Simulation of 2-D nonlinear waves using finite element method with cubic spline approximation

The estimation of forces and responses due to the nonlinearities in ocean waves is vital in the design of offshore structures, as these forces and responses would result in the extreme loads. Simulation of such events in a laboratory is quite laborious. Even for the preparation of the driving signals for the wave boards, one needs to resort to numerical models. In order to achieve this task, the two-dimensional time domain nonlinear problem has received considerable attention in recent years, in which a mixed Eulerian and Lagrangian method (MEL) is being used. Most of the conventional methods need the free surface to be smoothed or regridded at a particular/every time step of the simulation due to Lagrangian characteristics of motion even for a short time. This would cause numerical diffusion of energy in the system after a long time. In order to minimize this effect, the present study aims at fitting the free surface using a cubic spline approximation with a finite element approach for discretizing the domain. By doing so, the requirement of smoothing/regridding becomes a minimum. The efficiency of the present simulation procedure is shown for the standing wave problem. The application of this method to the problem of sloshing and wave interaction with a submerged obstacle has been carried out.     

Evolution of wide-spectrum unidirectional wave groups in a tank: an experimental and numerical study

Evolution of unidirectional nonlinear wave groups with wide spectra is studied experimentally and numerically. As an example of such an evolution, focusing of an initially wide wave train that is modulated both in amplitude and in frequency, to a single steep wave at a prescribed location along the laboratory wave tank is investigated. When numerous frequency harmonics arrive at the focusing location in phase, a very wave steep single emerges. The experimental study was carried out in two wave flumes that differ in size by an order of magnitude: a 330 m long Large Wave Channel in Hanover, and in 18 m long Tel-Aviv University wave tank. The spatial version of the Zakharov equation was applied in the numerical simulations. Detailed quantitative comparison is carried out between the experimental results and the numerical simulations. Spectra of the 2nd order bound waves are calculated using the theoretical model adopted. It is demonstrated that with the contribution of bound waves accounted for, a very good agreement between experiments and simulations is achieved.     

Pseudo-Spectral simulation of 1D nonlinear propagation in heterogeneous elastic media

In this work, a 1D Pseudo-Spectral Time Domain (PSTD) algorithm has been developed for solving elastic wave equation in nonlinear heterogeneous solids using FFTs for calculation of the spatial differential operator on staggered grid. The solver uses a staggered fourth order Adams–Bashforth method, by which stress and particle velocity are updated at alternating half time steps, to integrate forward in time. To circumvent wraparound inherent to FFT-based pseudo-spectral simulation, Convolution Perfectly Matched Layer (CPML) boundary condition has been used to eliminate implementation problems linked in classical PML to the introduction in nonlinear elasticity of a time dependent bulk modulus. Different kinds of nonlinear elastic models (quadratic and cubic nonlinearity, Nazarov hysteretic nonlinearity, bi-modular nonlinearity, PM-Space nonlinearity) have been implemented. The present study will focus on the comparison of nonlinear signature (harmonics generation, shock, frequency shift and attenuation) of these different kinds of nonlinearity for rod resonance, shock wave generation. These results are expected to be useful in helping to determine the predominant nonlinear mechanism in a specific experiment.     

Numerical simulation of fully nonlinear irregular wave tank in three dimension

A fully nonlinear irregular wave tank has been developed using a three‐dimensional higher‐order boundary element method (HOBEM) in the time domain. The Laplace equation is solved at each time step by an integral equation method. Based on image theory, a new Green function is applied in the whole fluid domain so that only the incident surface and free surface are discretized for the integral equation. The fully nonlinear free surface boundary conditions are integrated with time to update the wave profile and boundary values on it by a semi‐mixed Eulerian–Lagrangian time marching scheme. The incident waves are generated by feeding analytic forms on the input boundary and a ramp function is introduced at the start of simulation to avoid the initial transient disturbance. The outgoing waves are sufficiently dissipated by using a spatially varying artificial damping on the free surface before they reach the downstream boundary. Numerous numerical simulations of linear and nonlinear waves are performed and the simulated results are compared with the theoretical input waves. Copyright © 2006 John Wiley & Sons, Ltd.     

