深海极限波浪运动特性的简便算法

深海极限波浪运动特性的简便算法 J.MarineSci,App1.(200918:27—32 DoI:10.1007/sl1804—009—8046—8 Asimplifiedmodelforextreme--wavekinematicsindeepsea TENGBinandNINGDe.zhi StateKeyLaboratoryofCoastalandOffshoreEngineering,DalianUniversityofTechnology,Dalian116023,China Abstract:Basedonthefiffh-orderStokesregularwavetheory.asimplifiedmodelforextreme—wave kinematicsindeepseawasdeveloped.Inthismode1.fromthewaverecordstheaverageofo neighboringwaveperiodsfortheextremecrestortroughwasdefinedastheperiodoftheStokeswave bytheupanddownzero— crossingmethods.Thentheinputwaveamplitudewasdeducedbysubstituting thewaveperiodandextremecrestortroughintotheexpressionforthefifth— orderStokeswaveelevation. Thusthecorrespondingformulaforthewavevelocitycanbeusedtodescribekinematicsbeneaththe cxtrcvf~cwave.Bvcomparisonwiththepublishednumericalmodelsandexperimentaldata.theproposed modelisvalidatedtobeabletocalculatetheextremewavevelocityrathereasilyandaccurately. Keywords:extremewave:deepsea;fifth.orderStokesregularwave;kinematics;velocityfield CLCnumber:P731.22Documentcode:AArticleID:1671—9433(2009)0l一0027—06 1Introduction Becauseoftheeriectsofhurricanesandearthquakes, thereoftenoccursomeextremewavestodevastateocean structures,suchasoffshoreplatforlns,risersandshipsetc., inthecomplicatedandseveredeepsea.Thevelocitiesof extremewavesarerequiredforrelatedcalculationand analysisofloadsonstructures.Itwillbedirectlyrelated totheeconomicsandsafeWofoceanstructuredesigning. Becauseofthecomplexityofdeepseaenvironments,fast andaccuratedescriptionofthedistributionof extreme—wavevelocityisstillachallengeatpresent. Manyresearchesonextremewavekinematicshavebeen carriedoutnumericallyandexperimentallyinthepast twodecades.Forexample,Baldocketal【,Johannessen etal[, Sheetal1]andGrueetal[]physicallystudiedthe kinematicsofextremewavesindeepwater,andFentonet alI. Johannessenetale】andBatemanetal[]researched theextremewavekinematicsusingvariousnumerical models.Althoughallabovemodelscandescribethewave kinematicswel1.itisdi所culttoapplythemdirectlyto engineeringpracticebecausemanygivenconditionsare neededbeforephysicalornumericalsimulation.suchas frequencyrangeandamplitudedistributionofwave components,etc. Asimpleandaccuratemodelwasdevelopedbasedonthe fifth-orderStokeswavetheoryforfastcalculatingthe extreme—wavevelocityfieldindeepsea.Theonlyknown conditionwasthetimeseriesofextremewavesurface, Receiveddate:2008—07一O3. Foundationitem:SupportedbytheNSFC(underGrantNos.5070900and 10772040)andtheNationalHighTechResearchandDevelopmentProgram ofChina(2006AA09A109--3). CorrespondingauthorEmaihdzning@dlut.edu.cn whichcouldbeeasilyrecordedinpractice.Thekey problemsaretodefineproperwaveperiodand appropriateinputwaveamplitudefortheStokeswave whichcanrepresentextremewavesaboveorbelowthe meanwatersurface.Intheproposedmodel,theupand downzero.crossingmethodisadoptedtodefinethe averageoftwoneighboringwaveperiodscontainingthe extremecrestortroughastheperiodoftheStokeswave. Thustheinputwaveamplitudecanbededucedby substitutingthevalueofextremewavecrestortrough intothefcIrmulaforthefiftl1.orderStokeswaveelevation. Theproposedmodelisvalidatedbycomparisonwiththe publishednumericalresultsandexperimentaldata.Itis alsocomparedwiththelinearandsecond.orderirregular analyticalsolutions. 