采用Green-Naghdi方程数值研究浅水船波
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摘要
随着造船工业的发展,船舶的尺度、航速和动力不断增大。很多情况下,船舶在浅水区运行,如江、湖、河流,近海等区域。对于浅水船波问题,非线性、水面扰动、能量耗散、边界反射、船与结构物、以及多船波浪干涉等问题很重要。复杂地形地貌,变化水深使浅水船波问题更加困难。浅水船波和波浪力不仅对船舶的运行和操纵性能有重要的影响,而且对浅水区域的水环境和工程建设项目产生重要的,甚至不可逆转的影响。因此研究复杂环境条件下的浅水船波有着重要的理论和实际意义。
本文在研究浅水船波时,采用Green-Naghdi方程作为解决问题的控制方程,而把船舶看成为移动在自由水面的压力扰动。因为Green-Naghdi方程中的频率散射项能合理地解释有限水深对船波的影响。通过求解含有移动压力项的Green-Naghdi方程,我们得到了当船舶在浅水域中航行时的三维兴波波形,波浪阻力和横向力。浅水船波的水深波长比值一般都比较小,允许把它归入浅水波范畴来研究。事实上,船波是多频率混合的水波,其中可能含有某些值有限的波浪分量,因此水波频率散射的影响不能笼统忽略。Kirby对各种类型的Boussinesq方程作了综述,这些方程在不同程度上考虑了频率散射项的影响。Green-Naghdi方程是最典型的Boussinesq方程之一,它全面考虑频率散射和非线性的影响。它没有无旋条件,没有波高对水深比值的限制。
从形式上看,Green-Naghdi方程除了附加的频率散射项外,和无粘性的二维Navier-Stokes方程形式类似。它的数值解是计算流体力学领域最困难的问题之一,其数值解常常出现数值振荡。波动方程法能够有效地消除数值振荡,稳定数值解,是求解浅水方程的有效方法之一,并已经有了许多成功地应用。本文采用了波动方程/有限元法 (WE/FEM) 求解Green-Naghdi方程。论文最后以Series 60 (CB= 0.6) 船为算例,给出不同运行条件和不同环境下的三维自由面波高、兴波阻力和横向力的变化规律。
全文共分七章。第一章,介绍船舶兴波和有限水深对船波的影响,简要讨论浅水波方程。在第二章中,介绍各类Boussinesq型方程,主要介绍了Wu所提出的一类Boussinesq型方程,此外,也分析了Boussinesq型方程中的频率散射项对静水压力假定的修正。在第三章中给出Green-Naghdi方程,着重介绍了Green-Naghdi方程和Boussinesq型方程之间的关系,并对线性Green-Naghdi方程所包含的水波频率散射做
了分析;在第四章中,详细介绍了Green-Naghdi方程的数值解法。采用波动方程/有限元法 (WE/FEM) 求解Green-Naghdi方程,本章讨论了Green-Naghdi形式的波动方程的构造方法。在第五章以Series 60 (CB= 0.6) 船为算例,计算了当船舶在浅水区域以不同的速度均匀航行时的三维兴波波形和波浪阻力。通过与浅水方程的结果比较,说明了在亚临界速度区,频率散射对浅水船波有显著的影响,在超临界速度区,频率散射的影响可以忽略。对于多船舶集体运行时的波浪干涉问题,采用移动网格的有限元法计算了两船平行、前后跟随和三船品字形集合运行时的兴波波形、波浪阻力和横向力特性。计算结果发现:当两艘船并行航行时,两船承受侧向吸引力,同时波浪阻力稍有增加;当两船前后跟随运动时,两船的波浪阻力都减小;当三船品字形航行时,前船的阻力减小,后船的阻力增加,同时后面两船之间的吸力减小。第六章,我们计算了船舶经过变深度浅水区域,及两船相向交错航行时的非定常波浪特性。在船舶经过变深度浅水区域时,以Series 60 (CB= 0.6) 船为算例,给出自由面波高,波浪阻力在船舶经过一个水下凸包或凹坑时的变化结果,并与浅水方程的结果进行了比较。计算结果表明:当船舶经过凸包时,波浪阻力先增加,后减少,并逐渐趋于正常。同时发现,当船速小于临界速度时,Green-Naghdi方程给出的船后尾波波高比浅水方程的结果略大,波浪阻力也比浅水方程的结果有所提高,频率散射必须考虑;当船速大于临界速度时,Green-Naghdi方程的计算结果与浅水方程差别不大,频率散射的影响可以忽略。当船舶经过凹坑时有类似的结论。对于两船相向交错航行的问题,计算结果发现:当两艘船相向航行时,波浪阻力先增加,后减少;随着两艘船的相互离去,波浪阻力又增加,最后逐渐趋于正常。同时发现,两船之间的侧向力有类似的变化规律。侧向力先是相互排斥,后相互吸引,接着又相互排斥,最后逐渐趋于正常。两船间的侧向吸引力剧烈变化,对船舶的操纵性及稳定性有重要的影响,也是近年来河道频频发生撞船事故的重要原因之一。
With increasing of ship size, speed and power, shipbuilding industry has achieved a great advance in recent years. A great number of ships moved in shallow water, such as rivers, lakes and estuaries. Nonlinearity, frequency dispersion, free surface disturbance, energy dissipation and boundary reflection may play important roles to ship waves in shallow water. Topography complexity, such as irregular boundaries, varying water depth, makes ship waves more difficult. The shallow water ship waves are of great importance to water environments, engineering applications as well as the ships themselves. It is of great importance to investigate the dynamic behavior of the ship waves in shallow water.