Multiple scales analysis of wave�Cwave interactions in?a?cubically nonlinear monoatomic chain

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass?Cspring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave?Cwave interaction problems.     

Simulation of multi-cracks in solids using nonlinear elastic wave spectroscopy with a time-reversal process

In non-classical nonlinear media, much characteristic information is contained in their dynamic elastic responses. A method combining nonlinear elastic wave spectroscopy (NEWS) with a time-reversal (TR) process is used in this numerical study, in which the presence of one defect and two defects acting with non-classical nonlinearity in an attenuating medium is simulated. Nonlinear defect behavior is introduced using a modified Preisach–Mayergoyz (PM) model. Two methods are used to determine retrofocal quality: harmonic filtering and modulated wave filtering. In the simulation, the nonlinear signal is filtered from the received continuous wave, then reversed and re-sent; a crack image can be obtained from the nonlinear signal in a lossy solid. By comparison with the actual defect, the image can reflect the distribution of one or two flaws, which show the feasibility and value of the NEWS–TR methodology for microdamage imaging of two defects. These results also show that images obtained with different harmonic and modulated frequencies can reflect the presence of defects. With increasing frequency, the crack positions obtained from the image change due to the influence of solid loss and interaction with sound waves.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

孤立波与多个潜水障碍物相互作用的数值研究

本文以完整二维Navier-Stokes方程为控制方程,采用VOF界面跟踪技术和差分方法,数值计算孤立波与多个淹没水下物体相互作用的问题。本文对潜水物体的高度接近水深时,孤立波通过水下孤立直立方柱、两个间距较大的水下直立方柱和两个间距较小的水下直立方柱等三种情况分别进行了计算,给出了波形随时间的演化图,可以看到反射波、前传波和跟随振荡型小波列的生成。对孤立波通过水下孤立直立方柱情形的计算结果,与实验结果和势流理论结果进行了比较。     

Large eddy simulation of boundary layer flow under cnoidal waves

Water waves in coastal areas are generally nonlinear, exhibiting asymmetric velocity profiles with different amplitudes of crest and trough. The behaviors of the boundary layer under asymmetric waves are of great significance for sediment transport in natural circumstances. While previous studies have mainly focused on linear or symmetric waves, asymmetric wave-induced flows remain unclear, particularly in the flow regime with high Reynolds numbers.Taking cnoidal wave as a typical example of asymmetric waves, we propose to use an infinite immersed plate oscillating cnoidally in its own plane in quiescent water to simulate asymmetric wave boundary layer. A large eddy simulation approach with Smagorinsky subgrid model is adopted to investigate the flow characteristics of the boundary layer. It is verified that the model well reproduces experimental and theoretical results. Then a series of numerical experiments are carried out to study the boundary layer beneath cnoidal waves from laminar to fully developed turbulent regimes at high Reynolds numbers, larger than ever studied before.Results of velocity profile, wall shear stress, friction coefficient, phase lead between velocity and wall shear stress, and the boundary layer thickness are obtained. The dependencies of these boundary layer properties on the asymmetric degree and Reynolds number are discussed in detail.     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.     

基于Boussinesq方程的波浪模型

从欧拉方程出发,提供了另一种推导完全非线性Boussinesq方程的方法,并对方程的 线性色散关系和线性变浅率进行了改进. 改进后方程的线性色散关系达到了一阶Stokes波 色散关系的Pad\'{e}[4,4]近似,在相对水深达1.0的强色散波浪时仍保持较高的准确性,并且方程的非线性和线性 变浅率都得到了不同程度的改善. 方程的水平一维形式用预估-校正的有限差分格式求解, 建立了一个适合较强非线性波浪的Boussinesq波浪数值模型. 作为验证,模拟了波浪在潜 堤上的传播变形,计算结果和实验数据的比较发现两者符合良好.     

