非线性波浪作用下污染物输移扩散研究和波浪传播的抛物缓坡方程模型
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摘要
本文讨论了二阶 Stokes 非线性波浪作用下流体微团的平动和变形对污染物
输移及混合扩散的影响，推导了相关公式，得出了波浪场作用下污染物输移及混
合扩散的基本规律，为进一步研究波浪场中的纵向离散系数提供了一定的理论依
据。建立了基于有限元方法（FEM）的对流扩散数学模型，对理论推导的结果进
行了验证。研究结果表明：在非线性波浪场中，污染物的混合扩散除了受漂移速
度影响外，具有明显各向异性的特征。
本文在高阶抛物缓坡方程的基础上，综合考虑了波浪长距离传播的底摩阻问
题和风的影响，还考虑了近岸区波浪多次破碎问题以及弱非线性项对计算结果的
影响，以扩大抛物缓坡方程在解决实际工程问题中的适用范围。此模型适用于没
有建筑物的自然地形条件下的大面积波浪场的折、绕射推算。
本文建立了一个基于普通贴体正交网格系统的抛物型缓坡方程模型，并给出
了其在保角坐标系下的特殊形式，利用该模型研究了扩张防波堤和环形渠道内波
浪的传播这两个典型算例。对每一个算例，首先采用解析的方法构造了普通正交
网格系统和正交保角网格系统进行计算。计算结果表明，对相同的算例，在正交
保角网格系统下得到的结果明显优于普通正交网格下得到的结果。为增强模型的
实用性，本文将所建立的模型与一种保角网格系统的数值生成方法结合起来，并
在数值网格系统上对上面的两个算例进行了重新计算。计算结果表明，将该数值
网格生成技术形成的网格系统与本文所建立的缓坡方程模型相结合，能得到可靠
的计算结果。
用抛物缓坡方程模型进行了工程实例计算。模拟了东海大桥工程区域的波高
分布，从而得到建筑物设计波高。
In this paper the transport and diffusion of contaminants in nonlinear Stokes
water wave due to shift and rotation of the micro fluid mass is discussed, and some
characters of the transport and diffusion of contaminants in wave field have been
obtained. A numerical transport-diffuse model based on Finite Element Method (FEM)
is established to verify the theoretical results, and good agreements are obtained. The
results show that in wave filed the transport and diffusion of contaminants is
anisotropic, apart from being effected by drift velocity.
Based on the extended parabolic mild-slope equation, the bottom friction and
wind effect on the wave propagating through long distance are considered. This
numerical method can calculate the wave field after wave breaking more than once.
The equation includes the weak nonlinear term and high order differential term that
can forecast wave height distribution more accurately. In this numerical model, the
application of the parabolic mild-slope equation is expanded in solving the practical
engineering problem. This refined model is suitable for calculating the wave
refraction and diffraction in large water area without structures.
In this paper a numerical model for simulating propagation and transform of
water waves in coastal area is set up by using parabolic mild-slope equation in
orthogonal coordinates, and the particular format of this equation in conformal system
is also set up. Two typical examples namely expanded breakwaters and circular
channel are studied to validate the model. At first the examples were studied by using
the common orthogonal coordinates. Then the same examples were computed by
using the conformal system. The computational results show that much better result
can be obtained by using the conformal coordinates than that by using the common
orthogonal system. A numerical technique for generating conformal grid is combined
with the numerical model to improve the practicability of the model. And this
combination has a good result.
The applications of model of parabolic mild-slope equation in practical
engineering projects are carried out. The model is used in the Project of the Donghai
Bridge to calculate the wave distribution in the project area.