2Mathematicalmodel Fig.1givesthesketchesofextremewavecrestandtrough Theyaregenerallygeneratedbythemethodofwave focusing,inwhichallwavecomponentsareconsidered. SomeresearchersattemptedtousetheStokeswaveto describeex仃emewavesforconvenience.Thefih.order Stokeswaveisamuchhighernonlinearregularwave solutionatpresent.Althoughitjustdescribestheperiodic wave,itcanaccuratelypredictextremewavecrestor troughvalueandthecorrespondingvelocityfieldif properwaveperiodandinputamplitudearechosen.The fth—orderStokeswavevelocityfordeepwatercanbe expressedasfollowsLsI: 一一 )e~cos(+ 船ezbc.s2(一)+lkcSe3acos3(一 刎)】.?.() 28 Thecorrespondingwaveelevationexpressionis 0.I2 0O8 004 - ()04 . 008 0.12 0.12 0.08 004 — 0.04 O.O8 . 0l2 6 l}s (a)Crest 89 6789 (b,Trough Fig.1Sketchofextremewaveelevationhistories = ?Lcosc一+ (12十j1)c.s2(一)+(33+297,)c.s3(一)+ L~ncos4(kx一)+—12—5ssc.s5(kx一)] 3384 (21 wherewaveslope占=kA.kiswavenumber,Ainput waveamplitudeandangularfrequency.Thefollowing dispersionrelationissatisfied: . 24go 28x/gk Asacomparison,thelineartheoreticalwavevelocityand elevationindeepseaarealsogivenasfollows: v(x,y,z,f)=Agoecos(/~一cot)(3) and rffx,y,z,t)=Acos(~一got)(4) whereandcosatisfythelineardispersionrelation 2. FromEqs.(1),(4),itcanbeseenthatthetimeseriesof waveelevationandvelocityandtheirextremevaluescan beobtainedoncethewaveperiodTandinputwave TENGBin.etalAsimplifiedmodelforextreme—wavekinematicsindeepsea amplitudeAaregiven.Similarly,thelinearfifth—order inputwaveamplitudescanalsobeobtainedfromEqs.(2) and(4)ifthewaveperiodTandextremesurfacevalues AnI8xorAminareknown. TodescribetheextremewavesasshowninFig.1using theStokesregularwavetheory,twowaveperiodsT1and /'2,bothcontainingtheperiodofextremewave crest/trough,aredefinedbytheupanddown zero.crossingmethodsrespectivelyandtheiraverageis takenastheStokeswaveperiod.FromthestudiesofNing eta1[9]andBatemanetal{们.theshapeofextremewaveis symmetrica1abouttheextremecrestortroughifthewave nonlineariWisratherweakerandthecaseisclosetothe linearonei.e.:/'2.However,suchsymmetrywillbe deterioratedandT1willnotequaltoT2againwiththe increaseofnonlinearity.Therefore.theaverageofT1and T2istakenastheStokeswaveperiodinthispaperto considerthenonlineareffects.Theproposedmethodis similartothatmentionedbyGrueetalusingthetime intervaloftwoneighboringtroughsofextremecrestas thewaveperiod.Butthe1atter'sdisadvantageisthatthe minima1locationcannotbefoundeasilybecausethe troughwil1bemuchflatterowingtostrongerwave nonlinearity.Therefore.itisnotsuggestedhere.Thusthe inputwaveamplitudecanbecalculatedbysubstituting thewaveperiodTandtheextremewaveamplitudeAmax orAiintoEqs.(2)and(4)respectively. 3Numericalresultsanddiscussions Tovalidatetheproposedmodel,theinputparametersof extremewavesinaninfinitelydeepwaterusedbyNinget altaretakenasanexampleandshowninFig.2and Table1.Thefullynonlinearnumerica1wavetankbased onatime..domainhigher..orderboundaryelementmethod [91isadoptedtosimulatethecorrespondingextremewave timeseriesandvelocityfieldbeneaththeextremecrestor trough.Bycomparisonwithit,theeffectofthepresent regularwavemodelforpredictingextremewavecanbe validated. 