基于曲线坐标系下双曲型缓坡方程模拟近岸波浪及波生流
|
摘要
波浪以及破碎所产生的波生流场是近岸海域重要的水动力现象,是规划设计海岸工程,了解泥沙输运、海岸变形和近岸污染物扩散输移等近海问题的最基本因素。但是在近岸区域,地形和岸线往往比较复杂,采用规则矩形网格建模的常规方法易造成网格大小无法随地形变化,而且计算域边界与实际边界不吻合,从而增加计算量,并减少计算精度。而在实际工程中,工程师们通常更关心不规则岸线附近波浪和流场的分布,如河口、岛屿、码头和防波堤附近。
本文的宗旨是建立一种适用于近海区域复杂地形和多变边界的数值模型,实现对近海区域水动力因素更精确的数值模拟。适体曲线坐标可以实现对曲折边界的无缝拟合,可以根据地形的变化调整网格的大小,而且曲线坐标转换一般不会对数值求解方法产生太大的影响,是目前计算流体力学中较为常见的处理复杂边界的数学方法。综合这些认识,本文基于坐标转换的基本理论,建立了正交曲线坐标系下的双曲型缓坡方程数值模型和基于缓坡方程的近岸波生流数值模型。
首先,本文通过全面分析缓坡方程数学模型的研究进展,选择了数值计算方便,可以综合考虑波浪在大区域传播变形的两类双曲型模型作为控制方程,分别建立了正交曲线坐标系下的波浪数值模型。模型采用高效的ADI (Alternating Direction Implicit)有限差分法,进行离散求解。
将所建立的波浪模型应用到若干个物理模型实验中,数值模拟的结果分别和实验值、理论解以及直角坐标系下的数值结果做了比较和分析；着重验证了模型对复杂地形和复杂边界、对扭曲的正交网格和对实际海域的适用性；本文还对两类双曲型缓坡方程的数值计算效率和精度做了适当的分析。
本文通过波浪模型提供辐射应力等驱动力的概念,基于二维浅水方程建立了正交曲线坐标系下的近岸波生流数值模型。流场模型采用与波浪模型相同的正交网格和计算方法离散求解,即连续性方程采用隐格式,而动量方程交替采用显格式计算。
应用所建立的波生流数值模型验证了波浪在曲线边界附近破碎产生沿岸流和裂流的情况,由波浪模型得到的辐射应力,通过源项的形式加入到流场模型中,为波流模型提供驱动力。通过物理模型实验验证结果表明,本文建立的基于缓坡方程的曲线坐标系下波流模型计算方便,可适用对复杂地形和多变边界地区波生流的数值计算。
The nearshore waves and current induced by breaking waves are key hydrodynamic factors to coastal waters; understanding their rules of movement is the basic way to understand issues of the design of coastal projects, sediment transport, the deformation of coastlines and the diffusion of pollutants et al. However, the topography and coastline in the coastal regions are always complicated; therefore the conventional numerical modeling in rectangular grid could easily lead to the state that the boundary of computational domain does not match with the actual border, and thus reduce the accuracy of numerical results. While in practical engineering, engineers are usually more concerned about the distribution of waves and wave-induced current near irregular coastlines, such as, in the vicinity of estuaries, islands, piers and breakwaters.
It is the object of this paper to establish a numerical model for coastal regions with complex terrain and variable boundaries, and to realize a more accurate hydrodynamic simulation of waves and current in these areas. Three reasons make the curvilinear coordinates transformation to be an outstanding mathematical method in dealing with complex boundary in fluid dynamics. They are being that firstly boundary-fitted grid can achieve a seamless fit to the curve border; secondly, the size of the grids can be adjusted according to changes in topography; and lastly coordinate transformation will not change the numerical method, in most cases. Based on this understanding, the present paper shows numerical models of the hyperbolic mild-slope equation and shallow water equation in orthogonal curvilinear coordinates.
First of all, a comprehensive review of the development of nearshore wave modeling using the mild-slope equation was given. To facilitate numerical calculation, two different hyperbolic approximations of mild-slope equation, which are capable to describe wave transformation in large areas, were chosen to study under orthogonal curvilinear coordinates. The established wave models were discreted with a space-staggered grid and solved by the efficient ADI method (Alternating Direction Implicit).
Then, wave models were applied to several physical experiments; the numerical results were compared to experimental data, analytical results, and numerical results from Cartesian coordinates, available. The numerical models were validated by their applicability to the complicated terrain and boundaries, to the distortion of orthogonal grids, and to the practical engineering application. Appropriate analysis was also given on the numerical accuracy and efficiency of the two kinds of hyperbolic mild-slope equations.
