Directional energy distribution of wind waves

Two-dimensional ocean wave spectrum developing under the atmospheric surface pressure fluctuations is linearly correlated with that of wind pressure itself, so that angular distribution of energy of ocean surface waves can be determined by directional properties of surface pressure fluctuations with the same frequency to the surface wave.From empirically determined spectral formula of the atmospheric surface pressure fluctuations the coefficients of Fourier series expanded around mean direction of wind are analytically integrated, from which r.m.s. angular distribution, spectral peakedness and long-crestedness are calculated, compared with previously proposed empirical formulae and observations carried out by ultrasonic current meter.     

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。     

The energy cycle of wind waves on the sea surface

The problems of wind-induced waves on the sea surface are considered. To this end, the empirical fetch laws that determine variations in the basic periods and heights of waves in relation to their fetch are used. The relation between the fetch and the physical time is found, as are the dependences of the basic characteristics of waves on the time of wind forcing. It is found that about 5% of wind energy dissipated in the near-water air layer contributes to the growth of wave heights, i.e. wave energy, although this quantity depends on the age of waves and the exponent in the fetch laws. With consideration for estimates of the probability distribution functions for the wind over the world ocean [11], it is found that the rate of wind-energy dissipation in the near-water air layer is on the order of 1 W/m². The calculations of wind waves [19] for the world ocean for 2007 have made it possible to assess the mean characteristics of the cycle of wave development and their seasonal variations. An analysis of these calculations [19] shows that about 20% of wind energy is transferred to the water surface. The remaining amount (80%) of wind energy is spent on the generation of turbulence in the near-water air layer. About 2%, i.e., one tenth of the energy transferred to water, is spent on turbulence generation due to the instability of the vertical velocity profile of the Stokes drift current and on energy dissipation in the surf zones. Of the remaining 18%, 5% is spent directly on wave growth and 13% is spent on the generation of turbulence during wave breaking and on a small-scale spectral region. These annually and globally mean estimates have a seasonal cycle with an amplitude on the order of 20% in absolute values but with a smaller amplitude in relative values. According to [19] and to the results of this study, the annually mean height of waves is estimated as 2.7 m and their age is estimated as 1.17.     

Nonlinear properties of laboratory wind waves at energy containing frequencies

Nonlinear properties of wind waves in a wind-wave tunnel are investigated by measuring the probability density distribution of surface elevation. The surface elevation distribution of raw records are found to have a positive skewness (K ₃=0.21 to 0.43) and a negative kurtosis (K ₄=–0.74 to –0.41) with magnitude depending of fetch and wind speed. The values of skewness are in qualitative agreement with a prediction of the weak interaction theory for a random wave field incorporating the effects of second harmonics (Tayfun, 1980), but the values of kurtosis are different in sign from the prediction.To examine the nonlinear properties of energy containing components, higher harmonic components are excluded from the wave records by using a kind of a band-pass filter. The surface elevation distributions of the filtered waves show a sharp decrease in skewness , but the distributions remain highly non-Gaussian with a large negative kurtosis almost independent of the fetch and wind speed . It is concluded that the negative kurtosis is due to the non-random character of the phase and amplitude among the energy containing components, and that nonlinear interactions occur amongst the energy containing frequencies.     

Fluctuations of peak energy and peak frequency of local wavelet energy spectrum for wind waves

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Some new results on statistical properties of wind waves

For wind waves modelled by a stationary Gaussian process ζ(t) (ζ = height above m.w.l. of one point of the free surface) it is shown that, in a time interval including an instant t_m where a maximum ζ_m occurs, the ratio between ζ(t) and ζ_m tends with probability approaching one, to the ratio Ψ(t ? t_m)/Ψ(O), as $\text{ζ}{}_{\mathrm{m}}\text{/Ψ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(}\text{O}\text{)}$ tends to infinity, Ψ(t) being the autocovariance of the process.Starting from this result, it is possible to find analytically the characteristic periods of the highest waves in a sea with a given energy spectrum. These periods, calculated according to period definitions of three methods of wave record analysis, are found to be in remarkably good agreement with data from Bretschneider¹ and Svasek.²     

