Statistics of nonlinear wave crests and groups

Groups of large nonlinear waves with sharper higher crests can pose hazards to ships, induce harbor resonance and cause wave-overtopping of fixed and floating structures. Past interest in wave groups has mostly been focused on the statistics and modeling of linear wave groups. Studies on nonlinear wave groups are surprisingly few, and address deep water waves only. Here, statistics of nonlinear wave crests and wave-crest groups in deep and transitional water depths are considered, using an appropriate second-order representation for crest heights and the continuous wave-envelope approach. In particular, theoretical expressions describing the statistics of nonlinear wave crests and their groups are posed in the form of a simple second-order transformation of well-known results on linear waves. Predictions from the transformation so posed compare well with nonlinear wave data gathered in the North Sea, and demonstrate that nonlinearities do affect the statistics of large wave crests and their groups significantly.     

Forced convection accompanying wind waves

Wind-wave tunnel experiments reveal, by use of techniques of the flow visualization, that wind waves are accompanied by the wind drift surface current with large velocity shear and with horizontal variation of velocity relative to the wave profile. The surface current converges from the crest to a little leeward face of the crest, making a downward flow there, even though the wave is not breaking. Namely, wind waves are accompanied by forced convections relative to the crests of the waves. Since the location of the convergence and the downward flow travels on the water surface as the crest of the wave propagates, the motion as a whole is characterized by turbulent structure as well as by the nature of water-surface waves. In this meaning, the term of real wind waves is proposed in contrast with ordinary water waves. The study of real wind waves will be essential in future development of the study of wind waves.     

风压作用下非线性极端波浪的数值模拟

本文在时域非线性数值波浪水槽中,研究了不同风速条件下极端波浪的特性。采用推板造波的方式生成非线性波浪,基于Jeffrey遮蔽理论将风压项引入自由面动力学边界条件来模拟风压作用,通过高阶边界元法和混合欧拉-拉格朗日时间步进法来求解初边值问题。通过与已发表的聚焦波群实验结果对比验证了该数值模型的准确性,并研究了风压对极端波浪的最大波高、聚焦位置的偏移和波浪谱的演变等波浪性质的影响。本文进一步在数值波浪水槽中引入均匀水流,来模拟风生流对波浪演变的影响。结果表明,风压的存在会少量增大极端波浪的最大波高,波浪的聚焦和解焦过程伴随着明显的能量传递,并且风生流进一步导致了波浪聚焦位置的偏移。     

深水中极限波峰下速度场快速算法

基于五阶斯托克斯规则波理论,提出了一种快速求解深水极限波峰下速度场的数学模型.研究中,按照上跨零点和下跨零点的方法由计算或实测的极限波浪波面时间历程确定包含极限波峰的相邻两个周期的平均值为五阶斯托克斯规则波的波浪周期,然后根据极限波峰反推确定波浪入射波幅.通过与已有的数值结果和实验数据对比,验证了所建立的数值模型可以快速准确的计算出极限波峰下的速度场,相比其他模型,更适合于工程应用.     

Statistics of Wave Crest Characteristics

The statistical distribution of wave crest characteristics such as crest length, crest height, joint crest height and length are analyzed based on numerical simulation of 3-D random waves. The effects of directional functions and wave crest defining methods on crest characteristics are also studied.The results show that wave crests are no longer uniform and continuous in directional wave field; the distribution of crest length is obviously influenced by the directional function; the statistics of crest characteristics obtained by the two different methods are almost the same.     

Internal flow structure of short wind waves

The internal flow structure of wind waves in a wind-wave tunnel was investigated on the bases of the measured vorticity distributions, streamline patterns, internal pressure fields, and stress distributions at the water surface for some waves in the field. In part I the experimental method and the internal vorticity structure relative to the individual wave crests are described. The measured vorticity distributions of distinct waves (waves with waveheight comparable with or larger than that of significant wavesH _1/3) in the field indicate that the surface vorticity layer is extraordinarily thickened near the crest, and the vorticity near the water surface shows a particularly large value below the crest. The flow near the crest of distinct waves is found to be in excess of the phase speed in a very thin surface layer, and the tangential stress distribution has a dominant peak near the crest. It is argued that the occurrence of the region of high vorticity in distinct waves is associated with the local generation of vorticity near the crest by tangential stress which attains a peak, under the presence of excess flow.     