This work is focused on numerical study of waves generated by ships moving in shallow water based on Green-Naghdi equation. A moving ship is regarded as a moving pressure disturbance on free surface in this study. The moving pressure is incorporated into the Green-Naghdi equations to formulate forcing of ship waves in shallow water. Three-dimensional ship wave profiles, wave resistance and transverse forces are given. Generally speaking, ship waves in shallow water have small values, the ratio of water depth to wave length, and may be thought of as long waves. In fact, the ship wave is a combination of multi-frequency water waves. The wave constituents with finite values may have effects on ship waves. The frequency dispersion cannot be completely ignored. The frequency dispersion term of the Green-Naghdi equations accounts for the effects of finite water depth on ship waves. Kirby reviewed a variety of Boussinesq-type equations, which consider the frequency dispersion with different orders. The Green-Naghdi equation is a typical Boussinesq-type equation, which involves the nonlinearity and frequency dispersion. It does not require irrotational conditions and the limitation of the ratio of wave length to water depth.
The Green-Naghdi equations have an analogous form to the two-dimensional non-viscous Navier-Stokes equations with an additional frequency dispersion term. The continuity equation is the first-order hyperbolic differential equation. It is known that the numerical solution of the first-order hyperbolic differential equation had been a difficult
problem in computational fluid dynamics field. The numerical solutions of Green-Naghdi equations are easily spoiled by numerical oscillations. Wave equation model can eliminate numerical oscillation effectively and have better stability property. It had been successful used to solve shallow water questions in many practical applications. In this dissertation, a wave equation model and the finite element method (WE/FEM) are adopted to solve the Green-Naghdi equations. As the examples of Series 60 (CB = 60) ship, the numerical solutions of three-dimensional ship wave profiles, wave resistance and transverse forces are presented.