Inverse modeling of discrete interaction approximations for nonlinear interactions in wind waves

In the last two decades, the Discrete Interaction Approximation (DIA) has been the only economically feasible parameterization for nonlinear wave–wave interactions in operational wind wave models. Its major drawback is its limited accuracy. Several improvements to the DIA have been suggested recently. The present study summarizes these improvements and suggests some new modifications to the DIA. Using inverse modeling techniques, where the potential of various DIAs is assessed by optimal fitting to exact solutions, a comprehensive comparison of the potential of several such improvements is made. An in depth analysis of the behavior of DIAs in full wave models will be the subject of a second study, to be reported elsewhere. The major findings of this study are that: (i) An expanded definition of the representative quadruplet with additional degrees of freedom is necessary for an accurate representation of the exact interactions; (ii) Slowly varying the free parameters in such a DIA as a function of the spectral frequency f results in a (mostly qualitative) improvement; (iii) A DIA with expanded quadruplet definition and with four representative quadruplets is found to reproduce the exact source term accurately; (iv) Adding additional tunable constants to the equation for the strength of the interactions has little impact on the quality of the DIA.     

Incorporating irregular nonlinear waves in coupled simulation and reliability studies of offshore wind turbines

Design of an offshore wind turbine requires estimation of loads on its rotor, tower and supporting structure. These loads are obtained by time-domain simulations of the coupled aero-servo-hydro-elastic model of the wind turbine. Accuracy of predicted loads depends on assumptions made in the simulation models employed, both for the turbine and for the input wind and wave conditions. Currently, waves are simulated using a linear irregular wave theory that is not appropriate for nonlinear waves, which are even more pronounced in shallow water depths where wind farms are typically sited. The present study investigates the use of irregular nonlinear (second-order) waves for estimating loads on the support structure (monopile) of an offshore wind turbine. We present the theory for the irregular nonlinear model and incorporate it in the commonly used wind turbine simulation software, FAST, which had been developed by National Renewable Energy Laboratory (NREL), but which had the modeling capability only for irregular linear waves. We use an efficient algorithm for computation of nonlinear wave elevation and kinematics, so that a large number of time-domain simulations, which are required for prediction of long-term loads using statistical extrapolation, can easily be performed. To illustrate the influence of the alternative wave models, we compute loads at the base of the monopile of the NREL 5MW baseline wind turbine model using linear and nonlinear irregular wave models. We show that for a given environmental condition (i.e., the mean wind speed and the significant wave height), extreme loads are larger when computed using the nonlinear wave model. We finally compute long-term loads, which are required for a design load case according to the International Electrotechnical Commission guidelines, using the inverse first-order reliability method. We discuss a convergence criteria that may be used to predict accurate 20-year loads and discuss wind versus wave dominance in the load prediction. We show that 20-year long-term loads can be significantly higher when the nonlinear wave model is used.     

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.     

An experimental study of the relations between bispectra and nonlinear apparent features of wind waves

To investigate the nonlinear properties of wind waves, experiments are carried out in a wind-wave flume with slope bottom at different wind speeds and fetches. Both the internal structure and apparent features of the nonlin-earity of wind waves are studied by using bispectral and statistical analysis of surface elevations. The relations between bispectra and nonlinear apparent characteristics of wind waves are established and confirmed.     

风浪局域小波谱峰的涨落

将湍流研究中已得到有效应用的局域小波能谱概念引入到风浪研究.分析了实验室不同风速下风浪的局域小波能谱.结果表明,局域小波谱峰值存在着显着的涨落,涨落的方差随风速的增加而增大.这不仅体现了风浪的局域结构,还意味着通常风浪谱的应用中除了要考虑平均意义下的频谱峰值外,还必须考虑局域小波谱峰值的涨落方差.     