Thirdly, the two-dimensional shallow water equations were transformed in orthogonal curvilinear coordinates based on the concept of the radiation stress theory. The numerical solving method of the transform current model was the same as wave model, i.e., the continuity equation being calculated by the implicit scheme and the momentum equations by alternating explicit scheme.
Last but not the least, the regular wave-induced longshore currents and rip currents were calculated through the transformed current model. Here, the established hyperbolic mild-slope wave model was employed as a driving force to flow field. The good agreement of numerical results with experimental data and with published numerical results showed that the numerical model in this paper is applicable to simulate wave-induced current in areas with complicated topography and boundary conditions.
本文的宗旨是建立一种适用于近海区域复杂地形和多变边界的数值模型,实现对近海区域水动力因素更精确的数值模拟。适体曲线坐标可以实现对曲折边界的无缝拟合,可以根据地形的变化调整网格的大小,而且曲线坐标转换一般不会对数值求解方法产生太大的影响,是目前计算流体力学中较为常见的处理复杂边界的数学方法。综合这些认识,本文基于坐标转换的基本理论,建立了正交曲线坐标系下的双曲型缓坡方程数值模型和基于缓坡方程的近岸波生流数值模型。
首先,本文通过全面分析缓坡方程数学模型的研究进展,选择了数值计算方便,可以综合考虑波浪在大区域传播变形的两类双曲型模型作为控制方程,分别建立了正交曲线坐标系下的波浪数值模型。模型采用高效的ADI (Alternating Direction Implicit)有限差分法,进行离散求解。
将所建立的波浪模型应用到若干个物理模型实验中,数值模拟的结果分别和实验值、理论解以及直角坐标系下的数值结果做了比较和分析；着重验证了模型对复杂地形和复杂边界、对扭曲的正交网格和对实际海域的适用性；本文还对两类双曲型缓坡方程的数值计算效率和精度做了适当的分析。
本文通过波浪模型提供辐射应力等驱动力的概念,基于二维浅水方程建立了正交曲线坐标系下的近岸波生流数值模型。流场模型采用与波浪模型相同的正交网格和计算方法离散求解,即连续性方程采用隐格式,而动量方程交替采用显格式计算。
应用所建立的波生流数值模型验证了波浪在曲线边界附近破碎产生沿岸流和裂流的情况,由波浪模型得到的辐射应力,通过源项的形式加入到流场模型中,为波流模型提供驱动力。通过物理模型实验验证结果表明,本文建立的基于缓坡方程的曲线坐标系下波流模型计算方便,可适用对复杂地形和多变边界地区波生流的数值计算。
The nearshore waves and current induced by breaking waves are key hydrodynamic factors to coastal waters; understanding their rules of movement is the basic way to understand issues of the design of coastal projects, sediment transport, the deformation of coastlines and the diffusion of pollutants et al. However, the topography and coastline in the coastal regions are always complicated; therefore the conventional numerical modeling in rectangular grid could easily lead to the state that the boundary of computational domain does not match with the actual border, and thus reduce the accuracy of numerical results. While in practical engineering, engineers are usually more concerned about the distribution of waves and wave-induced current near irregular coastlines, such as, in the vicinity of estuaries, islands, piers and breakwaters.
It is the object of this paper to establish a numerical model for coastal regions with complex terrain and variable boundaries, and to realize a more accurate hydrodynamic simulation of waves and current in these areas. Three reasons make the curvilinear coordinates transformation to be an outstanding mathematical method in dealing with complex boundary in fluid dynamics. They are being that firstly boundary-fitted grid can achieve a seamless fit to the curve border; secondly, the size of the grids can be adjusted according to changes in topography; and lastly coordinate transformation will not change the numerical method, in most cases. Based on this understanding, the present paper shows numerical models of the hyperbolic mild-slope equation and shallow water equation in orthogonal curvilinear coordinates.
First of all, a comprehensive review of the development of nearshore wave modeling using the mild-slope equation was given. To facilitate numerical calculation, two different hyperbolic approximations of mild-slope equation, which are capable to describe wave transformation in large areas, were chosen to study under orthogonal curvilinear coordinates. The established wave models were discreted with a space-staggered grid and solved by the efficient ADI method (Alternating Direction Implicit).
Then, wave models were applied to several physical experiments; the numerical results were compared to experimental data, analytical results, and numerical results from Cartesian coordinates, available. The numerical models were validated by their applicability to the complicated terrain and boundaries, to the distortion of orthogonal grids, and to the practical engineering application. Appropriate analysis was also given on the numerical accuracy and efficiency of the two kinds of hyperbolic mild-slope equations.
