预报发生在双峰交叉海况下的畸形波.pdf

Forecasting Rogue Wave Occurrence in Bimodal Crossing SeasWANG Ying-guang1a,1b,XU Zhi-ting2,LI Rui-yao1a,1b(1a.State Key Laboratory of Ocean Engineering;b.School of Naval Architecture,Ocean and Civil Engineering,Shanghai Jiao Tong University,Shanghai 200240,China;2.Marine Design and Research Institute ofChina,Shanghai 200011,China)Abstract:A novel transformed linear simulation method for the estimation of wave crest amplitudesdistribution and rogue wave occurrence in bimodal crossing seas was proposed and implemented byconstructing a transformation model expressed in a monotonic exponential function to match the firstthree moments of the original true process with the corresponding moments of the transformed model.The proposed novel simulation method was applied to forecast the rogue wave occurrence in two bimodal crossing sea states,one with a directional wave spectrum based on the measured surface elevation data at the coast of Yura,and another one with a typical directional bimodal Ochi-Hubble wavespectrum.It is shown that the proposed method is more efficient than the nonlinear simulation method,and can offer more accurate forecasting results than those obtained from the traditional linear simulation method or those by using other theoretical(or empirical)distribution models.Key words:rogue wave;bimodal crossing sea;transformed linear simulation;exponential functionCLC number:O352Document code:Adoi:10.3969/j.issn.1007-7294.2023.06.0040 IntroductionRogue waves(also known as freak waves or abnormal waves)are unusually large ocean surfacewaves that can be extremely dangerous,even to large ships such as ocean liners.Rogue waves present considerable danger because they can impact with tremendous force.Although modern shipsare designed to tolerate a breaking wave of 15 t/m2,a rogue wave can dwarf this figure with a breaking pressure of 100 t/m2.Rogue waves have now been proven to be the cause of the sudden loss ofsome ocean-going vessels.Well-documented instances include the freighter MS Mnchen lost in1978,the merchant vessel El Faro sunk in 2015(Fedele et al1)and the MV Derbyshire lost in 1980,the largest British ship ever lost at sea.A rogue wave has been implicated in the loss of other vessels including the Ocean Ranger,which was a semi-submersible mobile offshore drilling unit thatsank in Canadian waters on Feb.15,1982.In oceanography,rogue waves are more precisely defined as waves whose height is more thantwice the significant wave height(Hs),which is itself defined as the mean of the largest third of第27卷第6期船舶力学Vol.27 No.62023年6月Journal of Ship MechanicsJun.2023Article ID：1007-7294（2023）06-0816-19Received date:2022-12-19Foundation item:Supported by the National Natural Science Foundation of China(51979165)Biography:WANG Ying-guang(1967-),male,Ph.D.,MSc.advisor,E-mail:;XU Zhi-ting(1985-),male,senior engineer;LI Rui-yao(1996-),male,master student.waves in a wave record.Therefore,rogue waves are not necessarily the biggest waves found on thewater;they are,rather,unusually large waves for a given sea state.Once considered mythical andlacking hard evidence for their existence,rogue waves are now proven to exist and known to be anatural ocean phenomenon.Eyewitness accounts from mariners and damage inflicted on ships havelong suggested their occurrence.The first scientific evidence of the existence of rogue waves camewith the recording of a freak wave by the Gorm platform in the central North Sea in 1984.A stand-out wave was detected with a wave crest height of 11 m in a relatively low sea state(Sand et al2).However,the wave that caught the attention of the scientific community was the digital measurement of the Draupner wave,a rogue wave at the Draupner E platform in the North Sea on January1,1995,with a maximum wave height of 25.6 m(crest peak elevation of 18.5 m).During that event,minor damage was also inflicted on the platform,far above sea level,confirming that the readingwas valid(Haver3).Haver and Andersen4presented a paper at the 2000 ISOPE conference,which collated evidence that rogue waves were not the rare realizations of a typical or slightly non-Gaussian sea surface population,but rather they were the typical realizations of a rare and strongly non-Gaussiansea surface population of waves.An interesting explanation was given earlier by Dean5to indicatethat both non-linearity and directionality are primary possible causes of rogue waves.Fedele et al6found that the main generation mechanism for rogue waves is the constructive interference of elementary waves enhanced by second-order bound nonlinearities and not the modulational instability.Laboratory studies have demonstrated that rogue waves can be generated through non-linearwave-wave interactions in a two-dimensional wave flume(Stansberg7).In their research Mori etal8showed that a well-defined rogue wave may occur in the developed wind-wave condition withsingle peak directional spectra.The crest and trough amplitude distributions of