Morletdezhi.com

NumericalmodellingofnonlinearextremewavesinpresenceofwindNINGDezhi1,2,DUJun2,4,BAIWei3,ZHANGChongwei2*,TENGBin21StateKeyLaboratoryofHydrology-WaterResourcesandHydraulicEngineering,HohaiUniversity,Nanjing210098,China2StateKeyLaboratoryofCoastalandOffshoreEngineering,DalianUniversityofTechnology,Dalian116023,China3SchoolofComputing,MathematicsandDigitalTechnology,ManchesterMetropolitanUniversity,ManchesterM15GD,UK4ChinaCommunicationsPlanningandDesignInstituteforWaterTransportationLtd.
,Beijing100007,ChinaReceived4August2017;accepted10October2017ChineseSocietyforOceanographyandSpringer-VerlagGmbHGermany,partofSpringerNature2018AbstractAnumericalwaveflumewithfullynonlinearfreesurfaceboundaryconditionsisadoptedtoinvestigatethetemporalcharacteristicsofextremewavesinthepresenceofwindatvariousspeeds.
Incidentwavetrainsarenumericallygeneratedbyapiston-typewavemaker,andthewind-excitedpressureisintroducedintodynamicboundaryconditionsusingapressuredistributionoversteepcrests,asdefinedbyJeffreys'shelteringmechanism.
Aboundaryvalueproblemissolvedbyahigher-orderboundaryelementmethod(HOBEM)andamixedEulerian-Lagrangiantimemarchingscheme.
Theproposedmodelisvalidatedthroughcomparisonwithpublishedexperimentaldatafromafocusedwavegroup.
Theinfluenceofwindonextremewaveproperties,includingmaximumextremewavecrest,focalpositionshift,andspectrumevolution,isalsostudied.
Toconsidertheeffectsofthewind-drivencurrentsonawaveevolution,thesimulationsassumeauniformcurrentovervaryingwaterdepth.
Theresultsshowthatwindcausesweakincreasesintheextremewavecrest,andmakesthenonlinearenergytransfernon-reversibleinthefocusinganddefocusingprocesses.
Thenumericalresultsalsoprovideacomparisontodemonstratetheshiftsatfocalpoints,consideringthecombinedeffectsofthewindsandthewind-drivencurrents.
Keywords:extremewaves,fullynonlinearnumericalwaveflume,higher-orderboundaryelement,wavefocusing,Jeffreys'shelteringmechanismCitation:NingDezhi,DuJun,BaiWei,ZhangChongwei,TengBin.
2018.
Numericalmodellingofnonlinearextremewavesinpresenceofwind.
ActaOceanologicaSinica,37(9):90–98,doi:10.
1007/s13131-018-1268-31IntroductionUnderactualoceanconditions,strongnonlinearextremewaves,whichareidentifiedbytheirexceptionallylargeheight,steepshape,asymmetricwaveform,andunpredictability,canposeaseriousthreattoshipsandoffshorestructures.
Currently,thereisnoconsensusonauniquedefinitionforextremewaveevents.
Onedefinitionthatisoftenusedisbasedontheheight.
Thewaveisconsideredtobeextremeifitsheightisgreaterorequalto2.
2timesthesignificantwaveheight(Kharifetal.
,2008).
Severalmechanismshavebeensuggestedasthepossiblecausesfortheextremewaves.
Thefirstmechanismishigh-ordernonlin-earity(higherthanthethirdorder),causingextremewavesinthedeepwater.
ThenonlinearinteractionscantransferenergyamongtheFouriermodesandexcitechaoticmodeevolutions,whichcangenerateasingleextremelylargewavewithanout-standingcrestheight,suchasaroguewave(Morietal.
,2002).
Thesecondmechanismismodulation,orBenjamin–Feirin-stability(BenjaminandFeir,1967),fortheextremewavesformedbyanarrow-bandanddeep-waterwavetrain.
Thismechanismhasbeeninvestigatedextensivelybothanalyticallyandnumeric-ally(Osborneetal.
,2000;Onoratoetal.
,2001,2002).
Addition-ally,dispersivespatial-temporalfocusinghasbeenverifiedtoef-fectivelyinducetheextremewavesthroughthesuperpositionofdifferentfrequencywavecomponentsataspecifictimeandposi-tion(Kharifetal.
,2001).
Thethirdpossiblemechanismfortheextremewavegenerationmaylieintheenergyfocusinginasmallspatialareaduringashorttime,thusgeneratinganabnor-mallylargewave(JohannessenandSwan,2001;BrandiniandGrilli,2001;FuhrmanandMadsen,2006).
Overall,thesestudiesprovidedagoodunderstandingofthemechanismsofextremewaveformation.
Onthebasisoftheabovemechanisms,numerousexperi-mentsandnumericalinvestigationshavebeenconductedre-gardingthephysicalcharacteristicsoftheextremewaves.
