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。
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.
引文
[1] Borgman L E, Chappelear J E. The use of the Stokes-Struik approximation for waves of finite height. Proc. 6th Coast. Eng. Conf., 1958:252-280P
[2] Skjelbreia L. Gravity waves, Stokes third order approximations, tables of functions. Council on Wave Research, Eng. Foundation. Berkeley: Univ. of California, 1959.
[3] Skjelbreia L, Hendrickson J A. Fifth order gravity wave theory. Proc. 7th Conf. Coast. Eng., 1961: 184-196P
[4]邹志利.水波理论及其应用.北京:科学出版社, 2005
[5]陶建华.水波的数值模拟.天津:天津大学出版社, 2005
[6] Laiton E V. The second approximation to cnoidal and solitary waves. J. Fluid Mech., 1961, 9: 430-444P
[7] R Grimshaw. The solitary wave in water of variable depth. Part 2. J. Fluid Mech., 1971, 46: 611-622P
[8] Fenton J. A high-order cnoidal wave theory. J. Fluid Mech., 1979, 94: 129-161P
[9] Peregrine D H. Long waves on a beach. J. Fluid Mech., 1967, 27: 815–827P
[10] Madsen P A, Murray R, Sorensen O R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Engineering, 1991, 15: 371-388P
[11] Madsen P A, Soresen O R. A new form of Boussinesq equations with improved linear dispersion characteristics. Part 2, A slowly-varying bathymetry. Coastal Engineering, 1992, 18: 183-204P
[12] Nwogu O. Alternative form of Boussinesq equations for nearshore wave propagation. Journal of Waterways, Port, Coastal and Ocean Engineering, 1993, ASCE 119 (6): 618–638P
[13] Schaffer H A, Madsen P A. Further enhancements of Boussinesq-typeequations. Coastal Engineering, 1995, 26: 1–14P
[14] Wei G, Kirby J T, Grill S T, Subramanya R. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech., 1995, 294: 71–92P
[15] Gobbi M F, Kirby J T. A fourth order Boussinesq-type model. Proceedings of the 25th International Conference on Coastal Engineering, Orlando, FL, 1996: 1116–1129P
[16] Madsen P A, Bingham H B, Liu H. A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech., 2002, 462: 1-30P
[17] Madsen P A, Bingham H B, Schafer H A. Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves. derivation and analysis. Proc. R. Soc. Lond, 2003, A 459: 1075-1104P
[18] Madsen P A, Fuhrman D R, Wang B. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Eng., 2006, 53: 487-504P
[19]张洪生,洪广文. Boussinesq型方程对三维非线性波传播的数值模拟.华东师范大学学报(自然科学版), 2002 (03): 88-94页
[20]朱良生,洪广文.任意水深变化Boussinesq型方程非线性波数值计算.海洋工程. 2000 (02), 29-37页
[21]王本龙,刘桦.一种适用于非均匀地形的高阶Boussinesq水波模型.应用数学和力学. 2005 (06): 714-722页
[22]柳淑学,俞聿修,赖国璋,李毓湘.数值求解Boussinesq方程的有限元法.水动力学研究与进展A辑.2000 (4): 399-410页
[23]张殿新,陶建华.一种改善了非线性和色散性的Boussinesq方程模型.应用数学和理学. 2008 (7): 813-824页
[24]邹志利.高阶Boussinesq水波方程的改进.中国科学(E辑), 1999 (1): 87-96页
[25]邹志利.适合复杂地形的高阶Boussinesq水波方程.海洋学报, 2001 (1): 109-119页
[26]詹杰民,李毓湘.数值水池的一种有效的数值消波方法.水动力学研究与进展A辑, 2002 (01): 46-52页
[27]房克照.四阶完全非线性Boussinesq水波方程及其简化模型.大连理工大学, 2008.
[28]刘亚男,郭晓宇,王本龙,刘桦.基于RANS方程的海堤越浪数值模拟.水动力学研究与进展A辑, 2007 (6): 682-688页
[29]谢伟松,陶建华.用Level Set算法模拟孤立波与前台阶的相互作用.应用数学和力学, 2000 (7): 686-692页
[30]袁德奎,陶建华.用Level Set方法求解具有自由表面的流动问题.力学学报, 2000 (3).
