Relationship between wave steepness and wave age in the course of wind wave growth

It is traditionally assumed that the relationship between wave steepness and wave age is inde- pendent of the wind wave growth state. In fact, the traditional relationship can not describe the whole course of wind wave growth. This paper assumes that the relationship between wave steepness and wave age changes with the variety of dimensionless fetch. Based on the relationship proposed by Hou and Wen (1990), a new relation- ship in the course of wind wave growth is revealed. Comparisons between the present study and other previous relationships show that this new relationship explains better the observations than the other existing relationships. In the case of small fetch, wave age value increases more quickly than other models while it is in opposition to that in the case of large fetch. The result in present paper can clearly reflect the whole course of wind wave growth, it is an improvement for traditional results.Key words: wave steepness, wave age, relationship between wave steepness and wave age

A relationship between wave steepness and wave age for wind waves in deep water

Studying the relationship between wave steepness and wave age is important for describing wind wave growth with energy balance equation of significant waves. After invoking the dispersion rela- tion of surface gravity wave in deep water, a new relationship between wave steepness and wave age is revealed based on the “3/2-power law” (Toba, 1972), in which wave steepness is a function of wave age with a drag coefficient as a parameter. With a given wave age, a larger drag coefficient would lead to larger wave steepness. This could be interpreted as the result of interaction between wind and waves. Comparing with previous relationships, the newly proposed one is more consistent with observational data in field and laboratory.

Joint Distribution of Wave Heights and Wavelengths and Distribution of Wave Steepness

A joint probability density is derived for wavelengths and wave heights. It is asymmetric and depends only on the spectral bandwidth epsilon defined by Cartwright and Longuet-Higgins (1956). After that a theoretical probability density for wave steepness is obtained. It tends to Rayleigh distribution as epsilon --> 0. A comparison between theoretical steepness distribution and laboratory experiment result shows good agreement.

Dependence of Wave Height Distribution on Spectral Width and Wave Steepness

In this paper experimental wind wave data are analyzed. It is found that differences in spectral width will give rise to differences in wave height distribution. The effect of spectral width on the distribution is mainly in the high wave range. The effect of wave steepness is in low, medium and high wave ranges. In the high wave range the effect of spectral width is comparable to that of wave steepness. Differences in spectral width in the observations may give rise to discrepancies in the result when wave steepness is the only parameter in the distribution.

Wave steepness retrieved from scatterometer data in a genetic algorithm

Wave steepness is an important characteristic of a high sea state, and is widely applied on wave propagations at ports, ships, offshore platforms, and CO2 circulation in the ocean. Obtaining wave steepness is a difficult task that depends heavily on theoretical research on wavelength distribution and direct observations. Development of remote-sensing techniques provides new opportunities to study wave steepness. At present, two formulas are proposed to estimate wave steepness from QuikSCAT and ERS-1/2 scatterometer data. We found that wave steepness retrieving is not affected by radar band, and polarization method, and that relationship of wave steepness with radar backscattering cross section is similar to that with wind. Therefore, we adopted and modified a genetic algorithm for relating wave steepness with radar backscattering cross section. Results show that the root-mean-square error of the wave steepness retrieved is 0.005 in two cases from ERS-1/2 scatterometer data and from QuikSCAT scatterometer data.

A New Probability Distribution Function of Wave Steepness

Wave steepness is an important characteristic describing the severity of sea state in ocean engineering. In the existing theoretical and experimental studies,wave steepness is often substituted by some related quantities. In this paper,a new probability density function(pdf) of steepness,which is a pdf of the steepness in its original definition,is obtained for narrowband Gaussian processes. The drawback inherent in the previous theoretical pdfs of steepness,that is,the probability density at zero steepness is nonzero,has been eliminated. Laboratory experiments were conducted in a wind-wave flume to measure the wave steepness distribution. Comparisons among laboratory measurements and some theoretical pdfs of steepness show that the new pdf generally fits the data better than the one proposed by Zheng et al.(1999) .