The contents of this dissertation are summarized as follows. Chapter 1 introduces the effects of finite water depth on ship waves, and briefly discusses the theory of shallow water equations. Chapter 2 introduces some kinds of Boussinesq-type equations, and briefly discusses the Boussinesq equations. We have also analyzed how the frequency dispersion term in the Boussinesq equations modify the hydrostatical pressure in wate depth. Chapter 3 is devoted to the relation between the Green-Naghdi equations and Boussinesq-type equations. We also discuss frequency dispersion of the linear Green-Naghdi equations. In chapter 4, we present numerical solutions of the Green-Naghdi equations. A wave equation model and finite element method (WE/FEM) are adopted to solve shallow water questions, and the Green-Naghdi equations. Furthermore, the implementation of boundary conditions is also discussed in this chapter. Three-dimensional ship wave profiles and wave resistance when a Series 60 () ship moves at different speed in shallow water are presented in chapter 5. A comparison between the numerical results predicted by the Green-Naghdi equations and the s
本文在研究浅水船波时,采用Green-Naghdi方程作为解决问题的控制方程,而把船舶看成为移动在自由水面的压力扰动。因为Green-Naghdi方程中的频率散射项能合理地解释有限水深对船波的影响。通过求解含有移动压力项的Green-Naghdi方程,我们得到了当船舶在浅水域中航行时的三维兴波波形,波浪阻力和横向力。浅水船波的水深波长比值一般都比较小,允许把它归入浅水波范畴来研究。事实上,船波是多频率混合的水波,其中可能含有某些值有限的波浪分量,因此水波频率散射的影响不能笼统忽略。Kirby对各种类型的Boussinesq方程作了综述,这些方程在不同程度上考虑了频率散射项的影响。Green-Naghdi方程是最典型的Boussinesq方程之一,它全面考虑频率散射和非线性的影响。它没有无旋条件,没有波高对水深比值的限制。
从形式上看,Green-Naghdi方程除了附加的频率散射项外,和无粘性的二维Navier-Stokes方程形式类似。它的数值解是计算流体力学领域最困难的问题之一,其数值解常常出现数值振荡。波动方程法能够有效地消除数值振荡,稳定数值解,是求解浅水方程的有效方法之一,并已经有了许多成功地应用。本文采用了波动方程/有限元法 (WE/FEM) 求解Green-Naghdi方程。论文最后以Series 60 (CB= 0.6) 船为算例,给出不同运行条件和不同环境下的三维自由面波高、兴波阻力和横向力的变化规律。
全文共分七章。第一章,介绍船舶兴波和有限水深对船波的影响,简要讨论浅水波方程。在第二章中,介绍各类Boussinesq型方程,主要介绍了Wu所提出的一类Boussinesq型方程,此外,也分析了Boussinesq型方程中的频率散射项对静水压力假定的修正。在第三章中给出Green-Naghdi方程,着重介绍了Green-Naghdi方程和Boussinesq型方程之间的关系,并对线性Green-Naghdi方程所包含的水波频率散射做
了分析;在第四章中,详细介绍了Green-Naghdi方程的数值解法。采用波动方程/有限元法 (WE/FEM) 求解Green-Naghdi方程,本章讨论了Green-Naghdi形式的波动方程的构造方法。在第五章以Series 60 (CB= 0.6) 船为算例,计算了当船舶在浅水区域以不同的速度均匀航行时的三维兴波波形和波浪阻力。通过与浅水方程的结果比较,说明了在亚临界速度区,频率散射对浅水船波有显著的影响,在超临界速度区,频率散射的影响可以忽略。对于多船舶集体运行时的波浪干涉问题,采用移动网格的有限元法计算了两船平行、前后跟随和三船品字形集合运行时的兴波波形、波浪阻力和横向力特性。计算结果发现:当两艘船并行航行时,两船承受侧向吸引力,同时波浪阻力稍有增加;当两船前后跟随运动时,两船的波浪阻力都减小;当三船品字形航行时,前船的阻力减小,后船的阻力增加,同时后面两船之间的吸力减小。第六章,我们计算了船舶经过变深度浅水区域,及两船相向交错航行时的非定常波浪特性。在船舶经过变深度浅水区域时,以Series 60 (CB= 0.6) 船为算例,给出自由面波高,波浪阻力在船舶经过一个水下凸包或凹坑时的变化结果,并与浅水方程的结果进行了比较。计算结果表明:当船舶经过凸包时,波浪阻力先增加,后减少,并逐渐趋于正常。同时发现,当船速小于临界速度时,Green-Naghdi方程给出的船后尾波波高比浅水方程的结果略大,波浪阻力也比浅水方程的结果有所提高,频率散射必须考虑;当船速大于临界速度时,Green-Naghdi方程的计算结果与浅水方程差别不大,频率散射的影响可以忽略。当船舶经过凹坑时有类似的结论。对于两船相向交错航行的问题,计算结果发现:当两艘船相向航行时,波浪阻力先增加,后减少;随着两艘船的相互离去,波浪阻力又增加,最后逐渐趋于正常。同时发现,两船之间的侧向力有类似的变化规律。侧向力先是相互排斥,后相互吸引,接着又相互排斥,最后逐渐趋于正常。两船间的侧向吸引力剧烈变化,对船舶的操纵性及稳定性有重要的影响,也是近年来河道频频发生撞船事故的重要原因之一。
With increasing of ship size, speed and power, shipbuilding industry has achieved a great advance in recent years. A great number of ships moved in shallow water, such as rivers, lakes and estuaries. Nonlinearity, frequency dispersion, free surface disturbance, energy dissipation and boundary reflection may play important roles to ship waves in shallow water. Topography complexity, such as irregular boundaries, varying water depth, makes ship waves more difficult. The shallow water ship waves are of great importance to water environments, engineering applications as well as the ships themselves. It is of great importance to investigate the dynamic behavior of the ship waves in shallow water.
