Efficient Numerical Solution of the Modified Mild-Slope Equation

An efficient numerical model for wave refraction, diffraction and reflection is presented in this paper. In the model the modified time-dependent mild-slope equation is transformed into an evolution equation and an improved ADI method involving a relaxation factor is adopted to solve it. The method has the advantage of improving the numerical stability and convergence rate by properly determining the relaxation factor. The range of the relaxation factor making the differential scheme unconditionally stable is determined by stability analysis. Several verifications are performed to examine the accuracy of the present model. The numerical results coincide with the analytic solutions or experimental data very well and the computer time is reduced.

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. The non-linearity of waves is widely discussed due to its high precision in application. But there are still some problems in dealing with the nonlinear waves in practice. In this paper, a modified form of mild-slope equation 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 solve nonlinear effect of waves. The model is tested against the laboratory measurement for the case of a submerged elliptical shoal on a slope beach given by Berkhoff et al. The present numerical results are also compared with those obtained through linear wave theory. Better agreement is obtained as the modified mild-slope equation is employed. And the modified mild-slope equation can reasonably simulate the weakly nonlinear effect of wave propagation from deep water to coast.

Numerical Modeling of the Hyperbolic Mild-Slope Equation in Curvilinear Coordinates

The mild-slope equation is familiar to coastal engineers as it can effectively describe wave propagation in nearshore regions. However, its computational method in Cartesian coordinates often renders the model inaccurate in areas with irregular shorelines, such as estuaries and harbors. Based on the hyperbolic mild-slope equation in Cartesian coordinates, the numerical model in orthogonal curvilinear coordinates is developed. The transformed model is discretized by the finite difference method and solved by the ADI method with space-staggered grids. The numerical predictions in curvilinear co- ordinates show good agreemenl with the data obtained in three typical physical expedments, which demonstrates that the present model can be used to simulate wave propagation, for normal incidence and oblique incidence, in domains with complicated topography and boundary conditions.

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

A time-dependent mild-slope equation for the extension of the classic mild-slope equation of Berkhoff is developed for the interactions of large ambient currents and waves propagating over an uneven bottom, using a Hamiltonian formulation for irrotational motions. The bottom topography consists of two components the slowly varying component which satisfies the mild-slope approximation, and the fast varying component with wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter describing the mild-slope condition. The theory is more widely applicable and contains as special cases the following famous mild-slope type equations: the classical mild-Slope equation, Kirby's extended mild-slope equation with current, and Dingemans's mild-slope equation for rippled bed. Finally, good agreement between the classic experimental data concerning Bragg reflection and the present numerical results is observed.

Numerical Simulation of Wave-Induced Currents Combined with Parabolic Mild-Slope Equation in Curvilinear Coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.

New Numerical Scheme for Simulation of Hyperbolic Mild-Slope Equation

The original hyperbolic mild-slope equation can effectively take into account the combined effects of wave shoaling, refraction, diffraction and reflection, but does not consider the nonlinear effect of waves, and the existing numerical schemes for it show some deficiencies. Based on the original hyperbolic mild-slope equation, a nonlinear dispersion relation is introduced in present paper to effectively take the nonlinear effect of waves into account and a new numerical scheme is proposed. The weakly nonlinear dispersion relation and the improved numerical scheme are applied to the simulation of wave transformation over an elliptic shoal. Numerical tests show that the improvement of the numerical scheme makes efficient the solution to the hyperbolic mild-slope equation, A comparison of numerical results with experimental data indicates that the results obtained by use of the new scheme are satisfactory.

An Extended Mild-Slope Equation

On the assumption that the vortex and the vertical velocity component of the current are small, a mild-slope equation for wave propagation on non-uniform flows is deduced from the basic hydrodynamic equations, with the terms of (V(h)h)(2) and V(h)(2)h included in the equation. The terms of bottom friction, wind energy input and wave nonlinearity are also introduced into the equation. The wind energy input functions for wind waves and swells are separately considered by adopting Wen's (1989) empirical formula for wind waves and Snyder's observation results for swells. Thus, an extended mild-slope equation is obtained, in which the effects of refraction, diffraction, reflection, current, bottom friction, wind energy input and wave nonlinearity are considered synthetically.

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 Enler predietor-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.

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 ofwave propagation and that of current are computed based on the wave dispersion relation. The current field iscomputed by solving the depth averaged shallow water equations. The wave field is computed by solving the mild-slope equation which has taken the currents effect into account. A numerical model is established using a finiteelement method for simulating the wave shoaling and diffraction in current over a mild-slope, and the numericalresults are reasonable to compare with the experimental data.