The observed analysis on the wave spectra of Hurricane Juan (2003)

Hurricane Juan provides an excellent opportunity to probe into the detailed wave spectral patterns and spectral parameters of a hurricane system, with enough wave spectral observations around Juan＇s track in the deep ocean and shallow coastal water. In this study, Hurricane Juan and wave observation stations around Juan＇s track are introduced. Variations of wave composition are discussed and analyzed based on time series of one-dimensional frequency spectra, as well as wave steepness around Juan＇s track： before, during, and after Juan＇s passing. Wave spectral involvement is studied based on the observed one-dimensional spectra and two-dimensional spectra during the hurricane. The standardization method of the observed wave spectra during Hurricane Juan is discussed, and the standardized spectra show relatively conservative behavior, in spite of the huge variation in wave spectral energy, spectral peak, and peak frequency during this hurricane. Spectral widths＇ variation during Hurricane Juan are calculated and analyzed. A two-layer nesting WW3 model simulation is applied to simulate the one-dimensional and two-dimensional wave spectra, in order to examine WW3＇s ability in simulating detailed wave structure during Hurricane Juan.

On the Fifth-Order Stokes Solution for Steady Water Waves

This paper presents a universal fifth-order Stokes solution for steady water waves on the basis of potential theory. It uses a global perturbation parameter, considers a depth uniform current, and thus admits the flexibilities on the definition of the perturbation parameter and on the determination of the wave celerity. The universal solution can be extended to that of Chappelear （1961）, confirming the correctness for the universal theory. Furthermore, a particular fifth-order solution is obtained where the wave steepness is used as the perturbation parameter. The applicable range of this solution in shallow depth is analyzed. Comparisons with the Fourier approximated results and with the experimental measurements show that the solution is fairly suited to waves with the Ursell number not exceeding 46.7.

Simplified Method for Calculating Standing Wave Pressure on Vertical Breakwater

Through numerical modeling, a kind of simplified calculating method for standing wave pressure on vertical face breakwater have been put forward. Not only the formulas proposed in this paper are simple in form and very easy in use, but also they possess continuity on the full range of standing wave. And more, the precision requiremennts of calculation can be satisfied to a certain extent in engineering practice.

Improvements on Mean Free Wave Surface Modeling

Some new results of the modeling of mean free surface of waves or wave set-up are presented. The stream function wave theory is applied to incident short waves. The limiting wave steepness is adopted as the wave breaker index in the calculation of wave breaking dissipation. The model is based on Roelvink (1993), but the numerical techniques used in the solution are based on the Weighted-Average Flux (WAF) method (Watson et al., 1992), with Time-Operator-Splitting (TOS) used for the treatment of the source terms. This method allows a small number of computational points to be used, and is particularly efficient in modeling wave set-up. The short wave (or incident primary wave) energy equation is solved by use of a traditional Lax-Wendroff technique. The present model is found to be satisfactory compared with the measurements conducted by Stive (1983).

Influence of impulse waves generated by rocky landslides on the pressure exerted on bank slopes

Rocky landslides on river banks can result in the generation of ultra-high waves,which may destroy structures on the opposite bank.Existing methods to calculate the pressure on bank slopes under the effect of impulse waves generated by landslides are,however,few and of low precision.Therefore,in this study,a three-dimensional physical model test was conducted by taking into account factors such as landslide geometry parameters and the bank slope angle.The model test section was generalized on the basis of a certain section of the Three Gorges reservoir area as a prototype,after which the wave parameters and wave pressure acting on the bank slope were measured.Subsequently,the magnitude,acting point,and distribution of the pressure of the impulse waves generated by the rocky landslide upon the bank slope were determined.The distribution curve of the impact pressure was similar to that calculated using theСНиПⅡ57-75 formula,and the experimental pulsating pressure value was close to the value calculated using the Subgrade formula.Based on the test results,a power function of the relative pulsating pressure steepness with respect to the reciprocal of the wave steepness,relative water depth,and slope ratio was proposed.The acting point of the maximum pulsating pressure was found to be located near the still water level.Finally,an empirical formula for calculating the envelope of the maximum pulsating pressure distribution curve was proposed.These formulas can serve as a theoretical basis for the prediction of impulse wave pressure generated owing to landslides on bank slopes.

