Boussinesq方程模型的数值造波方法研究
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摘要
Boussinesq方程是用于描述非线性色散波在浅水中传播的方程,以Boussinesq方程为基础的数学模型可以模拟近岸海域复杂地形上波浪传播的非线性变形,如折射、绕射、反射、浅化等现象。数值造波方法是该数学模型的一个关键技术,研究数值造波方法和无反射边界处理具有重要的理论意义和实用价值。
本文综述了各类Boussinesq方程在色散性、非线性、浅化性等方面的改进,以及数值造波方法和无反射边界处理的研究进展。为方便研究数学模型和数值造波方法,建立了Madsen和S?rensen(1992)形式的Boussinesq方程为基础的一维数学模型。模型中采用了高精度紧致差分格式离散方程,使用了“窄缝法”处理动边界,并引入了能量耗散项处理波浪破碎问题。
数值造波方法分别采用了入射边界上造波和域内源项造波,开边界则采用吸收边界和辐射边界相结合的方法进行数值消波处理。入射边界上造波建立了线性造波机、椭圆余弦造波机和能处理造波板二次反射的可吸收式造波机;域内源项造波研究了线源造波和区域源造波方法,由理论推导证明了域内造波源函数中的波速应使用能量速度,即群速度。为了分析和比较不同的数值造波方法,分别对小振幅波、孤立波和椭圆余弦波进行了模拟,数值试验结果与理论解析解符合良好,边界处理效果也令人满意。
Boussinesq equation is a nonlinear dispersion waves equation in shallow water. The numerical models based on Boussinesq equation could simulate nonlinear wave deformations on uneven bottom, such as refraction, diffraction, reflection, shoaling etc. The numerical wave generation method is a key technology of the models. It is also significant to the study of wave generation and absorbing boundary.
In this paper, the improvement of the dispersive property, dissipative property, shoaling for kinds of Boussinesq equations, and the study of the numerical wave generation and absorbing boundary are reviewed. In order to investigate the model and the numerical wave generation method further, 1D models based on the extended Boussinesq equation of Madsen & S?rensen(1992) are set up. The compact difference scheme is applied in Boussinesq equation. The slot method is used to treat the moving boundary. A simple eddy viscosity-type model is added for wave breaking model.
Numerical wave paddle and source function method are used to generate waves. The open boundary, which combines the sponge layer and radiation boundary, is set to permit incident waves to be transmitted freely. Linear wave maker, cnoidal wave maker and absorbing wave maker are built up. Theoretical derivations of the line and distributed source function methods show that the energy transport which equals the group velocity should be used in the model. The numerical model is applied to simulate sinusoidal, solitary and cnoidal waves. The numerical results agree well with the analytical solutions and experimental data. The sponge layer for open boundary is good.
本文综述了各类Boussinesq方程在色散性、非线性、浅化性等方面的改进,以及数值造波方法和无反射边界处理的研究进展。为方便研究数学模型和数值造波方法,建立了Madsen和S?rensen(1992)形式的Boussinesq方程为基础的一维数学模型。模型中采用了高精度紧致差分格式离散方程,使用了“窄缝法”处理动边界,并引入了能量耗散项处理波浪破碎问题。
数值造波方法分别采用了入射边界上造波和域内源项造波,开边界则采用吸收边界和辐射边界相结合的方法进行数值消波处理。入射边界上造波建立了线性造波机、椭圆余弦造波机和能处理造波板二次反射的可吸收式造波机;域内源项造波研究了线源造波和区域源造波方法,由理论推导证明了域内造波源函数中的波速应使用能量速度,即群速度。为了分析和比较不同的数值造波方法,分别对小振幅波、孤立波和椭圆余弦波进行了模拟,数值试验结果与理论解析解符合良好,边界处理效果也令人满意。
Boussinesq equation is a nonlinear dispersion waves equation in shallow water. The numerical models based on Boussinesq equation could simulate nonlinear wave deformations on uneven bottom, such as refraction, diffraction, reflection, shoaling etc. The numerical wave generation method is a key technology of the models. It is also significant to the study of wave generation and absorbing boundary.
