抛物型缓坡方程的变分及数值模拟

对线性水波的折射一绕射问题应用变分原理,对非等深、具有缓坡和不连续的底被导出了一种修改的抛物型缓坡方程近似模型,可预测三维地形上波浪的折射一绕射。同抛物型缓坡方程的线性方程进行了对比。通过数值模拟方法进行数值求解,表明本方法可用于地形条件下的波浪折射一绕射问题。     

A Modified Form of Mild-Slope Equation with Weakly Nonlinear Effect

Nonlinear effect is of importance to waves propagating from deep water to shallow water.Thenon-linearity of waves is widely discussed due to its high precision in application.But there are still someproblems in dealing with the nonlinear waves in practice.In this paper,a modified form of mild-slope equa-tion with weakly nonlinear effect is derived by use of the nonlinear dispersion relation and the steady mild-slope equation containing energy dissipation.The modified form of mild-slope equation is convenient to solvenonlinear effect of waves.The model is tested against the laboratory measurement for the case of a submergedelliptical shoal on a slope beach given by Berkhoff et al,The present numerical results are also comparedwith those obtained through linear wave theory.Better agreement is obtained as the modified mild-slope e-quation is employed.And the modified mild-slope equation can reasonably simulate the weakly nonlinear ef-fect of wave propagation from deep water to coast.     

Wave shoaling and diffraction in current over a mild-slope

The wave relative frequency in the coordinate system moving with current and the angle between the direction of wave propagation and that of current are computed based on the wave dispersion relation. The current field is computed by solving the depth averaged shallow water equations. The wave field is computed by solving the mildslope equation which has taken the current‘s effect into account. A numerical model is established using a finite element method for simulating the wave shoaling and diffraction in current over a mild-slope, and the numerical results are reasonable to compare with the experimental data.     

Numerical implementation of the harmonic modified mild-slope equation

A numerical solver is presented of the modified time-independent mild-slope equation, which incorporates energy dissipation. Using a second-order parabolic approximation, the following external boundary conditions are modelled: open and fully transmitting to both incoming and outgoing waves; partially reflecting, and; fully absorbing. Discretisation of the governing equation and boundary conditions is by means of a second-order accurate central difference scheme. The resulting sparse-banded matrix is solved using an inexpensive banded solver with Gaussian elimination. The numerical predictions are in excellent agreement with the analytical solution for the interaction of non-breaking waves with an array of vertical surface-piercing circular cylinders on a horizontal bed. Results are compared with those for the same array on various seabed topographies. The model is robust and can be used for wave propagation in complex geometries. It has fewer restrictions associated with wave obliqueness at boundaries than traditional models based on the mild-slope equation.     

Linear wave reflection by trench with various shapes

Two types of analytical solutions for waves propagating over an asymmetric trench are derived. One is a long-wave solution and the other is a mild-slope solution, which is applicable to deeper water. The water depth inside the trench varies in proportion to a power of the distance from the center of the trench (which is the deepest water depth point and the origin of x-coordinate in this study). The mild-slope equation is transformed into a second-order ordinary differential equation with variable coefficients based on the longwave assumption [Hunt's, 1979. Direct solution of wave dispersion equation. Journal of Waterway, Port, Coast. and Ocean Engineering 105, 457–459] as approximate solution for wave dispersion. The analytical solutions are then obtained by using the power series technique. The analytical solutions are compared with the numerical solution of the hyperbolic mild-slope equations. After obtaining the analytical solutions under various conditions, the results are analyzed.     

A note on the accuracy of the mild-slope equation

The mild-slope equation is a vertically integrated refraction-diffraction equation, used to predict wave propagation in a region with uneven bottom. As its name indicates, it is based on the assumption of a mild bottom slope. The purpose of this paper is to examine the accuracy of this equation as a function of the bottom slope. To this end a number of numerical experiments is carried out comparing solutions of the three-dimensional wave equation with solutions of the mild-slope equation.For waves propagating parallel to the depth contours it turns out that the mild-slope equation produces accurate results even if the bottom slope is of order 1. For waves propagating normal to the depth contours the mild-slope equation is less accurate. The equation can be used for a bottom inclination up to 1:3.     

A Finite-Difference Approach to the Time-Dependent Mild-Slope Equation

A finite-difference approach is used to develop a time-dependent mild-slope equation incorporating the effects of bottom dissipation and nonlinearity.The Euler predictor-corrector method and the three-point finite-difference method with varying spatial steps are adopted to discretize the time derivatives and the two-dimensional horizontal ones,respectively,thus leading both the time and spatial derivatives to the second-order accuracy.The boundary conditions for the present model are treated on the basis of the general conditions for open and fixed boundaries with an arbitrary reflection coefficient and phase shift.Both the linear and nonlinear versions of the numerical model are applied to the wave propagation and transformation over an elliptic shoal on a sloping beach,respectively,and the linear version is applied to the simulation of wave propagation in a fully open rectangular harbor.From comparison of numerical results with theoretical or experimental ones,it is found that they are in reasonable agreement.     

