Fluid–structure interaction between a two-dimensional

JournalofFluidsandStructures24(2008)527–540

/locate/jfs

Fluid–structureinteractionbetweenatwo-dimensional

mat-typeVLFSandsolitarywavesby

theGreen–Naghditheory

D.Xiaa,R.C.Ertekinb,Ã,J.W.Kimc

b

INTECEngineeringPartnershipLtd.,15600JFKBlvd,Houston,TX77032,USA

DepartmentofOceanandResourcesEngineering,UniversityofHawaii,2540DoleSt.,HolmesHall402,Honolulu,HI96822,USA

c

TechnipUSA,11700OldKatyRd.,Suite150,Houston,TX77079,USA

Received1June2007;accepted22October2007

Availableonline7January2008

a

Abstract

Thetwo-dimensional,nonlinearhydroelasticityofamat-typeverylarge oatingstructure(VLFS)isstudiedwithinthescopeoflinearbeamtheoryforthestructureandthenonlinear,LevelIGreen–Naghdi(GN)theoryforthe uid.ThebeamequationandtheGNequationsarecoupledthroughthekinematicanddynamicboundaryconditionstoobtainanewsetofmodi edGNequations.Theseequationsrepresentlong-wavemotionbeneathanelasticplate.Asetofjumpconditionsthatarenecessaryforthecontinuity(orthematching)ofthesolutionsintheopenwaterregionandthatunderthestructureisderivedthroughtheuseofthepostulatedconservationlawsofmass,momentum,andmechanicalenergy.Theresultinggoverningequations,subjectedtotheboundaryandjumpconditions,aresolvedbythe nite-differencemethodinthetimedomain.Thepresentmodelisapplicable,forexample,tothestudyofthehydroelasticresponseofamat-typeVLFSundertheactionofasolitarywave,orafrontaltsunamiwave.Goodagreementisobservedbetweenthemodelresultsandotherpublishedtheoreticalandnumericalpredictions,aswellasexperimentaldata.Theresultsshowthatconsiderationofnonlinearityisimportantforaccuratepredictionsofthebendingmomentofthe oatingelasticplate.Itisfoundthattherigidityofthestructuregreatlyaffectsthebendingmomentanddisplacementofthestructureinthisnonlineartheory.r2007ElsevierLtd.Allrightsreserved.

Keywords:Green–Naghdiequations;Hydroelasticity;Mat-typeVLFS;Solitarywaves;Nonlinearshallow-waterwaves;Bendingmoment

1.Introduction

Averylarge oatingstructure(VLFS)witha‘small’draftbehaveslikeanelasticplate.ThefrontalwaveofatsunamiaroundaVLFScanbedescribedasasolitarywave.Inthiswork,weconsiderathinelasticplateundertheactionofasolitarywaveinshallowwatertoinvestigatethedynamicresponseofaVLFStoatsunami.

Stoker(1957)studiedlongwavesbeneatha oatingelasticplate rstbyfocusinghisattentiononthetransmissionofwavesbeneatha oatingbreakwater,withinthescopeoflinearwavetheory.Hisworkhasbeenextendedtothree

Correspondingauthor.Tel.:+18009566818;fax:+18009563498.

E-mailaddresses:Dingwu.Xia@(D.Xia),ertekin@hawaii.edu(R.C.Ertekin),JWKim@(J.W.Kim).0889-9746/$-seefrontmatterr2007ElsevierLtd.Allrightsreserved.doi:10.1016/j.j uidstructs.2007.10.009

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D.Xiaetal./JournalofFluidsandStructures24(2008)527–540

dimensionsbyEvansandDavies(1968).Forbes(1986,1988)investigatedtwo-dimensionalperiodicwavesbeneathanelasticplaterestingonthesurfaceofanin nitelydeep uid,byuseofahigh-orderseriesexpansiontechnique.Lu(1991)examinedthetransmissionandre ectionofasolitonbyatwo-dimensionalstructure.Heusedthematchedasymptotic-expansionmethodwhichgivesarelationbetweentheoutersolutiongovernedbytheBoussinesqequationsandtheinnersolutiongovernedbytheLaplaceequation.Takagi(1996,1997)derivedtheBoussinesq-classequationswhichrepresentalong-wavemotionbeneathanelasticplate,andthematchingconditionsbetweentheordinaryandthemodi edBoussinesqequationsbyemployingthematchedasymptotic-expansionmethod.ErtekinandKim(1999)studiedthehydroelasticbehaviorofathree-dimensionalmat-typeVLFSinobliquewavesbyuseofthelinearGreen–Naghdi(GN)theory.

Inthepresentwork,theLevelIGNequationsthatrepresentlong-wavemotionbeneathanelasticplatearederived,andthematchingconditionsaredevelopedbyemployingasetofjumpconditions.ThetheoryandthenumericalmethodusedarediscussedinSections2and3,respectively.Results,theimportanceofvariousparametersoftheproblem,andcomparisonswiththeavailableexperimentaldataareprovidedinSection4.2.Theory

Thecoordinatesystemandgeometryofthe uid–structuresystemofthetwo-dimensionalproblemaredepictedinFig.1.Thebeamistakenfromastripofunitwidthfromaplateoflargeaspectratio(beamlengthoverbeamwidthratioislarge).Thebeamisenvisionedasthestripofa oatingrunwaythatfreely oatsonthetopofRegionII,withadraftdlengthbandthicknesshp.InFig.1,misthemassperunitlengthofthebeam(thatis,ofunitwidth).ItsthicknessismuchsmallerthanitslengthsothattheKirchhoffthin-platetheorycanbeapplied.Thewavepropagatesparalleltothebeam-lengthdirectionsothatthehydroelasticphenomenonalsoistwo-dimensional.

Thewholedomainisdividedintothreeparts.RegionsIandIIIcontaintheordinaryshallow-waterwaveproblems.Anonlinear,long(solitary)wavecomesfromleftandexcitesthemotionofthebeam.Theinviscid uidmotionisassumedtobegovernedbytheLevelIGNtheory[see,e.g.,ErtekinandWehausen(1986)].Thesea ooris at.Thestill-waterdepthintheopenwaterareaish0,andinRegionII,itish0Àd.Wefurtherassumethattherunwayishorizontallyrestrainedsomehow,andthus,nohorizontalmotioninthexdirectionisallowed.2.1.Governingequations

Thegoverningequationsforthemotionofthe uidareprovidedbytheLevelIGNtheory[see,e.g.,GreenandNaghdi(1976)].Theycanbewritteninacompactform[see,e.g.,ErtekinandBecker(1998)]:

Ztþ

q½ðhþZÞu

¼0,

qx

^xp1

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