输移及混合扩散的影响，推导了相关公式，得出了波浪场作用下污染物输移及混
合扩散的基本规律，为进一步研究波浪场中的纵向离散系数提供了一定的理论依
据。建立了基于有限元方法（FEM）的对流扩散数学模型，对理论推导的结果进
行了验证。研究结果表明：在非线性波浪场中，污染物的混合扩散除了受漂移速
度影响外，具有明显各向异性的特征。
本文在高阶抛物缓坡方程的基础上，综合考虑了波浪长距离传播的底摩阻问
题和风的影响，还考虑了近岸区波浪多次破碎问题以及弱非线性项对计算结果的
影响，以扩大抛物缓坡方程在解决实际工程问题中的适用范围。此模型适用于没
有建筑物的自然地形条件下的大面积波浪场的折、绕射推算。
本文建立了一个基于普通贴体正交网格系统的抛物型缓坡方程模型，并给出
了其在保角坐标系下的特殊形式，利用该模型研究了扩张防波堤和环形渠道内波
浪的传播这两个典型算例。对每一个算例，首先采用解析的方法构造了普通正交
网格系统和正交保角网格系统进行计算。计算结果表明，对相同的算例，在正交
保角网格系统下得到的结果明显优于普通正交网格下得到的结果。为增强模型的
实用性，本文将所建立的模型与一种保角网格系统的数值生成方法结合起来，并
在数值网格系统上对上面的两个算例进行了重新计算。计算结果表明，将该数值
网格生成技术形成的网格系统与本文所建立的缓坡方程模型相结合，能得到可靠
的计算结果。
用抛物缓坡方程模型进行了工程实例计算。模拟了东海大桥工程区域的波高
分布，从而得到建筑物设计波高。
In this paper the transport and diffusion of contaminants in nonlinear Stokes
water wave due to shift and rotation of the micro fluid mass is discussed, and some
characters of the transport and diffusion of contaminants in wave field have been
obtained. A numerical transport-diffuse model based on Finite Element Method (FEM)
is established to verify the theoretical results, and good agreements are obtained. The
results show that in wave filed the transport and diffusion of contaminants is
anisotropic, apart from being effected by drift velocity.
Based on the extended parabolic mild-slope equation, the bottom friction and
wind effect on the wave propagating through long distance are considered. This
numerical method can calculate the wave field after wave breaking more than once.
The equation includes the weak nonlinear term and high order differential term that
can forecast wave height distribution more accurately. In this numerical model, the
application of the parabolic mild-slope equation is expanded in solving the practical
engineering problem. This refined model is suitable for calculating the wave
refraction and diffraction in large water area without structures.
In this paper a numerical model for simulating propagation and transform of
water waves in coastal area is set up by using parabolic mild-slope equation in
orthogonal coordinates, and the particular format of this equation in conformal system
is also set up. Two typical examples namely expanded breakwaters and circular
channel are studied to validate the model. At first the examples were studied by using
the common orthogonal coordinates. Then the same examples were computed by
using the conformal system. The computational results show that much better result
can be obtained by using the conformal coordinates than that by using the common
orthogonal system. A numerical technique for generating conformal grid is combined
with the numerical model to improve the practicability of the model. And this
combination has a good result.
The applications of model of parabolic mild-slope equation in practical
engineering projects are carried out. The model is used in the Project of the Donghai
Bridge to calculate the wave distribution in the project area.
引文
[1] Berkhoff, J.C.W., 1972, Computation of combined refraction diffraction,
Proceedings of the 13th International Coastal Engineering Conference, ASCE,
New York, 471-490
[2] Booij, N., 1983, A note on the accuracy of the mild slope equation. Coastal
Engineering, 7, 191-203
[3] Radder, A. C., 1979, On the parabolic equation method for water wave
propagation. Journal of Fluid Mechanics, 95(1), 159-176
[4] Yue&Mei, 1980, Forward diffraction of Stokes waves by a thin wedge, J.Fluid
Mech.,99(1), 33-52
[5] Kirby, J. T. and Dalrymple, R. A., 1983. A parabolic equation for the
combined refraction-diffraction of Stokes waves by mildly varying topography.
Journal of Fluid Mechanics, 136, 453-466.
[6] Kirby, J. T. and Dalrymple, R. A., 1984. Verification of a parabolic equation
for propagation of weakly nonlinear waves. Coastal Engineering, vol.8,
219-232.
[7] Liu, P. L.F. and Tsay, T.K., 1984, Refraction-diffraction model for weakly
nonlinear water waves. Journal of Fluid Mechanics, Vol. 141, 265-274.
[8] Kirby, J. T., 1986, Higher-order approximations in the parabolic equation for
water waves. Journal of Geophysical Research, 91(C1), 933-952.
[9] Dalrymple, R. A. Kirby, J.T., and Paul A.H., 1984, Wave diffraction due to
areas of energy dissipation. Journal of Waterway, Port, Coastal and Ocean
Engineering, 110(1), 67-78.
[10] Watanabe, A. and Maruyama, K., 1986. Numerical modeling of nearshore
wave field under combined refraction, diffraction and breaking. Coastal
Engineering in Japan, Vol. 29, 19-39.
[11] Gao, Q. and Radder, A.C., 1998. A refraction-diffraction model for irregular
waves. Proc.26th Coastal Eng. Confer. ASCE, Copenhagen, 367-379.
[12]Liu, P.L.-F., Boissevain, P.L., 1988. Wave propagation between two breakwaters.