This work is focused on numerical study of waves generated by ships moving in shallow water based on Green-Naghdi equation. A moving ship is regarded as a moving pressure disturbance on free surface in this study. The moving pressure is incorporated into the Green-Naghdi equations to formulate forcing of ship waves in shallow water. Three-dimensional ship wave profiles, wave resistance and transverse forces are given. Generally speaking, ship waves in shallow water have small values, the ratio of water depth to wave length, and may be thought of as long waves. In fact, the ship wave is a combination of multi-frequency water waves. The wave constituents with finite values may have effects on ship waves. The frequency dispersion cannot be completely ignored. The frequency dispersion term of the Green-Naghdi equations accounts for the effects of finite water depth on ship waves. Kirby reviewed a variety of Boussinesq-type equations, which consider the frequency dispersion with different orders. The Green-Naghdi equation is a typical Boussinesq-type equation, which involves the nonlinearity and frequency dispersion. It does not require irrotational conditions and the limitation of the ratio of wave length to water depth.
The Green-Naghdi equations have an analogous form to the two-dimensional non-viscous Navier-Stokes equations with an additional frequency dispersion term. The continuity equation is the first-order hyperbolic differential equation. It is known that the numerical solution of the first-order hyperbolic differential equation had been a difficult
problem in computational fluid dynamics field. The numerical solutions of Green-Naghdi equations are easily spoiled by numerical oscillations. Wave equation model can eliminate numerical oscillation effectively and have better stability property. It had been successful used to solve shallow water questions in many practical applications. In this dissertation, a wave equation model and the finite element method (WE/FEM) are adopted to solve the Green-Naghdi equations. As the examples of Series 60 (CB = 60) ship, the numerical solutions of three-dimensional ship wave profiles, wave resistance and transverse forces are presented.
The contents of this dissertation are summarized as follows. Chapter 1 introduces the effects of finite water depth on ship waves, and briefly discusses the theory of shallow water equations. Chapter 2 introduces some kinds of Boussinesq-type equations, and briefly discusses the Boussinesq equations. We have also analyzed how the frequency dispersion term in the Boussinesq equations modify the hydrostatical pressure in wate depth. Chapter 3 is devoted to the relation between the Green-Naghdi equations and Boussinesq-type equations. We also discuss frequency dispersion of the linear Green-Naghdi equations. In chapter 4, we present numerical solutions of the Green-Naghdi equations. A wave equation model and finite element method (WE/FEM) are adopted to solve shallow water questions, and the Green-Naghdi equations. Furthermore, the implementation of boundary conditions is also discussed in this chapter. Three-dimensional ship wave profiles and wave resistance when a Series 60 () ship moves at different speed in shallow water are presented in chapter 5. A comparison between the numerical results predicted by the Green-Naghdi equations and the s
引文
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Baker A J. A finite element penalty algorithm for the parabolic Navier-Stokes equations for turbulent three-dimensional flow. Computer Methods in Applied Mechanics and Engineering, 1984, 46(1): 277~293.