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.     

Intensification of nonlinear wind waves on clean water and in the presence of an oil film

The intensification of gentle plane waves at the initial stage of their generation by a steady wind flow in a laboratory setup is investigated. It is found that the wave form is changed by eddies that are formed in the viscous layer of a steady air flow on the leeward slope: the water surface rises under the action of eddies to form a complexly shaped slope. Calculated and measured data for gentle nonlinear waves on clean water are in good compliance with each other. It is shown that, in the presence of an oil film, the region of eddy separation encompasses the wave trough as well, because oil flows down and the film is thickened in passing from the slope to the trough. The water-surface rise by eddies in the trough restricts an increase in the amplitude and steepness of the wave.     

波-波间非线性能量传输的一种新计算方法

从最基本的Boltzmann六重积分出发,在借鉴其他作者研究经验的基础上,建立起一种新的精确计算模型。该模型不但获得了与前人相一致的结果,还能够清晰的表现出每一个参与相互作用的共振架构对能量传输的贡献大小,并且直观的反映出构成这些架构的共振波数的详细情况。该研究工作为深入研究不同海浪谱中非线性能量传输特征以及其近似计算方法改进提供了基础。     

Negative correlations of variations in near-water wind and surface wind waves

The relationship between the intensity of surface wind waves and near-water wind is analyzed. The data of measuring wind waves and near-water wind under natural conditions in the Black Sea (July 2004) and Norwegian Sea (June 2003, 16th cruise of the R/V Akademik Sergei Vavilov) are used. A phenomenon of negative correlations has been found between the intensity of wind waves and near-water wind in regions of substantial restructuring of wind waves in the field of inhomogeneous flows: wind-wave amplification during wind decay and vice versa. Examples of such observations are presented, a theoretical model is constructed for the observed phenomenon, and a good agreement is obtained between theory and experiment.     

The steepness and shape of wind waves

Variations are found in the shape and the steepness of wind-generated surface gravity waves between very young waves, such as seen in a laboratory tank, and larger waves of various wave ages encountered at sea as the result of wind stress over larger fetches. These differences in the characteristic shape of wind waves are presented as a function of the wave age. The wave steepness is also expressed as a function of wave age, the measurement of which is consistent with the 3/2-power law connecting wave height and characteristic period, normalized by the air friction velocity.     

Duality of turbulence and wave in wind waves

The three-seconds power law for wind waves of simple spectra, already derived byToba (1972 and 1973), may also be derived by introducing surface-wave properties into the form of the rate of energy dissipation in the theory of turbulence. The universal constantB, which was formerly determined empirically as 0.062 is here obtained asB=(2)^–3/2=0.0635. Thus wind waves have the duality of turbulence and wave.     

Peak and trough distributions of nonlinear waves

The exact and approximate probability density functions of wave peaks and troughs are derived for nonlinear, narrow-band waves. A comparison of these quantities with those of linear waves is given. It is shown that nonlinearity has significant effect on these quantities.     

A quasi-similarity between wind waves and solid surfaces in their roughness characteristics

A formulation for the aerodynamic roughness length of air flow over wind waves $$z_0 = \gamma {\text{ }}u_* /\sigma p$$ which was proposed by Toba (1979) and Toba and Koga (1986) from dimensional considerations with some data analysis, is shown to correspond with a formulation for irregular solid surfaces $$(z_0 /h) = a(h/l)^{1 + \beta } $$ which resulted from work by Woodinget al. (1973) and Kustas and Brutsaert (1986);u _* is the friction velocity,σ _p the spectral peak frequency of wind waves,h the mean height of the solid obstacles,l the mean distance between their crests, andα,Β, andγ are constants. This correspondence is reached by the existence of a statistical 3/2-power law and an effective dispersion relationship for wind waves. Because both approaches of parameterizingz ₀ were arrived at independently, they provide each other mutual reinforcement.