Thirdly, the two-dimensional shallow water equations were transformed in orthogonal curvilinear coordinates based on the concept of the radiation stress theory. The numerical solving method of the transform current model was the same as wave model, i.e., the continuity equation being calculated by the implicit scheme and the momentum equations by alternating explicit scheme.
Last but not the least, the regular wave-induced longshore currents and rip currents were calculated through the transformed current model. Here, the established hyperbolic mild-slope wave model was employed as a driving force to flow field. The good agreement of numerical results with experimental data and with published numerical results showed that the numerical model in this paper is applicable to simulate wave-induced current in areas with complicated topography and boundary conditions.
引文
[1]李玉成,滕斌.波浪对海上建筑物的作用[M].北京：海洋出版社,2002.
[2]孙涛,陶建华.波浪作用下近岸区污染物输移扩散的数学模型及其实验验证[J].海洋学报,2003,25(3)：104-112.
[3]唐军.近岸波流场的缓坡模型数值模拟及波流场中污染物输运的研究[D].大连：大连理工大学,2005.
[4]孙涛,韩光,陶建华.波生沿岸流数值模拟研究及其实验验证[J].水利学报,2002,11：1-9.
[5]Berkhoff J C W. Computation of combined refraction-diffraction [A]. Proceeding of the 13th Coastal Engineering Conference[C], Vancouver. ASCE,1972:471-490.
[6]Booij N. A note on the accuracy of the mild-slope equation [J]. Coastal Engineering, 1983,7:513-533.
[7]郑永红.波浪、潮流、污染物迁移扩散数学模型及其数值模拟[D].大连：大连理工大学博士后研究工作报告,2000.
[8]邹志利.水波理论及其应用[M].北京：科学出版社,2005.
[9]Radder A C. On the parabolic equation method for water wave propagation [J]. Journal of Fluid Mechanics,1979,95(1):159-176.
[10]Kirby J T. Higher order approximations in the parabolic equation method for water waves [J]. Journal of Geophysical Research,1986,91(C):933-952.
[11]Ito Y and Tanimoto K. A method of numerical analysis of wave propagation-application to wave diffraction and refraction [C]. Proceeding of the 13th International Conference on Coastal Engineering, Vancouver, ASCE,1972,26:503-522.
[12]Copeland G J M. A practical alternative to the Mild-Slope wave equation [J]. Coastal Engineering,1985,9:125-149.
[13]Madsen P A and Larsen J. An efficient finite-difference approach to the mild-slope equation [J]. Coastal Engineering,1987,11:329-351.
[14]Lee C and Suh K D. Internal generation of waves for time-dependent mild-slope equations [J]. Coastal Engineering,1998,34:35-57.
[15]Lee C, Park W S, Cho Y, et al. Hyperbolic mild-slope equations extended to account for rapidly varying topography [J]. Coastal Engineering,1998,34:243-257.
[16]Oliveira F S B F. Improvement on open boundaries on a time dependent numerical model of wave propagation in the nearshore region [J]. Ocean Engineering 2000,28:95-115.
[17]Suh K D, Lee C Park Y and Lee T H. Experimental verification of horizontal two-dimensional modified mild-slope equation model [J]. Coastal Engineering,2001, 44:1-12.
[18]Zheng Y H, Shen Y M, Qiu D H. New numerical scheme for simulation of hyperbolic mild-slope equation [J]. China Ocean Engineering,2001,15(2):185-194.
[19]赵明,滕斌.利用有限元方法求解双曲型缓坡方程[J].海洋工程,2002,20(3)：54-60.
[20]Zou Z L and Jin H. Hyperbolic mild slope equations with inclusion of amplitude dispersion effect:random waves [J]. China Ocean Engineering,2008,24(4):595-610.
[21]Peregrine D H. Surf zone currents [J]. Theoretical and Computational Fluid Dynamics, 1998,10:295-309.
[22]MacMahan J H, Thornton E B and Reniers A. Rip current review [J]. Coastal Engineering 2006,53:191-208.
[23]邹志利.海岸动力学[M].北京：人民交通出版社,2009.
[24]http://www. gsdkj. net/pro/view. php?id=21206
[25]Longuet-Higgins M S and Stewart R W. Radiation stress in water waves:a physical discussion with application [J]. Deep Sea Research,1964,11(5):529-562.
[26]Bowen A J, Inman D L and Simmons V P. Wave "set-down" and "set-up" [J]. Journal of Geophysical Research 1968,73(8):2569-2577.
[27]Bowen A J. The generation of longshore currents on a plane beach [J]. Journal of Marine Research,1969,27:206-215
[28]Longuet-Higgins M S. Longshore currents generated by obliquely incident sea waves, 1 [J]. Journal of Geophysical Research,1970,75(33):6778-6801.
