Wamit-theory-manual

MASSACHUSETTESINSTITUTEOFTECHNOLOGYDEPARTMENTOFOCEANENGINEERINGCambridge,MASS.02139WAMITTHEORYMANUALbyC.-H.LeeReportNo.95-2October1995ThisworkwasconductedundertheJointIndustryProject‘Waveeffectsonoffshorestructures’.ThesponsorshaveincludedChevron,Conoco,DavidTaylorModelBasin,DetNorskeVeritas,Exxon,Mobil,theNationalResearchCouncilofCanada,OffshoreTech-nologyResearchCenter,NorskHydroAS,Petrobr´as,SagaPetroleum,ShellDevelopmentCompany,andStatoilAS.Copyrightc
MassachusettsInstituteofTechnology1995CONTENTS1.INTRODUCTION1.1DescriptionoftheProblem2.THEFIRST-ORDERPROBLEM2.1BoundaryValueProblem2.1.1Multiplebodies2.1.2Bodydeformations2.1.3Bodiesnearwalls2.2IntegralEquations2.2.1Greenfunction2.2.2Linearsystem2.3RemovaloftheIrregularFrequencies3.THESECOND-ORDERPROBLEM3.1BoundaryValueProblem3.2Second-orderIncidentWaves3.3Second-orderScatteringWaves4.THEFIRST-ANDSECOND-ORDERFORCE4.1ForceandMoment4.1.1Coordinatesystem4.1.2Coordinatetransform4.1.3Pressureintegration4.1.4HydrostaticforceandmomentofO(1)4.1.5Linearforceandmoment4.1.5Second-orderforceandmoment4.2HaskindForceandtheSecond-orderIndirectForce4.3EquationsofMotionREFERENCESAREFERENCESBAPPENDICIESChapter1INTRODUCTIONThisTheoryManualdescribesthetheoreticalbackgroundofthecomputerprogramWAMIT(WaveAnalysisMIT).Itprovidesanoverviewofthetheoriesandthecomputa-tionalmethodologiesinaconcisemanner.Fordetailedinformationontheselectedsubjects,someofthepublishedandunpublishedarticlesareincludedintheAppendix.TheuseoftheprogramWAMITanditscapabilitiesaredescribedinaseparateUserManual.WAMITisapanelprogramdesignedtosolvetheboundary-valueproblemfortheinter-actionofwater-waveswithprescribedbodiesinfinite-andinfinite-waterdepth.Viscouseffectsofthefluidarenotconsideredthroughoutandthustheflowfieldispotentialwith-outcirculation.Aperturbationseriessolutionofthenonlinearboundaryvalueproblemispostulatedwiththeassumptionthatthewaveamplitudeissmallcomparedtothewavelength.Itisalsoassumedthatthebodystaysatitsmeanpositionand,ifitisnotfixed,theoscillatoryamplitudeofthebodymotionisofthesameorderasthewaveamplitude.Thetimeharmonicsolutionscorrespondingtothefirst-andsecond-termsoftheseriesexpansionaresolvedforagivensteady-stateincidentwavefield.Theincidentwavefieldisassumedtoberepresentedbyasuperpositionofthefundamentalfirst-ordersolutionsofparticularfrequencycomponentsintheabsenceofthebody.TheboundaryvalueproblemisrecastintointegralequationsusingthewavesourcepotentialasaGreenfunction.Theintegralequationisthensolvedbya‘panel’methodfortheunknownvelocitypotentialand/orthesourcestrengthonthebodysurface.Usingthelatter,thefluidvelocityonthebodysurfaceisevaluated.Forincidentwavesofspecificfrequenciesandwaveheadings,thelinearproblemissolvedfirst.Chapter2describesthelinearboundaryvalueproblemandthesolutionprocedure.Thequadraticinteractionoftwolinearsolutionsdefinestheboundaryconditionforthesecond-orderproblem.Chapter3isdevotedtothediscussionofthesecond-orderproblem.InChapter4,therigidbodymotionisdiscussedinthecontextoftheperturbationseriesandtheexpressionsforthehydrodynamicforcesarederived.Chapter5liststhereferencesquotedinthismanualandChapter6liststhereferencesillustratingthecomputationalresultsobtainedfromWAMIT.TheAppendixincludesthearticlesonthesubjectswhicharenotexplainedindetailinthismanual.Sincethefollowingwillfocusprimarilyonthehydrodynamicinteraction,itisappropri-1atetomentionhererestrictionsonthebodieswhichmaybeanalyzedusingWAMIT.Thebodiesmaybefixed,constrainedorneutrallybuoyantandtheymaybebottommounted,submergedorsurfacepiercing.Multibodyinteractionsbetweenanycombinationsofthesebodiescanbeanalyzed.Ifonlyfirst-orderquantitiesarerequired,thebodiesneednotberigid.Flexiblebodiescanbeanalyzediftheoscillatorydisplacementsofthebodysurfacearedescribedbyspecifiedmodeshapes.Anextremelythinsubmergedbodymaybedifficulttoanalyzewhenthesmallthicknessbetweenthetwofacingsurfacescomparedtotheotherdimensionsofthebodymayrendertheintegralequationill-conditioned1.However,athinbodyfloatingonthefreesurface(inotherwordsthedraftisverysmall)isnotincludedinthiscategory,sinceithasonlyonelargewettedsurface.Intheextremecaseofthelatter,abodytouchingthefreesurfacemaybeanalyzed.1.1DescriptionoftheProblemTheflowisassumedtobepotential,freeofseparationorliftingeffects,anditisgovernedbythevelocitypotentialΦ(x,t)whichsatisfiesLaplace’sequationinthefluiddomain:∇2Φ=0(1.1)Heretdenotesthetimeandx=(x,y,z)denotestheCartesiancoordinatesofapointinspace.xmaybeexpressedasthesumofthecomponentvectorsasx=xi+yj+zk.Theundisturbedfreesurfaceisthez=0planewiththefluiddomainz0.ThefluidvelocityisgivenbythegradientofthevelocitypotentialV(x,t)=∇Φ=∂Φ∂xi+∂Φ∂yj+∂Φ∂zk(1.2)ThepressurefollowsfromBernoulli’sequation:p(X,t)=−ρ(∂Φ∂t+12∇Φ·∇Φ+gz)(1.3)whereρisthedensityofthefluidandgisthegravitationalacceleration.Thevelocitypotentialsatisfiesthenonlinearfree-surfacecondition:∂2Φ∂t2+g∂Φ∂z+2∇Φ·∇∂Φ∂t+12∇Φ·∇(∇Φ·∇Φ)=0(1.4)appliedontheexactfreesurfaceζ(x,y)=−1g(∂Φ∂t+12∇Φ·∇Φ)z=ζ(1.5)1SeeMartinandRizzo(1993)2Withtheassumptionofaperturbationsolutionintermsofasmallwaveslopeoftheincidentwaves,thevelocitypotentiali