近岸水域波浪传播的数学模型
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摘要
外海深水区的风浪或涌浪在向近岸浅水区传播的过程中，由于受到水深、地形变化、能量耗散、障碍物、水流等因素的作用会发生浅水变形、折射、绕射、反射、破碎和非线性效应等现象，这些现象是近岸波浪传播的主要物理过程和特征。随着社会经济的迅速发展，对近岸水域波浪传播变形规律的研究日益重要和迫切。现有的各种近岸水域波浪传播的数学模型都还有各自的不足之处，亟待进一步发展和完善。本报告主要沿着适宜于中、小尺度空间的缓变水深水域波浪传播的数学模型这条主线，对近岸水域中波浪的传播进行研究。并通过和非线性长波的数学模型在具体应用中的对比分析，进一步深化了对近岸水域波浪传播数学模型特点的认识。
首先，基于曲线坐标系，建立了缓变水深水域波浪传播的数值模拟模型，模型适宜于任意变化的边界形状，克服了各种代数坐标变换的局限性。并在此基础上，建立了水流作用下波浪传播的数学模型。在建立模型时，将原始的椭圆型缓坡方程的近似型式——依赖时间变化的抛物型方程，作为控制方程；从将开边界条件、不同反射特性的固壁边界条件相统一的表达式出发，对边界条件进行处理；用ADI法数值求解控制方程，格式无条件稳定；节省了计算机内存和计算量。比较详细的模型验证与应用表明，模型的数值模拟结果与解析解、物模实验值吻合良好；可以较好地模拟波浪传播过程中的浅水变形、折射、绕射和反射等多种现象；能正确合理地反映水流对波浪传播的影响。模型可广泛应用于具有复杂边界的工程实际。
其次，将包含底摩阻耗散项的缓坡方程化为等价的控制方程组，采用Crank-Nicolson格式离散方程组，建立了适宜于大范围水域内波浪传播的数学模型。模型具有较高的精度，而且更加便于实际应用。模型的数值模拟结果和解析解与物模实验值吻合较好。将模型应用于地形复杂变化的长江口南港水域内波浪场的计算，计算结果说明，模型能较好地反映滩槽相间、急剧变化的地形的影响。模型可广泛应用于大范围水域内波浪传播的计算。
最后，鉴于Boussinesq型方程和缓坡方程是在不同的假设条件下推导而来，应用于描述近岸水域波浪的传播变形时具有不同的特点。本报告根据作者所建立的可以对任意水深点流场与波动净压力场进行求解、适宜水深任意变化水域非线性波传播的数学模型，提供了在较强非线性作用下波浪传播的数值模拟结果。并将非线性长波传播模型和缓坡方程，分别应用于非线性作用较
摘要
强、地形为平底与圆形暗礁的组合这一经典物模实验，比较了二者应用于小尺度水域范围内波浪
传播变形的具体差别。
As surface waves propagate from deep to shallow water, due to the effects of water depth, topography, sea bed friction, obstacles and ambient currents etc, many phenomena occur, such as shoaling transformation, refraction, diffraction, reflection, wave breaking, non-linearity and so on, which are the main physical processes and characteristics during wave propagation. There are several kinds of mathematical models of wave propagation in coastal area now, however, they should be developed and perfected for many deficiencies exist. In this report, mathematical models for combined refraction-diffraction waves in water of slowly varying topography are presented. At the same time, being compared with application of the model for non-linear long waves, the knowledge of characteristics of wave propagation models in near shore area is deepened further.
Firstly, under the curvilinear coordinates, mathematical model for wave propagation in water of slowly topography is presented. The model is suitable to arbitrary boundary shapes and overcomes the limitation of other models with algorithm transformation. The mathematical model for wave propagation on non-uniform currents is established also. In the models, the time dependent parabolic equations, deduced from the mild slope equations with currents or not, are used as the governing equations. Based on the general conditions for open and fixed natural boundaries with an arbitrary reflection coefficient and phase shift, the boundary conditions for the present models are treated. The alternative direction implicit method is used to solve the governing equations and the numerical schemes are unconditional stable. The required computer storage is reduced. The computer speed is speeded up. The numerical results of the present models are in agreement with the theoretical solution and those of physical models. Systematical numerical tests show that the present models can reasonably simulate the wave transformation, such as shoaling, refraction, diffraction, reflection, effect of currents and so on. So the present models are
able to be used in coastal engineering with complicated boundary shapes extensively.
Secondly, a mathematical model suitable to large coastal region is developed, whose governing equations are deduced from the mild slope equation with dissipation terms and discretized with Crank-Nicolson scheme. This model is accurate and easy to be applied. The numerical tests show that the results of numerical solution are consistent with those of corresponding analytical solution and physical models. Being utilized in the wave propagation for Nan Gang of Yangtze River Estuary, this model can give good results of numerical simulation by effectively reflecting the influence of complicated topography which is comprised of shoal-channel spaced in between. So it can be applied to large areas.