Lohner R, Morgan K, Zienkiewicz O C. An adaptive finite element procedure for compressible high-speed flows. Computer Methods in Applied Mechanics and Engineering, 1985, 51(2): 441~465.
Girault V, Raviart P A. Finite element method for Navier-Stokes equations. Berlin, Heideberg: Springerverlag, 1986.
Bristeau M O, Glowinski R, Periaux J. Numerical methods for the Navier-Stokes equations: Applications to the simulation of compressible and incompressible viscous flows. Computer Physics Reports, North-Holland Physics Publishing Division, 1987, 73~187.
李开泰, 黄艾香. 有限元方法及其应用-发展及应用. 西安: 西安交通大学出版社, 1988.
Bristeau M O, Glowinski R, Dutto L, et al. Compressible viscous flow calculations using compatible finite element approximations. International Journal for Numerical Methods in Fluids, 1990, 11(2): 719~749.
Babuska I. The finite element method with Lagrangian multiplies. Numerical Method in Mathematics, 1973, 20(1): 179~192.
Roache P J. Computational fluid dynamics. Hermosa Press, Albuquerque, N.M., 1972.
Zienkiewicz O C. Newtonian and non-newtonian viscous incompressible flow, temperature induced flows, finite element solution. The Mathematics of Finite Elements and Application II, London: Academic Press, 1975.
Heinrich J C, Zienkiewicz O C. An upwind finite element scheme for two-dimensional convective transport problem. International Journal for Numerical Method in Engineering, 1977, 11(1): 131~143.
Hughes T J R. A simple scheme for developing upwind finite elements. Int. Journal for Numerical Method in Engineering, 1978, 12(6): 1359~1365.
Brooks A N, Hughes T J R. Streamline Upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Method in Applied Mechanics and Engineerng, 1982, 32(1): 199~259.
Boris P.J. (张德华译). 计算流体动力学的新趋势. 力学进展, 1990.
Wu, T.Y. Long Waves in Ocean and Coastal Waters, J. Eng. Mech. Div. ASCE, No. EM3, Proc. paper 16346, June, 1981, 107: 501~522.
Wu, T. Y. Generation of Upstream Advancing Solitons by Moving Disturbances. J. Fluid Mech, 1987, 184: 75~99.
Lepelletier, T. G. Tsunamis-Harbor Oscillations Induced by Nonlinear Transient Long Waves. W. M. Keck Lab. Report No. KH-R41, Calif. Inst. Of Tech., Pasadena, Calif., 1981, xxiii+481.
Schember, H. R. A New Model for Three-Dimensional Nonlinear Dispersive Long Waves. Ph. D. Thesis, Calif. Inst. of Tech., Pasadena, Calif, 1982, vii + 141.
Zhu, J. Internal Soliton Generated by Moving Disturbances. Ph.D. Dissertation, California Institute of Technology, Pasadena, CA, 1986.
Mi, Z. Y. and Liu, Y. Z. Calculation of Solitary Waves Generated by a 2-D Body. Proc. Of International Symposium on Ship Resistance and Powering Performance, Shanghai, 1989, 209~215.
Boussinesq, J. The?orie de L'Intumescence Liquide Appele?e Onde Solitaire ou de Translation. Comtes Rendus Acad. Sci. Paris, 1871, 72: 755~759.
Peregrine D H. Long waves on a beach. J. Fluid Mech, 1967, 27: 815~827.