A horizontally curvilinear non‐hydrostatic model for simulating nonlinear wave motion in curved boundaries

A horizontally curvilinear non‐hydrostatic free surface model that embeds the second‐order projection method, the so‐called θ scheme, in fractional time stepping is developed to simulate nonlinear wave motion in curved boundaries. The model solves the unsteady, Navier–Stokes equations in a three‐dimensional curvilinear domain by incorporating the kinematic free surface boundary condition with a top‐layer boundary condition, which has been developed to improve the numerical accuracy and efficiency of the non‐hydrostatic model in the standard staggered grid layout. The second‐order Adams–Bashforth scheme with the third‐order spatial upwind method is implemented in discretizing advection terms. Numerical accuracy in terms of nonlinear phase speed and amplitude is verified against the nonlinear Stokes wave theory over varying wave steepness in a two‐dimensional numerical wave tank. The model is then applied to investigate the nonlinear wave characteristics in the presence of dispersion caused by reflection and diffraction in a semicircular channel. The model results agree quantitatively with superimposed analytical solutions. Finally, the model is applied to simulate nonlinear wave run‐ups caused by wave‐body interaction around a bottom‐mounted cylinder. The numerical results exhibit good agreement with experimental data and the second‐order diffraction theory. Overall, it is shown that the developed model, with only three vertical layers, is capable of accurately simulating nonlinear waves interacting within curved boundaries. Copyright © 2011 John Wiley & Sons, Ltd.     

Improvement of moving particle semi‐implicit method for simulation of progressive water waves

Precise simulation of the propagation of surface water waves, especially when involving breaking wave, takes a significant place in computational fluid dynamics. Because of the strong nonlinear properties, the treatment of large surface deformation of free surface flow has always been a challenging work in the development of numerical models. In this paper, the moving particle semi‐implicit (MPS) method, an entirely Lagrangian method, is modified to simulate wave motion in a 2‐D numerical wave flume preferably. In terms of consecutive pressure distribution, a new and simple free surface detection criterion is proposed to enhance the free surface recognition in the MPS method. In addition, a revised gradient model is deduced to diminish the effect of nonuniform particle distribution and then to reduce the numerical wave attenuation occurring in the original MPS model. The applicability and stability of the improved MPS method are firstly demonstrated by the calculation of hydrostatic problem. It is revealed that these modifications are effective to suppress the pressure oscillation, weaken the local particle clustering, and boost the stability of numerical algorithm. It is then applied to investigate the propagation of progressive waves on a flat bed and the wave breaking on a mild slope. Comparisons with the analytical solutions and experimental results indicate that the improved MPS model can give better results about the profiles and heights of surface waves in contrast with the previous MPS models. Copyright © 2017 John Wiley & Sons, Ltd.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

On the analysis of 2D nonlinear gravity waves with a fully nonlinear numerical model

A fully nonlinear numerical method, developed on the basis of Euler equations, is used to study the dynamics of nonlinear gravity waves, mainly in the aspects of the propagation of Stokes wave with disturbed sidebands, the evolution of one wave packet and the interaction of two wave groups. These cases have previously been studied with the higher order spectral method, which will be an approximately fully nonlinear scheme if the order of nonlinearity is not large enough, while the present method in the case of the 2D model has an integration scheme that is exact to the computer precision. As expected, in most cases the results are consistent between these two numerical models and it is confirmed again that this fully nonlinear numerical model is also capable of maintaining a high accuracy and good convergence, particularly in the long-term evolutionary process.     

