Second-order wavemaker theory for multidirectional waves

The present paper develops the complete second-order wavemaker theory for the generation of multidirectional waves in a semi-infinite basin. The theory includes superharmonics and subharmonics and is valid for a rotational as well as a translatory serpent-type wave-board motion. The primary goal is to obtain the second-order motion of the wave paddles required to get a prescribed multidirectional irregular wave field correct to second order, i.e. to suppress spurious free-wave generation. The wavemaker theory is a 3D extension of the full second-order wavemaker theory for wave flumes by Schäffer (1996).     

Establishment of Numerical Wave Flume Based on the Second-Order Wave-Maker Theory

With growing computational power, the first-order wave-maker theory has become well established and is widely used for numerical wave flumes. However, existing numerical models based on the first-order wave-maker theory lose accuracy as nonlinear effects become prominent. Because spurious harmonic waves and primary waves have different propagation velocities, waves simulated by using the first-order wave-maker theory have an unstable wave profile. In this paper, a numerical wave flume with a piston-type wave-maker based on the second-order wave-maker theory has been established. Dynamic mesh technique was developed. The boundary treatment for irregular wave simulation was specially dealt with. Comparisons of the free-surface elevations using the first-order and second-order wave-maker theory prove that second-order wave-maker theory can generate stable wave profiles in both the spatial and time domains. Harmonic analysis and spectral analysis were used to prove the superiority of the second-order wave-maker theory from other two aspects. To simulate irregular waves, the numerical flume was improved to solve the problem of the water depth variation due to low-frequency motion of the wave board. In summary, the new numerical flume using the second-order wave-maker theory can guarantee the accuracy of waves by adding an extra motion of the wave board. The boundary treatment method can provide a reference for the improvement of nonlinear numerical flume.     

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.     

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.     

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.     

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.     

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-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.     

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.     

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.     

Analysis of hydrodynamic behaviors of gravity net cage in irregular waves

Based on the lumped-mass method and rigid-body kinematics theory, a mathematical model of a gravity cage system attacked by irregular waves is developed to simulate the hydrodynamic response of cage system, including the maximum tension of mooring lines and the motion of float collar. The normalized response amplitudes (response amplitude operators) are calculated for the cage motion response in heave and surge, and the mooring line tension response, in regular waves. In addition, a statistical approach is taken to determine the motion and tension transfer functions in irregular waves. In order to validate the numerical model of a gravity cage attacked by irregular waves, numerical predictions have been compared with the experimental observations in the time and frequency domain. The effect of wave incident angle on the float collar motion, mooring line tension and net volume reduction of the gravity cage system in irregular waves is also investigated. The results show that at high frequencies, the cage system has no significant heave motion. It tends to contour itself to longer waves. The variation amplitude of mooring line forces decreases as the wave frequency increases. With the increasing of wave incident angle, the horizontal displacement of the float collar increases.     

The elemination of re-reflected waves by a porous medium of finite thickness

The elimination of re-reflected waves in a wave channel by installing a porous medium in front of the wavemaker is investigated. The thickness of the porous wall required to eliminate the re-reflected waves is shown to be related to th porosity, friction coefficient, and wave period, as well as to both the positions of the porous medium and the test structure. However, this study indicates that the goal of eliminating re-reflected waves can be achieved by simply varying the thickness of the porous medium according to the wave period, with all the other factors arbitrarily selected.Assuming that the oscillation amplitude of the wavemaker board is constant, the primitive wave amplitude, before reaching the porous medium, becomes smaller as the wave period is increased. In addition, the study found that the required thickness of the porous medium for eliminating the re-reflected wave becomes larger as the wave period is increased. This results in a trend which further reduces the wave amplitude after the wave passes through the porous medium. In consequence, the oscillation amplitude of a wavemaker board has to be adjusted in a larger scale if the wave period is to be increased.     

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.     

An Improved Coupling of Numerical and Physical Models for Simulating Wave Propagation简

An improved coupling of numerical and physical models for simulating 2D wave propagation is developed in this paper. In the proposed model, an unstructured finite element model （FEM） based Boussinesq equations is applied for the numerical wave simulation, and a 2D piston-type wavemaker is used for the physical wave generation. An innovative scheme combining fourth-order Lagrange interpolation and Runge-Kutta scheme is described for solving the coupling equation. A Transfer function modulation method is presented to minimize the errors induced from the hydrodynamic invalidity of the coupling model and/or the mechanical capability of the wavemaker in area where nonlinearities or dispersion predominate. The overall performance and applicability of the coupling model has been experimentally validated by accounting for both regular and irregular waves and varying bathymetry. Experimental results show that the proposed numerical scheme and transfer function modulation method are efficient for the data transfer from the numerical model to the physical model up to a deterministic level.     

An Improved Coupling of Numerical and Physical Models for Simulating Wave Propagation

An improved coupling of numerical and physical models for simulating 2D wave propagation is developed in this paper. In the proposed model, an unstructured finite element model (FEM) based Boussinesq equations is applied for the numerical wave simulation, and a 2D piston-type wavemaker is used for the physical wave generation. An innovative scheme combining fourth-order Lagrange interpolation and Runge-Kutta scheme is described for solving the coupling equation. A Transfer function modulation method is presented to minimize the errors induced from the hydrodynamic invalidity of the coupling model and/or the mechanical capability of the wavemaker in area where nonlinearities or dispersion predominate. The overall performance and applicability of the coupling model has been experimentally validated by accounting for both regular and irregular waves and varying bathymetry. Experimental results show that the proposed numerical scheme and transfer function modulation method are efficient for the data transfer from the numerical model to the physical model up to a deterministic level.     

A new method for the generation of second-order random waves

In order to prevent the generation of spurious free sub- and superharmonics of random waves in a laboratory channel, the control signal for the wave board has to be derived according to a higher-order wave theory. An expression for this control signal has been derived with the perturbation method of multiple scales. It is much less complex and requires less computation time than the expressions obtained from the full second-order theory. The new method for second-order subharmonics was verified experimentally for waves with bichromatic and continuous first-order spectra. The data were analysed with the complex-harmonic principal-component analysis to reduce the influence of noise.     

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.     

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.     

Nonlinear wave effect on the slow drift motion of a floating body

The slow drift motion of a floating body in a two-dimensional wave field has been investigated using a time-domain, fully nonlinear numerical model with non-reflective open boundaries. Preliminary computations were conducted for incident bichromatic waves, in which wave theories with different orders were applied in generating the waves required. The results show that the use of low-order theories generates undesirable free waves, and that fourth-order terms contribute markedly to low-frequency input. The motion of a rectangular floating body in response to nonlinear bichromatic waves was computed. The numerical results for small-amplitude incident waves agree reasonably well with the second-order approximation for both the steady and difference-frequency (Δσ) components in the body's motion. For relatively large waves, however, the 2Δσ component becomes predominant compared with the Δσ component. The motion of the body in irregular waves with different wave parameters has also been presented in order to discuss the validity range of a second-order approximation.