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AN IMPROVED NUMERICAL METHOD FOR CALCULATIONS IN WAVE KINEMATICS

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AN IMPROVED NUMERICAL METHOD FOR CALCULATIONS IN WAVE KINEMATICS

AN IMPROVED NUMERICAL METHOD

    FOR CALCULATIONS IN WAVE

    KINEMATICS

    826

    Availableonlineat,_^sc.ien~irect.oom

    ScienceDirect

    JournalofHydrodynamics

    2009,21(6):826834

    DOI:10.1016/S10016058(08)60219X

    |lrww.scieneedirect.cornt

    sciencourna]/10016058

    ANIMPRoVEDNUMERICALMETHoDFoRCALCULATIoNSINWAVE

    KINEMATICS

    SUNYiyan,LIUShuxue,LIJinxuan

    StateKayLaboratoryofCoastalandOffshoreEngineering,DalianUniversityofTechnology,Dalian116024,

    Email:sunyy2007@mail.dlut.edu.cn

    (ReceivedMarch30,2009,RevisedJuly21,2009)

    Abstract:Thepredictionofwavekinematicsisanecessarybasisforoceanengineeringdesign.Inthisarticle,animprovedmethod

    tocalculatethewavekinematicsespeciallyinthecrestregionispresented.Thismethodisbasedonthe1oca1approximationmethod.

    Accordingtoonemeasuredwavesurface.theproposedsolutionfocusesonaFourierseriesexpansionjnasomewhatlargersegment

    ofthewavesurfacerecord.Thehorizonta1andvertica1kinematicsunderthewavecrestcanbedirectlypredicted.Goodagreement

betweenthenumerica1resultsandthetheoretica1Stokeswavesverifiesthevalidityofthenu

    merica1mode1.Theeffectivenessofthe

    numericalmethodisalsoprovedbythecalculationofthewavekinematicsofthelaboratoryfoc

    usedwavegroups.

    Keywords:Fourierseriesexpansion,wavekinematics,focusedwave

    1.Introduction

    Thepredictionofwavecrestkinematicsinthe estimationofsedimenttransportandintheanalysesof thewave.inducedcirculationinthenearshorezone andotheraspectsisespeciallynecessary.Inorderto examinethesafetyandserviceabilityofoffshore structuresandmarinepipelines,manyresearcheson thecalculationofwavekinematicshavebeenca~ied outexperimentallyandnumerically.Forexample, Baoproposedariverstageforecastingmodelbased onthehydrodynamicmodels.Johannesseneta1.'J

    simulatedthewavesurfaceandthehydrodynamicsof twodimensionalextremewavesusingthe

    timesteppingmodelbaseduponanappropriate descriptionoftheunderlingwavespectrum.Later SmithandSwanLusedthesamemethodtounde~ake anumericalcalculationoftheextremewavesurface ProjectsupposedbytheNationalNatura1Science FoundationofChina(GrantNo.50379002),theProgramfor NewCenturyExcellentTalentsinUniversity(GrantNo. NCET0520282,.

    Biography:SUNYiyan(1981-),Female,Ph.D.Candidate andthehydrodynamics.Grueetal,adoptedthe

    third.orderStokeswave,fullynonlinearStokeswave andafullynonlinearsimulationtosimulatethe

    kinematicsofextremewaveindeepwater.Batemanet a1.outlinedanumericalmethodbaseduponan adaptationofanexistingapproximationtothe Dirichlet.Neumannoperatorandthencalculatedthe waterparticlekinematicsarisinginadirectionally spreadingwavefield.TengandNing/1proposeda simplifiedmodelbasedonthenRhorderStokes

    regularwave

    .

    theoryforestimatingtheextreme

    kinematics.Liadoptedhigh.orderspectralmethod tosimulatethefocusedwavesurfacesandthe hydrodynamics.ZhaoIfurther

    characteristicsoffocusingwaveswith

    spectralmethod.