A second order stationary solution of the three dimensional wave diffraction potential

A theory on the second order wave diffraction by a three dimensional body fixed in a regular sea has been developed in the present paper. By regarding the sinusoidal disturb potential as a stationary solution of an initial value problem, and using Laplace transformation method and Tauberian theorem, the boundary value problems of stationary solution of the first and second order diffraction potential have been derived in this paper. Furthermore, the explicit solution of the second order stationary diffraction potential has been obtained with the method of the double Fourier transformation. It is found that the asymptotic behaviour of the second order stationary solution at far field is dependent on two wave systems, the first is “free wave”, travelling independently of the first order wave system, the other is “phase locked waves”, which accompany the first order waves. At the same time, the radiation conditions of the second order diffraction problems are derived. We also find that one can still pursue a steady state formulation with the inclusion of Rayleigh damping. Finally, as an example, the second order wave forces upon a fixed vertical circular cylinder have been calculated, and the numerical results agree well with the experimental data.     

Nonlinear wave propagation and reflection — Comparing the numerics with the analytics

The accuracy of numerical methods needs always a special attention. In this paper, analytical and numerical methods have been compared to describe the initial stage of nonlinear propagation and reflection of longitudinal ultrasonic waves. The perturbation method has been used to derive the analytical solution and the finite difference scheme to find the numerical solution for multiple free-boundary reflections of a harmonic burst at ultrasonic frequencies. The comparison of results at relatively small nonlinearities reveals a good qualitative and quantitative agreement between the analytical and numerical solutions. The method for determining analytically the exact region of interaction for counter-propagating waves is outlined in detail. At higher frequencies and larger nonlinear effects some quantitative differences between analytical and numerical results appear. The results are applicable in modelling nonlinear wave motion, including NDT and nonlinear one-dimensional vibrations.     

Suppression of transients in wave-excited response testing

When a floating body in a wave tank has low hydrodynamic damping, for example in the heave mode, very long duration transient responses can arise if it is excited from a state of rest by sinusoidal waves. Such behaviour can be undesirable when steady state response characteristics are the object of investigation in a numerical tank, because of the consequential need for very long computations. The present paper develops a method for suppressing such transient behaviour in computational models. The success of the approach is demonstrated in the context of the heaving motion of a simple buoy. A linear model of such a buoy initially at rest in a wave tank, excited by propagating sinusoidal waves, is used here for a preliminary investigation of the removal of transients. The technique is then incorporated into a fully nonlinear potential flow simulation of the buoy, and the approach is shown to be effective.     

直立码头前船波浪力耦合计算模型

建立了外域用Boussinesq方程、内域用刚体运动方程的直立码头前二维船剖面波浪力的时 域计算耦合模型,内域与外域在交界面的匹配条件是流量连续和压力相等. 进行了相关模型 实验,并把计算结果与实验结果进行了对比. 推导了船体与水底和直立码头之间间隙内流体 运动的自振频率,研究了间隙内流体运动的共振现象.     

基于VPM-THINC/QQ模型的波浪高保真模拟

为实现波浪传播的高保真数值模拟,采用包含单元均值和点值(volume-average/point-value method,VPM)的有限体积法求解纳维-斯托克斯方程和具有二次曲面性质和高斯积分的双曲正切函数(THINC method with quadratic surface representation and Gaussian quadrature,THINC/QQ)方法来重构自由面,建立以开源求解库OpenFOAM底层函数库为基础的VPM-THINC/QQ模型. 在本模型中添加推板造波法实现波浪的产生功能,采用松弛法实现消波功能,构建高精度黏性流数值波浪水槽. 分别采用VPM-THINC/QQ模型和InterFoam求解器(OpenFOAM软件包中广泛使用的多相流求解器)开展规则波的数值模拟,重点探究网格大小和时间步长等因素对波浪传播过程的影响,定量地分析波高衰减程度;为验证本模型的适应性,对长短波进行模拟. 结果表明,在相同网格大小或时间步长条件下,VPM-THINC/QQ模型的预测结果与参考值吻合较好,波高衰减较少,且无相位差,在波浪传播过程的模拟中呈现出良好的保真性. 本文工作 为波浪传播的模拟研究提供了一种高精度的黏性数值波浪水槽模型.