A practical alternative to the “mild-slope” wave equation

The “mild-slope” equation which describes wave propagation in shoaling water is normally expressed in an elliptic form. The resulting computational effort involved in the solution of the boundary value problem renders the method suitable only for small sea areas. The parabolic approximation to this equation considerably reduces the computation involved but must omit the reflected wave. Hence this method is not suited to the modelling of harbour systems or areas near to sea walls where reflections are considerable. This paper expresses the “mild-slope” equation in the form of a pair of first-order equations, which constitute a hyperbolic system, without the loss of the reflected wave. A finite-difference numerical scheme is described for the efficient solution of the equations which includes boundaries of arbitrary reflecting power.     

Improvement on open boundaries on a time dependent numerical model of wave propagation in the nearshore region

An improvement on the simulation of outgoing waves on a time dependent numerical model for water wave propagation in the nearshore region is presented. The governing equations consist of a system of first order partial differential equations (PDEs), the equation of continuity and the equation of motion. A comparative study of first order radiation boundary conditions (BCs) and first order radiation BCs combined with sponge layers is presented for cases where outgoing waves leave the numerical domain of calculation through the open boundary. A reduction of spurious reflections from the numerical open boundaries can be obtained with an irrelevant increase in terms of computational cost.     

一般曲线坐标系下波浪传播的数值模拟

在曲线坐标系下,建立了缓变水深水域波浪传播的数值模拟模型.模型适宜于复杂变化的边界形状,克服了各种代数坐标变换的局限性.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,既克服了一般抛物近似方法的缺点,又便利了方程的求解;从开边界条件、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.对模型的验证表明,数值解与物模实验值吻合良好,模型对于具有复杂边界的工程实际有较强的适应性.     

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.     

Simulation of wave breaking effects in two-dimensional elliptic harbor wave models

A technique is developed for including the effects of dissipation due to wave breaking in two-dimensional elliptic models based on the mild-slope wave equation. This involves exploration of convergence properties pertaining to iteration due to presence of the nonlinear wave breaking parameter in the governing equations as well as new boundary conditions that include wave-breaking effects. Five wave-breaking formulations are examined in conjunction with the resulting model, which is applied to tests involving a sloping beach, a bar-trough bottom configuration, shore-connected and shore-parallel breakwaters on a sloping beach, and two real-world cases. Model results show that three of the formulations, when used within the context of the modeling scheme presented here, provide excellent results compared to data.     

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.     

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。     

Assessment of open boundary conditions on the elliptic formulation of the mild-slope equation

The performance of open boundaries in a finite differences scheme of the elliptic mild-slope equation is assessed. The wave propagation results show that lowest order parabolic radiation boundary conditions, unlike sponge layers combined with first order radiation boundary conditions, are an efficient alternative to first order radiation boundary conditions in order to improve the accuracy of the numerical solution of the problem.     

An energy-controlling boundary condition for partial wave reflections in the mild slope equation

An energy-controlling technique to actively manage the reflective property of waves from solid boundary is presented. As linear waves propagate through an energy-controlling area, a reduction in wave heights occurs due to energy dissipation, which can be placed under direct control through the imaginary part of the wavenumber and phase velocity. Based on this relationship, the present study investigates a new method to control reflected waves with desired heights in the mild slope equation model. The method is validated through numerical tests for various reflection coefficients and the results confirm the promising use of energy-controlling boundary condition for partial wave reflections.     

适用于曲边界的浅水非线性波浪数值模型

实际工程中存在大量的曲边界,因此在曲边界上的计算准确性可以考察出一个数值模型的实用价值。利用Beji的改进型Boussinesq方程建立了一个有限元方法的数值波浪模型。造波方面采用Fenton提出的非线性规则波浪解;在墙边界处,以求解法线方向和切线方向的速度和导数代替求解x、y方向的速度和导数,从而使边界条件直接适用、严格满足,保证了对曲边界计算的准确性。"重开始广义极小残量法"的使用保证了求解方程组的效率和精度,使造波和边界处理方法的有效性和准确性得到了合理地诠释。通过与试验数据、他人数值结果、解析解的比对,显示出该模型计算稳定、结果准确,真正体现出了有限元方法对曲边界适用的优势。     

潮汐和流影响下长江口波浪场数值计算

采用SWAN模型和REF/DIF模型进行嵌套计算的方法来获取长江口海域实际波浪场.其中设计一种根据入射波向即时生成计算网格的方法,解决REF/DIF模型对于波浪入射角的限制从而实现两种模型的嵌套.为考虑水流和潮位分布的空间差异对波浪传播变形的作用,利用二维流场模型计算长江口的水位和流场过程,在实际波浪计算中引入了水位和水流作用.计算结果与观测资料的对比表明:1)SWAN模型和REF/DIF模型的嵌套计算方法可以作为提高浅水区域波浪计算精度的一种有效途径;2)水位和水流对长江口波浪计算的影响显著,考虑了水位和水流条件后,尤其是在大潮期间,能比较显著地提高计算精度.     

非线性波浪波面追踪的一种新模式

基于Laplace方程的Green积分表达式和波面BemouUi方程所建立的非线性波动数学模型,是一个时域上具有初始值的边值问题,而精确地追踪自由表面的波动位置,给出波面运动瞬时的波面高度和波面势函数,是建立时域内非线性波浪数值模式的基础。本文采用0-1混合型边界元剖分计算域边界并离散Laplace方程的Green积分表达式,采用有限元剖分自由水面并推导满足自由表面非线性边界条件的波面有限元方程,联立计算域内以节点波势函数和波面位置高度的时间增量为未知量的线性方程组,通过时步内的循环迭代,给出每个时步上的波面位置和波面势函数,从而建立了一种新的非线性波浪波面追踪模式。数值造波水槽内的波浪试验表明,其数值模拟结果具有良好的计算精度。     

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.     

