孤立波的迎面碰撞和追赶碰撞问题进行了研究。结果表明浅水G-N理论能够准确地模拟两个孤立波的碰撞过程。对非平整海底引起的三维浅水波浪传播变形问题进行数值模拟,研究表明浅水G-N模型比一些全非线性Boussinesq模型能更准确地模拟波浪的传播变形过程。6.假定海底是时间的函数,即海底形状随时间变化,本文将浅水G-N模型应用于海啸问题的研究。对二维滑坡海啸、二维地震海啸和三维地震海啸进行研究。研究表明浅水G-N模型能够准确地模拟海啸的发生和传播过程。7.使用全非线性深水G-N理论对浪向角分别为0°、15°和45°的深水单色波进行数值模拟,深水G-N理论的计算结果与流函数波浪理论的解吻合得很好。最后,在一个封闭方形水池中对风压兴波问题进行了研究,深水G-N理论展示了很高的计算精度和计算效率。更多还原

【Abstract】 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.更多还原