POLITECNICO DI TORINO Design of a wavemaker for the ...

POLITECNICODITORINO

COLLEGIODIINGEGNERIAMECCANICA,AEROSPAZIALE,DELL’AUTOVEICOLOEDELLAPRODUZIONE

CorsodiLaureainIngegneriaMeccanica

TesidiLaureaMagistrale

DesignofawavemakerforthewatertankatthePolitecnicodiTorino

RelatoreLaureando prof.GiovanniBraccoMarioBeneduce Correlatorematricola:244380prof.GiulianaMattiazzo

ANNOACCADEMICO2018–2019

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Aimieigenitori

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Abstract

Wavemakersaredevicesused intest tankstogeneratewaves. Inthesetanksareusually testedmodels of ships, offshore and onshore structures andwave energy converter: in thisway, it ispossibletoevaluatetheresistanceofferedbythehullofashipduringthemotionandthendefinethe power to install on board of the real ship or studying its seakeeping, evaluating theconsequencesoftheactionofwavesoncoastalfacilityandtheenergyconversioncapabilitiesofawaveenergyconverter.Testingmodels in tank is fundamental in thesesectorsbecause itmakepossibletoavoidcriticalandexpensivemistakesintheearlyphasesoftheprojectandtostudythebehaviorofprototypes.ThisthesisaimstoanalyzethewavegenerationcapabilitiesofthetankpresentsinthelaboratoryofthedepartmentofMechanicalandAerospaceengineeringofPolitecnicodiTorinoandtomakeafirst dimensioningof the actuation system touse. The final goal is to understand if this testingenvironmentcouldbeinterestingforourneeds,mainlyfocusedontestsonwaveenergyconverterdevices.Thefirstchaptergivesanoverviewontesttanks.Itisreportedahistoricalintroductioninwhichissummarizedtheevolutionofthesetestsandadescriptionofthemainfeaturesofwavetanks;acollectionofthemost importantfacilitiesoftheworld,EuropeandItalyclosesthischapter.Thesecondonereportsthemeanfeaturesofthelinearwaterwavestheoryandthethirdonethefirstorderwavemakerproblemincludingtheobliquewavegeneration.Thefourthchapterreportsthestudymade in thepresentwork forour test tank.First, it ismadeananalysisof the idealwavegenerationcapabilities:inthisphaseinfact,itisconsideredanidealwavegeneratorthatmeansthatalltheproblemsduetothephysicalrealizationoftheactuationsystemarenotconsidered.Thenweanalyzedtheforcesactingonthepaddleandthevelocitiesduringourworkingcycles:thisisimportanttoobtaintheF-vcharacteristicthatweneedtorealizetogenerateourwaves.Withthisinputdata,wesearchedasuitableactuatorforourapplication;thechoicemadeintroducescutsinthewavegenerationcapabilitiesthatisitisnotpossibletogenerateallthewavesreportedinthefirstidealanalysis.ThisconstrainyieldstoanewF-vcharacteristicthatitisusedlikeinputdataforthemotorchoice.Thislastanalysis,madewithoutandwithaspeedreducer,anditsresultsclosesthechapter. In the last chapter there isa summaryof theworkmadeand themain resultsarepointedout.Therecommendationsforfurtherandinproceedingworks,thatistheCFDanalysisinwhichIcollaborate,closethislastchapter.

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Contents1Testtanks71.1Originsanddevelopoftesttanks71.2Wavetanks141.3Collectionoftesttanks171.3.1Testtanksintheworld171.3.2TesttanksinEurope201.3.3TesttanksinItaly232Linearwaterwavestheory242.1Linearwaterwavestheory242.2Simplificationsforshallowanddeepwaterwaves423Firstorderwavemakertheory453.1Firstorderwavemakertheory453.2Firstorderobliqueregularwavegeneration574Wavemakerdesignwithfirstorderwaterwavestheory634.1Problemdefinition634.2Analysisofwavegenerationpotentialitiesandfirstdimensioningoftheactuationsystem644.2.1Wavesideallygenerableinourtank644.2.2Velocities,forcesandpowerrequirement704.2.3Actuatordimensioning874.2.4Consequencesoftheactuatorchoice984.2.5Motorchoice1014.2.5.1Analysiswithgearratioi=11044.2.5.2Analysiswithgearratioi=21045Conclusionsandfutureworks1085.1Conclusions1085.2Futureworks109

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Chapter1Testtanks

Neltrattarconl’acque

consultaprial’esperienzaepoilaragione

LeonardodaVinci

Theaimofthischapteristointroducethetestsintank.Inthefirstpart,thereisahistoricalintroductiontobetterunderstandthesignificantchangesthatthisapproachintroducednotonlyinthenavalengineering;forthispart,themainreferenceis[10].Afterthis,thereisadescriptionofthemainfeaturesofatesttankandasummaryofthemaintypes.Thechapterisclosedbyacollectionofthemostimportantfacilities.