林建国. 波动力学的流面理论、Boussinesq类方程及其浮体波动问题的求解[博士论文]. 上海:上海交通大学 (LinJianguo., Flow Surface Theory Boussinesq type equations and numerical,Simulation to Floating Body in Wave. Ph. D Thesis of Shanghai Jiaotong University.(in Chinese), 1995,12). 1995, 12.
McCowan A D. The range of application of Boussinesq type numerical short wave models. Proc 22nd IAHR Conger, Switzerland, 1987, 279~384.
陶明德. 水波引论. 复旦大学出版社, 1990.
Wei G, Kirby J T, Grilli, S T, Snbramaanya H. A fully nonlinear Boussinesq model for surface waves. Part 1, Highly nonlinear unsteady waves, J Fluid Mech, 1995, 27: 815~827.
Madean P A, Schaffer H A. Higher order Boussinesq-type equations for surface gravity waves-Derivation and analysis. Philos Trans Roy Soc London A, 1998, 356:1~60.
黄虎. 海岸波浪场模型研究进展. 力学进展, 2001, 592~610.
Green A. E., Laws N, Naghdi P. M. On the theory of water waves. Proc Roy Soc London A, 1974, 338: 43~55.
Green, A. E. and Naghdi, P. M. Water Waves in a Non-homogeneous Incompressible Fluid. J. Fluid Mech, Part 2, 1977, 78: 237~246.
Green A. E., Laws N, Naghdi P. M. A nonlinear theory of water waves for finite and infinite depths. Philos Trans Roy Soc London A, 1986, 320: 37~70.
Green A. E., Laws N, Naghdi P. M. Further developments in a nonlinear theory of water waves for finite and infinite depths. Philos Trans Roy Soc London A, 1987, 324: 47~72.
Naghdi P. M. Fluid Jets and Fluid Sheets. Proc. 12th Symp. On Naval Hydrodyn., National Academy of Sciences, Wash., D.C., 1978, 500~515.
Naghdi P. M., Rubin M B. On the transition to planing of a boat. J. Fluid Mech, 1981, 103: 345~374.
Shields J. J., Webster W C. On direct methods in water-wave theory. J. Fluid Mech, 1988, 197: 171~199.
Marshall J S, Naghdi P M. Wave reflection and transmission by steps and rectangular obstacles in channels of finite depth. Theoretical and Computational Fluid Dynamics, 1990, 1:287~301.
Demirbilek Z, Webster W C. Evolution of time-dependent nonlinear shallow water waves. In: Advance in Hydro-Science & Engineering I, Washington D C, 1993, 1536~1544.
Demirbilek Z, Webster W C. Nonlinear theories of steady and unsteady shallow water waves. In: Invited presentation to the MEET 93 Conf. Univ of Virginia, Charlottesville, VA, 1993, 06~09.
Demirbilek Z, Webster W C. The Green-Naghdi theory of fluid sheets for shallow water waves. In: Herbich J B, ed. Developments in Offshore Engineering.Houston: Gulf Publishing Company, 1999, 1~54.
Zou G Y and Wu J X. The diffraction of the solitary wave by a hump at the sea bottom and practical comparisons of the different OBC. Chinese J. Computational Physics, 1994, 11(2): 185~194.
Y. Toda, F. Sterm, and J. Longo. Mean-Flow Measurements in the Boundary Layer and Wave Field of Series 60 CB =0.6 Ship Model. Part 1: Froude Number 0.16 and 0.316, J. Ship Research, 1992, 36(4): 360~377.
J. B. Keller. Surface waves on water of non-uniform depth. J. Fluid Mech, 1958, 4: 607~614.
M. C. Shen, J. B Keller. Uniform ray theory of surface, internal, and acoustic wave propagation in a rotating ocean or atmosphere. SIAM J. Appl. Math, 1975, 28: 857~875.
R. Grimshaw. The solitary wave in water of variable depth. J. Fluid Mech, 1970, 42: 639~656.
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姜贵庆, 童秉纲, 曹树声. 以有限元方法为主体的计算气动热力学. 力学与实践, 1992, 14(3): 1~8.