Longuet-Higgins(1952)wasoneoftheearliestpioneerstoinvest-igatethestatisticsoftheextremewaves,whothenclarifiedtheef-fectsoffinitebandwidthandnonlinearity(Longuet-Higgins,1980).
Baldocketal.
(1996)createdwavefocusingeventsusingmanysuperimposedregularwavetrainsbasedonalinearwavetheory,andexaminedtheeffectsofnonlinearwave–waveinter-Foundationitem:TheNationalNaturalScienceFoundationofChinaundercontractNos51679036,51490672and51709038;theFundamentalResearchFundsfortheCentralUniversitiesundercontractNosDUT17GJ202andDUT16RC(3)113;theOpenFoundationofStateKeyLaboratoryofHydrology-WaterResourcesandHydraulicEngineeringundercontractNo.
2016490111.
*Correspondingauthor,E-mail:chongweizhang@dlut.
edu.
cnActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–98DOI:10.
1007/s13131-018-1268-3http://www.
hyxb.
org.
cnE-mail:hyxbe@263.
netactionsonstructureinuni-directionalwavegroups.
Theysub-sequentlyintroducedtheconceptofagroupinversioninanex-perimentalcontexttoinvestigatethefreesurfaceprofileoffo-cusedwavegroups.
ThedirectionalfocusedwavegroupwasstudiedexperimentallybyJohannessenandSwan(2001),whoconcludedthatthedirectionalityofthewavefieldhasapro-foundeffectonthenonlinearityofalargewaveevent,andthatlargedirectionallyspreadwavesaremuchlessnonlinearthantheunidirectionalwaves.
Grueetal.
(2003)studiedthekinematicsofthefocusedwavesinthedeepwaterandfoundthatStokesdriftandacorrespond-ingreturnflowbeneathafocusedwavegroupwereinherentinallextremewaveevents.
Intermsofthenumericalsimulation,Ducrozetetal.
(2008)developedanefficientfullynonlinearpo-tentialflowmodelbasedonahigh-orderspectral(HOS)methodtosimulatethepropagationof3-Ddirectionalwavefields.
Twomethods,meshlesslocalPetrov-Galerkinmethod(MLPG_R)andquasi-arbitraryLagrangian-Eulerianfiniteelementmethod(QALE-FEM),werealsodevelopedandcomparedbyMa(2007).
HuandZhang(2014)usedaMorletwaveletspectrummethodtoanalyzenumericalandfieldmeasurementdataontheextremewaveprocess.
Onthebasisofacomparisonofenergycharacter-istics,itwasfoundthatroguewavegenerationdependednotonlyonthecontinuoustransferofthewavetrainenergytoacer-tainregionwhereitsmaximumenergyfinallyoccurs,butalsoonthedistinctshiftoftheconvergedenergytohigh-frequencycom-ponentsinaveryshorttime.
Nevertheless,noneofthesestudiesconsideredthedirecteffectsofwindonextremewaveevents.
Theextremewavesgenerallydonotexistinisolation,andarecommonlyobservedasbeingaccompaniedbywind(MoriandYasuda,2002).
Intheprocessofawavepropagation,thewinden-ergyistransferredtothewavegroup,whichhasastronginflu-enceonthewavepropagationandthenonlinearcharacteristics.
Therefore,itiscriticaltostudytheinfluenceofwindonthepropagationofextremewavesandtheirnonlinearcharacterist-ics.
Liuetal.
(2004)conductedanexploratoryobservationalstudyofthegenerationandpropagationoftheextreme,roguewavesinthesouthernIndianOcean,basedonwavemeasure-ments.
Touboul(2007)performedthenumericalsimulationsoftheextremewaveevolutioninwindusingahigh-orderspectralmethodbasedonJeffreys'shelteringmechanismandmodula-tioninstability.
Inaddition,somenumericalsimulationshavebeenestab-lishedbysolvingtheNavier-Stokesequations,asinSullivanetal.
(2000),SullivanandMcWilliams(2002)andFulgosietal.
(2003).
Kharifetal.
(2008)andToubouletal.
(2006)introducedanaddi-tionalairpressureatthefreesurfaceboundaryconditionsbyconsideringJeffreys'shelteringmechanism.
YanandMa(2011)presentedanimprovedmodelforevaluatingtheeffectsoftheairpressureon2-Dextremewavesbyanalyzingthepressuredistri-butionovertheextremewavesusingtheQALE-FEMandStarCDapproaches.
Theeffectsofwindontwo-dimensionaldispersivefocusingwavegroupswerealsostudiedbyTianandChoi(2013).
Thedirectcomparisonsofmeasurementsandsimulationsweremadebyincludingwind-drivencurrentsinthesimulations.
ZouandChen(2016)investigatedtheeffectsofwindontheevolutionofthe2-Ddispersivefocusingwavegroupsusingatwo-phaseflowmodel.