1.1OriginsanddevelopoftesttanksBirthanduseoftesttanksarestrictlylinkedtothenavalengineering.ItseemsthattowingtestsonshipmodelsweremadeinVenicein‘500;theaimofthistestswastodeterminetheresistanceofferedbythehulltothemotion.ItisproventhatshipsmodelswereusedtostudytheinteractionbetweenwaterandhullsinceXVIIcentury.Infact,itisknownthatinEnglandSamuelFortray(1622-1681)usedtesttanksinwhichtheverticalmotionofafallingweightwasused,withasimplepulleysystem,todragtheshipmodels.Howevertherestillwasnottheclassicalmechanictheory,sotestswerequalitative.In1687thepublicationofPhilosophiaenaturalisprincipiamatematicabyIsaacNewton(1642-1726)andthefollowingdiffusionofthelawofmechanicalsimilarity,gaveascientificbackgroundtothetests.Thefirstevidenceofscientificrigorisdatedbackto1720inthestudiesofEmanuel

Swedenborg(1688-1772),atheologian,philosopherandscientistfolloweroftheNewton’stheory.LikeFortray,heusedatankinwhichtheshipmodelwasdraggedbytheactionoffallingdownweights.Thegeneralprincipleofthesetestswastocomparetheresistanceofferedbydifferenthullsduringtheentrainment;inthisway,themotionofdifferentmodelswiththesamelengthwerecomparedinordertofindthehullwithlessresistanceabsolute.InFrancePierreBouguer(1698-1758),authorofTraitéduNavire(1746),madesimilartestswiththesamegoal.Hestillusedthesocalledgravitytankandmodelsinwhichthemaximumwidthwassettoonethirdofthelength.More

EmanuelSwedenborg(1688-1772)

PierreBouguer(1698-1758)

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attentionwaspayedtogetastraightpathforthemodels:thetowingcablescrossedallthetankandmodelshadhooksonthetopinordertomaintainthecorrecttrim.Thesetestsstatedthatthebesthullwastheonewiththebiggestprow.However,theseexperimentshadanimportantlimit.Infact,itwasquiteimpossibletoobtainadragwithaconstantvelocitybecausethedraggingweightissubjecttothegravityaccelerationandthisyieldstoanacceleratedmotion.Toovercomethisproblemoneshoulduseverylongtanksorverysmallmodels,butinthiswayseveralandimportantexperimentalmistakestakeover.Between1750and1760,FrederikHenrikafChapman(1721-1808),designeroftheRoyalswedishNavy,madehisexperimentalcampaign;herecordedhistestsinArchitecturaNavalisMercatoria(1768)andinTraitédelaconstructiondesvaisseaux(1781).Hestillusedgravitytanksbutthepolesfromwhichtheweightscamedownwere30mhigh,theshipmodelswereinweightedwoodandtheyreachedtherecordlengthof8.5m.Thedraggingcablewaslinkedbothforwardandaftformaintainingastraightpathalongthetank.Fromtheseexperimentshestatedthatthelongitudinalpositionofthemastersectiontoobtaintheminimumresistancevarieswiththetestingvelocityand,forthattime,thiswasaveryimportantconclusion.TheChapman’sstudiesestablishedthattheabsolutebesthulldoesn’texist;howevertherearevariouspossiblegeometrieseachoneofthesewithitsbestefficiencypointincertainworkingcondition.Theconsequenceofthesestudieswastheintroductionofmorehydrodynamichulls:thebowandthesternweresharpenedandthelengthstartedtobefiveorsixtimesthewidthinsteadthreetimes.Inthesameperiodprogressweremadeinhydraulicsandfluidmechanics.HenridePitot(1695-1771),DanielBernoulli(1700-1782),LeonardEuler(1707-1783)andothersgavethefirstinstrumentstounderstandthephysicsbehindthemotionofanobjectontheseparationsurfacebetweenairandwater.OthersimportantstudiesweremadebyJeandeRondd’Alambert(1717-1783),JosephLouisLagrange(1736-1813)andGiovanniBattistaVenturi(1746-1822).Allthesestudiesinfluencedthegeometriesofthehulls.Meanwhilealsothetestingmodalitieswereimproving.In1775d’Alambert,Condorcet(1743-1794)andBossut(1730-1814)builtagravitytankonrequestoftheacademyofParis:itwas32.5mlong,17.2mwideand2mdepth.Evenifthedraggingmethodwasthesameoftheothertanks,thespeedwasmeasuredtimingthepassageofthemodelthroughsomedoorspositionedatknowndistanceonefromtheother.