In this paper, the improvement of the dispersive property, dissipative property, shoaling for kinds of Boussinesq equations, and the study of the numerical wave generation and absorbing boundary are reviewed. In order to investigate the model and the numerical wave generation method further, 1D models based on the extended Boussinesq equation of Madsen & S?rensen(1992) are set up. The compact difference scheme is applied in Boussinesq equation. The slot method is used to treat the moving boundary. A simple eddy viscosity-type model is added for wave breaking model.
Numerical wave paddle and source function method are used to generate waves. The open boundary, which combines the sponge layer and radiation boundary, is set to permit incident waves to be transmitted freely. Linear wave maker, cnoidal wave maker and absorbing wave maker are built up. Theoretical derivations of the line and distributed source function methods show that the energy transport which equals the group velocity should be used in the model. The numerical model is applied to simulate sinusoidal, solitary and cnoidal waves. The numerical results agree well with the analytical solutions and experimental data. The sponge layer for open boundary is good.
引文
[1] Peregrine, D.H., Long waves on a beach [J], Fluid Mech., 1967, 27: 815~827
[2] Witting, J.M., A numerical model for the evolution of non-linear water waves [J], Comput Phys, 1984, 56: 203~236
[3] Madsen, P.A., Murray, R., S(?)rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Engineering, 1991, 15: 371~388
[4] Madsen, P.A., S(?)rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry, Coastal Engineering, 1992, 18: 183~204
[5] Nwogu, O., Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, 1993, Vol. 119, No. 6: 618~638
[6] Schaffer, H.A., Madsen, P.A., Further enhancements of Boussinesq-type equations, Coastal Engineering, 1995, 26: 1~14
[7] Gobbi, M.F., Kirby, J.T., A fourth order Boussinesq-type wave model, Coastal Engineering, 1996, Chapter 87: 1116~1129
[8] Wei, G., Kirby, J.T., Grilli, S.T., et al., A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves [J]. Fluid Mech., 1995, Vol. 294: 71~92
[9] Gobbi, M.F., Kirby, J.T., et al., A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)~4, J. Fluid Mech., 2000, Vol. 405: 181~210
[10] Madsen, P.A., Banijamali, B., Schaffer, H.A., et al., Boussinesq type equations with high accuracy in dispersion and nonlinear, Coastal Engineering, 1996, 8: 95~109
[11] Madsen, P.A., Binggham, H.B., Liu, H., A new Boussinesq method for fully nonlinear waves from shallow to deep water, J. Fluid Mech., 2002, 462: 1~30
[12] Biesel, F., Suquet, F., Les Appareils Generateurs de houle en laboratoire, Lahouille blanche, 1951, 6, No. 2, 4, 5, 7, No. 6
[13] Biesel, F. Suquet, F., Laboratory wave-generating apparatus, St. Anthony falls hydraulic laboratory Proj. Rep, 1954, No. 29
[14] Fontanet P., Theorie de la generation de la houle cylindrique par un batteur plan, La Huille Blanche, 1961, 16: 3~31 (part 1) , 174~196 (part 2)
[15] Madsen, 0.S., On the generation of long waves [J]. Geophys. Res., 1971, 76: 8672~8673
[16] Hudspeth, R., Sulisz, W., Stokes drift in two dimensional wave flumes [J]. Fluid Mech., 1991, 230: 209~139
[17] Sulisz, W., Hudspeth, R., Complete second-order solution for water waves generated in wave lfumes [J]. Fluids and Strudctures, 1993, 7: 253~268