三维波峰的数值模拟

给出了三维波峰的定义,描述了波峰临界值的确定过程,分析了三维随机海浪的方向谱和三维随机海浪的数学描述,在MATLAB环境下设计了随机海面波峰的数值模拟程序,并给出了深水波峰的三维空间模型及其在不同方向传播参数和不同方向函数下平面分布的变化。     

波峰特征量的统计分布

本文在作者另文数值模拟得到的三维海浪基础上 ,进一步分析给出了三维波峰长度、高度及方向角度等特征量的统计分布。发现考虑海浪的方向性质后波峰分布不再是均匀连续的 ;波峰的长度分布受方向函数影响 ,方向分布越宽 ,波峰的平均长度越短 ,波峰的方向角分布越宽 ;波峰高度和长度在波峰高度较小时有很大的相关性 ,而在波峰高度很大时无关     

Steepness and asymmetry of extreme waves and the highest waves in deep water

Traditional wave steepness s=H/L does not define steep asymmetric waves in a random sea uniquey. Three additional parameters characterising single zero-downcross waves in a time series are crest front steepness, vertical asymmetry factor and horizontal asymmetry factor. Results for steepness and asymmetry from zero-downcross analysis of wave data obtained from full scale measurements in deep water on the Norwegian continental shelf in 58 time series are presented. The analysis demonstrates clearly the asymmetry of both “extreme waves” and the highest waves. The period and height of the highest waves are also given together with their correlation to spectral parameters. The measured maximum wave heights are also compared with predicted values of maximum wave heights showing good agreement.     

Three-dimensional experiments on landslide generated waves at a sloping coast

This paper describes new three dimensional experiments on water waves generated by landslides. The landslide is reproduced by a rigid elliptical body, sliding along an inclined plane (slope of 1/3, 1 vertical, 3 horizontal). The generated water waves are free to propagate both offshore and alongshore, since the plan dimensions of the used wave tank are of at least one order of magnitude larger than the width of the landslide, which can be considered to be a scale of the wave length. The experimental study has been carried out reproducing both subaerial and partially submerged landslides. The wave generation process is studied by means of video records of the near field flow and measurement of the landslide movement; the properties of the waves propagating along the coast are described on the basis of runup gauges. The waves observed during the experiments always present first a crest and then a trough; as the first wave propagates away from the generation area the crest tends to become smaller than the trough and the maximum runup along the coast is given by the second or by the third wave. An important feature is that the observed runup along the coast firstly grows with the distance from the generation area, it reaches a maximum value at about two times the width of the landslide, and then decreases. An estimate of the celerity at which the waves propagate along the coast is given on the basis of gauge measurements; it results that the crests propagate faster than the troughs, and the wave period increases.     

Extreme wave crest distribution by direct numerical simulations of long-crested nonlinear wave fields

The wave crest height qualification checks are required during the wave calibration before the model test in wave basin. However, the reliable criteria of nonlinear wave crest probability distribution in 3-h duration (full-scale) has not been well established yet. We investigate wave crest-height statistics of long-crested nonlinear wave fields using high-order spectral (HOS) method, which can take the effects of both second-order bound waves and third-order free waves into account. The energy dissipation effects due to wave breaking were included by employing an eddy viscosity model. Sensitivity analyses to the wave breaking onset criterion have been performed. Validation is provided by comparing the obtained numerical results with the available calibration test data. Based on extensive and direct numerical simulations, semi-empirical single realization distributions for wave calibration have been developed through 3-parameter Weibull fitting and systematic regression analyses. Particular attention has been paid to the tail of upper bound of wave crest distributions. The effects of wave steepness and water depth on the maximum wave crest height in 3-h duration have been examined. It is found that with the increase of wave steepness, the extreme wave crest height increases until it reaches a critical value. In addition, for the scale water depth k_ph < 1.36, the maximum crest height decreases as the water depth increases, while in the opposite case the maximum crest height increases as the water depth increases. Moreover, it is confirmed that that the fourth-order nonlinearity does not have significant effects on the distribution of the wave crest height.     