AnevidenceofthegrowingimportanceoftesttanksinshipdesigningisgivenbythestudiesmadebytheScottishbrothersJamesandWilliamHallaround1830.SonsofAlexandreHall,animportantshipbuilderofAberdeen,theyusedatank3mlong,40cmwidthand25cmdepthfortestingseveralshipsmodels;thewallsofthetankweretransparentandtheyalsousedtracerliquidtobetterobservetheinteractionbetweenwaterandhull.Thesestudiesledtotheinventionoftherakedbowofships,thesocalledAberdeenbow;infact,fromthetestsitemergedthatthe

rakedbowcreatedsmootherwavesaroundthehull,asymptomofbetterinteractionbetweenwaterandship.ThiswasconfirmedbytheconstructionoftheschoonerScottishMaidin1839byAlexanderHallandCompany,inwhichspeedandsailingperformanceswereimproved.

FrederikHenrikafChapman(1721-1808)

Scottish Maid Prototype Half Model by James And William Hall, 1839

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However,evenifthefirstgoodresultsofmodelstestingwerearriving,stilltherewasnotacompleteandsolidtheorytouse;alsotherewasthewrongbeliefthatthemodelthatofferedlessresistanceinthetest,wouldhavebeenthebestalsoinrealscale.Forthesereason,designersdidnottrusttoomuchintheseexperimentsbuttimeswerereadyforadecisivechange.AroundthehalfofXIXcentury,JohnScottRussel(1808-1882),aScottishnavalengineer,madehisstudiesonwavesgenerationduetoshipmotion.In1860heformalizedhisresearchinwhichheconnectedthehulllengthandthegoalvelocitywiththewavesgeneratedbytheship.Inparticular,hestatedthatforeachhullthereisacriticalvelocityafterwhichtheresistancegrowsrapidly;thisphenomenoncanbereducedbyhullgeometriesthathelptheinteractionwiththewater.Afterthis,therewereallthebasisfordiscoverthelawofsimilaritywhichisoneofthemostimportantlawofnavalengineering.Inthesameperiod,WilliamFroude(1810-1879),arailwayengineer,startedtobeinterestedinhydrodynamicsandworkedwithRusseldesigningtwoimportantships,theGreatWesternandtheGreatEastern.Atthebeginning,hemadehisstudieswithasmallgravitytankbutheunderstood

allthelimitations,abovementioned,ofthisexperiment.Froudemadehisstudiesbetween1860and1870andformulatedhislawofsimilaritybasedonNewton’stheory.Thehypothesisofthislawisthattheresistanceofferedbywatertothehullmotioncanbedividedintotwocomponents:thefrictiononecausedbythecontactbetweenwaterandhullandageneralone,composedofmanycontributions,inwhichthemostimportantisthewavesgenerationduetothemotion.FroudestatedthatonlythegeneralcomponentfollowsthemechanicallawofsimilarityofNewton.Thefrictioncomponentinsteaditiscomparabletotheoneofaflatplate,ofnegligiblewidthandwithlengthandsurfaceequaltothestudiedhull,immersedandlongitudinallydirectedwithrespecttothemotiondirection.Steelnow,thesearethebasisfortestsinwater

tanksandFroudecarriedqualitativeexperimentstoasuperiorlevelwithanimportantscientificbackground.

