Second-order wavemaker theory for irregular waves

Through the last decade the theory for second-order irregular wave generation was developed within the framework of Stokes wave theory. This pioneering work, however, is not fully consistent. Furthermore, due to the extensive algebra involved, the derived transfer functions appear in an unnecessarily complicated form. The present paper develops the full second-order wavemaker theory (including superharmonics as well as subharmonics) valid for rotational as well as translatory wave board motion. The primary goal is to obtain the second-order motion of the wave paddle required in order to get a spatially homogeneous wave field correct to second order, i.e. in order to suppress spurious free-wave generation. In addition to the transfer functions developed in the line of references on which the present work is based, some new terms evolve. These are related to the first-order evanescent modes and accordingly they are significant when the wave board motion makes a poor fit to the velocity profile of the desired progressive wave component. This is, for example, the case for the high-frequency part of a primary wave spectrum when using a piston-type wavemaker. The transfer functions are given in a relatively simple form by which the computational effort is reduced substantially. This enhances the practical computation of second-order wavemaker control signals for irregular waves, and no narrow band assumption is needed. The software is conveniently included in a PC-based wave generation system—the DHI Wave Synthesizer. The validity of the theory is demonstrated for a piston type wavemaker in a number of laboratory wave experiments for regular waves, wave groups and irregular waves.     

Generation and propagation of transient nonlinear waves in a wave flume

A semi-analytical nonlinear wavemaker model is derived to predict the generation and propagation of transient nonlinear waves in a wave flume. The solution is very efficient and is achieved by applying eigenfunction expansions and FFT. The model is applied to study the effect of the wavemaker and its motion on the generation and propagation of nonlinear waves. The results indicate that the linear wavemaker theory may be applied to predict only the generation of waves of low steepness for which the nonlinear terms in the kinematic wavemaker boundary condition and free-surface boundary conditions are of secondary importance. For waves of moderate steepness and steep waves these nonlinear terms have substantial effects on wave profile and wave spectrum just after the wavemaker. A wave spectrum corresponding to a sinusoidally moving wavemaker possesses a multi-peak form with substantial nonlinear components, which disturbs or may even exclude physical modeling in wave flumes. The analysis shows that the widely recognized weakly nonlinear wavemaker theory may only be applied to describe the generation and propagation of waves of low steepness. This is subject to further restrictions in shallow and deep waters because the kinematic wavemaker boundary condition as well as the nonlinear interaction of wave components and the evolution of wave energy spectrum is not properly described by weakly nonlinear wavemaker theory. Laboratory experiments were conducted in a wave flume to verify the nonlinear wavemaker model. The comparisons show a reasonable agreement between predicted and measured free-surface elevation and the corresponding amplitudes of Fourier series. A reasonable agreement between theoretical results and experimental data is observed even for fairly steep waves.     

Approximate Stream Function wavemaker theory for highly non-linear waves in wave flumes

An approximate Stream Function wavemaker theory for highly non-linear regular waves in flumes is presented. This theory is based on an ad hoc unified wave-generation method that combines linear fully dispersive wavemaker theory and wave generation for non-linear shallow water waves. This is done by applying a dispersion correction to the paddle position obtained for non-linear long waves. The method is validated by a number of wave flume experiments while comparing with results of linear wavemaker theory, second-order wavemaker theory and Cnoidal wavemaker theory within its range of application.     

Deterministic combination of numerical and physical coastal wave models

A deterministic combination of numerical and physical models for coastal waves is developed. In the combined model, a Boussinesq model MIKE 21 BW is applied for the numerical wave computations. A piston-type 2D or 3D wavemaker and the associated control system with active wave absorption provides the interface between the numerical and physical models. The link between numerical and physical models is given by an ad hoc unified wave generation theory which is devised in the study. This wave generation theory accounts for linear dispersion and shallow water non-linearity. Local wave phenomena (evanescent modes) near the wavemaker are taken into account. With this approach, the data transfer between the two models is thus on a deterministic level with detailed wave information transmitted along the wavemaker.     

Second-order theory for coupling 2D numerical and physical wave tanks: Derivation,evaluation and experimental validation

A full second-order theory for coupling numerical and physical wave tanks is presented. The ad hoc unified wave generation approach developed by Zhang et al. [Zhang, H., Schäffer, H.A., Jakobsen, K.P., 2007. Deterministic combination of numerical and physical coastal wave models. Coast. Eng. 54, 171–186] is extended to include the second-order dispersive correction. The new formulation is presented in a unified form that includes both progressive and evanescent modes and covers wavemaker configurations of the piston- and flap-type. The second order paddle stroke correction allows for improved nonlinear wave generation in the physical wave tank based on target numerical solutions. The performance and efficiency of the new model is first evaluated theoretically based on second order Stokes waves. Due to the complexity of the problem, the proposed method has been truncated at 2D and the treatment of regular waves, and the re-reflection control on the wave paddle is also not included. In order to validate the solution methodology further, a series of nonlinear, periodic waves based on stream function theory are generated in a physical wave tank using a piston-type wavemaker. These experiments show that the new second-order coupling theory provides an improvement in the quality of nonlinear wave generation when compared to existing techniques.     

