Sea-Loads-on-Ships-and-Offshore-Structures---O-M-F

132SECOND-ORDERNON-LINEARPROBLEMSSECOND-ORDERNON-LINEARPROBLEMSI33Theorderofmagnitudeofmeanwaveforcesrelativetolinearfirst-orderwaveforcescanbeexemplifiedbyresultspresentedbyZhaoeial.(1988)forahemispherefloatinginthefree-surface.Incidentregularwaveswithwaveamplitudec,werestudiedinalimitedfrequencyrange.Bytranslatingtheresultstoafull-scalediameterof50mwefindthelinearwaveexcitationforcestobeabout104c,(kN)whilethehorizontaldriftforceisabout102tn2(kN).Forawaveamplitudeof1mthelinearwaveexcitationforcesare100timeslarger,andforawaveamplitudeof10mthefirst-orderforceisabout10timeslarger,thanthemeanwaveforces.Sincemeanwaveforcesandmoments--mhg~k.a_'esmall,ahighdegreeofaccuracyisneededbothinthecalculations~mhkq~andintheexperiments.Theresultsaresensitivetowaveheadinbbd%,1048,form,bodymotion,wavelengthandwaveheight.Inirregularseast:G&fiJu@.g10048second-orderloadsarealsosensitivetothewidthofthewavespectrum,Imhx$qRPi.e.thewavefrequencyrangewithsignificantwaveenergy.09BEfB@&--Inapotential-flow-modelwavedriftforcesareduetoastructure'sabil$y~ocausewaves.Theconsequenceofthisisthatdriftforcesaresmallinapotentialflowmodelwhenmassforcesdominate.Thisoccursforsemi-submersibles.However,viscouseffectsmayalsocontributetodriftforces.Thisisathird-ordereffectthatwillbrieflybediscussedlaterinthechapter.Letustrytogiveasimpleexplanationofwhywegetameanwaveforceonastructureinregularincidentharmonicallyoscillatingwaves.Forasurfacepiercingbodyamajorcontributiontothehorizontalmeanwaveforceisduetotherelativeverticalmotionbetweenthestructureandthewaves.Thiscausessomeofthebodysurfacetobepartofthetimeinthewaterandpartofthetimeoutofthewater.Examiningthepressureinoneofthepointsinthissurfacezone,itisobviousfromFig.5.1thattheresultisanon-zeromeanpressureeveninregularharmonicallyoscillatingwaves.Iftherelativeverticalmotiondiffersaroundthewaterline,theresultisanon-zeromeanforce.Thisoccursforlarge-volumestructureswheretheincidentwavesaremodifiedbythestructure.However,therearealsoothercontributions.WegetoneofthembyaveragingthequadraticterminBernoulli'sequation.PressurePartlyinoroutofWater71Non-zero3meanforce-TimeOutofthewaterJFig.5.1.Horizontalmeanwaveforcecontributionduetopressureforcesonthefree-surfacezoneofastructure.Asimplewayto.illustratethepresenceofnon-linearwaveeffectsistoconsiderthequadraticvelocityterminBernoulli'sequationforthefluidpressure.WecanwritethistermaswhereV=(V,,V,,V,)isthefluidvelocityvector.Weemphasizethatequation(5.1)providesonlyoneofthenon-lineareffects.Othercontributionsmaybeequallyimportant.Consideranidealizedseastateconsistingoftwowavecomponentsofcircularfrequenciesoland02.Anapproximationforthex-componentofthevelocity'canbewrittenformallyasByalgebraitnowfollowsthatThismeansthatwehavethepresenceofaconstanttermrepresentedby-0.5~(A12/2+AZ2/2)andapressuretermwhichoscillateswiththedifferencefrequencyol-02.Foramorerealisticrepresentationoftheseaway,acdconsideringthewaveasthesumofNcomponentsofdifferentcircularfrequencieso,,wewillfindpressuretermswithdifferencefrequencieso,-ok,(k,j=1,N).Thesenon-linearinterac-tiontermsproduceslowly-varyingexcitationforcesa-ndmomentswhich-:----.maycauseresonanceoscillationsinthesurge,sway,andyawmotionsofamooredstructure.Typicalresonanceperiodsare1-2minutes.Equation(5.3)alsotellsusthatnon-lineareffectscancreateexcitationforces.-with--frequencies---higherthanthedominantfrequencycomponentsinawavespectrum.Thisisduetotermsoscillatingwithfrequencies201,20,and(o,+02)Thesemaybeimportantforexcitingtheresonanceoscillationsintheheave,pitchandrollforTLPs.Typicalresonanceperiodsare2-4s.TherestoringforcesfortheTLPareduetothetethersandthemassforcesduetothestructure.Inthefollowingtextwewillstartbydiscussingmeanwaveloads.