[18] Schaffer, H.A., Laboratory wave generation correct to second order, Wave kinematics and environmental forces, London, March 1993
[19] Schaffer, H.A., second-order wavemaker theory for irregular wave, Ocean Engineering, 1996, Vol. 23, No.1: 47~88
[20] Williams, J.M., Limiting gravity waves in water of finite depth, Phil. Trans. Roy. Soc. Lond. A, 1981, 302: 139~188
[21] Hammack, J.L., Segur, H., The KdV equation and water waves. Part 2. Comparison with experiment, Journal of Fluid mechanics, 1974, Vol. 65: 289~314
[22] Goring D.G., Tsunamis-the propagation of long waves onto a shelf. Doctoral Dissertation, Report. W.M. Keck laboratory of hydraulics and water resources. California, 1979: 9~45
[23] Goring D.G., Raichlen, F., The generation of long waves in the laboratory, Proceedings of the 17th Intenrational Conference on Coastal Engineering, ASCE, 1980, Vol. 1: 763~783
[24] Madsen, O.S., A three-dimensional wavemaker, its theory and application [J]. Hydralulics Res., 1974, Vol. 12, No.2
[25] Gilbert, G., Generation of oblique waves, Technical report 18, Hydraulic Research Station, Wallingford, 1976
[26] Takayama, T., Theory of oblique waves generated by serpent-type wavemaker, Coastal Engineering in Japan, 1984, 27: 1~19
[27] Takayama, T., Hiraishi, T., Fundamental characteristics of oblique regular waves and diretional random waves generated by a serpent-type wave generator, Report of Port and Harbor Research Institute, Japan, 1987, 26(5): 101~136
[28] Dalrymple, R.A., Greenberg, M., Directional wave makers. In: Dalrymple, R.A. (Ed.), Physical Modelling in Coastal Engineering. A.A. Balkema, Rotterdam, 1985: 67~81
[29] Suh, K., Dalrymple, R., Directional wavemaker theory: a special approach, Proceedings of 22th IAHR conference, Lausanne, Switzerland, 1987: 389~395
[30] Steenberg, C.M., Schaffer, H.A., second-order wave generation in laboratory basins, Proceedings of the 27th Intenrational Conference on Coastal Engineering, Sidney, Australia, July, 2000
[31] Schaffer, H.A., Steenberg, C.M., second-order wavemaker theory for multi directional waves, Ocean Engineering, 2003, 30: 1203~1231
[32] 王永学,无反射造波数值波浪水槽,水动力学研究与进展 A 辑,1994(2),205~214
[33] Larsen, J., Dancy, H., Open boundaries in short wave simulations – a new approach, Coast. Eng., 1983, 7: 285~297
[34] Brorsen M., Larsen J., Source generation of nonlinear gravity waves with the boundary intergral equation method, Costal Engineering, 1987, 11: 93~113
[35] Wei G., Kirby J. T., Sinha A., Generation waves in Boussinesq models using a source function method, Coastal Eng., 1999, 36: 271~299
[36] 钦文婷,Boussinesq 方程数学模型的改进及其工程应用: [硕士学位论文],天津;天津大学,2003
[37] 高学平,曾广冬等,不规则波浪数值水槽的造波和阻尼消波,海洋学报,2002,24(2): 127~132
[38] KOJI, Kawasaki, Numerical simulation of breaking and post-breaking wave deformation process around a submerged breakwater, Coastal Engineering, 1999, 41: 201~223
[39] 刘长根,用 URANS 方程模型模拟非线性水波与海工结构物的相互作用,[博士学位论文],天津;天津大学,2003
[40] Wenting Qin, High accuracy numerical scheme of Boussinesq equation for a shallow water wave model, XXIX IAHR Congress Proceedings, John F. KennedyStudent Paper Competition, 2001: 248~256
[41] 陶建华,波浪在岸滩上的爬高和破碎的数学模拟,海洋学报,1984, 6(5):692~700
[42] Madsen, P.A., S(?)rensen, O.R., Schaffer, H.A., Surf zone dynamics simulated by a Boussinesq type model. Part Ⅱ: surf beat and swash oscillations for wave groups and irregular waves, Coastal Engineering, 1997, 32: 289~319