In the end, in view of the fact that Boussinesq-type equations and the mild slope equations are deduced from different hypothesis conditions and behave differently in simulation of wave propagation, the numerical results of wave propagation effected by strong non-linearity are given by the nonlinear three-dimensional mathematical model which was established for the calculation of 3-D wave particle velocity and wave pressure and suitable to small size waters of arbitrarily varying depth. The model for non-linear long wave and the mild slope equation are respectively applied to simulation wave propagation
on a classical topography for small size waters-submerged shoal with concentric
contours. The differences between them in wave propagation are got through comparing the numerical solutions. And the results are accordant with actual cases.
首先，基于曲线坐标系，建立了缓变水深水域波浪传播的数值模拟模型，模型适宜于任意变化的边界形状，克服了各种代数坐标变换的局限性。并在此基础上，建立了水流作用下波浪传播的数学模型。在建立模型时，将原始的椭圆型缓坡方程的近似型式——依赖时间变化的抛物型方程，作为控制方程；从将开边界条件、不同反射特性的固壁边界条件相统一的表达式出发，对边界条件进行处理；用ADI法数值求解控制方程，格式无条件稳定；节省了计算机内存和计算量。比较详细的模型验证与应用表明，模型的数值模拟结果与解析解、物模实验值吻合良好；可以较好地模拟波浪传播过程中的浅水变形、折射、绕射和反射等多种现象；能正确合理地反映水流对波浪传播的影响。模型可广泛应用于具有复杂边界的工程实际。
其次，将包含底摩阻耗散项的缓坡方程化为等价的控制方程组，采用Crank-Nicolson格式离散方程组，建立了适宜于大范围水域内波浪传播的数学模型。模型具有较高的精度，而且更加便于实际应用。模型的数值模拟结果和解析解与物模实验值吻合较好。将模型应用于地形复杂变化的长江口南港水域内波浪场的计算，计算结果说明，模型能较好地反映滩槽相间、急剧变化的地形的影响。模型可广泛应用于大范围水域内波浪传播的计算。
最后，鉴于Boussinesq型方程和缓坡方程是在不同的假设条件下推导而来，应用于描述近岸水域波浪的传播变形时具有不同的特点。本报告根据作者所建立的可以对任意水深点流场与波动净压力场进行求解、适宜水深任意变化水域非线性波传播的数学模型，提供了在较强非线性作用下波浪传播的数值模拟结果。并将非线性长波传播模型和缓坡方程，分别应用于非线性作用较
摘要
强、地形为平底与圆形暗礁的组合这一经典物模实验，比较了二者应用于小尺度水域范围内波浪
传播变形的具体差别。
As surface waves propagate from deep to shallow water, due to the effects of water depth, topography, sea bed friction, obstacles and ambient currents etc, many phenomena occur, such as shoaling transformation, refraction, diffraction, reflection, wave breaking, non-linearity and so on, which are the main physical processes and characteristics during wave propagation. There are several kinds of mathematical models of wave propagation in coastal area now, however, they should be developed and perfected for many deficiencies exist. In this report, mathematical models for combined refraction-diffraction waves in water of slowly varying topography are presented. At the same time, being compared with application of the model for non-linear long waves, the knowledge of characteristics of wave propagation models in near shore area is deepened further.
Firstly, under the curvilinear coordinates, mathematical model for wave propagation in water of slowly topography is presented. The model is suitable to arbitrary boundary shapes and overcomes the limitation of other models with algorithm transformation. The mathematical model for wave propagation on non-uniform currents is established also. In the models, the time dependent parabolic equations, deduced from the mild slope equations with currents or not, are used as the governing equations. Based on the general conditions for open and fixed natural boundaries with an arbitrary reflection coefficient and phase shift, the boundary conditions for the present models are treated. The alternative direction implicit method is used to solve the governing equations and the numerical schemes are unconditional stable. The required computer storage is reduced. The computer speed is speeded up. The numerical results of the present models are in agreement with the theoretical solution and those of physical models. Systematical numerical tests show that the present models can reasonably simulate the wave transformation, such as shoaling, refraction, diffraction, reflection, effect of currents and so on. So the present models are
able to be used in coastal engineering with complicated boundary shapes extensively.
Secondly, a mathematical model suitable to large coastal region is developed, whose governing equations are deduced from the mild slope equation with dissipation terms and discretized with Crank-Nicolson scheme. This model is accurate and easy to be applied. The numerical tests show that the results of numerical solution are consistent with those of corresponding analytical solution and physical models. Being utilized in the wave propagation for Nan Gang of Yangtze River Estuary, this model can give good results of numerical simulation by effectively reflecting the influence of complicated topography which is comprised of shoal-channel spaced in between. So it can be applied to large areas.
In the end, in view of the fact that Boussinesq-type equations and the mild slope equations are deduced from different hypothesis conditions and behave differently in simulation of wave propagation, the numerical results of wave propagation effected by strong non-linearity are given by the nonlinear three-dimensional mathematical model which was established for the calculation of 3-D wave particle velocity and wave pressure and suitable to small size waters of arbitrarily varying depth. The model for non-linear long wave and the mild slope equation are respectively applied to simulation wave propagation
on a classical topography for small size waters-submerged shoal with concentric
contours. The differences between them in wave propagation are got through comparing the numerical solutions. And the results are accordant with actual cases.
引文
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