[29]Iwata N. A note on the wave wet-up, longshore currents and undertows [J]. Journal of the Oceanographical Society of Japan,1970,26(4):233-236.
[30]Mei C C.A note on the averaged momentum balance in two-dimensional water waves [J]. Journal of Marine Research,1973,31 (2):97-104.
[31]Bettess P and Bettess J. A generalization of the radiation stress tensor [J]. Applied Mathematical Modelling,1982,6:145-150.
[32]Svendsen I A, Hass K and Zhao Q. Quasi-3D Nearshore Circulation Model SHORECIRC version 2.4, User Manual. Technical Report, University of Delaware,2002.
[33]Yoon S B, Cho Y, and Lee C. Effects of breaking-induced currents on refraction-diffraction of irregular waves over submerged shoal [J]. Ocean Engineering, 2004,31:633-652.
[34]唐军,沈永明,邱大洪.结合椭圆型缓坡方程模拟近岸波流场[J].海洋学报,2006,28(1)：146-150.
[35]Choi J, Lim C, Jeon Y, and Yoon S B. Numerical simulation of irregular wave transformation due to wave-induced current [J]. Journal of Hydro-environment Research, 2007,1:133-142.
[36]Choi J, Lim C, Lee J, and Yoon S B. Evolution of waves and currents over a submerged laboratory shoal [J]. Coastal Engineering,2009,56:297-312.
[37]魏美芳.非结构网格下椭圆型缓坡方程及近岸波生流数值模拟[D].大连：大连理工大学,2009.
[38]Kirby J T, Wei G, Chen Q, Kennedy A B and Dalrymple R A. Fully nonlinear Boussinesq wave model, User Manual. Technical Report CACR-98-06, University of Delaware,1998.
[39]Chen Q, Kirby J T, Dalrymple R A, Kennedy A B and Chawla A. Boussinesq modeling of wave transformation, breaking, and runup. II:2D [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering,2000,126(1),48-56.
[40]李绍武,黄筱云.用Boussinesq方程计算沿岸流的数值方法[J].天津大学学报,2004,37(12)：1059-1062.
[41]李绍武,连怡贞.完全非线性波浪破碎模型沿岸流数值模拟[J].海洋工程,2007,25(3)：46-52.
[42]卢吉,余锡平.基于Boussinesq方程的近岸波流统一模型[J].水动力学研究与进展,2008,23(3)：314-320.
[43]Kuriyama Y, Itob Y, Yanagishima S. Cross-shore variation of long-term average longshore current velocity in the nearshore zone [J]. Continental Shelf Research, 2008,28:491-502.
[44]Castelle B, Bonneton P, Senechal N, Dupuis H, Butel R and Michel D. Dynamics of wave-induced currents over an alongshore non-uniform multiple-barred sandy beach on the Aquitanian Coast, France [J]. Continental Shelf Research,2006,26:113-131.
[45]Rozynski G and Reeve D. Multi-resolution analysis of nearshore hydrodynamics using discrete wavelet transforms [J]. Coastal Engineering,2005,52:771-792.
[46]邹志利,常梅,邱大洪,王凤龙,董国海.沿岸流的实验研究[J].水动力研究与进展(A),2002,17(2)：174-180.
[47]Shi F Y, Dalrymple R A, Kirby J T, Chen Q and Kennedy A. A fully nonlinear Boussinesq model in generalized curvilinear coordinates [J]. Coastal Engineering,2001,42: 337-358.
[48]Li Y S, Liu S X, Yu Y X and Lai G Z. Numerical modeling of Boussinesq equations by finite element method [J]. Coastal Engineering,1999,37:97-122.
[49]Yuan D K, Lin B L and Xiao H. Parabolic wave propagation modeling in orthogonal coordinate systems [J]. China Ocean Engineering,2004,18(3):445-456.
[50]Shi F Y and Kirby J T. Curvilinear parabolic approximation for surface wave transformation with wave-current interaction [J]. Journal of Computational Physics, 2005,204:562-586.
[51]Zhang H S, Zhu L S and You Y X. A numerical model for wave propagation in curvilinear coordinates [J]. Coastal Engineering,2005,52:513-533.
[52]张洪生,尤云祥,朱良生,张军.曲线坐标系下波浪传播的数学模型及其比较与验证[J].水动力学研究与进展,A辑,2005,20(1)：106-117.
[53]张洪生,洪广文,丁平兴.任意曲线边界条件下缓变水深水域波浪传播的数值模拟[J].海洋学报,2002,24(1)：108-116.