Tide simulation using the mild-slope equation with Coriolis force and bottom friction

Since the mild-slope equation was derived by Berkhoff （1972）,the researchers considered various mechanism to simplify and improve the equation,which has been widely used for coastal wave field calculation.Recently,some scholars applied the mild-slope equation in simulating the tidal motion,which proves that the equation is capable to calculate the tide in actual terrain.But in their studies,they made a lot of simplifications,and did not consider the effects of Coriolis force and bottom friction on tidal wave.In this paper,the first-order linear mild-slope equations are deduced from Kirby mild-slope equation including wave and current interaction.Then,referring to the method of wave equations’ modification,the Coriolis force and bottom friction term are considered,and the effects of which have been performed with the radial sand ridges topography.Finally,the results show that the modified mild-slope equation can be used to simulate tidal motion,and the calculations agree well with the measurements,thus the applicability and validity of the mild-slope equation on tidal simulation are further proved.

An extended time-dependent numerical model of the mild-slope equation with weakly nonlinear amplitude dispersion

In the present paper, by introducing the effective wave elevation, we transform the extended elliptic mild-slope equation with bottom friction, wave breaking and steep or rapidly varying bottom topography to the simplest time-dependent hyperbolic equation. Based on this equation and the empirical nonlinear amplitude dispersion relation proposed by Li et al. （2003）, the numerical scheme is established. Error analysis by Taylor expansion method shows that the numerical stability of the present model succeeds the merits in Song et al. （2007）＇s model because of the introduced dissipation terms. For the purpose of verifying its performance on wave nonlinearity, rapidly varying topography and wave breaking, the present model is applied to study： （1） wave refraction and diffraction over a submerged elliptic shoal on a slope （Berkhoff et al., 1982）; （2） Bragg reflection of monochromatic waves from the sinusoidal ripples （Davies and Heathershaw, 1985）; （3） wave transformation near a shore attached breakwater （Watanabe and Maruyama, 1986）. Comparisons of the numerical solutions with the experimental or theoretical ones or with those of other models （REF/DIF model and FUNWAVE model） show good results, which indicate that the present model is capable of giving favorably predictions of wave refraction, diffraction, reflection, shoaling, bottom friction, breaking energy dissipation and weak nonlinearity in the near shore zone.

EXTENDED MILD-SLOPE EQUATION

The Hamiltonian formalism for surface waves and the mild-slope approximation were empolyed in handling the case of slowly varying three-dimensional currents and an uneven bottom, thus leading to an extended mild-slope equation. The bottom topography consists of two components: the slowly varying component whose horizontal length scale is longer than the surface wave length, and the fast varying component with the amplitude being smaller than that of the surface wave. ne frequency of the fast varying depth component is, however, comparable to that of the surface waves. The extended mild-slope equation is more widely applicable and contains as special cases famous mild-slope equations below: the classical mild-slope equation of Berkhoff, Kirby's mild-slope equation with current, and Dingemans's mild-slope equation for rippled bed. The extended shallow water equations for ambient currents and rapidly varying topography are also obtained.

High Order Numerical Code for Hyperbolic Mild-slope Equations with Nonlinear Dispersion Relation

Based on the hyperbolic mild-slope equations derived by Copeland (1985), a numerical model is established in unstag- gered grids. A composite 4 th-order Adam-Bashforth-Moulton (ABM) scheme is used to solve the model in the time domain. Terms involving the first order spatial derivatives are differenced to O ( Δx )4accuracy utilizing a five-point formula. The nonlinear dispersion relationship proposed by Kirby and Dalrymple (1986) is used to consider the nonlinear effect. A numerical test is performed upon wave propagating over a typical shoal. The agreement between the numerical and the experimental results validates the present model. Biodistribution and applications are also summarized.

COMPARISON BETWEEN BOUSSINESQ EQUATIONS AND MILD-SLOPE EQUATIONS MODEL

In this paper, the Poussinesq equations and mild-slope equation of wave transformation in near-shore shallow water were introduced and the characteristics of the two forms of equations were compared and analyzed. Meanwhile, a Boussinesq wave model which includes effects of bottom friction, wave breaking and subgrid turbulent mixing is established, slot technique dealing with moving boundary and damping layer dealing with absorbing boundary were estab lished. By adopting empirical nonlinear dispersion relation and including nonlinear term, the mild-slope equation model was modified to take nonlinear effects into account. The two types of models were validated with the experiment results given by Berkhoff and their accuracy was analysed and compared with that of correlated methods.