[31] Green A E, Laws N, Naghdi P M. On the theory of water waves. Proc. Roy. Soc. London, 1974, A 338: 43-55P
[32] Green A E, Naghdi P M. A nonlinear theory of water waves for finite and infinite depths. Philos. Trans. Roy. Soc. London, 1986, A 320: 37-70P
[33] Ertekin R C, Webster W C, Wehausen J V. Waves caused by a moving disturbance in a shallow channel of finite width. J. Fluid Mech., 1986, 169: 272-292P
[34] Ertekin R C, Wehausen J V. Some soliton calculations. Proc. 16th Symp. Naval Hydrodyn. Berkeley, 1986: 167-184P
[35] Green A E, Naghdi P M. Further developments in a nonlinear theory of water waves for finite and infinite depths. Philos. Trans. Roy. Soc. London 1987, A 324: 47-72P
[36] Shields J J, Webster W C. On direct methods in water-wave theory. J. Fluid Mech, 1988, 197: 171-199P
[37] Webster W C, Shields J J. Applications of high-level, Green-Naghdi theory to fluid flow problems. IUTAM Symposium on Marine Dynamics, Brunel University London, Elsevier Science Publisher, 1990.
[38] Demirbilek Z, Webster W C. Application of the Green-Naghdi theory of fluid sheets to shallow-water wave problems. U.S. Army Corps of Engineers, Vicksburg, Mississippi, Technical Report, 1992a, CERC-92-11:1-45P
[39] Demirbilek Z, Webster W C. User’s manual and examples for GNWAVE. U.S. Army Corps of Engineers, Vicksburg, Mississippi, Technical Report, 1992b, CERC-92-13, 23 pp. + Appendix, 1-32P
[40] Webster W C, Kim D Y. The dispersion of large-amplitude gravity waves in deep water. Proc. 18th Symp. On Naval Hydrodynamics, 1990, 397-415P
[41] Xu Q, Pawlowski J S, Baddour R E. Simulation of nonlinear, irregular waves by Green-Naghdi theory. 8th International Workshop on Water Waves and Floating Bodies, St. John's, Canada, 1993a.
[42] Xu Q, Pawlowski J S, Baddour R E. Development and application of uni- and multi-directional Green-Naghdi method for nonlinear, irregular wave propagation. Proceedings of the 2nd Marine Dynamics Conference, Vancouver, Canada, 1993b.
[43] Xu Q, Pawlowski J S, Baddour R E. Development of Green-Naghdi models with a wave-absorbing beach for nonlinear, irregular wave propagation. J. of Marine Sci. & Tech., 1997. 21–34P
[44] Wu J K, Chen B. Unsteady ship waves in shallow water of varying depth based on Green-Naghdi equation. Ocean Engineering, 2003a, 30: 1899-1913P
[45]吴建康,陈波.采用Green-Naghdi方程和有限元法计算浅水船波.中国造船, 2003b (1): 17-23页
[46]陈波,吴建康.变深度浅水域中非定常船波.力学学报, 2003 (1): 64-68页
[47]刘元高,刘耀儒. Mathematica4.0实用教程.北京:国防工业出版社. 2000
[48]裘宗燕. Mathematica数学软件系统的应用及其程序设计.北京:北京大学出版社. 1994
[49]丁大正.科学计算强档Mathematica4教程.北京:电子工业出版社. 2002
[50] Wei G, Kirby J T. Time-dependent numerical code for extended Boussinesqequations. J Waterway Port Coastal and Ocean Engineering, 1995, ASCE 121: 251-261P
[51] Rienecker M M, Fenton J D. A Fourier approximation method for water waves. J. Fluid Mech., 1981, 104: 119-137P
[52] Hammack J, Henderson D, Guyenne P, et al. Solitary-wave collisions. OMAE 2004, June 21-22, Vancouver, BC CANADA.