Measurement of skin friction distribution along the surface of wind waves

Measurements of local values of the skin friction have been made at many points along the surface of representative wind wave crests in a wind wave tunnel, by use of the distortion of hydrogen-bubble lines. The results obtained at 2.85-m fetch under 6.2 m s^–1 mean wind speed show that the intensity of the skin friction varies greatly along the surface of wind waves as a function of the phase angle. It increases rather continuously at the windward surface toward the crest, attains a value of about 12 dyn cm^–2 near the crest, decreases suddenly just past the crest, and the value at the lee surface is substantially zero Values of the skin friction thus determined along the representative wind waves give an average value of 3.6 dyn cm^–2, rather exceeding the overall stress value of 3.0 dyn cm^–2, which has been estimated from the wind profile. The results are interpreted as that the skin friction bears most of the shearing stress of wind, and that it exerts most intensively around the representative wave crests at their windward faces.     

Gravity torques for surface waves

The effects of the gravity torques acting on the angular momentum of surface gravity waves are calculated theoretically. For short crested waves the gravity torque is caused by the force of gravity on the orbiting fluid particles acting down the slopes of the crests and troughs and in the direction parallel to the crests and troughs. The gravity torque tries to rotate the angular momentum vectors, and thus the waves themselves, counterclockwise in the horizontal plane, as viewed from above, in both hemispheres. The amount of rotation per unit time is computed to be significant assuming reasonable values for the along-crest and trough slopes for waves in a storm area. The gravity torque has a frequency which is double the frequency of the waves. For long crested waves the gravity torque acts in the vertical plane of the orbit and tries to decelerate the particles when they rise and accelerate them when they fall. By disrupting the horizontal cyclostrophic balance of forces on the fluid particles (centrifugal force versus pressure force) the gravity torque accounts qualitatively for the three characteristics of breaking waves: that they break at the surface, that they break at the crest, and that the crest breaks in the direction of wave propagation.     

On the form of crests in limiting gravity waves

Properties of surface singularities and the form of wave crests of limiting gravity waves in steady-state flows of an ideal liquid are considered by analyzing the kinematic boundary condition. It is shown that, for rotational waves, the angle at the crest can have any value from 0° to 180°, while it has the only value 90° in the case of irrotational waves. Two inferences are made from Bernoulli’s integral and the properties of singularities: (i) the Stokes wave is a rotational wave and (ii) no angular points can appear on the profiles of capillary-gravity and capillary waves.     

破碎波冲击直立桩柱的大比尺试验研究

波浪破碎是海洋中最常见的现象之一,其能够对海洋中的结构物产生巨大的波浪力作用。本文在大比尺波浪水槽通过聚焦波的方法生成了极端波浪和不同破碎阶段的破碎波浪,并对其冲击桩柱过程中的点压力进行了测量,进而采用连续小波变换的方法,对桩柱上点压力的分布及大小进行了细致分析。结果表明,多次重复试验下,相比非破碎极端波浪,破碎极端波浪产生的点压力离散性更强;波浪破碎程度越大,测点位置越靠近波峰,则点压力离散程度越大;破碎波的最大点压力出现在1.2倍的最大波面附近,且其大小可达3倍的最大静水压力;基于点压力小波谱,不同破碎阶段破碎波产生冲击作用不同,对于波浪作用桩柱前波浪已经发生破碎的情况,其冲击区域更大,点压力分布更复杂;而对于桩面破碎的情况,其造成的波浪总力更大。     

Laboratory modeling of the propagation of periodic internal waves over bottom slopes

The experimental investigation of the run-up of periodic internal waves in a two-layer fluid on the coastal slope is performed in an open hydrochannel at the Physical Department of the Lomonosov Moscow State University. The waves are produced by a wave generator. We study the transformation of waves, the vertical structure of the field of velocities of mass transfer, and the behavior of the parameters of internal waves propagating over the sloping bottom. It is shown that the run-up and breaking of internal waves are accompanied by periodic emissions of portions of the heavier fluid from the bottom layer upward along the slope. The Stokes drift velocity changes its sign as a function of depth. Moreover, both the wave length (the horizontal distance between the neighboring crests) and the height of waves over the sloping bottom (the elevation of the crest over the slope along the vertical) decrease as the wave approaches the coast.     