57
参考文献
J. of Waterway, Port, Coastal, and Ocean Eng., 114(2): 237-247
[13] Kirby, J.T., 1988. Parabolic wave computations in non-orthogonal coordinate
systems. J. of Waterway, Port, Coastal, and Ocean Eng., 114(6): 673-685.
[14] Kirby, J.T., Dalrymple, R.A. & Kaku, H., 1994. Parabolic approximations for
water waves in conformal coordinate systems. Coastal Eng., 23: 185-213.
[15] Smith R., Scott C.F.. Mixing in the tidal environment. Journal of Hydraulic
Engineering, 1997, 123: 332-340.
[16] Tao J.H., Han G.. Effects of water wave motion on pollutant transport in shallow
coastal water. Science in China (Series E), 2002, 45: 593-605.
[17] Pearson J.M., Guymer I., et al. Effect of wave height on cross-shore solute
mixing. Journal of Waterway, Port, Coastal and Ocean Engineering, 2002, 128:
10-20.
[18] Fisher H.B., List E.J., et al. Mixing in Inland and Coastal Waters. London:
Academic Press, 1979.
[19] Giarrusso C.C., Carratelli E.P. & Spulsi G... On the effects of wave drift on the
dispersion of floating pollutants. Ocean Engineering, 2001, 28: 1339-1348.
[20] Law A.W.K.. Taylor dispersion of contaminants due to surface waves. Journal of
Hydraulic Research, 2000, 38: 41-48.
[21] 苏翼林, 材料力学［M］, 北京：高等教育出版社, 1987.
[22] J.T.Kirby, 1986, Rational approximation in the parabolic equation method for
water waves, Coastal Engineering, 10, 355-378
[23] Kirby, J. T.，Dalrymple, R. A.，A parabolic equation for the combined
refraction-diffraction of Stokes waves by mildly varying topography，Journal of
Fluid Mechanics，1983，136, 453～466
[24] Dalrymple, R. A.，Kirby, J. T.，1988,Models for very wide-angle water waves and
wave diffraction, J. Fluid Mech., 192,33～50
[25] Berkhoff, J.C.W., Booij, N., and Radder, A.C., 1982. Verification of numerical
wave propagation models for simple harmonic water waves. Coastal Engineering,
Vol.6, 255-279.
58
参考文献
[26] Dally, W.R., Dean, R.G. and Dalrymple, R.A., 1985,Wave height variation across
beaches of arbitrary profile. Journal of Geophysical Research, 90 (C6),
11917-11927.
[27] Watanabe, A. and Maruyama, K., 1988. A numerical model of wave deformation
in surf zone.Proc. 21st Coastal Eng.Conf. ASCE, Malaga, 578-587.
[28] Johnson, H.K. and Poulin, S. 1998. On the accuracy of parabolic wave models.
Proc.26th Coastal Eng. Confer. ASCE, Copenhagen, 352-365.
[29] Vogel, J.A., Radder, A.C. and de Reus, J.H. 1988. Verification of numerical wave
propagation models in tidal inlets. Proc. 21st Coastal Eng.Conf. ASCE, Malaga,
433-447.
[30] Gao, Q. and Radder, A.C., 1998. A refraction-diffraction model for irregular
waves. Proc.26th Coastal Eng. Confer. ASCE, Copenhagen, 367-379.
[31] Lin, B.L., Chandler-Wilde S.N., 1996. A depth-integrated 2D coastal and
estuarine model with conformal boundary-fitted mesh generation. International
journal for numerical methods in fluids, 23: 819-846.
59
Proceedings of the 13th International Coastal Engineering Conference, ASCE,
New York, 471-490
[2] Booij, N., 1983, A note on the accuracy of the mild slope equation. Coastal
Engineering, 7, 191-203
[3] Radder, A. C., 1979, On the parabolic equation method for water wave
propagation. Journal of Fluid Mechanics, 95(1), 159-176
[4] Yue&Mei, 1980, Forward diffraction of Stokes waves by a thin wedge, J.Fluid
Mech.,99(1), 33-52
[5] Kirby, J. T. and Dalrymple, R. A., 1983. A parabolic equation for the
combined refraction-diffraction of Stokes waves by mildly varying topography.
Journal of Fluid Mechanics, 136, 453-466.
[6] Kirby, J. T. and Dalrymple, R. A., 1984. Verification of a parabolic equation
for propagation of weakly nonlinear waves. Coastal Engineering, vol.8,
219-232.