Engelman M S, Strang G, Bathe K J. The application of quasi-newton method in fluids mechanics. International Journal for Numerical Method in Engineering, 1981, 17(5): 707~718.
Baker A J. A finite element penalty algorithm for the parabolic Navier-Stokes equations for turbulent three-dimensional flow. Computer Methods in Applied Mechanics and Engineering, 1984, 46(1): 277~293.
Lohner R, Morgan K, Zienkiewicz O C. An adaptive finite element procedure for compressible high-speed flows. Computer Methods in Applied Mechanics and Engineering, 1985, 51(2): 441~465.
Girault V, Raviart P A. Finite element method for Navier-Stokes equations. Berlin, Heideberg: Springerverlag, 1986.
Bristeau M O, Glowinski R, Periaux J. Numerical methods for the Navier-Stokes equations: Applications to the simulation of compressible and incompressible viscous flows. Computer Physics Reports, North-Holland Physics Publishing Division, 1987, 73~187.
李开泰, 黄艾香. 有限元方法及其应用-发展及应用. 西安: 西安交通大学出版社, 1988.
Bristeau M O, Glowinski R, Dutto L, et al. Compressible viscous flow calculations using compatible finite element approximations. International Journal for Numerical Methods in Fluids, 1990, 11(2): 719~749.
Babuska I. The finite element method with Lagrangian multiplies. Numerical Method in Mathematics, 1973, 20(1): 179~192.
Roache P J. Computational fluid dynamics. Hermosa Press, Albuquerque, N.M., 1972.
Zienkiewicz O C. Newtonian and non-newtonian viscous incompressible flow, temperature induced flows, finite element solution. The Mathematics of Finite Elements and Application II, London: Academic Press, 1975.
Heinrich J C, Zienkiewicz O C. An upwind finite element scheme for two-dimensional convective transport problem. International Journal for Numerical Method in Engineering, 1977, 11(1): 131~143.
Hughes T J R. A simple scheme for developing upwind finite elements. Int. Journal for Numerical Method in Engineering, 1978, 12(6): 1359~1365.
Brooks A N, Hughes T J R. Streamline Upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Method in Applied Mechanics and Engineerng, 1982, 32(1): 199~259.
Boris P.J. (张德华译). 计算流体动力学的新趋势. 力学进展, 1990.
Wu, T.Y. Long Waves in Ocean and Coastal Waters, J. Eng. Mech. Div. ASCE, No. EM3, Proc. paper 16346, June, 1981, 107: 501~522.
Wu, T. Y. Generation of Upstream Advancing Solitons by Moving Disturbances. J. Fluid Mech, 1987, 184: 75~99.
Lepelletier, T. G. Tsunamis-Harbor Oscillations Induced by Nonlinear Transient Long Waves. W. M. Keck Lab. Report No. KH-R41, Calif. Inst. Of Tech., Pasadena, Calif., 1981, xxiii+481.
Schember, H. R. A New Model for Three-Dimensional Nonlinear Dispersive Long Waves. Ph. D. Thesis, Calif. Inst. of Tech., Pasadena, Calif, 1982, vii + 141.
Zhu, J. Internal Soliton Generated by Moving Disturbances. Ph.D. Dissertation, California Institute of Technology, Pasadena, CA, 1986.
Mi, Z. Y. and Liu, Y. Z. Calculation of Solitary Waves Generated by a 2-D Body. Proc. Of International Symposium on Ship Resistance and Powering Performance, Shanghai, 1989, 209~215.
Boussinesq, J. The?orie de L'Intumescence Liquide Appele?e Onde Solitaire ou de Translation. Comtes Rendus Acad. Sci. Paris, 1871, 72: 755~759.
Peregrine D H. Long waves on a beach. J. Fluid Mech, 1967, 27: 815~827.
林建国. 波动力学的流面理论、Boussinesq类方程及其浮体波动问题的求解[博士论文]. 上海:上海交通大学 (LinJianguo., Flow Surface Theory Boussinesq type equations and numerical,Simulation to Floating Body in Wave. Ph. D Thesis of Shanghai Jiaotong University.(in Chinese), 1995,12). 1995, 12.