Simulation of directional waves in a numerical basin by a desingularized integral equation approach

A numerical solution is developed to investigate the generation and propagation of small-amplitude water waves in a semi-infinite rectangular wave basin. The three-dimensional wave field is produced by the prescribed “snake-like” motion of an array of segmented wave generators located along the wall at one end of the tank. The solution technique is based on the boundary element approach and uses an appropriate three-dimensional Green function which explicitly satisfies the tank-wall boundary conditions. The Green function and its derivatives which appear in the integral equation formulation can be shown to be slowly convergent when the source and field points are in close proximity. Therefore, when computing the velocity potentials on the wave generators, the source points are chosen outside the fluid domain, thereby ensuring the rapid convergence of these functions and rendering the integral equations non-singular. Numerical results are shown which illustrate the influence of the various wavemaker and basin parameters on the generated wave field. Finally, the complete wave field produced by the diffraction of oblique waves by a vertical circular cylinder in a basin is presented.     

Breakpoint generated surf beat induced by bichromatic wave groups

This paper presents new experimental data on 2-D surf beat generation by a time-varying breakpoint induced by bichromatic wave groups. The experimental investigation covers a broad range of wave amplitudes, short wave frequencies, group frequencies and modulation rates. The data include measurements of incident and outgoing wave amplitudes, breakpoint position, shoreline run-up and the cross-shore structure of both the short and long wave motion. Surf beat generation is shown to be in good agreement with theory [Symonds, G., Huntley, D.A., Bowen, A.J., 1982. Two dimensional surf beat: long wave generation by a time-varying breakpoint. J. Geophys. Res. 87, 492–498]. In particular, surf beat generation is dependent on the normalised surf zone width, which is a measure of the phase relationship between the seaward and shoreward breakpoint forced long waves, and linearly dependent on the short wave amplitude. The cross-shore structure of the long wave motion is also consistent with theory; at maximum and minimum surf beat generation, the mean breakpoint coincides with the nodal and anti-nodal points, respectively, for a free long wave standing at the shoreline. A numerical solution, using measured data as input, additionally shows that the phase relationship between the incident bound long wave and the outgoing breakpoint forced wave is consistent with the time-varying breakpoint mechanism.     

The initialization of nonlinear water waves in a numerical wave flume

This study investigates the initialization of nonlinear free-surface simulations in a numerical wave flume.Due to the mismatch between the linear input wavemaker motion and the kinematics of fully nonlinear waves,direct numerical simulations of progressive waves,generated by a sinusoidally moving wavemaker,are prone to suffering from high-frequency wave instability unless the flow is given sufficient time to adjust.A time ramp is superimposed on the wavemaker motion at the start that allows nonlinear free-surface simulations to be initialized with linear input.The duration of the ramp is adjusted to test its efficiency for short waves and long waves.Numerical results show that the time ramp scheme is effiective to stabilize the wave instability at the start of the simulation in a wave flume.     

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.     

Second-Oorder Analytic Solutions of Nonlinear Interactions of Edge Waves on A Plane Sloping Bottom

Based on the full water-wave equation, a second-order analytic solution for nonlinear interaction of short edge waves on a constant plane sloping bottom is presented in this paper. For special case of slope angle b=p/2, this solution can be reduced to the same order solution of deep water gravity surface waves traveling along parallel coastline. Interactions between two edge waves including progressive, standing and partially reflected standing waves were also discussed. The unified analytic expressions with transfer functions for kinematic-dynamic elements of edge waves were also discussed. The random model of the unified wave motion processes for linear and nonlinear irregular edge waves is formulated, and the corresponding theoretical autocorrelation and spectral density functions of the first and second orders are derived. The boundary conditions for the determining determination of the parameters of short edge wave are suggested, that may be seen as one special simple edge wave excitation mechanism and an extension to the sea wave refraction theory. Finally some computation results are demonstrated.     

斜向和多向不规则波对直立堤平均越浪量研究

通过三维波浪模型试验研究了斜向和多向不规则波对直立堤的越浪量。分别按平均越浪量和单波最大越浪量进行研究,探讨了平均越浪量随相对堤高、波浪方向、波浪方向分布宽度、波陡和相对水深等影响因素的变化规律,导得了斜向和多向不规则波作用于直立堤上的平均越浪量的计算公式。     

A Piston-Type Active Absorbing Wavemaker System with Delay Compensation

Physical model tests with highly reflective structures often encounter a problem of multiple reflections between the structures and the wavemaker. This paper presents a piston-type active absorbing wavemaker system which can absorb most of the reflections. Based on the first-order wavemaker theory, a frequency domain absorption transfer function is modeled. Its time realization can be achieved by designing an IIR digital filter, which is used to control the absorbing wavemaker system. In a real system, time delays often exist in the wave making process. Thus a delay compensation term to the transfer function is proposed. Experimental results show that the system performs well for both regular and irregular waves with periods from 0.6 s to 2.0 s, and the absorption capability is larger than 96.5% at target wave fields.     