[43] Kennedy, A.B., Chen, Q., Kirby, J.T., et al., Boussinesq modeling of wave transformation breaking, and runup. Ⅰ: 1D, Journal of Waterway, Port, Coastal and Ocean Engineering, 2000, 126(1): 39~47
[44] Lee, C., Cho, Y.-S., Yum, K., Internal generation of waves for extended Boussinesq equations. Coast. Eng., 2001, 42: 155~162
[45] Schaffer, H.A., S(?)rensen, O.R., On the internal wave generation in Boussinesq and mild-slope equations. Coast. Eng., 2006, 53: 319~323
[46] Laitone, E.V., The second approximation to Cnoidal and Solitary waves, J. Fluid Mech., 1961, 9: 430~444
[47] 陶建华,水波的数值模拟,天津:天津大学出版社,2005
[2] Witting, J.M., A numerical model for the evolution of non-linear water waves [J], Comput Phys, 1984, 56: 203~236
[3] Madsen, P.A., Murray, R., S(?)rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Engineering, 1991, 15: 371~388
[4] Madsen, P.A., S(?)rensen, O.R., A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry, Coastal Engineering, 1992, 18: 183~204
[5] Nwogu, O., Alternative form of Boussinesq equations for nearshore wave propagation, Journal of Waterway, Port, Coastal and Ocean Engineering, 1993, Vol. 119, No. 6: 618~638
[6] Schaffer, H.A., Madsen, P.A., Further enhancements of Boussinesq-type equations, Coastal Engineering, 1995, 26: 1~14
[7] Gobbi, M.F., Kirby, J.T., A fourth order Boussinesq-type wave model, Coastal Engineering, 1996, Chapter 87: 1116~1129
[8] Wei, G., Kirby, J.T., Grilli, S.T., et al., A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves [J]. Fluid Mech., 1995, Vol. 294: 71~92
[9] Gobbi, M.F., Kirby, J.T., et al., A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)~4, J. Fluid Mech., 2000, Vol. 405: 181~210
[10] Madsen, P.A., Banijamali, B., Schaffer, H.A., et al., Boussinesq type equations with high accuracy in dispersion and nonlinear, Coastal Engineering, 1996, 8: 95~109
[11] Madsen, P.A., Binggham, H.B., Liu, H., A new Boussinesq method for fully nonlinear waves from shallow to deep water, J. Fluid Mech., 2002, 462: 1~30
[12] Biesel, F., Suquet, F., Les Appareils Generateurs de houle en laboratoire, Lahouille blanche, 1951, 6, No. 2, 4, 5, 7, No. 6
[13] Biesel, F. Suquet, F., Laboratory wave-generating apparatus, St. Anthony falls hydraulic laboratory Proj. Rep, 1954, No. 29
[14] Fontanet P., Theorie de la generation de la houle cylindrique par un batteur plan, La Huille Blanche, 1961, 16: 3~31 (part 1) , 174~196 (part 2)
[15] Madsen, 0.S., On the generation of long waves [J]. Geophys. Res., 1971, 76: 8672~8673
[16] Hudspeth, R., Sulisz, W., Stokes drift in two dimensional wave flumes [J]. Fluid Mech., 1991, 230: 209~139
[17] Sulisz, W., Hudspeth, R., Complete second-order solution for water waves generated in wave lfumes [J]. Fluids and Strudctures, 1993, 7: 253~268
[18] Schaffer, H.A., Laboratory wave generation correct to second order, Wave kinematics and environmental forces, London, March 1993
[19] Schaffer, H.A., second-order wavemaker theory for irregular wave, Ocean Engineering, 1996, Vol. 23, No.1: 47~88
[20] Williams, J.M., Limiting gravity waves in water of finite depth, Phil. Trans. Roy. Soc. Lond. A, 1981, 302: 139~188
[21] Hammack, J.L., Segur, H., The KdV equation and water waves. Part 2. Comparison with experiment, Journal of Fluid mechanics, 1974, Vol. 65: 289~314
[22] Goring D.G., Tsunamis-the propagation of long waves onto a shelf. Doctoral Dissertation, Report. W.M. Keck laboratory of hydraulics and water resources. California, 1979: 9~45