Comprehensivenumericalstudyoftheevolutionofnonlin-earextremewavesunderwindforcingisbynomeanscomplete,however,newunderstandingsofthisphenomenonarestillre-quiredforthepurposeofaidingengineeringdesignsinharsherenvironments.
Inthepresentstudy,theeffectsofsomeimport-antparameters,suchaswindspeed,inputwaveamplitude,andspectrumbandwidthontheformationofextremewavesandtheircorrespondingtemporal–spatial–spectralevolutionarefur-therevaluated.
Inadditiontothis,thecombinedeffectofwindandwind-drivencurrentsarecomparedtoaddressfocalpointshifts.
Inthispaper,adetaileddescriptionofthenumericalmod-elispresentedinSection2.
Ahigher-orderboundaryelementmethod(HOBEM)basedonthepotential-flowtheoryisadoptedinthisstudy.
ComparedwiththemethodsdescribedabovethatrelyonsolvingtheNavier-Stokesequationsthepresentnumeric-almodelhasclearadvantageswithrespecttocomputationeffi-ciency.
Additionally,regardingthesimulationoffreesurfacewaves,thepresentmethodhasfewernumericaldissipationsthanthosebasedontheNavier-Stokesequationsforlongtimesimula-tion.
Theproposednumericalmodelisfurthervalidatedbycom-parisonwithpublishedexperimentaldatainSection3,andthenumericalresultsarediscussedinSection4.
Finally,conclusionsareprovidedinSection5.
2NumericalmodelTheinteractionsbetweentheextremewavesandwindwithvelocityuinatwo-dimensional(2-D)fluiddomainaredescribedinFig.
1.
Thefreesurface,wavemaker,seabedandtankendaredenotedbyΓf,Γi,ΓdandΓr,respectively.
ACartesiancoordinatesystem,Oxz,isusedsothattheoriginislocatedoverthestillwa-terlevelattheleftendofthedomain,andthez-axisispositiveintheupwarddirection.
Itisassumedthatthefluidisincompress-ible,inviscid,andtheflowmotionirrotationalsothatavelocitypotentialexistsinthefluiddomain.
Consideringthattherearecurrentsinducedbywindandassumingthecurrentsareuni-formlydistributedalongthewaterdepth,thetotalvelocitypoten-tialinthefluiddomaincanthenbeexpressedasΦ=u0x+(x,z,t),whereu0isthesteadyuniformcurrentvelocityand(x,z,t)istheperturbationpotential.
Inthisstudy,themagnitudeoftheuni-formcurrentisempiricallydefinedas0.
9%ofthefree-streamwindspeedu,i.
e.
,u0=0.
9%u,thesamevalueusedbyTianandChoi(2013)andZouandChen(2016).
Boththetotalvelocitypo-tentialandperturbationpotentialsatisfytheLaplaceequationinthecomputationaldomainΩ.
Giventheboundaryconditions,thevelocitypotentialcanbedeterminedbysolvingthefollowingboundaryintegralequa-tionbasedonGreen'ssecondidentity(BrebbiaandWalker,1980;Anderson,1984):whereΓrepresentstheentirecomputationalboundary;pandqarethesourcepoint(x0,z0)andfieldpoint(x,z),respectively;αisFig.
1.
Sketchofthenumericalwaveflume.
NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–9891thesolidangle;andGistheGreenfunctionconsideringanim-ageoftheRankinesourceabouttheseafloor,andcanbewrittenasG(p,q)=lnr+lnr1,whereandr1=.
OntheinstantaneousfreesurfaceГf,thefullynonlinearkin-ematicanddynamicboundaryconditionsaresatisfied.
Theso-calledmixedEulerian-Lagrangianmethodisusedtodescribeatime-varyingfreesurface.
Towardstheendofthecomputationaldomain,anartificialdampingbeachisappliedtothefreesurfacesothatthewaveenergyisgraduallydissipatedinthedirectionofwavepropagation.
Theprofileandmagnitudeofartificialdamp-ingmustminimizepossiblewavereflectionattheleadingedgeofthedampingzonewhilemaximizingwaveenergydissipationinthedampingzone.
Inthepresentstudy,both-andη-typedampingtermsareintroducedinthefreesurfaceboundarycon-ditions,whichcanbeexpressedintheLagrangianexpressionasfollows:wheregistheaccelerationduetothegravity;pisthepressure;ηistheinstantaneousfreesurfaceelevation;D/Dtisthematerialderivative;andx0isthestartingpositionofthedampinglayer.
Thedampingcoefficientfunctionμ(x)isdefinedaswhereωmindenotestheminimumangularfrequencyofthewavecomponents;andLbisthelengthofthedampinglayerandsetas1.
5λmax(whereλmaxdenotesthemaximumwavelengthofallwavecomponents)inthepresentstudy.
Inordertoconsiderthepressureofwind,followingworkbyKharifetal.