10 , , 一 , 6 4 2 f/ltz Fig.2Inputwavespectra&frequencies JournalofMarineScienceandApplicaton(2009,8:27—32 Table1Parametersofinputwavegroup Frequencyrangef(Hz)0.6_厂1.5 Peakfrequency0.80 SumofinputamplitudesAi(m)0.04375,0.0875,0.120 Numberofwavecomponents30 Waveslopes岛0.130,0.283,0.356 Figs.3and4givethetimeseriesofextremewavecrest andthecorrespondingvelocityfieldbeneaththehighest crestwithA,=0.04375m,respectively,inwhichthe resultsobtainedbythefullynonlinearnumericalwave tanktechniouearemarkedas'FN—NWT'.The comparisonswiththefifth.orderStokesregularwave resultsresolvedbythepresentmodelarealsoshownin thefigures.Thelinearregularwavevelocityfieldisalso giveninFig.4andmarkedas'linear—RWT'.Fromthe fullynonlinearcrestwaveelevationhistoryinFig.3.it canbededucedthattheperiodandinputamplitude correspondingtothefifth.orderStokesregularwaveare T=I.186sandA0.043l5m.respectively.Andthe linearinputamplitudeisAI=A=0.04587m.FromFig.4, itcanbeseenthattherearegoodagreementsbetweenthe fullynonlinearresultsandthefih.ordersolutions.and thelinearsolutionisa1ittlegreaterthanthetwoformers. O08 0.06 . O02 . 0.04 , 0.06 6789lo t/m Fig.3TimeseriesofextremewavecrestwithAI0.04375m 0 . 0.2 ? O.4 O.6 . O+8 . 1.O 00.10.20,304 v/ms Fig.4Distributionoftheextremevelocityfieldbeneaththe highestcrestwithAI=0.04375m Figs.5and6describetheextremewavetroughcasewith F.0.04375m.Inthiscase.theperiodandinput amplitudeofthefifth—orderStokeswaveare1.197s. A.0.04403mrespectively.Thecorresponding1inear inputamplitudeis1=min=-0.04123m.Similar conclusionswiththoseinFigs.3and4canbeobtained exceptthatthelinearkinematicresultisalittlesmaller thantheothers.Theproposedmodelshowsagood descriptionforextremetroughbelowthemeanwater 1eve1anditsvelocityfield. tls Fig.5Timeseriesofextremewavetroughwith,=一0.04375m O . O.2 毫O-4 0.6 . 0.8 . 1 00.IO.20.30.4 vI/ms Fig.6Distributionoftheextremevelocityfieldbeneaththe lowesttroughwith,=一0.04375m Figs.7and8showtheex仃emewavecrestcaseasthe inputamplitudesumA,isincreasedto0.0875m.From thefullynonlinearnumericalresultsinFig.7.thewave periodandinputamplitudecorrespondingtothe fifu1.orderStokeswaveareT=I.135sandA5=0.089l1m respectively,andthelinearinputwaveamplitudeis A1=Amax=0.1028m.FromFig.8,itcanbeseenthatthe linearkinematicsismuchgreaterthantheotherswiththe increasingofwavenonlineariespeciallynearthe instantaneousfreesurface.However,thethreeresults agreewellwitheachanotherwhenthewaterdepth exceedsabout0.6m.whichindicatesthatthenonlinearity isdecreasedtothelinearcasebelowthedepth-0.6m. Figs.9and10aretheextremewavetroughcaseasthe inputamplitudesumA1isdecreasedto-0.0875m.From O OO g 30 thefullynonlinearnumericalresultsinFig.9.thewave periodandinputamplitudecorrespondingtothe fifth—orderStokeswaveareT-1.195sandA':.0.09431 mrespectively,andthe1inearinputwaveamplitudeis A1max=0.1028m.Fromthefigure,itcanbeseenthat thedifferencebetween1inearkinematicsandtheothersis notasgreatasthatintheextremecrestcase.Thereason isthattheinstantaneOusfeesurfaceis1owerthanthe staticfreesurfaceandthenonlinearityisdecreasedby exponentialindexwithdecreasingofwater1eve1. ().12 008 ; 0.04 0 — 004 . (){)8 . 0.I2 56 tfs l() Fig.7TimeseriesofextremewavecrestwithA10.0875m 0 . 02 — 04 E06 . 0.8 ? I.f) . I2 . I4 , - , T RWl O()20406(}.8 ',l/m/s Fig.8Distributionoftheextremevelocityfieldbeneaththe highestcrestwithAt=0.0875m 0?l2 0.08 0.04 堇0 - (I.04 . 0.08 . 0l2 Fig.9TimeseriesofextremewavetroughwithAI=一0.0875m Figs.11and12aretheextremecrestcaseastheinput amplitudeisfurtherincreasedtoAI=0.12m.Inthiscase, TENGBin.elalAsimplifiedmodelforextreme-wavetdnematicsindeepsea ,,-/Ills Fig.10Distributionoftheextremevelocityfieldbeneaththe lowesttroughwith,=一O.0875m theperiodandinputamplitudeofthefifth.orderStokes regularwaveareT=1.1655sandA=0.1221m respectively,andthelinearinputamplitudeisAI=A= 0.148m.FromFig.11.itcanbeseenthatthepresent modeldescribestheextremecrestverywel1.Fig.12gives thecomparisonsofthevelocityfieldbeneaththegreatest cresttothefullynonlinearnumericalresults.thepresent. 