[23] Goring D.G., Raichlen, F., The generation of long waves in the laboratory, Proceedings of the 17th Intenrational Conference on Coastal Engineering, ASCE, 1980, Vol. 1: 763~783
[24] Madsen, O.S., A three-dimensional wavemaker, its theory and application [J]. Hydralulics Res., 1974, Vol. 12, No.2
[25] Gilbert, G., Generation of oblique waves, Technical report 18, Hydraulic Research Station, Wallingford, 1976
[26] Takayama, T., Theory of oblique waves generated by serpent-type wavemaker, Coastal Engineering in Japan, 1984, 27: 1~19
[27] Takayama, T., Hiraishi, T., Fundamental characteristics of oblique regular waves and diretional random waves generated by a serpent-type wave generator, Report of Port and Harbor Research Institute, Japan, 1987, 26(5): 101~136
[28] Dalrymple, R.A., Greenberg, M., Directional wave makers. In: Dalrymple, R.A. (Ed.), Physical Modelling in Coastal Engineering. A.A. Balkema, Rotterdam, 1985: 67~81
[29] Suh, K., Dalrymple, R., Directional wavemaker theory: a special approach, Proceedings of 22th IAHR conference, Lausanne, Switzerland, 1987: 389~395
[30] Steenberg, C.M., Schaffer, H.A., second-order wave generation in laboratory basins, Proceedings of the 27th Intenrational Conference on Coastal Engineering, Sidney, Australia, July, 2000
[31] Schaffer, H.A., Steenberg, C.M., second-order wavemaker theory for multi directional waves, Ocean Engineering, 2003, 30: 1203~1231
[32] 王永学,无反射造波数值波浪水槽,水动力学研究与进展 A 辑,1994(2),205~214
[33] Larsen, J., Dancy, H., Open boundaries in short wave simulations – a new approach, Coast. Eng., 1983, 7: 285~297
[34] Brorsen M., Larsen J., Source generation of nonlinear gravity waves with the boundary intergral equation method, Costal Engineering, 1987, 11: 93~113
[35] Wei G., Kirby J. T., Sinha A., Generation waves in Boussinesq models using a source function method, Coastal Eng., 1999, 36: 271~299
[36] 钦文婷,Boussinesq 方程数学模型的改进及其工程应用: [硕士学位论文],天津;天津大学,2003
[37] 高学平,曾广冬等,不规则波浪数值水槽的造波和阻尼消波,海洋学报,2002,24(2): 127~132
[38] KOJI, Kawasaki, Numerical simulation of breaking and post-breaking wave deformation process around a submerged breakwater, Coastal Engineering, 1999, 41: 201~223
[39] 刘长根,用 URANS 方程模型模拟非线性水波与海工结构物的相互作用,[博士学位论文],天津;天津大学,2003
[40] Wenting Qin, High accuracy numerical scheme of Boussinesq equation for a shallow water wave model, XXIX IAHR Congress Proceedings, John F. KennedyStudent Paper Competition, 2001: 248~256
[41] 陶建华,波浪在岸滩上的爬高和破碎的数学模拟,海洋学报,1984, 6(5):692~700
[42] Madsen, P.A., S(?)rensen, O.R., Schaffer, H.A., Surf zone dynamics simulated by a Boussinesq type model. Part Ⅱ: surf beat and swash oscillations for wave groups and irregular waves, Coastal Engineering, 1997, 32: 289~319
[43] Kennedy, A.B., Chen, Q., Kirby, J.T., et al., Boussinesq modeling of wave transformation breaking, and runup. Ⅰ: 1D, Journal of Waterway, Port, Coastal and Ocean Engineering, 2000, 126(1): 39~47
[44] Lee, C., Cho, Y.-S., Yum, K., Internal generation of waves for extended Boussinesq equations. Coast. Eng., 2001, 42: 155~162
[45] Schaffer, H.A., S(?)rensen, O.R., On the internal wave generation in Boussinesq and mild-slope equations. Coast. Eng., 2006, 53: 319~323
[46] Laitone, E.V., The second approximation to Cnoidal and Solitary waves, J. Fluid Mech., 1961, 9: 430~444
[47] 陶建华,水波的数值模拟,天津:天津大学出版社,2005