Refraction-Diffraction Model for Irregular Wave and Directional Spectrum Considering Current

- A mild-slope equation for combined refraction and diffraction of random waves in the independent time current is derived using Kirby equation(1984). In the derivation, the frequency of random waves is represented based on the time series concept, by a typical frequency and its deviation part. Numerical results, compared with those of laboratory experiments, demonstrate that new set of irregular mild slope equation with current is of good adaptability.     

Verification of A Numerical Harbour Wave Model

A numerical model for wave propagation in a harbour is verified by use of physical models.The extended time-dependent mild slope equation is employed as the governing equation,and the model is solved by use of ADI method containing the relaxation factor.Firstly,the reflection coefficient of waves in front of rubble-mound breakwaters under oblique incident waves is determined through physical model tests,and it is regarded as the basis for simulating partial reflection boundaries of the numerical model.Then model tests on refraction,diffraction and reflection of waves in a harbour are performed to measure wave height distribution.Comparative results between physical and numerical model tests show that the present numerical model can satisfactorily simulate the propagation of regular and irregular waves in a harbour with complex topography and boundary conditions.     

港域波浪数学模型的改进与验证

通过物理模型对改进的港内波浪传播变形数学模型进行验证。该数学模型以推广的时变缓坡方程为控制方程,采用含松弛因子的ADI法求解,并对波浪反射和透射边界模拟方法进行改进。先通过物理模型试验确定斜向浪入射条件下抛石防波堤前的波浪反射系数,作为数学模型中部分反射边界模拟的依据。然后进行了一个典型港口内波浪折射、绕射和反射的模型试验,测量港内波浪分布。对比模型试验和数学模型计算的结果表明,数学模型可较好地模拟港内复杂地形和边界条件下规则波和不规则波的传播变形。     

波浪谱形对不规则波数值模拟的影响

通过数值模拟分析了波浪谱形对不规则波浪数值模拟结果的影响.采用不同参数的JONSWAP谱模拟入射波要素,基于抛物型缓坡方程模拟不规则波浪的传播,分析了波浪谱形状对波浪数值模拟结果的影响.结果表明,采用抛物型缓坡方程模拟不规则波浪时,入射波浪谱形对模拟结果影响不明显;但由于模型中非线性项的影响,采用不规则波模拟的波高分布和采用规则波模拟的结果略有差别.     

Combined refraction and diffraction of short waves using the finite element method

A two-dimensional finite element numerical model is presented that calculates combined refraction and diffraction of short waves. The wave equation solved governs the propagation of periodic, small amplitude surface gravity waves over a variable depth seabed of mild slope. An efficient computational scheme is employed that allows the solution of practical problems that typically require large computational grids. Comparisons are presented between the finite element model calculations and an analytical solution, a two-dimensional numerical solution, a three-dimensional numerical solution, and measurements from a hydraulic model.     

Numerical Simulation of Sand Shoreline Evolution Under Action of Regular Waves and Irregular Waves

Based on the singleline theory, a numerical simulation is presented to predict the shoreline evolution on sand beach. A parabolic equation of longshore sediment transport and boundary conditions are proposed. The combined effect of wave diffraction and refraction on the shoreline evolution on the downdrift side of the breakwater is taken into account and is calculated using the theory of regular waves and irregular waves. The present model is verified by the field observation data of erosion for half a year on the downdrift side of a harbor, and compared with some experimental results. The numerical results are in good agreement with the field measured and experimental data.     

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.     

Refraction and diffraction of random waves through breakwater

A physical model study of combined refraction and diffraction of waves through a breakwater gap at different incident angles was conducted. Both regular and random waves with narrow and broad frequency and direction spreading were studied. Besides the presence of a mild bottom slope in the lee of the breakwater, the distribution of wave heights across the width of a navigation channel inside the model harbor was also simulated. In addition to contributing to an understanding of the phenomenon of refraction and diffraction of random waves, the relatively complete set of data obtained can serve as a benchmark for testing of numerical models.     

基于抛物型缓坡方程模拟近岸植被区波浪传播

植被对波浪传播运动有重要影响。考虑近岸波浪在植被区传播中的折射、绕射、破碎及植被引起的波能耗损效应,基于抛物型缓坡方程建立了模拟近岸植被区波浪传播的数学模型,对模型进行了数值模拟验证,采用数值模拟试验分析了植被对波浪传播的影响。数值模拟结果表明,波浪在近岸植被区传播时,随着植被密度和植被高度的增加,波浪传播中的波高衰减增大,波能耗损增加;不同周期波浪在植被区传播中的波高衰减过程也明显不同。     

港池中波浪折射、绕射的数值模拟方法

从流体运动方程和动量方程出发,引入海底摩擦力和港池不完全反射作用,推导出了Berkhoff折射、绕射方程。在深海部分解析方法用精确解(含待定系数)、在复杂地形用数值离散方法求解,中间过度段用的光滑匹配将离散数值(有限元)和解析解(精确解)一同求解。通过极值问题建立泛函,利用泛函的驻定性将海岸(港湾)问题进行数值离散,建立了可行的数值模拟模型。     

Improved boundary equations for elliptic water wave models

Approximate equations for the elliptic mild slope equation are derived based on Padé approximation and used as absorbing boundaries. The new boundary equations can absorb the incident waves for high range of wave incidence angle. In addition, the new boundary equations can be used iteratively to refine the solution by eliminating reflections from the boundaries. An iterative conjugate gradient scheme has been used to solve the elliptic water wave equation and the new boundary equations. The model developed can accommodate for wave diffraction, refraction and reflections from structures with wide range of wave angles at the boundaries. The new model has been tested for several cases. The model compares very well with other models.     