(2008)andToubouletal.
(2006),thepressureontheinterfacez=η(x,t)isrelatedtothelocalwaveslope.
Inthepresentstudy,athresholdforthelocalwaveslopeηxisintroduced,abovewhichanenergytransferfromwindtowaveoccurs.
Thecriticalvalueoftheslopeηxcissetat0.
35(Toubouletal.
,2006)andthepressurecanbecalculatedbythefollowingexpression:wheretheconstantsistheshelteringcoefficientwithavalueof0.
5basedonexperimentaldata,uisthewindspeed,isthemaximumlocalwaveslope,cisthewavephasevelocity,andρaistheatmosphericdensity.
AttheoutflowboundaryГr,therigidandimpermeableboundaryconditionissatisfiedasAttheinflowboundaryГi,fluidmotionisgeneratedbyapis-tonwavemaker,andforthefocusedwavethedisplacementSandvelocityupofthewavemakercanbespecifiedas(Ningetal.
,2015)whereNisthenumberofwavecomponents;ai,kiandωiaretherespectivelinearwaveamplitude,wavenumberandangularfre-quencyoftheithcomponentsatisfyingthelinearDoppler-shif-teddispersionrelationship(ωi–kiu0)2=gkitanhkih.
xpandtpde-notethefocalpositionandfocaltimeaspredictedbylinearwavetheory.
Tr=4sinh2(kih)/[2kih+sinh(2kih)]isthetransferfunctionforthepistonwavemakerandhisthestaticwaterdepth.
Astheaboveboundaryvalueproblemissolvedinthetimedomain,theinitialwatersurfaceconditionsareappliedinthisstudy:Inaddition,thewavemakerpropertiesontheinflowbound-aryГiareimposedgraduallyusingarampingfunction,whichsat-isfiescalmwaterconditionsandsmoothlyapproachesunityasthesimulationproceeds.
TherampingfunctionisgivenbywhereTmisspecifiedasthelengthoftimeforwhichtheinputwaveisramped,herechosenastwicethemaximumwaveperiod(i.
e.
,2Tmax)amongallthewavecomponentsinthefocusedwavegroup.
Inthisstudy,theboundarysurfaceisdiscretizedbythree-nodeisoparametricelements,bywhichEq.
(1)inthediscretizedformcanbeexpressedasfollows:whereξrepresentsthelocalintrinsiccoordinates,Misthenum-berofdiscretizedelementsonthesurface,andJ(ξ)istheJacobi-anmatrixrelatingthephysicalcoordinatestothelocalintrinsiccoordinateswithinanelement.
Eventually,thediscretizedinteg-ralequationistransformedintoasystemoflinearalgebraicequations.
Aftersolvingtheboundaryvalueproblemandobtainingfluidvelocitiesandthenormalvectoronthefreesurface,thefreesur-faceboundaryconditionsinEq.
(2)areadvancedintimeasde-scribedbyNingandTeng(2007).
Forthispurpose,afourth-or-derRunge-Kutta(RK4)schemeisadopted.
Thefluiddomainisremeshedateachtimesteptopreventfree-surfacenodesfrompilingupatcertainpositions.
Basedonthehorizontalcoordin-atesofnewnodesobtainedthroughmeshgeneration,thevertic-alpositionandpotentialcouldbeinterpolatedusingthequadrat-icshapeequation.
Tofindwhicholdlineelementthenewnodebelongsto,thefollowingcriterionwasused:92NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–98whereL0isthelengthoftheoldlineelement,Liisthelengthofasub-elementconsistingofonenodeintheoldelementandthenewnodebeingconsidered,andM1isthenumberofsub-ele-mentssurroundingthenode:hereM1=2.
3ValidationtestsInordertovalidatethepresentmodel,numericalresultsarecomparedwiththeexperimentaldatainKharifetal.
(2008)forthecaseofa2Dextremewaveunderwindaction.
ThewindspeedissetasU=0.
Thetankis40minlengthand2.
6minheight,withawaterdepthof1m.
Theextremewaveisgeneratedbyawavemakerwithmotiondefinedbyasinefunction.
Thefre-quencyofthesinefunctionvarieslinearlyfromthemaximumfrequency(fmax=1.
85Hz)totheminimumfrequency(fmin=0.
8Hz)overadurationofT=23.
5s.
Themotionofthewavemakerisgov-ernedbywhereaistheexpectedwaveamplitude,whichisgivenas0.
007,andFisthetransferfunctionforthewavemaker(Ma,2007),writ-tenasFigure2displaysthewaveelevationatx=1minthephysicalexperimentandnumericalsimulation.
Therewasgenerallygoodagreementandthediscrepancyduringtheinitialperiodwasduetotheuseofdifferentrampingfunctions.
Figure3showsthesur-faceelevationatseveralpositions,measuredexperimentallyandcomputednumerically.