[54]Spaulding M L. A vertically averaged circulation model using boundary-fitted coordinates [J]. Journal of Physical Oceanography,1984,14:973-982.
[55]Kirby J T. Parabolic wave computations in non-orthogonal coordinate systems [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering,1988,114 (6):673-685.
[56]Kirby J T, Dalrymple R A and Kaku H. Parabolic approximations for water waves in conformal coordinate systems [J]. Coastal Engineering,1994,23:185-213.
[57]Shi F, Svendsen I A, Kirby J T and Smith J M. A curvilinear version of a quasi-3D nearshore circulation model [J]. Coastal Engineering,2003,49:99-124.
[58]Tang J, Shen Y M, Zheng Y H et al. An efficient and flexile computational model for solving the mild-slope equation [J]. Coastal Engineering,2004(2),51:143-154.
[59]柳淑学,孙冰.考虑边界波浪方向的缓坡方程自适应求解模型[J].海洋工程,2007,25(1)：35-56.
[60]魏美芳,唐军,沈永明.非结构化网格下椭圆型缓坡方程的数值求解[J].海洋学报,2009,31(2)：159-164.
[61]Dally W R, Dean R G and Dalrymple R A. Wave height verification across beaches of arbitrary profile [J]. Journal of Geophysical Research,1985,90(C6):11917-11927.
[62]Thompson J F, Warsi Z U A and Mastin C W. Numerical Grid Generation:Foundations and Applications [M]. North Holland,1985.
[63]Thompson J F, Thames F C, and Mastin C W. Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies [J]. Journal of Computational Physics,1974,15(3):299-319.
[64]Thompson J F, Warsi Z U A and Mastin C W. Body-fitted coordinate system for numerical solution of partial differential equations [J]. Journal of Computational Physics, 1982,47(1):1-108.
[65]魏文礼,张宗孝,刘玉玲,刘明琴.采用Poisson方程生成曲线网格时源项P、Q选择的研究[J].应用力学学报,2000,17(3)：87-91.
[66]孙建国,蒋丽丽.用于起伏地表条件下地球物理场数值模拟的正交曲网格生成技术[J].石油地球物理勘探,2009,44(4)：494-500.
[67]http://coastal. udel. edu/-fyshi/coastgrid/coastgrid. html
[68]http://en. wikipedia. org/wiki/Alternating_direction_implicit_method
[69]王永学.有限差分法讲义.大连理工大学土木水利学院内部讲义,2008.
[70]Berkhoff J C W, Booij N and Radder A C. Verification of numerical wave propagation models for simple harmonic linear waves [J]. Coastal Engineering,1982,6(3):255-279.
[71]Beji S and Nadaoka K. Fully dispersive nonlinear water wave model in curvilinear coordinates [J]. Journal of Computational Physics,2004,198:645-658.
[72]Dalrymple R A, Kirby J T and Martin P A. Spectral methods for forward-propagating water waves in conformally-mapped channels [J]. Applied Ocean Research,1994,16: 249-266.
[73]Isobe M. A parabolic refraction-diffraction equation in the ray-front coordinate system, in:Proceedings of the 20th International Conference on Coastal Engineering, Taipei,1986, pp.306-317.
[74]Pan J N, Hong G W, Zuo Q H and Wang D T. Verification of a numerical harbor Wave Model. China Ocean Engineering [J].2007,21(4):647-658.
[75]潘军宁,左其华,王登婷.港域波浪数学模型的改进与验证[J].海洋工程,2008,26(2)：34-42.
[76]Memos C D. water waves diffracted by two breakwaters [J]. Journal of Hydraulic Research, 1980,18(4):343-357.
[77]Copeland G J M. Practical radiation stress calculations connected with equations of wave propagation [J]. Coastal Engineering,1985,9:195-219.
[78]Lu Y J, Zuo L Q, Shao X J, Wang H C and Li H L. A 2D mathematical model for sediment transport by waves and tidal currents [J]. China Ocean Engineering,2005,19(4), 571-586.
[79]Hamm L. Directional nearshore wave propagation over a rip channel:an experiment [J]. Proceeding of the 23rd International Conference of Coastal Engineering, Venice, Italy, 1992, pp.226-239.
[80]S(?)rensen 0 R, Schaffer H A and Madsen P A. Surf zone dynamics simulated by a Boussinesq type model. III. Wave-induced horizontal nearshore circulations [J]. Coastal Engineering,1998,33:155-176.
[81]S(?)rensen 0 R, Schaffer H A and S(?)rensen L S. Boussinesq-type modelling using an unstructured finite element technique [J]. Coastal Engineering,2004,50:181-198.