    studiedthe

    thehighorder

    However,theabovementionedresearcheswere

    oftenbasedonthesimulatedwavesurfacewhichis calculatedfirstthroughsomeassumptionsandknown parameters.Butinmanysituations.thetimeevolution ofwatersurfaeeatafixedlocationismeasured. Becausethespatialmeasurementsarerareandhardt0 beperformed,theaccuratepredictionofthewave kinematicsfromonemeasuredwavesurfaceis

    becomingimportantforwaveanalysis.Togetthe wavekinematicsusingmeasuredwavesurfaces,two approximationtechniquesareoftenemployed.Oneis calledtheglobalapproximation,whichfocusesona

completemeasuredwavesurfacedefinedby

    consecutivezeroup.crossings.Thetypicalmode1is calledthedoubleFourierseriessolutiondevelopedby BaldockandSwan.Inthismode1.theFourierseries wasgiveninthespatialandtemporaldomainsand thenthewavekinematicswassimulatedbasedonone globalwatersurfacetimehistory.Anotheriscalled thelocalapproximationsthatfoCUSonasomewhat smallersegmentofthewavesurfacerecord.This approximationgenerallyplacesmoreemphasisonthe freesurfaceboundaryconditionsanddonot compromiselocalfidelityintheglobalinterest.soit hasconsiderablepotentialintheestimationofwave kinematics.Relativelyspeaking,thelocal approximationissimpleinmathematicaland computationalcalculationthanglobalapproximation. Wheeler"everproposedoneforlTloflocal

    approximationwhichissocalled"stretching"method.

    Inthisarticle.anempiricaltransformationonthelocal elevationwasintroduced.Variationsonthe"Wheeler stretching"havebeenintroducedbyChakrabarti"J. GudmestadandConnort13J.

    LoandDeantand

    RodenbuschandForristal[15.

    Nielsen[16,17and

    FentonJfurther1ocalizedthedefinitionofamplitude. frequencyandphaseforamovingwindowofthree consecutivewaterSill"faceobservations.Sobey proposedalpealFourierapproximationmethodto estimatethewavekinematics.Itintroducedamoving

    windowinwhichitsupposesthatthesurfaceprofileis locallysteady.Althoughthissolutionhasbeenshown tobeingoodagreementwithhigherordersteadystate

    solutionsithasnotasyetbeenusedtopredictthe kinematicsbeneathahighlynonlinearandunsteady (ortransient)waveevent.

    Inaddition,realoceanwavesareirregularand nonlinear,andsomeextremewavesoftenoccurinthe complicatedseafield.Nowadays.thefocusedwaves generatedbythesuperpositionofcomponentwaves withdifferentfrequenciesatafixedplaceandtimeare alwaysusedtorepresentthelargesttransientwavein ashorttime.Somanyresearcheshavebeencarried outonthewavepropertiesorthekinema

    

    ticsof

    focusedwaves.Forexample.Baldocketa1.Land Suneta1.performedlaboratorystudiesonthe

    Two.dimensionalfocusedwavecharacteristics. Johannesseneta1.andLiueta1.generated

    focusingwavewithdirectionalwavesandinvestigated themultidirectionalwavefocusingpropertiesandthe wavebreakingcharacteristics.

    Inthisarticle.the1ocalFourierapproximation methodpresentedbySobeyLJisemployed.The 827

    mathematicalequationsandthe"local"timewindow definedinthemodelareimproved.Comparisonsof numericalresultswiththeoreticalStokeswavesverifv theusabilityofthismodelandtheappropriate

    parametersareproposedaccordingly.Atthesame time,twodimensionalfocusedwaveswithdifferent wavefactorsaregeneratedinawaveflume.The improvedmodelisalsousedtosimulatethe kinematicsoffocusedwavesanditseffectivenessis furtherverified.

    2.Numericaimodel

    ForacoordinatesystemshowninFig.1.suppose thataunidirectionalirregularwavetrainis propagatinginanlnVlSClO,homogeneousand incompressiblefluidofconstantdepth,thenthefluid motioncanbeassumedtobeirrotationalandthe conservationofmasscanbegivenas

    +=.aZ(1)

    whereisthevelocitypotentia1.Thevelocity components(,w)inthe(,z)directionsaregiven by

    a

    ,

    d

    a

    w=

    dz

    Ifthewaterbedisimpermeable,thevertical velocitymustbezeroatthewaterbed,i.e., w=0atz=-h

    Thekinematicanddynamicfreesurface boundaryconditionsarerespectivelygivenas 一一:

0z=r/wat一——一——U

    a

    +"z+z+g77一一::r/wB0atzr/+"'+~'+g77==

    a22..