Thephasesandamplitudesofthenu-mericalandexperimentalwavetrainswereingoodagreement,demonstratingtheefficiencyofthepresentnumericalmethodincorrectlyreproducingthenonlinearevolutionofwavegroupsduringthefocusing-defocusingcycle.
4ThenumericalresultsanddiscussionNumericalsimulationswerenextcarriedoutforthefocusedwavegroupinteractionwithwindandwind-drivencurrents.
Theeffectsofthewindvelocity,thewavespectrabandwidthandtheinputwavegroupamplitudewerestudied.
4.
1EvolutionofwavegroupsunderwindforcingTheparametersforthiscaseincludedstaticwaterdepthh=0.
4m,waveperiod0.
8s≤T≤1.
2s(definedasthenarrow-bandcase),0.
6s≤T≤1.
4s(definedasthewide-bandcase),andtotalinputgroupamplitudeAt=0.
05mandAt=0.
06m.
Inaddi-tion,thewaveamplitudewaskeptconstantamongthetotalof29wavecomponents,andthedesiredfocusingeventoccurredatxp=6.
5λminandtp=16.
5Tmin(λminandTmindenotetheshortestwavelengthandsmallestwaveperiodamongallwavecompon-ents,respectively).
Forthepurposeofeasiercomparison,inthefollowingfiguresboththefocalpositionandfocaltimewereshif-tedto0ontheaxesbysubtractingthecorrespondingcoordin-ateswithxpandtp.
Figure4showsthemaximumfocusingwaveamplitudeun-derdifferentwindspeedconditionswithboththewide-andnar-row-bandwidthspectra.
Thewaveamplitudewasnon-dimen-sionalizedbyAt,whichincreasedasthewindspeedincreasedduetothefactthatmoreenergyistransferredtothewavegroup.
Inaddition,thewavespeedseemedtohaveagreaterinfluenceonlargerwaves:theincreaseinthewaveamplitudeforlargerwaveswasmoresignificantwiththeincreaseinthewindspeed.
Frequencybandwidthwasalsoanimportantfactorthataffectedtheextremewavecharacteristics.
Inthecaseofthenarrow-band-widthspectrum,thewaveamplitudegrewatthesamewindspeed,exhibitingstrongernonlinearity.
Figure5showsthesurfaceelevationwhenthefocusingeventoccurredatwindspeedsof0,2,4,6and8m/satAt=0.
06m.
Themaximumfocusingamplitudeclearlyincreasedwiththein-creaseinthewindspeed.
Thisfigurealsoshowstheeffectofwindbyshiftingthefocalpositiondownstream,mostobviouslyforthenarrow-bandcase.
Forexample,inFig.
5athefocalpositionshif-ted0.
55mdownstreamatu=8m/s,whileatthesamespeedtheshiftinfocalpositionincreasedto1.
34mforthenarrow-bandcase,asshowninFig.
5b.
Figure6showsthedeviationinfocalpositionasafunctionofthewindvelocity.
At=0.
05m,theshiftofthefocalpositiondidnotFig.
2.
Surfaceelevationasafunctionoftimeatx=1m.
Fig.
3.
Surfaceelevationasafunctionoftimeatx=21,18and11m.
Solidlinerepresentstheexperimentaldataanddashedlinenu-mericalsimulation.
NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–9893appeartobesensitivetothewindspeedandhardlychangedasthewindspeedincreased,whileAt=0.
06m,thewindcausedaweakdownstreamshiftatthefocalpoint.
Thesamephenomen-onwasalsoobservedinstudiesbyKharifetal.
(2008)andToubouletal.
(2006),whichwasduetocurrentsbeinginducedbythewind.
TheJeffreys'shelteringmechanismdescribesairflowseparationoverwaves.
Thismechanismisnotremarkableformilderwaves.
However,forsteepwavesitiswellknownthattheairflowseparationresultsinamuchhigherenergytransferfromwindtowaves(Toubouletal.
,2008).
Figure7showstheamplificationfactorHmax/A,asafunctionofspaceforthewavegroupunderfourdifferentwindforcingconditions(u=0,2,4and6m/s).
HereHmaxisthemaximumheightbetweentheconsecutivecrestandtroughinthetransientgroup.
Incontrastwiththecasewithoutwind,therewasanasymmetricprofilethatappearedbetweenthefocusinganddefo-cusingstages.
Particularlyduringthedefocusingstage,itwasob-servedthatHmax/Aincreasedmarkedlywiththeincreaseinthewindspeedforthenarrow-bandcase.
InFig.
7a,whenu=6m/sthemaximumHmax/Awasaround1.
93,butinFig.
7dthemaxim-umHmax/Areached2.
62atthesamewindspeed.
Thespectralenergyevolutionforwavefocusinganddefocus-ingisshownforcaseswithandwithoutwindactioninFig.
8.
AninputgroupamplitudeofA=0.