the observed seawaves including rogue waves are different from the Rayleigh distribution.Mori et al8further concluded that the exceedance probabilities of the crest amplitude of the rogue wave are underestimated by the Rayleigh distribution.Therefore,accurate calculation of the exceedance probabilities ofthe wave crest amplitudes is critical for the prediction of the occurrence of rogue waves in a nonlinear(or non-Gaussian)sea.In a real world ocean engineering project,the known information regarding the environmentalconditions is typically a wave spectrum corresponding to a short-term,stationary sea state.Basedon this specific wave spectrum,there are several approaches to calculate the exceedance probabilities of the wave crest amplitudes.The first and most straightforward way is to resort to some empirical(or theoretical)models9-16.Chakrabarti17mentioned the Rayleigh model of wave crest amplitude distribution of random ocean waves.However,the previous research work of Wang18hasshown that the Rayleigh model systematically underestimates the wave crest amplitude distributionof nonlinear random ocean waves.From a specific wave spectrum,there is another way to calculatethe exceedance probabilities of the wave crest amplitudes.Based on the wave spectrum at the Gullfaks C platform,Wang and Xia19calculated the wave crest amplitude distribution using a linearsimulation method.However,their calculation results showed that the linear simulation method willpredict overly non-conservative probability distributions of the wave crest amplitudes in an actually第6期WANG Ying-guang et al:Forecasting Rogue Wave Occurrence in 817nonlinear sea state.This will result in the designing of unsafe ocean engineering structures.Wangand Xia19used a nonlinear simulation method to compute the wave crest amplitude distribution forthe nonlinear irregular waves.They had verified that their nonlinearly simulated wave crest amplitude distribution is more accurate than that calculated by using the linear simulation method.However,the nonlinear simulations performed by Wang and Xia19are too time-consuming,and thisdrawback will certainly affect the applicability of the nonlinear simulation method in some time-constrained ocean engineering projects.Wang20proposed a transformed Rayleigh method for calculating the wave crest amplitude distribution of shallow water nonlinear waves.It was demonstrated inRef.20 that the transformed Rayleigh method has a higher efficiency and equivalent accuracy,incomparison with the nonlinear simulation method.However,the transformed Rayleigh method developed in Wang20is only applicable to a sea state with an ideal 2D(long crested)wave spectrum,andthis drawback will certainly hinder its usefulness in solving some real world ocean engineering problems.The prediction method of the wave crest amplitude distribution presented in Ref.20 assumesthat wave energy travels in a specific direction,commonly considered traveling in the same direction as the wind.In this respect,the wave spectrum in Ref.20 may be considered as a long-crestedspectrum or a uni-directional spectrum.In reality,however,wind-generated wave energy does notnecessarily propagate in the same direction as the wind;instead,the energy usually spreads overvarious directions.Thus,for an accurate description of random seas,it is necessary to clarify thespreading status of energy.The wave spectrum representing energy spreading over multi-directionsis called a multi-directional spectrum or a short crested(3D)spectrum.Information on wave directionality is extremely significant for the design of marine systems such as ships and ocean engineering structures.This is because not only the responses of a system in a seaway computed using a unidirectional(2D long crested)wave spectrum are overestimated,but also the associated coupled responses induced by waves from other directions are disregarded.Furthermore,all the above-mentioned wave spectra(no matter a 2D spectrum or a 3D spectrum)are unimodal wave spectra.It is well known that not all sea states have unimodal wave spectraor narrow(or finite)spectral bandwidth.Frequently,sea states are due to the coexistence of variouswave systems.In particular,local wind waves often develop in the presence of some background lowfrequency swell coming from distant storms,and the resulting mixed sea states will have bimodalwave spectra(Wang and Xia21)and are called crossing sea states.Crossing sea states are commonsituations in the ocean and occur when two wave systems exist simultaneously.Analysis of a database of ship accidents(Toffoli et al22)found that many had occurred in crossing sea conditions.Thesinking of the oil tanker Prestige in 2002,which has been suggested to be due to a rogue wave,happened in a crossing sea state(Trulsen et al23).An accident that was analyzed in the Extreme Seasproject,in which the cruise ship Louis Majesty was hit by a giant rogue wave,took place in a crossing sea(Cavaleri et al24).Therefore,it is of vital