    (4)

    whereistheBernoulliconstant.Hereitisnota freeparameter.Itcanbecalculatedbythefollowing equation:

    =

    12(cu

    j/

    n

    cA,

    //

    /

    828

    Fig.1Thedefinitionofthecoordinatesystem Foragivenwavesurfacetimehistoryata

    specifiedposition,thetimespancanbedividedinto severalsmallwindows.Thedefinitionofthetime windowgivenbySobeyisshowninFig.2.The velocitypotentia1adoptedbySobey'srepresentation jneachtimewindowis

    (,f):?Acoshjk(+)'cosh1si.nj(kxoJr)(7)

    wherehisthewaterdepth,AaretheFourier

    

    coemcientskisthewavenumber.isthewave frequencyand(,z)isthespatialpositioninthefixed flame.Togetthesolutionsineachwindow,thewave canbeassumedlocallysteady.Thespecificequations

    definingthewindowsolutiongivenfromEqs.f41and f51canbewrittenasfollowsateachfleesurfacenode: ---=

    .

    D--

    警十22."

    (8)

    Hencethekinematicanddynamicfreesurface boundaryconditionsarecalculatedseparatelyateach timepointofthewindowasSobeymentioned.In

    Fig.2,thesuperscriptsKandDrepresentthe kinematicanddynamicfreesurfaceboundary conditions,respectively.Thenumberrepresentsthe numberofequations.

    D7D2D6D4D8

    Fig.2ThedistributionoflocalpointsdefinedbySobey[. Inthisarticle,toimprovethenumerical

    computation,boththekinematicanddynamic boundaryconditionsarefirstnondimensionalized

    withrespecttotheunknownflequency?,andthe

    formoftheleastsquaressumisusedtominimizethe errorsattheboundaries,i.e.,

    ecc=

    ().+(.]=.ct.,

    Thedisibutionoflocalpointsandfreesurface boundaryconditionsinthetimewindowforthe minimizationcalculationjsshowninFig.3.where?f

    isthetimeinterva1.Ateachtimepointofthewindow. twoboundaryconditionsareconsideredadequately.A largertimewindowcanjncludemorewave

jnf0rmation.

    ?l_———————一

    …….

    .2At-A,

    ———————

    ?

    Att=-2At……..

    Fvec)Fvec(2)Fvec(1)Fvec(3)Fvec(5)

    Fig.3Theimproveddistributionoflocalpointsandboundary

    conditions

    Theprimitiveunknownsinthedefinedwindows areco,k,andA.ofwhichtherearetotallyJ J

    Thespatialphasealwaysoccursinthecombination kx.Therefore.thenumbersofunknownsare3+J. Thenumbersofboundaryequationsshouldsatisfy M?3+J.Thefinalsolutionscanbeobtainedusing Eq.f10)withastandarditerationminimization procedure.Butinitialsolutionsshouldbegivenfirst. Basically,theAiryapproximationstothefreesurface boundaryconditionscanprovidethewellinitial solutionestimates.UsingEq.r5)withJ=1and solvingforAandjntermsofandkgive

    4:3t+

    =tan

    3rl

    a

    ktanhkh

    g77

    o9

(12)

    Theinitialestimateofthewavefrequencyisset as2n/T(Tisthezerocrossingperiod),withk

    estimatedfromfobytheAirydispersionrelationship. InitialestimatesofthehigherorderFourier

    coefficientscanbecalculatedasAj4/10 respectively.

    Togetareasonablesolutionforaninterested wave.themostimportantproblemistochoosethe appropriatenumbersofunknowns,thewindowwidth .andthetimeintervalAff0raspecifiedwave surface.Sobeysuggestedthatthereisno discernableadvantageinadoptingJvaluesin excessof3.Therelationshipbetweenthese parameterswillbeanalyzedanddiscussedinthe followingverification.

    3.Verificationofthenumericalmethod 3.1Comparisonwiththeoreticalkinematics Toevaluatetheeffectivenessofthisimproved method.numericalresultsarefirstcomparedwitha secondorderStokeswavewhichhasaperiodTof 8.0s.awaterdepthhof30.0mandawaveheightH of4.0m.ThetimeintervalAtistakenas0.4s. 0

    0

    _

    0

    0

    Q

    O

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