In1870FroudeconvincedtheBritishadmiraltytobuildawatertankandin1871itwasestablishedinTorquay,inthesouthwestofEngland.Thistestingfacilityitisconsideredthefirstmodernwatertankfortheplantitselfandfortheinnovativetestingmethodsused.Thetankwas85mlong,11mwidthand3mdepth;overthetankinlongitudinaldirectiontherewererailsonwhichacartcanrunfromonesideofthetanktotheother.Thecartwasmovedatthedesiredvelocitybyametalcablethatwindedaroundarotatingdrumputinrotationbyasteamengine;thiswasveryimportantbecausebeforeitwasn’tpossibletodragthemodelwithaconstantvelocitywithnotnegligibleeffectsontheresistance

John Scott Russel (1808-1882)

William Froude (1810-1879)

William Froude and the admiralty’s first naval test tank at Torquay 1872

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evaluation.Thehullmodelwasconnectedtothecart,itwasbetween3mand4manditwasmadewithparaffin,muchmoreworkablethanwood.Betweencartandmodelthereweretheinstrumentsfortheresistancemeasureandonthetopofthecartthereweredevicesthattookoverthedynamicandtrimvariationofthemodel.Thesolutionofthetowingcartwithinstruments,usedforthefirsttimebyFroudeinTorquay,issteelnowpresentineverywatertank.ThereliabilityoftheFroude’smethodwasascertainedbytowingtestsinrealscaleinPortsmouth:theresultsoftestsintankwerecomparedwiththeresistanceevaluatedduringthedraggingofasmallshipbyabiggeroneandthisyieldedverysimilarconclusions.Thegoodresultsofthesefirsttestsledtotheconstructionofanewtesttankin1894inPorthsmouth,byFroude’ssoon:itwas110mlong,6.1mwidthand2.8mdepth.MeanwhiletheseexperimentswerespreadingalsooutofUK.Tests,steelwithgravitytankbutusingFroude’stheory,weremadeinHolland,ItalyandFrance;inparticular,inthearsenalof

Brest,werealsomadestudiesofnavalpropulsionstudyingthepushgavetothemodelsbyhelix.VerysoontheimportanceofthesetestingfacilitieswasclearinallEuropeandintherestoftheworld.In1894intheRussianEmpirewasestablishedwhatitisnowcalledtheKrylovStateResearchCentre,thatisnowoneofthemostimportantinstitutionoftheworldinshipresearchanddesign.In1894wasthefirstfacilityintheRussianEmpireandthesixthintheworldforshipmodelstests.ThisinstitutiontakesthenameofAlekseyNikolaevichKrylov,aRussiannavalengineerthatdevelopedthestudiesofFroude.

In1898wasrealizedthefirsttesttankofUSAunderthesupervisionofDavidWatsonTaylor,navalarchitectandengineeroftheUnitedStatesNavywhichstillnowdetainstherecordofhighestgradeaverageinthehistoryofU.S.NavalAcademy;thefacilityinCarderock,Maryland,inwhichthereismaybethemostimportanttesttankoftheworld,theMASK,retainshisname.Militarynavyandshipbuildersfirmsplayedanimportantroleinspreadingthisdesignmethod;theyunderstoodhowmuchimportantwouldhavebeenmodelstestingforresearchandtoavoidveryexpensivemistakes:itisclearthatfromthisdependboththemilitarypowerofanationandthebusinesssuccessofacompany.Likeevidenceofthisfastspread,beforeWWItherewereabouttwentytestingtanksallovertheworld.Thisfastdiffusionwasduetotwotechnicalreasonmainly.First,thesteamengineandthehelixpropulsionsimplifiedthemodelingandthehulloptimization:infact,thepropulsiondirectionwasnowfixedrespecttothelongitudinalsymmetryplaneoftheshipandnotvariablelikeinsailpropulsion.Second,butnotlessimportant,itwasnowfundamentaltoknowwithagoodaccuracythepowerrequiredbypropulsiontoavoidcriticalandexpensivemistakes.AfterWWI,thenumberofwatertanksallovertheworldincreased.AnevidenceofthegrowingimportanceofthesetestsintheseyearsisthemeetingoftheInternationalHydro-mechanical

David Watson Taylor(1864-1940)