A numerical model for wave propagation in curvilinear coordinates

Using the perturbation method, a time dependent parabolic equation is developed based on the elliptic mild slope equation with dissipation term. With the time dependent parabolic equation employed as the governing equation, a numerical model for wave propagation including dissipation term in water of slowly varying topography is presented in curvilinear coordinates. In the model, the self-adaptive grid generation method is employed to generate a boundary-fitted and varying spacing mesh. The numerical tests show that the effects of dissipation term should be taken into account if the distance of wave propagation is large, and that the outgoing boundary conditions can be treated more effectively by introduction of the dissipation term into the numerical model. The numerical model is able to give good results of simulating wave propagation for waters of complicatedly boundaries and effectively predict physical processes of wave propagation. Moreover, the errors of the analytical solution deduced by Kirby et al. (1994) [Kirby, J.T., Dalrymple, R.A., Kabu, H., 1994. Parabolic approximation for water waves in conformal coordinate systems. Coastal Engineering 23, 185–213.] from the small-angle parabolic approximation of the mild-slope equation for the case of waves between diverging breakwaters in a polar coordinate system are corrected.     

Numerical Modeling of Wave Diffraction-Refraction in Water of Varying Current and Topography

—A numerical model for wave diffraction-refraction in water of varying current and topogra-phy is proposed,and time-dependent wave mild-slope equation with a dissipation term and correspondingequivalent governing equations are presented.Two different expressions of parabolic approximations forthe case of the absence of current are also given and analyzed.The influence of current on the results ofsimulation of waves is discussed.Some examples show that the present model is better than others in simu-lating wave transformation in large water areas.And they also show that the influence of current shouldbe taken into account,on numerical modeling of wave propagation in water of strong current and coastalareas,otherwise the modeling results will be largely distorted.     

A finite difference method for effective treatment of mild-slope wave equation subject to non-reflecting boundary conditions

A numerical model for coastal water wave motion that includes an effective method for treatment of non-reflecting boundaries is presented. The second-order one-way wave equation to approximate the non-reflecting boundary condition is found to be excellent and it ensures a very low level of reflection for waves approaching the boundary at a fairly wide range of the incidence angle. If the Newman approximation is adopted, the resulting boundary condition has a unique property to allow the free propagation of wave components along the boundary. The study is also based on a newly derived mild-slope wave equation system that can be easily made compatible to the one-way wave equation. The equation system is theoretically more accurate than the previous equations in terms of the mild-slope assumption. The finite difference method defined on a staggered grid is employed to solve the basic equations and to implement the non-reflecting boundary condition. For verification, the numerical model is then applied to three coastal water wave problems including the classical problem of plane wave diffraction by a vertical circular cylinder, the problem of combined wave diffraction and refraction over a submerged hump in the open sea, and the wave deformation around a detached breakwater. In all cases, the numerical results are demonstrated to agree very well with the relevant analytical solutions or with experimental data. It is thus concluded that the numerical model proposed in this study is effective and advantageous.     

非结构化网格下椭圆型缓坡方程的数值求解

椭圆型缓坡方程是一种用线性波浪理论研究近岸波浪传播变形的有效波浪数学模型。非结构化网格下的有限容积法不仅对复杂边界的适应性好,还能保证迭代求解过程的守恒性。建立了非结构化网格下的椭圆型缓坡方程数值模型。在模型中采用非结构化网格下的有限容积法对椭圆型缓坡方程进行了数值离散,结合GPBiCG(m,n)算法求解离散方程。数值计算结果表明,该数值模型可有效地用于模拟近岸缓坡区域复杂边界下波浪的传播。     

Generation of random waves in time-dependent extended mild-slope equations using a source function method

We develop techniques of numerical wave generation in the time-dependent extended mild-slope equations of Suh et al. [1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Engineering 32, 91–117] and Lee et al. [2003. Extended mild-slope equation for random waves. Coastal Engineering 48, 277–287] for random waves using a source function method. Numerical results for both regular and irregular waves in one and two horizontal dimensions show that the wave heights and the frequency spectra are properly reproduced. The waves that pass through the wave generation region do not cause any numerical disturbances, showing usefulness of the source function method in avoiding re-reflection problems at the offshore boundary.     