[82]Gourlay M R. Wave generated currents [D]. University of Queensland, Australia,1978.
[83]Nielsen C and Apelt C. The application of wave induced forces to a two-dimensional finite element long wave hydrodynamic model [J]. Ocean Engineering,2003,30: 1233-1251.
[2]孙涛,陶建华.波浪作用下近岸区污染物输移扩散的数学模型及其实验验证[J].海洋学报,2003,25(3)：104-112.
[3]唐军.近岸波流场的缓坡模型数值模拟及波流场中污染物输运的研究[D].大连：大连理工大学,2005.
[4]孙涛,韩光,陶建华.波生沿岸流数值模拟研究及其实验验证[J].水利学报,2002,11：1-9.
[5]Berkhoff J C W. Computation of combined refraction-diffraction [A]. Proceeding of the 13th Coastal Engineering Conference[C], Vancouver. ASCE,1972:471-490.
[6]Booij N. A note on the accuracy of the mild-slope equation [J]. Coastal Engineering, 1983,7:513-533.
[7]郑永红.波浪、潮流、污染物迁移扩散数学模型及其数值模拟[D].大连：大连理工大学博士后研究工作报告,2000.
[8]邹志利.水波理论及其应用[M].北京：科学出版社,2005.
[9]Radder A C. On the parabolic equation method for water wave propagation [J]. Journal of Fluid Mechanics,1979,95(1):159-176.
[10]Kirby J T. Higher order approximations in the parabolic equation method for water waves [J]. Journal of Geophysical Research,1986,91(C):933-952.
[11]Ito Y and Tanimoto K. A method of numerical analysis of wave propagation-application to wave diffraction and refraction [C]. Proceeding of the 13th International Conference on Coastal Engineering, Vancouver, ASCE,1972,26:503-522.
[12]Copeland G J M. A practical alternative to the Mild-Slope wave equation [J]. Coastal Engineering,1985,9:125-149.
[13]Madsen P A and Larsen J. An efficient finite-difference approach to the mild-slope equation [J]. Coastal Engineering,1987,11:329-351.
[14]Lee C and Suh K D. Internal generation of waves for time-dependent mild-slope equations [J]. Coastal Engineering,1998,34:35-57.
[15]Lee C, Park W S, Cho Y, et al. Hyperbolic mild-slope equations extended to account for rapidly varying topography [J]. Coastal Engineering,1998,34:243-257.
[16]Oliveira F S B F. Improvement on open boundaries on a time dependent numerical model of wave propagation in the nearshore region [J]. Ocean Engineering 2000,28:95-115.
[17]Suh K D, Lee C Park Y and Lee T H. Experimental verification of horizontal two-dimensional modified mild-slope equation model [J]. Coastal Engineering,2001, 44:1-12.
[18]Zheng Y H, Shen Y M, Qiu D H. New numerical scheme for simulation of hyperbolic mild-slope equation [J]. China Ocean Engineering,2001,15(2):185-194.
[19]赵明,滕斌.利用有限元方法求解双曲型缓坡方程[J].海洋工程,2002,20(3)：54-60.
[20]Zou Z L and Jin H. Hyperbolic mild slope equations with inclusion of amplitude dispersion effect:random waves [J]. China Ocean Engineering,2008,24(4):595-610.
[21]Peregrine D H. Surf zone currents [J]. Theoretical and Computational Fluid Dynamics, 1998,10:295-309.
[22]MacMahan J H, Thornton E B and Reniers A. Rip current review [J]. Coastal Engineering 2006,53:191-208.
[23]邹志利.海岸动力学[M].北京：人民交通出版社,2009.
[24]http://www. gsdkj. net/pro/view. php?id=21206
[25]Longuet-Higgins M S and Stewart R W. Radiation stress in water waves:a physical discussion with application [J]. Deep Sea Research,1964,11(5):529-562.
[26]Bowen A J, Inman D L and Simmons V P. Wave "set-down" and "set-up" [J]. Journal of Geophysical Research 1968,73(8):2569-2577.
[27]Bowen A J. The generation of longshore currents on a plane beach [J]. Journal of Marine Research,1969,27:206-215
[28]Longuet-Higgins M S. Longshore currents generated by obliquely incident sea waves, 1 [J]. Journal of Geophysical Research,1970,75(33):6778-6801.
[29]Iwata N. A note on the wave wet-up, longshore currents and undertows [J]. Journal of the Oceanographical Society of Japan,1970,26(4):233-236.
[30]Mei C C.A note on the averaged momentum balance in two-dimensional water waves [J]. Journal of Marine Research,1973,31 (2):97-104.