Nonlinear Numerical Analysis of Vortex-Induced Vibration of A Three-Dimensional Deepwater Steep Wave Riser with Large Deformation Features

The cross-flow(CF)vortex-induced vibration(VIV)of a deepwater steep wave riser(SWR)subjected to uniform or shear flow loads is investigated numerically.The model is based on a three-dimensional(3D)nonlinear elastic rod theory coupled with a wake oscillator model.In this numerical simulation,the nonlinear motion equations of the riser with large deformation features are established in a global coordinate system to avoid the transformation between global and local coordinate systems,and are discretized with the time-domain finite element method(FEM).A wakeoscillator model is employed to study the vortex shedding,and the lift force generated by the wake flow is described in a van der Pol equation.A Newmark-βiterative scheme is used to solve their coupling equation for the VIV response of the SWR.The developed model is validated against the existing experimental results for the VIV response of the top-tension riser(TTR).Then,the numerical simulations are executed to determine VIV characteristics of the SWR.The effects of both flow velocity and the spanwise length of the flow field on the drag coefficient in the inline(IL)direction and the lift coefficient in the CF direction are investigated systematically.The results indicate that compared with TTR,the low frequency and multi-modal vibration are the main components of the SWR due to the large deformation and flexible characteristics.For shear flow,the multi-frequency resonance dominates the VIV response of the SWR,especially at the hang-off segment.

Nonlinear Dynamic Analysis and Fatigue Study of Steep Wave Risers Under Irregular Loads

As a reliable alternative option for traditional steel catenary risers(SCRs),steep wave risers(SWRs)have been widely applied to deepwater oil and gas production.However,the nonlinear dynamic analysis of SWRs is more complicated than that of traditional SCRs due to their special configuration and significant geometric nonlinearity.Moreover,SWRs are highly susceptible to fatigue failure under the combined excitation of irregular waves and top floater motions(TFMs).In this study,considering irregular waves and TFMs,a numerical SWR model with an internal flow is constructed based on the slender rod model and finite element method.The Newmark-βmethod is adopted to solve the dynamic behavior of SWR.Moreover,the Palmgren-Miner rule,a specified S-N curve,and rainflow counting method are applied to estimate the fatigue damage.An efficient numerical computation procedure,i.e.,DRSWR,is programmed with MATLAB in this study.Calculation results are compared with those of OrcaFlex to verify the accuracy of the DRSWR.The nonlinear dynamic response and fatigue damage of an SWR under the combined excitation of irregular waves and TFMs are obtained,and a comprehensive parametric analysis is then conducted.The analysis results show that the buoyancy section undergoes the highest level of stress and fatigue damage under the combined excitation of irregular waves and TFMs.An internal flow with high velocity and high density produces a high level of fatigue damage.The buoyancy factor and length of the buoyancy section should be set moderately to reconcile the reduction of the top tension with increased fatigue life.These results are expected to provide some reference significance for the engineering design of SWR.

Nonlinear Analysis of Bidirectional Vortex-Induced Vibration of A Deepwater Steep Wave Riser Subjected to Oblique Currents

An improved three-dimensional(3D)time-domain couple model is established in this paper to simulate the bidirectional vortex-induced vibration(VIV)of a deepwater steep wave riser(SWR)subjected to oblique currents.In this model,the nonlinear motion equations of the riser are established in the global coordinate system based on the slender rod theory with the finite element method.Van der Pol equations are used to describe the lift forces induced by the x-and y-direction current components,respectively.The coupled equations at each time step are solved by a Newmark-βiterative scheme for the SWR VIV.The present model is verified by comparison with the published experimental results for a top-tension riser.Then,a series of simulations are executed to determine the influences of the oblique angle/velocity of the current,different top-end positions and the length of the buoyancy segment on the VIV displacement,oscillating frequency as well as hydrodynamic coefficients of the SWR.The results demonstrate that there exists a coupled resonant VIV corresponding to x-direction and y-direction,respectively.However,the effective frequency is almost identical between the vibrations at the hang-off segment along x and y directions.The addition of the buoyancy modules in the middle of the SWR has a beneficial impact on the lift force of three segments and simultaneously limits the VIV response,especially at the decline segment and the hang-off segments.Additionally,the incident current direction significantly affects the motion trajectory of the SWR which mainly includes the fusiform and rectangle shapes.

Mathematical simulation of steep waves at a focus point

Mathematical models simulating steep waves at a focus point are presented in this paper. Simulations of extreme waves in a model basin were used to determine the loads on floating structures induced by the waves. Based on a new wave theory, numerical test results show that the simulation procedure is effective and the induced motion of water particles in the front of waves is an important factor influencing impact loads on floating bodies.