Second-Order Analytic Solutions of Nonlinear Interactions of Edge Waves on A Plane Sloping Bottom

Based on the full water-wave equation,a second-order analytic solution for nonlinear interaction of short edge waves on a plane sloping bottom is presented in this paper.For special case of slope angle β=π/2,this solution can reduced to the same order solution of deep water gravity surface waves traveling along parallel coastline.Interactions between two edge waves including progressive,standing and partially reflected standing waves are also discussed.The unified analytic expressions with transfer functions for kinematic-dynamic elements of edge waves are also given.The random model of the unified wave motion processes for linear and nonlinear irregular edge waves is formulated,and the corresponding theoretical autocorrelation and spectral density functions of the first and the second orders are derived.The boundary conditions for the determination of the parameters of short edge wave are suggested,that may be seen as one special simple edge wave excitation mechanism and an extension to the sea wave refraction theory.Finally some computation results are demonstrated.     

Dynamic Responses of Pontoons Connecting Process of Mobile Offshore Base

The dynamic responses of two pontoons while connected with each other in irregular waves are calculated by means of three-dimensional (3-D) potential flow theory.The computation is to find the optimal status for connection at a certain sea state.On the basis of the relative motion of two pontoons in irregular waves,Visual FORTRAN programming language,as well as OpenGL (Open Graphics Library),is used to develop a set of virtual reality system of the relative motion of two pontoons,which is fully interactive with realistic effect.The transfinite interpolation scheme is applied for the mesh generation of wave surface,and the wave motion is simulated by surface elevation and calculated by 3-D potential flow theory.     

Extreme Wave Simulation with Iterative Adaptive Approach in Numerical Wave Flume

Extreme wave is highly nonlinear and may occur due to diverse reasons unexpectedly.The simulated results of extreme wave based on wave focusing,which were generated using high order spectrum method,are presented.The influences of the steepness,frequency bandwidth as well as frequency spectrum on focusing position shift were examined,showing that they can affect the wave focusing significantly.Hence,controlled accurate generation of extreme wave at a predefined position in wave flume is a difficult but important task.In this paper,an iterative adaptive approach is applied using linear dispersion theory to optimize the control signal of the wavemaker.The performance of the proposed approach is numerically investigated for a wide variety of scenarios.The results demonstrate that this approach can reproduce accurate wave focusing effectively.     

Fortran与Delphi混合编程及其在造波控制系统中的应用

在介绍实验室造波原理的基础上,提出了使用混合编程进行造波系统控制的思路,并且给出了一个应用Fortran和Delphi混合编程方法实现波浪水槽中不规则波生成的实验实例,结果表明,利用多语言集成方法实现海洋测量设备的自动控制具有较大优势.     

基于物理造波和主动消波的黏性数值波浪水槽开发

基于推板造波理论和摇板造波理论,在Open FOAM平台上采用重叠网格技术建立黏性数值波浪水槽,并使用一种结合SIMPLE算法和PISO算法的PIMPLE算法对数值模型进行求解。利用开发的数值模型通过数值收敛性测试和网格独立性测试分别重点研究了时间步长、库朗数和网格尺寸对数值精度和计算效率的影响。并对比研究了此数值模型分别嵌入层流模型和湍流模型的计算精度和计算效率。实现的规则波和二阶有限振幅波与理论结果和试验结果吻合,验证了此黏性数值波浪水槽的造波和主动消波功能。基于二维数值波浪水槽,进一步研究了三维数值造波,数值计算结果与理论结果吻合良好。研究结果不仅验证了重叠网格在二维和三维两相流体域中求解运动物体与流场交互的可靠性和正确性,而且为使用此黏性数值波浪水槽解决更复杂的海洋工程问题提供了依据。     

分段式造波机生成波浪的方向谱分析

采用声学多普勒流速仪(ADV)和多点测波仪的实验资料,对分段式造波机生成的多向随机波浪进行方向谱分析.结果表明,分段式造波机生成的多向随机波浪的时间平稳性优于空间均匀性;贝叶斯方法(BDM)适合于流速实验资料和阵列波浪资料的方向谱分析,给出的方向谱满足随机波浪的内禀性质(如光滑性、连续性等);流速资料和阵列波浪资料给出的方向谱具有相当的精度,但与测速的深度有关;三参量(如三点波浪、两个方向的速度分量和一点波浪等)可给出具有相当精度的方向谱;给出了衡量方向谱精度的初步标准.     

基于二阶斯托克斯波理论的辐射应力垂向分布

基于二阶斯托克斯波理论推导了辐射应力的垂向分布表达式,通过算例讨论了辐射应力在深水和有限水深条件下的垂向分布规律,并与基于微幅波理论的辐射应力进行了比较.结果表明,在波浪非线性不强时,基于二阶斯托克斯波理论的辐射应力与基于微幅波理论的辐射应力表达式计算结果接近;而当水深较浅波浪非线性较强时,基于二阶斯托克斯波理论的辐射应力在近表面处明显大于基于微幅波理论的辐射应力.采用二阶斯托克斯波理论推导的波浪辐射应力更为合理地反映了波浪非线性效应.