1inearandthird—ordersolutions.Thethird.orderStokes solutionisobtainedfromthemethodofGrueetal【刮and markedas'3rd—SRWT'inthefigure.inwhichtheperiod iscorrespondingto1.14s.Exceptthelinearsolutions.the othersagreewellwithoneanotherandonlyalittle diflferenceexistsbetweenthethird.orderoneandthefully nonlinearresultsinthescope0.2ms<v<0.5ms,. O.12 OO8 O-O4 詈0 - O-O4 _ OO8 Ol2 Fig.1lTimeseriesofextremewavecrestwithA,=0.12m I~/IllS Fig.12Distributionoftheextremevelocityfieldbeneaththe highestcrestwithA,:0.12m JournalofMarineScienceandApplicaton(2009)8:27—32 Furtherworkiscarriedouttovalidatetheproposedmodel bycomparisonwithexperimenta1data【JJ.Thephysical modelwasmadeinaflumewithwaterdepth0.7m.input waveperiodsvaryingfrom0.8sto1.2s:thewavegroup iscomposedof29wavecomponents.TWocasesfkh>3.01 withinputamplitudeAI=0.038mand0.055m respectivelyareconsidered.Fromthetimeseriesof experimentalwaveelevations(inFigs.6(b)and(c)from Ref.tu),thecorrespondingperiodsandinputamplitudes ofthefifth.orderStokeswavearef=0.9521s.Af 0.04268m)and(T=0.943s,A5=0.06257m)respectively. Theexperimentalresultswereeverusedtovalidatethe linearandsecond.orderirregularanalyticalsolutionsby BaldocketalUJ.Theirregularsolutionsarealsousedto comparewiththeproposedmodeIinthepresentstudy. Figs.13-16givethecomparisonsofwaveelevation historyandvelocityfieldbeneaththeextremecrestfor thesetwocases.FromFigs.13and15.itcanbeseenthat theproposedmodeldescribestheextremewavecrest abovemeanwaterlevelverywel1.InFig.14.theresults ontheproposedmode1andtwoirregularwavetheoriesall agreewellwithexperimenta1resultsbecauseofsmaller wavenonlinearitY.InFig.16.theirregularwavetheories cannotpredictextremevelocityfieldwell,especially nearthefreesurface,withstrongernonlinearity.However, theproposedmodelstillshowsagoodagreementwith experiment. 006 O.O2 O - O.O2 AA V }Ij}IIllIIJ{liIIIl;Ijl' 005I.01.S t/s Fig.13TimeseriesofextremewavecrestwithAI=0,038m v/ms Fig.14Distributionoftheextremevelocityfieldbeneaththe highestcrestwithAt=0.038m O.O6 O?O2 0 - 0.02 . O.06 }AA 一 1.51O-O.5OO.51.O1.5 t/s Fig.15TimeseriesofextremewavecrestwithAt=0.055m v/ms. Fig.16Distributionoftheextremevelocityfieldbeneaththe highestcrestwithAt0.055m 4Conclusions Thispaperproposesasimplifiedmodelwhichis developedbasedonthefiRh.orderStokeswavetheory, forpredictingtheextreme.wavevelocityfieldindeepsea. Inthemodelupanddownzero.crossingmethodsare usedtoobtaintwoneighboringwaveperiodscontaining extremewavecrestor~oughandtheaverageofthemis takenastheregularwaveperiodrepresentingtheextreme wave.Thusthecorrespondinginputwaveamplitudecan bededucedbysubstitutingthemeasuredextreme.wave valueintothefifth—orderStokeswaveelevationformula sothatthevelocityfieldbeneaththeex仃emewavecanbe directlycalculatedfromthefifth.orderStokeswave velocityexpression.Thecomparisonswiththefully nonlinearnumericalresults.theexperimentaldataand someotheranalyticalsolutionsshowthattheproposed modelcandescribeextremewaveverywellandhasa goodpredictionfortheextremewavevelocityfield.The proposedmodelresortsonlytothewaverecord,which makesitmoreeasilyapplicabletooceanengineering practice. 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