McCowan A D. The range of application of Boussinesq type numerical short wave models. Proc 22nd IAHR Conger, Switzerland, 1987, 279~384.
陶明德. 水波引论. 复旦大学出版社, 1990.
Wei G, Kirby J T, Grilli, S T, Snbramaanya H. A fully nonlinear Boussinesq model for surface waves. Part 1, Highly nonlinear unsteady waves, J Fluid Mech, 1995, 27: 815~827.
Madean P A, Schaffer H A. Higher order Boussinesq-type equations for surface gravity waves-Derivation and analysis. Philos Trans Roy Soc London A, 1998, 356:1~60.
黄虎. 海岸波浪场模型研究进展. 力学进展, 2001, 592~610.
Green A. E., Laws N, Naghdi P. M. On the theory of water waves. Proc Roy Soc London A, 1974, 338: 43~55.
Green, A. E. and Naghdi, P. M. Water Waves in a Non-homogeneous Incompressible Fluid. J. Fluid Mech, Part 2, 1977, 78: 237~246.
Green A. E., Laws N, Naghdi P. M. A nonlinear theory of water waves for finite and infinite depths. Philos Trans Roy Soc London A, 1986, 320: 37~70.
Green A. E., Laws N, Naghdi P. M. Further developments in a nonlinear theory of water waves for finite and infinite depths. Philos Trans Roy Soc London A, 1987, 324: 47~72.
Naghdi P. M. Fluid Jets and Fluid Sheets. Proc. 12th Symp. On Naval Hydrodyn., National Academy of Sciences, Wash., D.C., 1978, 500~515.
Naghdi P. M., Rubin M B. On the transition to planing of a boat. J. Fluid Mech, 1981, 103: 345~374.
Shields J. J., Webster W C. On direct methods in water-wave theory. J. Fluid Mech, 1988, 197: 171~199.
Marshall J S, Naghdi P M. Wave reflection and transmission by steps and rectangular obstacles in channels of finite depth. Theoretical and Computational Fluid Dynamics, 1990, 1:287~301.
Demirbilek Z, Webster W C. Evolution of time-dependent nonlinear shallow water waves. In: Advance in Hydro-Science & Engineering I, Washington D C, 1993, 1536~1544.
Demirbilek Z, Webster W C. Nonlinear theories of steady and unsteady shallow water waves. In: Invited presentation to the MEET 93 Conf. Univ of Virginia, Charlottesville, VA, 1993, 06~09.
Demirbilek Z, Webster W C. The Green-Naghdi theory of fluid sheets for shallow water waves. In: Herbich J B, ed. Developments in Offshore Engineering.Houston: Gulf Publishing Company, 1999, 1~54.
Zou G Y and Wu J X. The diffraction of the solitary wave by a hump at the sea bottom and practical comparisons of the different OBC. Chinese J. Computational Physics, 1994, 11(2): 185~194.
Y. Toda, F. Sterm, and J. Longo. Mean-Flow Measurements in the Boundary Layer and Wave Field of Series 60 CB =0.6 Ship Model. Part 1: Froude Number 0.16 and 0.316, J. Ship Research, 1992, 36(4): 360~377.
J. B. Keller. Surface waves on water of non-uniform depth. J. Fluid Mech, 1958, 4: 607~614.
M. C. Shen, J. B Keller. Uniform ray theory of surface, internal, and acoustic wave propagation in a rotating ocean or atmosphere. SIAM J. Appl. Math, 1975, 28: 857~875.
R. Grimshaw. The solitary wave in water of variable depth. J. Fluid Mech, 1970, 42: 639~656.
R. Grimshaw. The solitary wave in water of variable depth. Part 2, J. Fluid Mech. 1970, 46: 611~622
P. Milewki. Long wave interaction over varying topography. Physica D, 1998, 123: 36~47