海岸悬沙运移数学模型

根据适合大范围缓变地形情况下,不恒定、非均匀流场中随机波折绕射联合数值模式及波浪作用下的二维浅水环流方程和悬沙扩散方程,建立了波浪、潮流共同作用下二维悬沙数学模型,并将该模型应用于渤海湾北部海域,进行了波、流共同作用下航道疏浚弃土的悬沙扩散、运移及海床演变的数值模拟,为工程单位决策提供科学依据     

Determination of wave energy dissipation factor and numerical simulation of wave height in the surf zone

Based on the time-dependent mild slope equation including the effect of wave energy dissipation, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation. The wave height of regular and irregular waves is numerically simulated by use of the parabolic mild slope equation considering the energy dissipation due to wave breaking. Comparison of numerical results with experimental data shows that the expression for the energy dissipation factor is reasonable. The effects of the wave breaking coefficient on the breaking point and the distribution of wave height after breaking are discussed through the study of a specific experimental topography.     

On the modeling of wave propagation on non-uniform currents and depth

By transforming two different time-dependent hyperbolic mild slope equations with dissipation term for wave propagation on non-uniform currents into wave-action conservation equation and eikonal equation, respectively, shown are the different effects of dissipation term on the eikonal equation in the two different mild slope equations. The performances of intrinsic frequency and wave number are also discussed. Thus the suitable mathematical model is chosen in which the wave number vector and intrinsic frequency are expressed both more rigorously and completely. By using the perturbation method, an extended evolution equation, which is of time-dependent parabolic type, is developed from the time-dependent hyperbolic mild slope equation which exists in the suitable mathematical model, and solved by using the alternating direction implicit (ADI) method. Presented is the numerical model for wave propagation and transformation on non-uniform currents in water of slowly varying topography. From the comparisons of the numerical solutions with the theoretical solutions of two examples of wave propagation, respectively, the results show that the numerical solutions are in good agreement with the exact ones. Calculating the interactions between incident wave and current on a sloping beach [Arthur, R.S., 1950. Refraction of shallow water waves. The combined effects of currents and underwater topography. EOS Transactions, August 31, 549–552], the differences of wave number vector between refraction and combined refraction–diffraction of waves are discussed quantitatively, while the effects of different methods of calculating wave number vector on numerical results are shown.     

Numerical study on water waves and wave-induced longshore currents in Obaky coastal water

In this paper, the water waves and wave-induced longshore currents in Obaky coastal water which is located at the Mediterranean coast of Turkey were numerically studied. The numerical model is based on the parabolic mild-slope equation for coastal water waves and the nonlinear shallow water equation for the wave-induced currents. The wave transformation under the effects of shoaling, refraction, diffraction and breaking is considered, and the wave provides radiation stresses for driving currents in the model. The numerical results for the water wave-induced longshore currents were validated by the measured data to demonstrate the efficiency of the numerical model. Then the water waves and longshore currents induced by the waves from main directions were numerically simulated and analyzed based on the numerical results. The numerical results show that the movement of the longshore currents was different while the wave propagated to a coastal zone from different directions.     

Analytic study to wave scattering by a general Homma island using the explicit modified mild-slope equation

In this paper, an exact analytic solution in terms of Taylor series to the explicit modified mild-slope equation (EMMSE) for wave scattering by a general Homma island is constructed and the convergence of the series solution is analyzed. To validate the new analytic solution, comparisons are made against the existing solutions including analytic solutions to both the long-wave equation and Helmholtz equation, approximate analytic solutions to the modified mild-slope equation, numerical solutions to the mild-slope equation and experimental solutions. Because of the use of the governing equation EMMSE together with mass-conserving matching conditions along the toe of the shoal, the present model is valid for not only waves in the whole spectrum from long waves to short waves but also bathymetries with the maximal seabed slope being as high as 4.27:1. Since the general Homma island is an extension of the original Homma island, the present solution can be very conveniently used to study the effects of bottom topography on combined refraction and diffraction. It is found that the larger the shoal size is, the more significant the wave amplification against the cylinder is.     

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.     

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.     

Experimental study of the hydrodynamic performance of an onshore wave power device in the presence of an underwater mound

The hydrodynamic functioning of an oscillating water column (OWC) in the presence of an underwater tri-dimensional mound (UTDM) through large-scale ocean engineering basin experiments is described. Experiments are carried out with both regular and irregular waves and are compared to numerical models. The analysis is based on the measurements of the wave amplification in the water column for the OWC performance and on surface deformation upwave and over the UTDM for the wave transformation due to both UTDM and OWC. A significant increase of the capture-width ratio due to wave focusing above the mound is observed experimentally. This wave focusing is also well described numerically with a refraction–diffraction model. The wave amplification in the water column for both regular and irregular waves is compared to results from a linear potential model based on an integral matching method. Linear behaviour of the hydrodynamic response of the device is verified for both open and partially closed conditions, in particular for irregular waves.     

Parabolic Approximation of the Weakly Nonlinear Mild Slope Equation with Bottom Friction

This paper presents a refined parabolic approximation model of the mild slope equation to simu-late the combination of water wave refraction and diffraction in the large coastal region.The bottom frictionand weakly nonlinear term are included in the model.The difference equation is established with the Crank-Nicolson scheme.The numerical test shows that some numerical prediction results will be inaccurate in com-plicated topography without considering weak nonlinearity;the bottom friction will make wave height damp-ing and it can not be neglected for calculation of wave field in large areas.