[31]Bettess P and Bettess J. A generalization of the radiation stress tensor [J]. Applied Mathematical Modelling,1982,6:145-150.
[32]Svendsen I A, Hass K and Zhao Q. Quasi-3D Nearshore Circulation Model SHORECIRC version 2.4, User Manual. Technical Report, University of Delaware,2002.
[33]Yoon S B, Cho Y, and Lee C. Effects of breaking-induced currents on refraction-diffraction of irregular waves over submerged shoal [J]. Ocean Engineering, 2004,31:633-652.
[34]唐军,沈永明,邱大洪.结合椭圆型缓坡方程模拟近岸波流场[J].海洋学报,2006,28(1)：146-150.
[35]Choi J, Lim C, Jeon Y, and Yoon S B. Numerical simulation of irregular wave transformation due to wave-induced current [J]. Journal of Hydro-environment Research, 2007,1:133-142.
[36]Choi J, Lim C, Lee J, and Yoon S B. Evolution of waves and currents over a submerged laboratory shoal [J]. Coastal Engineering,2009,56:297-312.
[37]魏美芳.非结构网格下椭圆型缓坡方程及近岸波生流数值模拟[D].大连：大连理工大学,2009.
[38]Kirby J T, Wei G, Chen Q, Kennedy A B and Dalrymple R A. Fully nonlinear Boussinesq wave model, User Manual. Technical Report CACR-98-06, University of Delaware,1998.
[39]Chen Q, Kirby J T, Dalrymple R A, Kennedy A B and Chawla A. Boussinesq modeling of wave transformation, breaking, and runup. II:2D [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering,2000,126(1),48-56.
[40]李绍武,黄筱云.用Boussinesq方程计算沿岸流的数值方法[J].天津大学学报,2004,37(12)：1059-1062.
[41]李绍武,连怡贞.完全非线性波浪破碎模型沿岸流数值模拟[J].海洋工程,2007,25(3)：46-52.
[42]卢吉,余锡平.基于Boussinesq方程的近岸波流统一模型[J].水动力学研究与进展,2008,23(3)：314-320.
[43]Kuriyama Y, Itob Y, Yanagishima S. Cross-shore variation of long-term average longshore current velocity in the nearshore zone [J]. Continental Shelf Research, 2008,28:491-502.
[44]Castelle B, Bonneton P, Senechal N, Dupuis H, Butel R and Michel D. Dynamics of wave-induced currents over an alongshore non-uniform multiple-barred sandy beach on the Aquitanian Coast, France [J]. Continental Shelf Research,2006,26:113-131.
[45]Rozynski G and Reeve D. Multi-resolution analysis of nearshore hydrodynamics using discrete wavelet transforms [J]. Coastal Engineering,2005,52:771-792.
[46]邹志利,常梅,邱大洪,王凤龙,董国海.沿岸流的实验研究[J].水动力研究与进展(A),2002,17(2)：174-180.
[47]Shi F Y, Dalrymple R A, Kirby J T, Chen Q and Kennedy A. A fully nonlinear Boussinesq model in generalized curvilinear coordinates [J]. Coastal Engineering,2001,42: 337-358.
[48]Li Y S, Liu S X, Yu Y X and Lai G Z. Numerical modeling of Boussinesq equations by finite element method [J]. Coastal Engineering,1999,37:97-122.
[49]Yuan D K, Lin B L and Xiao H. Parabolic wave propagation modeling in orthogonal coordinate systems [J]. China Ocean Engineering,2004,18(3):445-456.
[50]Shi F Y and Kirby J T. Curvilinear parabolic approximation for surface wave transformation with wave-current interaction [J]. Journal of Computational Physics, 2005,204:562-586.
[51]Zhang H S, Zhu L S and You Y X. A numerical model for wave propagation in curvilinear coordinates [J]. Coastal Engineering,2005,52:513-533.
[52]张洪生,尤云祥,朱良生,张军.曲线坐标系下波浪传播的数学模型及其比较与验证[J].水动力学研究与进展,A辑,2005,20(1)：106-117.
[53]张洪生,洪广文,丁平兴.任意曲线边界条件下缓变水深水域波浪传播的数值模拟[J].海洋学报,2002,24(1)：108-116.
[54]Spaulding M L. A vertically averaged circulation model using boundary-fitted coordinates [J]. Journal of Physical Oceanography,1984,14:973-982.
[55]Kirby J T. Parabolic wave computations in non-orthogonal coordinate systems [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering,1988,114 (6):673-685.
[56]Kirby J T, Dalrymple R A and Kaku H. Parabolic approximations for water waves in conformal coordinate systems [J]. Coastal Engineering,1994,23:185-213.
