2-D numerical analysis of near-field flow at low-crested permeable breakwaters

This paper describes the capability of a numerical model named COrnell BReaking waves And Structures (COBRAS) [Lin, P., Liu, P.L.-F., 1998. A numerical study of breaking waves in the surf zone. Journal of Fluid Mechanics 359, 239–264; Liu, P.L.-F., Lin, P., Chang, K.A., Sakakiyama, T., 1999. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal and Ocean Engineering 125, 322–330, Liu, P.L.-F., Lin, P., Hsu, T., Chang, K., Losada, I.J., Vidal, C., Sakakiyama, T., 2000. A Reynolds averaged Navier–Stokes equation model for nonlinear water wave and structure interactions. Proc. Coastal Structures '99, 169–174] based on the Reynolds Averaged Navier–Stokes (RANS) equations to simulate the most relevant hydrodynamic near-field processes that take place in the interaction between waves and low-crested breakwaters. The model considers wave reflection, transmission, overtopping and breaking due to transient nonlinear waves including turbulence in the fluid domain and in the permeable regions for any kind of geometry and number of layers. Small-scale laboratory tests were conducted in order to validate the model, with different wave conditions and breakwater configurations. In the present study, regular waves of different heights and periods impinging on a wide-crested structure are considered. Three different water depths were tested in order to examine the influence of the structure freeboard. The experimental set-up includes a flow recirculation system aimed at preventing water piling-up at the lee of the breakwater due to overtopping. The applicability and validity of the model are examined by comparing the results of the numerical computations with experimental data. The model is proved to simulate with a high degree of agreement all the studied magnitudes, free surface displacement, pressure inside the porous structure and velocity field. The results obtained show that this model represents a substantial improvement in the numerical modelling of low-crested structures (LCS) since it includes many processes neglected previously by existing models. The information provided by the model can be useful to analyse structure functionality, structure stability, scour and many other hydrodynamic processes of interest.     

Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part II: Experimental validation in two-dimensions

This paper provides an experimental validation of the second-order coupling theory outlined by Yang et al. (Z. Yang, S. Liu, H.B. Bingham and J. Li., 2013. Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part I: Formulation, implementation and numerical properties, submitted for publication) using 2D irregular waves. This work provides a second-order dispersive correction for the physical wavemaker signal which improves the nonlinear transfer of information between the numerical and physical models compared to the first-order method of Zhang et al. (2007). The important nonlinear parameters and numerical performance were theoretically investigated in Part I. In the present Part II, careful experimental validation is carried out using a sequence of progressively more complex analytical and numerical target waves. The results demonstrate clearly that improved performance is achieved by using the second-order correction. When controlling with a second-order coupling signal, two key points are notable: (i) The higher harmonics underlying the numerical waves are accurately captured and transferred into the physical model. (ii) The second-order behavior leads to an unwanted spurious freely propagating second harmonic that is substantially reduced when compared to an identical wave paddle operating with a first-order coupling signal. Using nonlinear regular (monochromatic), bi-chromatic and irregular wave cases as well as varying coupled wave tank bathymetries, both these aspects are verified over a broad range of wave frequencies and shown to be extensively applicable to physical wave tanks.     

强非线性和色散性Boussinesq方程数值模型检验

采用同位网格有限差分法,建立了强非线性和色散性Boussinesq方程数值计算模型。以稳恒波Fourier近似解给定入射波边界条件,对均匀水深深水和浅水域不同非线性的行进波、缓坡地形上深水至浅水域的浅水变形波、以及缓坡和陡坡地形上的波浪水槽实验进行了数值计算,并将计算结果与解析解、解析数值解以及实验值进行了较为详细的比较,从而检验了模型的色散性、非线性以及不同底坡下非线性波的浅水变形性能。     

Performance and limitations of the Hilbert–Huang transformation (HHT) with an application to irregular water waves

This paper relates to the newly developed Hilbert–Huang transformation (HHT). An overview of this time-frequency analysis technique and its applications are given. Key elements of the numerical procedure and principles of the Hilbert transformation (HT) are established. A simple parameter study with trigonometric functions to get an idea about the numerical performance of the empirical mode decomposition (EMD) is performed. The main results of estimating relative standardized errors made between analytically exact defined sine waves and disintegrated intrinsic functions as well as their specific influence on each other are determined. Practical applications are carried out next to evaluate computed nonlinear irregular water waves based on Stokes perturbation expansion approach and measurements on fully nonlinear irregular water waves recorded in a laboratory wave flume. Correspondence between simulated and recorded wave trains is given for narrow-banded fundamental components. Deviations are unveiled when carrier and riding waves get broad banded. Time-dependent spectral representation shows signs of an interesting phenomenon as instantaneous frequencies and amplitudes exhibit strong correlations with water surface elevations of both numerical and measured data series.     