06mwasconsideredforboththenarrow-bandandwide-bandcases.
Fiverepresentativespatialpoints,includingupstreampoints,theactualfocalpointxf,anddownstreampointsareplotted.
Thesolidlineindicatesthedens-ityspectrumatthefirstupstreamreferencepoint,andthedashedlinesdenotethoseattheothermarkedpointsinthefigures.
Asthewavegroupapproachedthefocalposition,thetransferofspectralenergyfromtheprimaryfrequencytohigherfrequen-ciescouldclearlybeseen.
Thewaveenergywasthentransferredfromthehighfrequenciesbacktothefundamentalone,andthecorrespondingspectragraduallyreturnedtotheiroriginalrefer-Fig.
4.
Plotsoffocalcrestelevationagainstwindspeedatdifferentwaveamplitudesandspectra.
a.
0.
6s≤T≤1.
4sandb.
0.
8s≤T≤1.
2s.
Fig.
5.
Spatialdistributionofwaveelevationatfocaltimewithwindvelocitiesof0,2,4,6and8m/s.
a.
0.
6s≤T≤1.
4sandb.
0.
8s≤T≤1.
2s.
Fig.
6.
Plotoffocalpositiondeviationagainstwindspeedatdifferentwaveamplitudesandspectra.
a.
0.
6s≤T≤1.
4sandb.
0.
8s≤T≤1.
2s.
94NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–98Fig.
7.
EvolutionofHmax/Aasafunctionofspaceforvariouswindspeedsatdifferentwaveamplitudesandspectra.
a.
0.
6s≤T≤1.
4s,b.
0.
6s≤T≤1.
4s,c.
0.
8s≤T≤1.
2sandd.
0.
8s≤T≤1.
2s.
Fig.
8.
Energyspectrumatdifferentpointsforcaseswithandwithoutwind.
a.
0.
6s≤T≤1.
4s,u=0m/s;b.
0.
6s≤T≤1.
4s,u=6m/s;c.
0.
8s≤T≤1.
2s,u=0m/s;andd.
0.
8s≤T≤1.
2s,u=6m/s.
NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–9895encevaluesduringthewavedefocusingprocessinthecasewithoutwind.
Thismeansthatthenonlinearenergytransferwasreversibleinthefocusinganddefocusingprocessesandtheef-fectsofthewavetowaveinteractionsweregraduallydiminished.
Incontrast,whenthewindvelocityuwas6m/s,theenergytrans-ferredtothehighfrequenciescouldnotrecovertoitsinitialrefer-encelevelasshowninFig.
8.
Energytransfertothehighfrequen-cieswasstillvisibleatthefocalpoint.
4.
2Wind-drivencurrentsThepresenceofwindforcingintroducesathinsurfacedriftlayer,whichmayhaveimportanteffectsontheevolutionofthewavegroups(BannerandPhillips,1974).
Thislayerhashighvor-ticityandthevelocityprofiledependsstronglyonthewaterdepth(PhillipsandBanner,1974);however,forsimplicity,thelayercanbemodeledasauniformsurfacecurrent(Kharifetal.
,2008)withamagnitudethatistypicallyafewpercentofthewindspeed.
Fig-ure9comparesthedistributionofthefocusedcrestelevationun-derwindactionwithandwithoutwind-drivencurrents,withlin-earwindspeedpredictionsofu=2and4m/s.
Itshowsthat,asthewaveamplitudeincreases,themaximumwaveelevationin-creasesanddeviatesfromthelinearsolution.
Furthermore,themaximumcrestelevationincreasedmorerapidlyforthecasewithwindonly,butlesssignificantlyforthecasewiththewind-drivencurrentsduetoadecreasingwavenonlinearity.
Figures10and11showthetemporalhistoryofthewaveelev-ationatthefocusingpositionandthespatialdistributionofthewaveelevationatthefocusingtimeforfourdifferentcases,i.
e.
,purewave(u=0m/s,u0=0m/s),windaction(u=6m/s,u0=0m/s),andthedualactionofwindsandinducedcurrents(u=6m/s,u0=0.
054m/s).
Itcanbeseenthatthefocusingtimedelaysandthefocusingpositionshifteddownstreamcomparedwiththoseinthepurewavesfortwocases(windonlyandwind-drivencur-rents).
Inparticular,thepostponementofthefocaltimeandthefocalpositionwasmostsignificantforthewind-drivencurrentscase,becauseboththeinfluenceofthewindandthewind-driv-encurrentsweretakenintoaccount.
Ontheotherhand,duetothenonlineareffect,thedelaysinthefocalpositionandthefocaltimeweremoreobviousinthenarrow-bandspectrum.
Forex-ample,inFig.
10dthedelayinfocaltimeforthecaseofu=6m/s,u0=0.
054m/swas2.
9s,whileinFig.
11dthedelayinfocalposi-tionforthesamecasewas4.
14m.