[7] Liu, P. L.F. and Tsay, T.K., 1984, Refraction-diffraction model for weakly
nonlinear water waves. Journal of Fluid Mechanics, Vol. 141, 265-274.
[8] Kirby, J. T., 1986, Higher-order approximations in the parabolic equation for
water waves. Journal of Geophysical Research, 91(C1), 933-952.
[9] Dalrymple, R. A. Kirby, J.T., and Paul A.H., 1984, Wave diffraction due to
areas of energy dissipation. Journal of Waterway, Port, Coastal and Ocean
Engineering, 110(1), 67-78.
[10] Watanabe, A. and Maruyama, K., 1986. Numerical modeling of nearshore
wave field under combined refraction, diffraction and breaking. Coastal
Engineering in Japan, Vol. 29, 19-39.
[11] Gao, Q. and Radder, A.C., 1998. A refraction-diffraction model for irregular
waves. Proc.26th Coastal Eng. Confer. ASCE, Copenhagen, 367-379.
[12]Liu, P.L.-F., Boissevain, P.L., 1988. Wave propagation between two breakwaters.
57
参考文献
J. of Waterway, Port, Coastal, and Ocean Eng., 114(2): 237-247
[13] Kirby, J.T., 1988. Parabolic wave computations in non-orthogonal coordinate
systems. J. of Waterway, Port, Coastal, and Ocean Eng., 114(6): 673-685.
[14] Kirby, J.T., Dalrymple, R.A. & Kaku, H., 1994. Parabolic approximations for
water waves in conformal coordinate systems. Coastal Eng., 23: 185-213.
[15] Smith R., Scott C.F.. Mixing in the tidal environment. Journal of Hydraulic
Engineering, 1997, 123: 332-340.
[16] Tao J.H., Han G.. Effects of water wave motion on pollutant transport in shallow
coastal water. Science in China (Series E), 2002, 45: 593-605.
[17] Pearson J.M., Guymer I., et al. Effect of wave height on cross-shore solute
mixing. Journal of Waterway, Port, Coastal and Ocean Engineering, 2002, 128:
10-20.
[18] Fisher H.B., List E.J., et al. Mixing in Inland and Coastal Waters. London:
Academic Press, 1979.
[19] Giarrusso C.C., Carratelli E.P. & Spulsi G... On the effects of wave drift on the
dispersion of floating pollutants. Ocean Engineering, 2001, 28: 1339-1348.
[20] Law A.W.K.. Taylor dispersion of contaminants due to surface waves. Journal of
Hydraulic Research, 2000, 38: 41-48.
[21] 苏翼林, 材料力学［M］, 北京：高等教育出版社, 1987.
[22] J.T.Kirby, 1986, Rational approximation in the parabolic equation method for
water waves, Coastal Engineering, 10, 355-378
[23] Kirby, J. T.，Dalrymple, R. A.，A parabolic equation for the combined
refraction-diffraction of Stokes waves by mildly varying topography，Journal of
Fluid Mechanics，1983，136, 453～466
[24] Dalrymple, R. A.，Kirby, J. T.，1988,Models for very wide-angle water waves and
wave diffraction, J. Fluid Mech., 192,33～50
[25] Berkhoff, J.C.W., Booij, N., and Radder, A.C., 1982. Verification of numerical
wave propagation models for simple harmonic water waves. Coastal Engineering,
Vol.6, 255-279.
58
参考文献
[26] Dally, W.R., Dean, R.G. and Dalrymple, R.A., 1985,Wave height variation across
beaches of arbitrary profile. Journal of Geophysical Research, 90 (C6),
11917-11927.
[27] Watanabe, A. and Maruyama, K., 1988. A numerical model of wave deformation
in surf zone.Proc. 21st Coastal Eng.Conf. ASCE, Malaga, 578-587.
[28] Johnson, H.K. and Poulin, S. 1998. On the accuracy of parabolic wave models.
Proc.26th Coastal Eng. Confer. ASCE, Copenhagen, 352-365.
[29] Vogel, J.A., Radder, A.C. and de Reus, J.H. 1988. Verification of numerical wave
propagation models in tidal inlets. Proc. 21st Coastal Eng.Conf. ASCE, Malaga,
433-447.
[30] Gao, Q. and Radder, A.C., 1998. A refraction-diffraction model for irregular
waves. Proc.26th Coastal Eng. Confer. ASCE, Copenhagen, 367-379.
[31] Lin, B.L., Chandler-Wilde S.N., 1996. A depth-integrated 2D coastal and
estuarine model with conformal boundary-fitted mesh generation. International
journal for numerical methods in fluids, 23: 819-846.
59