[57]Shi F, Svendsen I A, Kirby J T and Smith J M. A curvilinear version of a quasi-3D nearshore circulation model [J]. Coastal Engineering,2003,49:99-124.
[58]Tang J, Shen Y M, Zheng Y H et al. An efficient and flexile computational model for solving the mild-slope equation [J]. Coastal Engineering,2004(2),51:143-154.
[59]柳淑学,孙冰.考虑边界波浪方向的缓坡方程自适应求解模型[J].海洋工程,2007,25(1)：35-56.
[60]魏美芳,唐军,沈永明.非结构化网格下椭圆型缓坡方程的数值求解[J].海洋学报,2009,31(2)：159-164.
[61]Dally W R, Dean R G and Dalrymple R A. Wave height verification across beaches of arbitrary profile [J]. Journal of Geophysical Research,1985,90(C6):11917-11927.
[62]Thompson J F, Warsi Z U A and Mastin C W. Numerical Grid Generation:Foundations and Applications [M]. North Holland,1985.
[63]Thompson J F, Thames F C, and Mastin C W. Automatic numerical generation of body-fitted curvilinear coordinate system for field containing any number of arbitrary two-dimensional bodies [J]. Journal of Computational Physics,1974,15(3):299-319.
[64]Thompson J F, Warsi Z U A and Mastin C W. Body-fitted coordinate system for numerical solution of partial differential equations [J]. Journal of Computational Physics, 1982,47(1):1-108.
[65]魏文礼,张宗孝,刘玉玲,刘明琴.采用Poisson方程生成曲线网格时源项P、Q选择的研究[J].应用力学学报,2000,17(3)：87-91.
[66]孙建国,蒋丽丽.用于起伏地表条件下地球物理场数值模拟的正交曲网格生成技术[J].石油地球物理勘探,2009,44(4)：494-500.
[67]http://coastal. udel. edu/-fyshi/coastgrid/coastgrid. html
[68]http://en. wikipedia. org/wiki/Alternating_direction_implicit_method
[69]王永学.有限差分法讲义.大连理工大学土木水利学院内部讲义,2008.
[70]Berkhoff J C W, Booij N and Radder A C. Verification of numerical wave propagation models for simple harmonic linear waves [J]. Coastal Engineering,1982,6(3):255-279.
[71]Beji S and Nadaoka K. Fully dispersive nonlinear water wave model in curvilinear coordinates [J]. Journal of Computational Physics,2004,198:645-658.
[72]Dalrymple R A, Kirby J T and Martin P A. Spectral methods for forward-propagating water waves in conformally-mapped channels [J]. Applied Ocean Research,1994,16: 249-266.
[73]Isobe M. A parabolic refraction-diffraction equation in the ray-front coordinate system, in:Proceedings of the 20th International Conference on Coastal Engineering, Taipei,1986, pp.306-317.
[74]Pan J N, Hong G W, Zuo Q H and Wang D T. Verification of a numerical harbor Wave Model. China Ocean Engineering [J].2007,21(4):647-658.
[75]潘军宁,左其华,王登婷.港域波浪数学模型的改进与验证[J].海洋工程,2008,26(2)：34-42.
[76]Memos C D. water waves diffracted by two breakwaters [J]. Journal of Hydraulic Research, 1980,18(4):343-357.
[77]Copeland G J M. Practical radiation stress calculations connected with equations of wave propagation [J]. Coastal Engineering,1985,9:195-219.
[78]Lu Y J, Zuo L Q, Shao X J, Wang H C and Li H L. A 2D mathematical model for sediment transport by waves and tidal currents [J]. China Ocean Engineering,2005,19(4), 571-586.
[79]Hamm L. Directional nearshore wave propagation over a rip channel:an experiment [J]. Proceeding of the 23rd International Conference of Coastal Engineering, Venice, Italy, 1992, pp.226-239.
[80]S(?)rensen 0 R, Schaffer H A and Madsen P A. Surf zone dynamics simulated by a Boussinesq type model. III. Wave-induced horizontal nearshore circulations [J]. Coastal Engineering,1998,33:155-176.
[81]S(?)rensen 0 R, Schaffer H A and S(?)rensen L S. Boussinesq-type modelling using an unstructured finite element technique [J]. Coastal Engineering,2004,50:181-198.
[82]Gourlay M R. Wave generated currents [D]. University of Queensland, Australia,1978.
[83]Nielsen C and Apelt C. The application of wave induced forces to a two-dimensional finite element long wave hydrodynamic model [J]. Ocean Engineering,2003,30: 1233-1251.