不均匀海底地形影响下柔性板与斜向来浪的相互作用

The present work analyzes the interaction of oblique waves by a porous flexible breakwater in the presence of a step-type bottom.The physical models for both scattering and trapping cases are considered and developed within the framework of small amplitude water-wave theory.Darcy’s law is used to model the wave interaction with the porous medium.It is assumed that the varying bottom extends over a finite interval,connected by a finite length of uniform bottom near an impermeable wall,and a semi-infinite length of bottom in the open water region.The boundary value problem is solved using the eigenfunction expansion method in the uniform bottom regions,while a modified mild-slope equation(MMSE)is used for the region with the varying bottom.Additionally,a mass-conserving jump condition is employed to handle the solution at slope discontinuities in the bottom.A system of equations is derived by matching the solutions at interfaces.The reflection coefficient and force on the breakwater and impermeable wall are plotted and analyzed for various parameters,such as the length of the varying bottom,depth ratio,angle of incidence,and flexural rigidity.It is observed that moderate values of flexural rigidity and depth ratio significantly contribute to an optimum reflection coefficient and reduce the wave force on the wall and breakwater.Remarkably,the outcomes of this study are assumed to be applicable in the construction of this type of breakwater in coastal regions.

缓坡泄洪放空洞掺气坎水力特性试验分析

高速水流掺气是减免流道空蚀破坏的有效措施,然而在缓底坡、大水位变幅条件下,常规掺气坎难以形成稳定无积水的掺气空腔。本文在缓底坡(3.3%)条件下,采用模型试验方法,对比分析了常规连续坎、异型坎及组合坎的水流特性和掺气效果,认为Fr数、掺气坎坡度与水舌落点处底板坡度是影响水流入射角的主要因素;在此底坡条件下,掺气坎尺寸变化的影响要远大于体型变化的影响,采用异型掺气坎亦难有效解决掺气空腔积水问题;经依托工程模型试验论证,组合坎掺气流态良好,适应大变幅的运行水头,可较好地改善缓底坡空腔回流积水问题。

斜坡叠加起伏海床对波浪反射的修正缓坡方程有限差分解

基于修正缓坡方程,建立了方程求解的有限差分模型,研究了斜坡叠加起伏海床对波浪的反射.首先,对波浪入射平面斜坡以及水平海床上单周期、双周期正弦沙纹的反射问题进行了验证,结果与他人的数值解、解析解及实验数据吻合较好,证明了该模型的正确性.接着,讨论了抛物形斜坡的上凸高度及下凸深度,斜坡叠加单周期正弦沙纹的数量、高度及长度,斜坡叠加双周期正弦沙纹的高度对波浪反射系数的影响.结果表明,反射系数随抛物形斜坡上凸高度的增大而减小,随下凸深度的增大而增大.斜坡叠加单周期正弦沙纹对波浪反射的规律与水平海床基本一致,相比水平海床Bragg共振相位下移的幅度随沙纹数量增加,先减小后保持不变的情况,斜坡叠加沙纹共振相位下移的幅度随沙纹数量增加先减小后增大.波浪入射斜坡叠加双周期正弦沙纹地形,随叠加的两个沙纹高度分别增加,Bragg共振峰值均相应地增加,共振带宽几乎不受影响,主振峰值的相位下移幅度减小,这与单周期沙纹中随沙纹高度增加,共振相位下移幅度更大的情况相反.对比发现:当固定沙纹的数量、长度、高度,不论是水平海床还是斜坡海床,双周期正弦沙纹对波浪的反射强度以及激发的共振带宽均大于单周期正弦沙纹,共振相位下移的幅度则小于单周期沙纹;斜坡叠加正弦沙纹对波浪的反射强度要大于水平海床,零反射的现象不再存在,主振峰值的相位下移幅度较水平海床更大.

大尺度圆柱墩群周围的波流场的数值模拟

本文对波流共同作用下大尺度圆柱墩群周围的波流场进行了数值研究。利用波浪弥散关系的迭代计算求得波向与流向的夹角以及波浪的相对频率。流场通过求解浅水环流方程得到,波浪场通过求解含流的缓坡方程得到,通过二者的迭代计算得到大尺度圆柱墩群周围的波流场的耦合解。用有限元法建立了数值模型,并将本文的计算数据与试验数据以及其他学者计算数据进行了比较,结果较为合理。

大尺度圆柱周围的波流场的耦合计算

本文利用波浪弥散关系的迭代计算求得波向与流向的夹角,并用有限元法求解含流的缓坡方程,得到在缓变地形和定常流场共同影响下的大尺度圆柱周围的波流场的耦合解。

近岸波浪传播变形数学模型的研究与进展

对近岸波浪传播变形的各种数学模型进行了归纳评述和总结,以期对本研究方向的发展起到一定的引导和促进作用。