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.     

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.     

弱非线性斯托克斯波在非平整海底上传播的新型抛物型方程

由于缓坡方程计算量大和其本身的缓坡假定而在实际应用中受到了限制,故对斯托克斯波在非平整海底(适用于缓坡和陡坡地形)上传播的Liu和Dingemans的三阶演化方程进行抛物逼近,得到一个新的非线性抛物型方程,它能够包含同类方程未曾考虑的二阶长波效应.通过数值计算结果与Berkhoff等人的经典实验数据的比较,证明所提出的抛物型模型理论具有较高的精度.     

波浪沿斜面传播过程的数值模拟

精确模拟非线性波沿斜面传播过程非常困难,为此论文从势函数的边界积分方程出发,建立了一种时域内二维波浪模拟的数值模型,主要用来模拟完全非线性波浪的传播变形过程。论文的数值模型使用高阶二维边界元方法,采用可调节时间步长的基于二阶显式泰勒展开的混合欧拉-拉格郎日时间步进来求解带自由表面的线性或完全非线性波浪传播问题。在计算区域一端造出线性或非线性的周期性波浪,另一端采用消除反射波的人工粘性吸收边界。通过与现有理论比较证明了论文数值方法所得结果是准确可靠的。     

Numerical Model of Internal Solitary Wave Evolution on Impermeable Variable Seabed in A Stratified Two-Layer Fluid System

1.IntroductionIn mathematics and physics,a soliton is a self-reinforcing solitary wave caused by nonlinear ef-fectsinthe medium.Solitons are found in many physical phenomena(Chou and Shih,1996;Chouand Quyang,1999;Chouet al.,2003;Chenet al.,2004;Wang,2004;…     

A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.     

Stokes波的谱特性分析

应用理论推导及数值计算方法，对Stokes随机波的谱特性进行了分析。首先将波面方程，海水质点水平速度用一阶波面分量的非线性组合表示，应用平稳随机高阶短的降阶计算法则，得到了波面方程及海水质点水平速度与一阶波面分量的自相关函数之间的关系，从而确定了Stokes随机波浪的波浪谱密度及海水质点水平速度和加速度谱密度，进而求得有关波浪要素的均方根值。文章还应有数值计算方法，分析了波浪基本参数对均方根值的影响。     

Numerical simulations using momentum source wave-maker applied to RANS equation model

For Navier-Stokes equation model using the VOF scheme, Lin and Liu (Lin, P. and Liu, P.L.-F. (1999). Internal wave-maker for Navier-Stokes equations models. J. Waterw. Port Coast. Ocean Eng., 125 (4), 207–215.) developed an internal wave-maker method for which a mass source function of the continuity equation was used to generate target wave trains. Using this internal wave-maker method, various numerical experiments have been conducted without any problems due to waves reflected by a wave-maker. In this study, an internal wave-maker method using a momentum source function was proposed. Various numerical simulations in two and three dimensions were performed using the momentum source wave-maker applied to the RANS equation model in a CFD code, FLUENT. To verify their applicability in 2 dimensions, the computational results obtained using the momentum source wave-maker in a channel of constant depth were compared with the results obtained by using the mass source wave-maker and with the analytical solutions. And the results of the present numerical simulations of hydraulic experiments, which represent nonlinear waves on a submerged shoal and breaking waves on a plane beach, were compared with measurements. The comparisons showed good agreements between them. To see their applicability into 3-dimensional cases, the present results in a basin of constant depth were compared with the analytical solutions, and they agreed well with each other. In addition, vertical variation of longshore current was presented by using the 3-dimensional simulation results.     

A new kind of nonlinear mild-slope equation for combined refraction-diffraction of multifrequency waves

This paper presents the numerical solution of a new nonlinear mild-slope equation governing waves with different frequency components propagating in a region of varying water depth. There are two new nonlinear equations. The linear part of the equations is the mild-slope equation, and one of the models has the same non-linearity as the Boussinesq equations. The new equations are directly applicable to the problems of nonlinear wave-wave interactions over variable depth. The equations are first simplified with the parabolic approximation, and then solved numerically with a finite difference method. The Crank-Nicolson method is used to discretize the models. The numerical models are applied to a set of published experimental cases, which are nonlinear combined refraction-diffraction with generation of higher harmonic waves. Comparison of the results shows that the present models generally predict the measurements better than other nonlinear numerical models which have been applied to the data set.     

Spectral analysis of Stokes waves

The spectral properties of Stokes waves are shown in this paper by theoretical and numerical methods. This is done by expressing wave profiles and velocities of water particles as nonlinear combinations of the first order component of wave profiles. Under the assumption of the first order wave profiles being zero mean Gaussian processes, the relationship between autocorrelation functions of wave profiles and velocities of water particles and the first order component of wave profiles is established using the nonlinear spectral analysis. The spectral densities of nonlinear random waves, the velocities and accelerations of water particles are then obtained. Numerical computations are carried out to analyze the effect of fundamental parameters of waves. The results indicate that wave height is the most sensible parameter to the root mean squares related and wave depth is the least sensible one of all.     