Aleksey Nikolaevich Krylov

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Congress,heldinHamburgin1932;thiswasthefirststepoftheorganizationthatwenowcallInternationalTowingTankConference.TheaimofITTCistogatherthescientificinstitutionsthatoperatetestsintanks,definestandardtestingmethodologyandlinkresearcher.Inthisatmosphereofconstantdevelopthenumberoftesttanksincreased,reachingaboutfortybeforeWWII.Intheseyearstestsintankdevelopedandwatertankswereusedlikealternativetowindtunnelandtotestportandoffshorestructure.DuringXXcenturyimportantresearchcenterwithseveralwatertanksfordifferenttestswerebuiltintheUSA,URSS,China,Japanandlotsofresearcherdailyworktoimprovescientificknowledgeinfluiddynamic.Testsinwatertanksdevelopedbothintheoryandfacilities:forexample,wavemakingtheorygaveusthepossibilitytostudytheshipbehaviorinroughseaandwecannowuseicedtanktotesticebreakership.InthelastyearsthedevelopofCFDgavealsoanimportantboostinthissector,givingusthepossibilitytostudyfluiddynamicsproblemsinalltheircomplexity.Recently,thenewdevelopofseaenergyconvertersgaveanotherimpulsetotestsintank.Bytheway,itcanbementionedtheEUMARINETproject:itsgoalistocoordinateresearchanddevelopmentofseaenergyconverters,startingfrommodelsuntilopenseatests,andtoeasetheuseoftestingfacilitiesforresearcher.ItisinterestingtorememberherethedevelopoftanktestinginItaly.In1886theministerofthenavy,admiralBenedettoBrin,decidedtofinancetheconstructionofawatertank.Inthisway,Italywasthefirstnation,afterUK,tobuildamoderntesttank.TheplantwasbuiltinthearsenalofLaSpeziaandbecameactivebetween1889and1890.ThedesignwasmadefollowingtheFroude’sfacilitiesprojectsanditwasdirectedbyGiuseppeRota(1860-1953),promisingengineerofthemilitarynavy,whoselifewouldhavebeenbraidedwithtestingtanks.Concludedtheconstruction,hewasalsotheplantfirstdirector.Thetankwas150x6x3m,ithadawoodentowingcartwithdynamometeranditwasmovedbyasteamengine,exactlyliketheoneinTorqaytank,withamaximumvelocityof5m/s.Theactivityofthisplantledtoimportantdiscoveriesaboutwaterdepthinfluenceonthemotionresistanceandthepropulsionwithtwohelixeswiththesameaxisbutrotatinginoppositedirections.Aboutthisexperience,RotawroteLavascaperleesperienzediarchitetturadelR.ArsenalediSpezia(Genova1898);thispublicationhadagoodsuccessbetweennavalengineers.AfterWWI,thefacilityofLaSpeziaprovedtobenotsufficienttomeetnewtechnicalneeds:therewastherequestofbetterexperimentalapparatusandbiggertanks.Moreover,wasdifficulttosatisfytheincreaseofdemandsfortestsbyprivateshipbuilderscompanies:infact,thetankwasamilitaryfacilityandmilitarynavytestshadthepriority.Inthissituation,theactionofRota,intheseyearsmemberofSenate,convincedthegovernmentofthe“necessityandurgency”togivetotheItalianindustrythepossibilityofmakingnavaltestsinItaly.Forthesereasons,in1927theItaliangovernmentestablishedinRometheVascaNazionaleperStudiedEsperienzediArchitetturaNavale.ThisnewplantwascreatedinRomefortestingthesolutionoftheComitatoprogettidellenaviandforitsthecentralposition,quiteeasytoreachfromallItaly.ThenameofthisinstitutionwouldhavebeenthenINSEAN,IstitutoNazionaleperStudiedEsperienzediArchitetturaNavale.Thetankwas275x12.5x6.3mandthetowingcartcouldreach10m/s.Theinstitutionstartedtheactivitiesin1929;Rota,likeinLaSpezia,becamethefirstdirectorandin1931publishedLavascanazionaleperleesperienzediarchitetturanavale

Benedetto Brin (1833-1898)

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(Roma1931).Thefacilitywassoonusedintensivelyformanytests;classicalexperimentsofnavalengineering,liketowingandpropulsiontests,wereputbesidesbyaerodynamicsstudiesforothersengineeringfields.Inparticular,FIATAutoorderedaerodynamicstestsondifferentcarbodies,includingthepopular“Balilla”,andFerroviedelloStatotestedtheshapeof“littorina”railcar.Theseexperimentsweremadedraggingonatablethecompletelyimmersedmodelinthewater;evenifcarsandtrainsdon’tmoveinwater,testsaresolidsbecauseairandwaterarebothfluidsanddifferencesbetweenthemiscorrectedincalculations.