An experiment at sea on the reflection of the wind waves

Before a reflecting wall, the wind waves strongly differ from the periodic waves. The nodes and the antinodes tend to disappear starting one wave length from the wall. The wave period, as a function of the distance from the wall, exhibits some negative discontinuities at the nodes. Those properties were predicted from an analysis of theoretical spectra and were confirmed by an experiment which employed 30 wave gauges at sea. The experiment revealed also a characteristic pressure-fall under the crests of the highest waves at the wall. The phenomenon was related to high water jets detected at the centers of the wave fronts.     

Effects of high-order nonlinear interactions on unidirectional wave trains

Numerical simulations of gravity waves with high-order nonlinearities in two-dimensional domain are performed by using the pseudo spectral method. High-order nonlinearities more than third order excite apparently chaotic evolutions of the Fourier energy in deep water random waves. The high-order nonlinearities increase kurtosis, wave height distribution and H_max/H_1/3 in deep water and decrease these wave statistics in shallow water. Moreover, they can generate a single extreme high wave with an outstanding crest height in deep water. High-order nonlinearities (more than third order) can be regarded as one cause of freak waves in deep water.     

Breaking wave forces and velocity fields

A new technique for measuring velocities under breaking wave crests using laser-doppler anemometry has been developed. Results may be obtained at positions up to 4 millimeters from the crest.A wave field is presented for a vertically fronted wave and comparison is made with an appropriate numerical study. The velocities obtained are far in excess of the predictions of linear theory.Measurements have also been made of the forces produced by this wave on a horizontal cylinder. Comparisons with large regular waves indicate that forces due to breaking waves can be up to five times greater for similar wave heights.     

Accuracy of asymptotic series for the kinematics of high solitary waves

The accuracy of several asymptotic series expansions for wave speed and particle velocity under the crest of a solitary wave (on a fluid at rest) up to maximum height is investigated. The very accurate numerical results of Williams (1985) are the measure for our comparisons. The results are based on a scaling of calculated properties of long periodic waves to the case of solitary waves.For wave speeds the classical Boussinesq–Rayleigh expression gives good agreement up to a relative wave height of, say, 0.3. An asymptotic fourth-order expression based on Fenton (1990) can be used up to a relative wave height of 0.7, whereas the corresponding fifth-order expression is slightly less accurate.The Eulerian particle velocity profile under the wave crest is examined using a cnoidal wave expression from Fenton (1990) in the limit of the solitary wave. For low waves a `consistent' (i.e. properly truncated) fifth-order expression and an `inconsistent' ditto both coincide with Williams' results. Beginning at medium high waves, the consistent expression surprisingly exhibits oscillations in the velocity profile, and the oscillations become stronger as the wave gets higher. The inconsistent expression, however, yields the same shape as Williams' profile, but is displaced parallel to this, resulting in slightly larger velocities. For high waves also the inconsistent expression begins to differ in shape from Williams' profile, and asymptotic theory fails. Only for low waves `lowest order theory' gives acceptable results. We show analytically that for the highest wave the particle velocity profile has a horizontal tangent at the water surface; this is corroborated by Williams' numerical results.We also study the particle velocity at the wave crest as a function of wave height. It is shown that the variation has a vertical tangent for the highest wave. Two fifth-order asymptotic series for this velocity, based on the wave speed through the Bernoulli equation, show very good agreement with Williams up to a relative wave height of about 0.6.It is finally shown that it is possible to produce very accurate rational-function approximations to Williams' results for the wave speed as well as for the particle velocity at the wave crest.