Time-Domain Nonlinear Wave-Current Interaction with A Steep Wave Riser Considering Internal Flow Effect

The nonlinear dynamic response induced by the wave-current interaction on a deepwater steep wave riser(SWR)is numerically investigated based on a three-dimensional(3 D)time-domain finite element method(FEM).The governing equation considering internal flow is established in the global coordinate system.The whole SWR consists of three segments:the decline segment,buoyancy segment and hang-off segment,in which the buoyancy segment is wrapped by several buoyancy modules in the middle section,leading to the arch bend and sag bend.A Newmark-β iterative scheme is adopted for the accurate analysis to solve the governing equation and update the dynamic response at each time step.The proposed method is verified through the published results for the dynamic response of steel catenary riser(SCR)and static configuration of steel lazy wave riser(SLWR).Simulations are executed to study the influence of wave height,current velocity/direction,internal flow density/velocity and top-end pressure on the tension,configuration and bending moment of the SWR.The results indicate that the influence of the current on the configuration and mechanical behavior of the SWR is greater than that of the wave,especially in the middle section.With increasing current velocity,the suspending height of the middle section drops,meanwhile,its bending moment decreases accordingly,but the tension increases significantly.For a fixed external load,the increasing internal flow density induces the amplification of the tension at the hang-off segment and the mitigation at the decline segment,while the opposite trend occurs at the bending moment.

Numerical method for wave height distribution within the artificial harbor with water depth of steep variation

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半潜船波浪诱导运动与载荷预报数值方法比较研究

Numerical simulation tools based on potential-flow theory and/or Morison’s equation are widely used for predicting the hydrodynamic responses of floating offshore wind platforms.In general,these simplified approaches are used for the analysis under operational conditions,albeit with a carefully selected approach to account for viscous effects.Nevertheless,due to the limit hydrodynamic modelling to linear and weakly nonlinear models,these approaches severely underpredict the low-frequency nonlinear wave loads and dynamic responses of a semisubmersible.They may not capture important nonlinearities in severe sea states.For the prediction of wave-induced motions and loads on a semisubmersible,this work systematically compares a fully nonlinear viscous-flow solver and a hybrid model combining the potential-flow theory with Morison-drag loads in steep waves.Results show that when nonlinear phenomena are not dominant,the results obtained by the hybrid model and the high-fidelity method show reasonable agreement,while larger discrepancies occur for highly nonlinear regular waves.Specifically,regular waves with various steepness over different frequencies are focused in the present study,which supplements the understanding in applicability of these two groups of method.

On the parameterization of drag coefficient over sea surface

Six parameterization schemes of roughness or drag coefficient are evaluated on the basis of the data from six experiments. They present great consistency with measurement when friction velocity u＊〈0.5 m/s （ap- proximately corresponding to 10 m wind speed U10〈 12 m/s） and large deviation from measurement when u＊≥0.5 m/s （approximately U10 ≥ 12 m/s）. In order to improve the deviation, a new parameterization of drag coefficient is derived on the basis of the similarity theory, Charnock relationship and Toba 3/2 power law. Wave steepness and wind-sea Reynolds number are considered in the new parameterization. Then it is test- ed on the basis of the measurements and shows significant improvement when u＊≥0.5 m/s. Its standard errors are much smaller than the ones of the other six parameterizations. However, the new parameteriza- tion still needs more tests especially for high winds.

Wind speed scaling and the drag coefficient

Wind speed scaling in similarity law in wind-generated waves and the drag coefficient are studied. In analyzing the data in the wind wave channel, it is found that the u＊ scaling greatly reduces the scatter in the U10 scaling. The u＊ scaling has much less scatter than the scaling using other wind speeds. The friction velocity seems to play a distinctive role in wave growth. The result is important in the applications of the similarity law and in wave modeling. In theory it gives an insight into the mechanism of wind wave interaction. It is found that wave steepness is important in influencing the drag coefficient. The variability of the coefficients in the currently widely used drag form can be explained by the differences in wave steepness in the observations. A drag coefficient model with wind speed and wave steepness as parameters is proposed. An explanation for Kahma＇s result that the u, scaling does not reduce the scatter in the U10 scaling is given.