In1935anothertankwasbuiltinGuidoniaintheplantofDirezioneSuperioreStudiedEsperienze(DSSE)byAeronauticaMilitare.Thetankwas475x6x3.5mandhadtwomeasuringcartsforhighspeedtestsonairfoilsandseaplanesmodels.Theslowestcartcouldreach20m/s,thefastest40m/s;inparticular,thefastestwascarinateandtowedthemodelatsufficientdistanceformitwithalonghorizontalarmforavoidingaerodynamicsinfluences.Thehistoryofthisresearchcenterisasimportantasforgotten.InGuidonia,undertheboostofwardevelopment,wascreatedanintegratedenvironmentinwhichItalianscientistcouldmakeresearchesandthentestsinlaboratory;inthesameareatherewasalsothepossibilitytobuildaircraftprototypesandtomakethemtestedbypilotsofAeronauticaMilitare.Theactivitiesendedin1943duetothewarbutineightyearsofworkremarkableresultswereobtainedandworldwiderecognized:wecanrememberherethescientistsAntonioFerriandLuigiCrocco,afterWWIIbothintheUSAinprestigiousresearchpositions,andLuigiBroglio,thefatherofItalianastronautics;manyothersresearchertookparttotheactivityofDSSEwithimportantresultsbutherearenotcited.

Test on locomotives in Rome

Direzione Superiore Studi ed Esperienze (DSSE)

DSSE, slowest cart DSSE, fastest cart

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Forwhathasbeensaid,beforeWWIIItalyclaimedthreebigwatertanksbutduringthewarlostthefacilitiesofLaSpeziaandGuidonia;theonlytankofRomesurvivedtothisperiod.ThereconstructionofItalyafterwardisasters,broughtalotofworkforINSEAN.Alreadyin1956,theItalianmerchantfleetwasoneofthefirstintheworld,thetransportfleetthesecondabsoluteandItaliansshipbuildersreceivedmanyordersfromallovertheworld.IfthiswaspossibleitwasalsofortheactivityofINSEAN;infact,lawstatedthatalltheshipstobuildhadtobepreventivelytested,withcorrespondingmodels,inthetankofRome.Forthisreasonwecansaythat,inthatperiod,therewasnotanyships,flyingItalianflag,thatwasnottestedbeforeatINSEAN.Thedataof1955areeloquent:85helixmodels,55hullmodelsand555experimentalcampaign.Evenifwarsparedthisfacility,structuralproblemsoccurred;in1974theactualINSEANplant,endowedoftwotanks,startedhisactivity.Thebiggertankis470x13.5x6.5mwithamaximumvelocitytestof15m/s;duringconstruction,itwasthelongesttankofallwesternEurope.Thesmalleroneis220x9x3.5mwithamaximumvelocitytestof10m/s;inthistankthereisalsoaflaptypewavemaker.INSEANhasalsoothersfacilitieslikethecirculatingwaterchannel,forcavitationandpressurefieldtests,andthemaneuveringbasin:thisoneislocatedontheshoreofNemi’slakewhereispossibletoperformmaneuveringoutdoortests.InItalytherearealsootherplants.Since1980,isalsoactivethetesttankofUniversitàdiNapoliFedericoIIthat,withthedimensionsof150x9x4m,isthebiggestuniversitywatertankofEurope;themaximumvelocityis10m/sandit’salsoendowedofaflaptypewavemaker.PolitecnicodiBarihasanimportantfacility:therearetwotanksofdimensions90𝑚𝑥50𝑚𝑥1.2𝑚and30𝑚𝑥50𝑚𝑥3𝑚andtwowaveflumes,bothofdimensions40𝑚𝑥2.4𝑚𝑥1.2𝑚,activesince2001.Thebiggertankismainlyusedforstudiesoncoastaldynamicandonshorestructuresdefensewhilethesmalleroneisusedforoffshoremodeltests.IntheUniversityofGenovathereisa60x2.5x3mtankwithameasuringcartthatcanreach3m/sbutitisnotequippedwithwavegenerator.UniversityofPadovahasawaveflumewithdimensionsof35𝑚𝑥1𝑚𝑥1.3𝑚andamarinetankof20.6𝑚𝑥17.8𝑚𝑥0.8𝑚;researchareaismainlyfocusedonwaveshydrodynamics,floatingstructures,waveenergyconverterandonshorestructures.TheLaboratoryofMaritimeEngineering(LABIMA)ofUniversitàdiFirenzehastwotanks:oneof37𝑚𝑥0.8𝑚𝑥0.8𝑚,operativesince1980andcompletelyrebuiltin2013,andanotheroneof52𝑚𝑥1.5𝑚𝑥2𝑚,recentlybuiltbyresearchersandcompletelyoperativefrom2020.Inbothtanks,wavesaregeneratedbyapistontypewavemakeranditisalsopossibletostudycurrentsactions.PolitecnicodiTorinohasatankof30x0.7x2.2mwithatowingcartthatcanrunfor23mwithvelocityincludedbetween0.004and2.8m/s.Forthethisfacilityitisinevaluationtheinsertionofawavemakerdeviceandthepresentworkwouldliketobethefirststepofthisproject.