5ConclusionsTheinfluenceofwindonthecharacteristicsoftheextremewaveswasinvestigatedusingafullynonlinearwindandwaveFig.
9.
Comparisonoffocuswaveamplitudeundertheactionofvarioussourcesatu=2m/s(a)and4m/s(b).
Fig.
10.
Comparisonoftimehistoryatfocalposition.
aandb.
0.
6s≤T≤1.
4s;candd.
0.
8s≤T≤1.
2s.
96NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–98mixing2-Dnumericaltankmodel.
Thewind-excitedpressurewasmodelledusingamodifiedJeffreys'shelteringmechanismmodel.
Throughaseriesofnumericalinvestigations,effectsofthewindpressureontheextremewaveweredescribedandcanbeclassifiedasfollows.
First,themaximumfocusingamplitudeoftheextremewavewasincreasedduetothepresenceofawindpressure.
Second,thecurrentinducedbythewindweaklyshif-tedthefocusingpositionoftheextremewaves.
Thecaseswithnarrow-bandspectraandlargerinputwaveamplitudesweremoresignificantlyinfluencedbywind.
Thirdly,unlikethecaseswithnowind,thewaveprofilesforcaseswithwindwereasym-metricbetweenthefocusinganddefocusingstages.
Duringtheprocessofdefocusinginparticular,aclearincreaseinHmax/Awasobservedasthewindspeedincreased,forthenarrow-bandcase.
Finally,thewindaffectedthespectralevolutionofthefo-cusingwavegroups.
Forthecasewithoutwind,asthewavegroupapproachedthefocusingposition,therewasacleartrans-ferofspectralenergyfromtheprimaryfrequencytohigherfre-quencies.
Thentherewasareversetransferofthewaveenergyandthecorrespondingspectragraduallyrecoveredtotheirori-ginalreferencevaluesduringthewavedefocusingprocess.
Incontrast,consideringwhenthewindvelocityis6m/s,theenergytransferredtothehigherfrequencieswasnotabletoreturntotheinitialreferencelevel.
Thedirectcomparisonoftheeffectsofwind,currents,andwind-drivencurrentsrevealsthatthemaxim-umcrestelevationincreasesmoreclearlyinthecaseofwindonly,andleastofallinthecasewithcurrentsonly.
Inaddition,becausetheinfluenceofbothwindsandcurrentswastakenintoaccountinthewind-drivencurrentscase,thefocusingtimeandthefocusingpositionweremostobviouslydelayed.
Itmustbenotedthatthepresentstudyisbasedonthenonlinearpotential-flowtheory.
Formoredetailontheinteractionsofwindandwaves,viscouseffectsshouldbeconsideredinthefutureinvest-igations.
ReferencesAndersonMP.
1984.
Theboundaryintegralequationmethodforpor-ousmediaflow.
EosTransactionsAmericanGeophysicalUni-on,65(9):76–76BaldockTE,SwanC,TaylorPH.
1996.
Alaboratorystudyofnonlin-earsurfacewavesonwater.
PhilosophicalTransactionsoftheRoyalSocietyofLondonA:Mathematical,PhysicalandEngin-eeringSciences,354(1707):649–676BannerML,PhillipsOM.
1974.
Ontheincipientbreakingofsmallscalewaves.
JournalofFluidMechanics,65(4):647–656BenjaminTB,FeirJE.
1967.
Thedisintegrationofwavetrainsondeepwater:Part1.
Theory.
JournalofFluidMechanics,27(3):417–430BrandiniC,GrilliS.
2001.
Modelingoffreakwavegenerationina3D-NWT.
In:11thInternationalOffshoreandPolarEngineeringConference,17-22June,Stavanger,NorwayBrebbiaCA,WalkerS.
1980.
BoundaryElementTechniquesinEngin-eering.
Amsterdam:ElsevierDucrozetG,BonnefoyF,FerrantP.
2008.
Analysisoffreakwavesformationwithlargescalefullynonlinearhighorderspectralsimulations.
In:18thInternationalOffshoreandPolarEngin-eeringConference,July6-11,Vancouver,BC,CanadaFuhrmanDR,MadsenPA.
2006.
Numericalsimulationofextremeeventsfromfocuseddirectionallyspreadwavefields.
In:Pro-ceedingsofthe30thInternationalConferenceonCoastalEn-gineering.
SanDiego,California,USA:WorldScientificFulgosiM,LakehalD,BanerjeeS,etal.
2003.
Directnumericalsimu-lationofturbulenceinashearedair-waterflowwithadeform-ableinterface.
JournalofFluidMechanics,482:319–345GrueJ,ClamondD,HusebyM,etal.
2003.
Kinematicsofextremewavesindeepwater.
AppliedOceanResearch,25(6):355–366HuJinpeng,ZhangYunqiu.
2014.
Analysisofenergycharacteristicsintheprocessoffreakwavegeneration.