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。     

A pretty good sponge: Dealing with open boundaries in limited-area ocean models

The problem of computing within a limited domain surrounded by open boundaries is discussed within the context of the shallow-water wave equations by comparing three different treatments, all of which surround the domain by absorbing zones intended to prevent reflections of outgoing waves. The first, which has attracted a lot of attention for use in electromagnetic and aeroacoustic applications, is intended to prevent all reflections. However, it has not yet been developed to handle the second important requirement of open boundaries, namely the ability to pass information about external conditions into the domain of interest. The other two treatments, which absorb differences from a specified external solution, allow information to pass through the open boundary in both directions. One, based on the flow relaxation scheme of [Martinsen, E.A., Engedahl, H., 1987. Implementation and testing of a lateral boundary scheme as an open-boundary condition in a barotropic ocean model. Coastal Eng. 11, 603–627] and termed here the “simple sponge,” relaxes all fields toward their external counterparts. The other, a simplification and generalization of the perfectly matched layer, referred to here as the “pretty good sponge,” avoids absorbing the component of momentum parallel to the open boundary. Comparisons for a case that is dominated by outgoing waves shows the pretty good sponge to perform essentially as well as the perfectly matched layer and better than the simple sponge. In comparisons for a geostrophically balanced eddy passing through open boundaries, the pretty good sponge out-performed the simple sponge when the only external information available was about the advecting flow, but when information about the nature of the eddy in the sponge zones was also available, the simple sponge performed better. For the case of an equatorial soliton passing through the boundary and no information provided about its nature outside the open domain, again the pretty good sponge performed better. Proving useful for situations governed by nonlinear equations forced by external conditions and being easy to implement, the pretty good sponge should be considered for use with existing limited-area ocean models.     

New periodic wave solutions of a time fractional integrable shallow water equation

In this paper, author employed Jacobi elliptic function expansion method to build the new wave solutions of time fractional modified Camassa–Holm equation which is completely integrable dispersive shallow-water equation. In ocean engineering, Camassa–Holm equation is generally used as a tool in computer simulation of the water waves in shallow sees, coastal and harbors. The obtained solutions show that the Jacobi elliptic function expansion method (JEFEM) which based on Jacobi elliptic functions is an efficient, reliable, applicable and accurate tool for analytic approximation of a wide variety of nonlinear conformable time fractional partial differential equations.     

Modeling nonlinear wave–wave interactions with the elliptic mild slope equation

An approach is developed to simulate wave–wave interactions using nonlinear elliptic mild-slope equation in domains where wave reflection, refraction, diffraction and breaking effects must also be considered. This involves the construction of an efficient solution procedure including effective boundary treatment, modification of the nonlinear equation to resolve convergence issues, and validation of the overall approach. For solving the second-order boundary-value problem, the Alternating Direction Implicit (ADI) scheme is employed, and the use of approximate boundary conditions is supplemented, for improved accuracy, with internal wave generation method and dissipative sponge layers. The performance of the nonlinear model is investigated for a range of practical wave conditions involving reflection, diffraction and shoaling in the presence of nonlinear wave–wave interactions. In addition, the transformation of a wave spectrum due to nonlinear shoaling and breaking, and nonlinear resonance inside a rectangular harbor are simulated. Numerical calculations are compared with the results from other relevant nonlinear models and experimental data available in literature. Results show that the approach developed here performs reasonably well, and has thus improved the applicability of this class of wave transformation models.     

结合椭圆型缓坡方程模拟近岸波流场

波浪向近岸传播的过程中,由波浪破碎效应所产生的近岸波流场是近岸海域关键的水动力学因素之一.结合近岸波浪场的椭圆型缓坡方程和近岸波流场数学模型对近岸波浪场及由斜向入射波浪破碎后所形成的近岸波流场进行了数值模拟.计算中考虑到波浪向近岸传播中由于波浪的折射、绕射、反射等效应使局部复杂区域波向不易确定,采用结合椭圆型缓坡方程所给出的波浪辐射应力公式来计算波浪产生的辐射应力,在此基础上耦合椭圆型缓坡方程和近岸波流场数学模型对近岸波流场进行数值模拟,从而使模型综合考虑了波浪的折射、绕射、反射等效应且避免了对波向角的直接求解,可以应用于相对较复杂区域的近岸波流场模拟.     

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.     

西沙湾工程波浪推算及波浪场数值模拟

本研究根据崇武水文站1976—2007年共31 a的波浪观测资料,使用皮尔逊Ⅲ型频率曲线进行拟合分析,推算出崇武在不同重现期条件下的特征波浪要素,再将其作为外海波浪要素输入,通过基于椭圆型缓坡方程的CGWAVE近岸波浪数值模型,模拟西沙湾的重现期波浪场,得到其设计波浪要素,得到较为合理的计算结果。西沙湾的海底地形强烈地影响近岸波浪,导致了波高和波向的不均匀分布特征,岛屿和礁石都是天然屏障,东侧水域受龟屿及闽台码头的掩护,波浪反射、绕射明显,对避风坞起到了有效的保护作用。此外,对于不同地质的边界,反射系数不同使得近岸波高分布有较大差异。