practical importance to develop an accurate and efficient method for predicting the wave crest amplitudes distribution and rogue wave occurrence inbimodal crossing seas.Motivated by the afore-mentioned facts,the present study will first propose a novel transformed linear simulation method for the estimation of wave crest amplitudes distribution and rogue818船舶力学第27卷第6期wave occurrence in 3D bimodal crossing seas.For implementing the proposed transformed linearsimulation method,a transformation model expressed in a monotonic exponential function will beconstructed so that the first three moments of the original true process match the corresponding moments of the transformed model.The proposed novel transformed linear simulation method will beapplied to forecast the rogue wave occurrence in two bimodal crossing sea states,one with a directional wave spectrum based on the measured surface elevation data at the coast of Yura,and another one with a typical directional bimodal Ochi-Hubble wave spectrum.The calculated exceedanceprobabilities of a specific critical wave crest amplitude are directly related to the occurrence ofrogue waves.It will be demonstrated that the novel transformed linear simulation method can makepredictions more accurate than those obtained from the traditional linear simulation method andfrom theoretical(or empirical)crest amplitude distribution models.It will also be shown that the efficiency of the novel transformed linear simulation method is higher than that of the nonlinear simulation method.1 Theories of the nonlinear random ocean wavesWaves in an idealized linear Gaussian random sea have crest-trough symmetry.However,it isknown that real world ocean surface elevation process deviates from the Gaussian assumption,i.e.,the wave crests are becoming steeper and higher and the wave troughs are becoming flatter and shallower than expected under the Gaussian assumption.If the longitudinal coordinate is denoted by xand time is denoted by t,the free surface elevation(x,t)for a nonlinear random sea state can thenbe written as:()x,t=()1()x,t+()2()x,t=Ren=1Ncnei()nt-knx+n+Rem=1Nn=1Ncmcnrmnei()mt-kmx+m+nt-knx+n+qmnei()mt-kmx+m-nt+knx-n(1)in which()1()x,tare the first order linear components and()2()x,tare the second order nonlinearcorrection components.In Eq.(1)N usually is chosen to be a sufficiently large positive integer,cndenotes a random complex valued wave amplitude that can be calculated based a specific wave spectrumS()and the angular frequencyn.For a short-crested sea state with a multi-directionalwave spectrumS(),in whichdenotes the wave direction,an equivalent frequency spectrumS()can be obtained by taking the directional(3D)wave spectrumS(),and integrating the energy over all the directions to give the total energy at each frequency.Furthermore,in Eq.(1)knis aspecific wave number related tonthrough the dispersion relation,andnis the uniformly-distributed phase angle between 0 and 2.Finally,the termsrmnandqmnin Eq.(1)are second order transfer functions that can be calculated(for a constant water depth d)by using the following equations:rmn=-()1g()14mn2()m+n()2n2m-knkmg2+n()4m-g2k2m+m()4n-g2k2n()m+n2cosh()|km+knd-g|km+knsinh()|km+knd第6期WANG Ying-guang et al:Forecasting Rogue Wave Occurrence in 819()m+ncosh()|km+knd-()14gmn()kmkng2-2n2m+()14g()2m+2n(2)qmn=-()1g()14mn2()m-n()2n2m+knkmg2-n()4m-g2k2m+m()4n-g2k2n()n-m2cosh()|km-knd-g|kn-kmsinh()|kn-kmd()n-mcosh()|kn-kmd-()14gmn()kmkng2+2n2m+()14g()2m+2n(3)Omitting the term()2()x,twhen implementing Eq.(1)will lead to linearly simulated wave timeseries,and this is called the linear simulation method.Implementing Eqs.(1)-(3)altogether willlead to nonlinearly simulated wave time series,and this is called the nonlinear simulation method.However,because Eqs.(2)-(3)contain N2sum frequencies components and another N2differencefrequencies components,the nonlinear simulation based on Eqs.(1)-(3)will become very time-consuming when N is large.In order to improve the simulation efficiency for generating shallow waternonlinear waves,in this paper we propose a novel transformed linear simulation method whose theoretical background will be explained in the next chapter.2 Theoretical background of the proposed transformed linear simulationmethodWithout loss of generality,we set x=0 in()x,tand write it in a simplified way as()t.Thenon-Gaussian wave elevation process()tcan be modeled as a function of a single Gaussian process()twith mean valueand variance2:()t=G()()t(4)In Eq.(4),G()is a continuously differentiable function with positive derivative.In the proposed transformed linear simulation method,the transformation model(the G()function)is chosento be a monotonic exponential function,calibrated such that the first three moments of the originaltrue process match the corresponding moments of the transformed model.After a lengthy derivationprocess as shown in Ref.25,Eq.(1)can be re-written in the following matrix notation:()t=sTX+XTQ+R X+YTQ-R Y(5)where Q and R are real s