四叉树网格下的椭圆型缓坡方程数值模型研究

波浪是近岸海域关键的水动力因素之一。考虑到近岸地形复杂、波浪演化显著的特点,建立了四叉树网格体系下的椭圆型缓坡方程数值模型,采用有限体积法对模型进行数值离散,应用GPBiCG(m, n)算法求解离散后的控制方程。模型中根据波浪波长布局计算网格,生成多层次四叉树网格,对复杂计算域有较好的适应性,并且在离散和方程求解中无需引入形函数、不产生复杂的交叉项,节约了存储空间和计算时间。将模型成功应用于物理模型实验及Acapulco海湾的波浪场数值模拟,结果表明该模型能够准确、高效地模拟近岸波浪场,可为近岸波浪场的模拟提供一定的理论和技术支持。     

Numerical solution of a mathematical model for water wavesin large coastal areas

on the evolution equation for water waves,a mathematical model for wave propagation in large mild-slope areas is derived.The model is solved by the finite difference method with the staggered grid system.The computational results are in good agreement with experimental data and show that the model can obtain better results with relatively coarser grids.The model can be used to simulate water wave propagation in large coastal areas and can be efficiently solved without much programming effort.     

考虑边界波浪方向的缓坡方程自适应求解模型

缓坡方程是描述近岸波浪运动较好的数学模型之一。在发展的自适应有限元求解缓坡方程的基础上,采用迭代求解的方法,确定波浪相对于边界的入射方向,从而对边界条件进行改进,建立了求解缓坡方程的数值计算模型。典型算例表明,考虑波浪相对于边界的入射角度后,模型可以更好地模拟吸收波浪边界,同时对多向波对双突堤的绕射进行了模拟研究,与试验结果比较表明,所建立的数值计算模型能够适用于多向不规则波传播过程的模拟研究。     

近岸大区域水波数学模型及其数值求解

从水波发展方程出发,导出了大区域缓坡水波数学模型,并采用交错网格系统下的有限差分法对该数学模型进行了数值求解,计算结果表明该数学模型在粗网格下也能得到与实验结果比较一致的结果,从而表明该模型可用于较大区域的水波问题.该模型具有编程简单、求解比较快速、经济的优点.     

A numerical model for combined refraction-diffraction of short waves on an island

A combined wave refraction-diffraction numerical model was developed to predict wave conditions around an arbitrary island. The methodology was based on the mild-slope equation, solved using a finite difference scheme with a marching procedure. The new model reduced the computer's memory demand considerably in comparison with finite-element, parabolic, error vector propagation and other finite difference approaches, and could therefore predict wave conditions for a large coastal area under given offshore boundary-wave conditions. Laboratory data on wave conditions under submerged circular and elliptical shoal conditions were selected to validate the numerical results. Good agreement was observed in all cases. Wave characteristics around an island were predicted using this model with the given deep-water wave condition. The model can predict wave conditions for any island with a mild-slope coastline.     

适于模拟不规则水域波浪的缓坡方程两种数值模型比较

本文分析比较了适于不规则水域波浪模拟的椭圆型缓坡方程两种数值模型。两种数值模型均采用有限体积法离散,分别基于四叉树网格和非结构化三角形网格建立。首先结合近岸缓坡地形上波浪传播的经典物理模型实验对两种数值模型分别进行了验证,并结合计算结果对比分析了两种模型的计算精度和效率。计算结果表明,两种数值模型均可有效地模拟近岸波浪的传播变形;相对非结构化三角形网格下的模型,基于四叉树网格建立的数值模型在数值离散和求解过程中无需引入形函数、不产生复杂的交叉项,离散简单,易于程序实现,且节约计算存储空间,计算效率高。     

A finite element model for wave refraction and diffraction

A two-dimensional hybrid finite element method is developed to study the scattering of water waves by an island and to calculate wave forces and moments on offshore structures. The offshore structure, which could be either semi-submerged or fully extended in the water, is assumed to be stationary. The numerical model is based on the mild-slope equation. It can be applied to both long-wave and short-wave problems. A special treatment for the problem with the semi-submerged structure is introduced. Comparisons are given with existing analytical solutions and other numerical results. The present model is shown to be an efficient and accurate method for the solution of wave refraction and diffraction problems.     

Computation of Wave Refraction-Diffraction with Error Vector Propagation Method

After modifying the basic computation model made by Panchang (1988), the error vector propagation (EVP) method has been adopted to compute the combined effects of water wave refraction and diffraction in the presence of reflection boundary. The results show that the present method is successful in restraining the noise in Panchang's solution. Compared to other numerical methods for the mild-slope wave equation, EVP method can both consider the influence of reflection and save computer memory and computing time.