基于非线性模型的畸形波模拟及其时频能量谱研究

采用VOF(volume offluid)方法实现了畸形波的数值模拟.将数值结果与线性理论(方程(5))计算结果进行了对比,发现数值模拟结果更能够反映非线性特征.使用小波分析方法研究了畸形波的时频能量谱,发现形成畸形波的过程中存在很强的波浪非线性相互作用,使得波浪的能量向高频端转化.变水深地形可以加强波浪的非线性相互作用.使得转化到高频端的能量更多,产生畸形程度更大的畸形波.     

浅水中浮体在波浪作用下运动的三维时域计算模型

港口中系泊船在波浪作用下运动问题的本质是浅水波浪与浮体的相互作用。与深水情况不同,浅水问题应当考虑水底、水域边界的影响及浅水波浪自身的特性,单一模型很难实现该模拟过程。为此,建立了Boussinesq方程计算入射波和Laplace方程计算散射波的全时域组合计算模型。有限元法求解的Boussinesq方程能使入射波充分考虑到水底、水域边界的影响和浅水波浪的特性;散射波被线性化,采用边界元法求解,并以浮体运动时的物面条件为入射波和散射波求解的匹配条件。该方法为完全的时域方法,计算网格不随时间变动,计算过程较为方便。通过与实验及其他数值方法的结果进行比较,验证了本模型对非线性波面、浮体的运动都有比较理想的计算结果,显示了本模型对非线性问题具有较好的计算能力。     

Interactions between fully nonlinear water waves and cylinder arrays in a wave tank

Fully nonlinear interactions between water waves and vertical cylinder arrays in a numerical tank are studied based on a finite element method (FEM). The three-dimensional (3D) mesh is constructed through an extension of a 2D Delaunay surface grid along the vertical line. The velocity potential is obtained by solving a linear matrix system of FEM, and a difference scheme is then used to calculate the velocity on the free surface to track its movement. Waves and hydrodynamic forces are obtained for both bottom mounted and truncated cylinders. The simulations have provided many results to show the nature of mutual interference between cylinders in arrays and its effects on waves and forces at the nearly trapped mode frequency. The effect of the tank wall on waves and forces has been investigated, and the nonlinear features of waves and forces have also been discussed.     

Boundary integral equation solutions for solitary wave generation, propagation and run-up

The boundary integral equation method (BIEM) is developed as a tool for studying two-dimensional, nonlinear water wave problems, including the phenomena of wave generation, propagation and run-up. The wave motions are described by a potential flow theory. Nonlinear free-surface boundary conditions are incorporated in the numerical formulation. Examples are given for either a solitary wave or two successive solitary waves. Special treatment is developed to trace the run-up and run-down along a shoreline. The accuracy of the present scheme is verified by comparing numerical results with experimental data of maximum run-up.     

An explicit finite difference model for simulating weakly nonlinear and weakly dispersive waves over slowly varying water depth

In this paper, a modified leap-frog finite difference (FD) scheme is developed to solve Non linear Shallow Water Equations (NSWE). By adjusting the FD mesh system and modifying the leap-frog algorithm, numerical dispersion is manipulated to mimic physical frequency dispersion for water wave propagation. The resulting numerical scheme is suitable for weakly nonlinear and weakly dispersive waves propagating over a slowly varying water depth. Numerical studies demonstrate that the results of the new numerical scheme agree well with those obtained by directly solving Boussinesq-type models for both long distance propagation, shoaling and re-fraction over a slowly varying bathymetry. Most importantly, the new algorithm is much more computationally efficient than existing Boussinesq-type models, making it an excellent alternative tool for simulating tsunami waves when the frequency dispersion needs to be considered.     

Nonlinear lifting theory for unsteady WIG in proximity to incident water waves. Part 1: Two-dimension

A nonlinear analysis is made for determining the two-dimensional unsteady potential-flow characteristics about a wing subject to wing-in-ground effect (WIG) operating above progressive water waves. The dynamic boundary condition requiring the constant pressure and the kinematic boundary condition prescribing the continuity in the vertical velocity are satisfied on the undisturbed free surface. The boundary conditions imposed on the free surface are linear, but the kinematic boundary condition satisfied on the foil surface is nonlinear. Through the derivation and evaluation of the time-domain Green's functions for two-dimensional singularities above a free surface, the influence of water waves on the lift performance of the two-dimensional WIG is addressed using the discrete vortex method. Furthermore, the roll-up of the wake vorticity is considered. The comparison of present work with other numerical results available in the literature shows the validation of the present approach.