ChinaOceanEngineer-ing,28(2):193–205JohannessenTB,SwanC.
2001.
AlaboratorystudyofthefocusingofFig.
11.
Comparisonofspatialdistributionatdifferentfocaltimes.
aandb.
0.
6s≤T≤1.
4s;candd.
0.
8s≤T≤1.
2s.
NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–9897transientanddirectionallyspreadsurfacewaterwaves.
Pro-ceedingsoftheRoyalSocietyA:Mathematical,PhysicalandEngineeringSciences,457(2008):971–1006KharifC,GiovanangeliJP,TouboulJ,etal.
2008.
Influenceofwindonextremewaveevents:experimentalandnumericalapproaches.
JournalofFluidMechanics,594:209–247KharifC,PelinovskyE,TalipovaT,etal.
2001.
Focusingofnonlinearwavegroupsindeepwater.
JournalofExperimentalandTheor-eticalPhysicsLetters,73(4):170–175LiuPC,MacHuchonKR,WuCH.
2004.
ExploringroguewavesfromobservationsinSouthIndianOcean.
Actesdecolloques-IFRE-MER,39:1–10Longuet-HigginsMS.
1952.
Onthestatisticaldistributionsofseawaves.
JournalofMarineResearch,11(3):245–265Longuet-HigginsMS.
1980.
Onthedistributionoftheheightsofseawaves:someeffectsofnonlinearityandfinitebandwidth.
JournalofGeophysicalResearch:Oceans,85(C3):1519–1523MaQingwei.
2007.
NumericalgenerationoffreakwavesusingMLPG_RandQALE-FEMmethods.
ComputerModelinginEn-gineering&Sciences,18(3):223–234MoriN,LiuPC,YasudaT.
2002.
Analysisoffreakwavemeasure-mentsintheSeaofJapan.
OceanEngineering,29(11):1399–1414MoriN,YasudaT.
2002.
Effectsofhigh-ordernonlinearinteractionsonunidirectionalwavetrains.
OceanEngineering,29(10):1233–1245NingDZ,TengB.
2007.
Numericalsimulationoffullynonlinearir-regularwavetankinthreedimension.
InternationalJournalforNumericalMethodsinFluids,53(12):1847–1862NingDezhi,ZhuoXiaoling,HouTiancong,etal.
2015.
Numericalin-vestigationoffocusedwavesonuniformcurrents.
Internation-alJournalofOffshoreandPolarEngineering,25(1):19–25OnoratoM,OsborneAR,SerioM,etal.
2001.
Freakwavesinrandomoceanicseastates.
PhysicalReviewLetters,86(25):5831–5834OnoratoM,OsborneAR,SerioM.
2002.
Extremewaveeventsindir-ectional,randomoceanicseastates.
PhysicsofFluids,14(4):L25–L28OsborneAR,OnoratoM,SerioM.
2000.
Thenonlineardynamicsofroguewavesandholesindeep-watergravitywavetrains.
Phys-icsLetters:A,275(5-6):386–393PhillipsOM,BannerML.
1974.
Wavebreakinginthepresenceofwinddriftandswell.
JournalofFluidMechanics,66(4):625–640SullivanPP,McWilliamsJC.
2002.
Turbulentflowoverwaterwavesinthepresenceofstratification.
PhysicsofFluids,14(3):1182–1195SullivanPP,McWilliamsJC,MoengCH.
2000.
Simulationofturbu-lentflowoveridealizedwaterwaves.
JournalofFluidMechan-ics,404:47–85TianZhigang,ChoiW.
2013.
Evolutionofdeep-waterwavesunderwindforcingandwavebreakingeffects:numericalsimulationsandexperimentalassessment.
EuropeanJournalofMechanics–B/Fluids,41:11–22TouboulJ.
2007.
Ontheinfluenceofwindonextremewaveevents.
NaturalHazardsandEarthSystemSciences,7(1):123–128TouboulJ,GiovanangeliJP,KharifC,etal.
2006.
Freakwavesundertheactionofwind:experimentsandsimulations.
EuropeanJournalofMechanics–B/Fluids,25(5):662–676TouboulJ,KharifC,PelinovskyE,etal.
2008.
OntheinteractionofwindandsteepgravitywavegroupsusingMiles'andJeffreys'mechanisms.
NonlinearProcessesinGeophysics,15(6):1023–1031YanShiqiang,MaQingwei.
2011.
Improvedmodelforairpressureduetowindon2Dfreakwavesinfinitedepth.
EuropeanJourn-alofMechanics–B/Fluids,30(1):1–11ZouQingping,ChenHaifei.
2016.
Numericalsimulationofwindef-fectsontheevolutionoffreakwaves.
In:26thInternationalOceanandPolarEngineeringConference.
ISOPE,635–64098NINGDezhietal.
ActaOceanol.
Sin.
,2018,Vol.
37,No.
9,P.
90–98