[53] Chawla A, Kirby J T. Wave transformation over a submerged shoal. CACR Rep. No. 96-03, Dept. of Civ. Engrg., University of Delaware, Newark, Del., 1996
[54] Berkhoff J C W, Booy N, Radder A C. Verification of numerical wave propagation models for simple harmonic linear waves. Coastal Engineering, 1982, 6(13): 255-279P
[55] Chen Q, Kirby J T, Dalrymple R A, Kennedy A B, Chawla A. Boussinesq modeling of wave transformation, breaking, and runup. II: 2D. J Waterway Port Coastal and Ocean Engineering, 2000, ASCE 126: 48-56P
[56] Wei G, Kirby J T, Sinha A. Generation of waves in Boussinesq models using a source function method. Coastal Engineering, 1999, 36: 271-299P
[57] Sue P. Modelling of tsunami generated by submarine landslides. Doctor of Philosophy in Civil Engineering, University of Canterbury, Christchurch New Zealand, 2007
[58] Hammack J. A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech., 1973, 60(4): 769-799P
[59] Hayir A. The near-field tsunami amplitudes caused by submarine landslides and slumps spreading in two orthogonal directions. Ocean Engineering, 2006, 33: 654-664P
[60] Maxworthy T. Experiments on collisions between solitary waves J. Fluid Mech., 1976, 76: 177-185P
[61] Su C H, Mirie R M. On head-on collision between two solitary waves J. Fluid Mech., 1980, 98: 509-525P
[62] Mirie R M, Su C H. Collisions between two solitary waves. Part 2. Anumerical study. J. Fluid Mech., 1982, 115: 476-492P
[63] Byatt-Smith J G B. The interaction of two solitary waves of unequal amplitude. J. Fluid Mech., 1989, 205: 573-579P
[64] Wu T Y S. Nonlinear waves and solitons in water. Physica D, 1998, 123: 48-63P
[65] Keulegan G H. Gradual damping of solitary waves. J. Res. Nat. Bur. Stand, 1948, 40: 487-498P
[66] French J A. Wave uplift pressures on horizontal platforms. W. M. Keck Lab. Hydraul. & Water Res. Calif. Inst. Tech. Rep, 1969, KH-R-19.
[67] Jiang L, LeBlond P H. The coupling of a submarine slide and the surface waves which it generates. Journal of Geophysical Research, 1992, 97(C8, 12): 731-744P
[68] Rzadkiewicz S, Mariotti C, Heinrich P. Numerical simulation of submarine landslides and their hydraulic effects. Journal of Waterway, Port, Coastal, and Ocean Engineering, 1997, 123(4): 149-157P
[69] Grilli S T, Watts P. Modeling of waves generated by a moving submerged body. Applications to underwater landslides. Journal of Engineering Analysis with Boundary Elements, 1999, 23: 645-656P
[70] Gotoh H, Sakai T, Hayashi M. Lagrangian two-phase flow model for the wave generation process due to large-scale landslides. Asian and Pacific Coastal Engineering, 2001, October, 18-21: 176-185P
[71] Panizzo A, Dalrymple R A. SPH modelling of underwater landslide generated waves. ICCE 2004 Conference, Lisbon, 2004.
[72] Iwasaki S. Experimental study of a Tsunami generated by a horizontal motion of a sloping bottom. Bull. Earthq. Res. Inst. Univ. Tokyo, 1982, 57: 239-262P
[73] Heinrich P. Nonlinear water waves generated by submarine and aerial landslides. Journal of Waterways, Port, Coastal, and Ocean Engineering, 1992, 118(3): 249-266P
[74] Watts P. Wave maker curves for Tsunamis generated by underwaterlandslides. Journal of Waterways, Port, Coastal, and Ocean Engineering, 1998, 124(3): 127-137P
[75] Ichiye T. A theory of the generation of tsunami by an impulse at the sea bottom. J. Oceanograph. Soc. Japan, 1958, 14: 41-44P
[76] Kajiura K. The leading wave of a Tsunami. Bull. Earthq. Res. Inst. Univ. Tokyo, 1963, 41: 535-571P
[77] Comer R P. Tsunami generation by earthquakes. PhD Thesis. Cambridge, Mass: Earth and Planetary Sciences, Mass. Inst. Tech, 1982
[78] Peregrine. Calculations of the development of an undular bore. J. Fluid Mech., 1966, 25: 321-330P
[79] Todorovska M I, Trifunac M D. Generation of Tsunamis by slowly spreading uplift of the sea floor. Soil Dynamics and Earthquake Engeneering, 2001, 21 (2): 167-180P
[80] Hayir A. Ocean depth effects on Tsunami amplitudes used in source models in linearized shallow-water wave theory. Ocean Engineering, 2004, 31: 253-361P