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Despitethepresenceoftheabovementionedfacilities,newtechnologieschallengesareimposingnewplants.Itisrecent(July2019)theconstructionannouncementofthefirstoceantankinItalyinthearsenalofLaSpezia,therewhereallstartedwithBrinandRotaattheendofXIXcentury.Aswesaw,thissectorofengineering,evenifmaybeabithidden,hasitshistoryanditissurelyadynamicenvironmentmadeofresearchers,newprojectsanddiscoveries.Thesetestsgivetodesignersthesecuritythattheirprojectwillsatisfyalltherequirement,allowingimportantresourcessavingWhatatthebeginningwasconsideredanexpensiveplayforthevariousintellectualslikeSvedemborg,Chapman,thebrothersHallandsteelthesameFroude,isnowadaysneedfulineveryshipandseafacilitiesdesign,alwayswiththesamebasicidea:studyingamodeltoforecastthereality.

1.2WavetanksAsprobablyunderstood,wavetanksareparticulartesttanksinwhichitispossibletogeneratewavesandthentostudythemodelbehaviorinamorerealisticseastate.Fromthehistoricalintroductionitisclearthattestingmodelsisfundamentaltobetterunderstandeventualdesignmistakes,tocorrectthembeforeitbecomesimpossibleandthensavingmoney.Therearethreefundamentalszonesinawavetank:thewavemaker,themodelandthebeachzone.

Thefirstonecontainsthewavemakerandit’suptothreetimesthewaterdepthfromthedevicetoallowtheextinctionofthelocaldisturbances;these,aswewillseeinthefollowingchapters,areduetothemismatchingbetweenthepaddlesandthewatermotion.Thoughthereareseveraltypesofwavemakers,themostcommonarethepistonandtheflaptypes;forthisreason,inthepresentworkwewillfocusononlythesetwodevices.Themodelzonehastohostthemodel,toallowmeasuresandithastobeenoughtohavethecorrectsimulationenvironment;forexample,referringtoatowingtank,weneedenoughspacetoacceleratethemodel,todragitataconstantvelocityandthentoslowdownandstop.

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Thelastzonehoststhebeach,thatistheelement,usuallyontheoppositesideofthewavemaker,thathastodissipatethewaveenergyavoidingwavereflection.Thishastobeavoidedbecausewavescanreachthemodelzonedisturbingthemeasurementsandmakingtestsnotrepeatable.Ifthebeachisafixedstructure,wecanhavetheslopedorthemeshstylebeach;otherwise,itispossibletohaveawavemakerthatworkslikeawaveabsorber,generatingwavesinphaseoppositionrespecttotheincomingones.Itisoftenpossibletohaveatowingcartwhichallowstodragmodelsinthewavetankexactlylikeinatowingtank;again,herewehavethedynamometerandothermeasurementsdevices.Wehaveseveraltypesofwavetanksandhereitisgivenageneraloverview.Thefrontalwavetankisthemostcommon;itusuallyhasarectangularshape,withthewavemakerthatoccupiesoneofthefoursidesandthebeachopposite.Ifthewidthissosmalltoallowthepresenceofonlyonepaddle,wecallthemwaveflumes.Otherwise,ifthereisthepossibilitytohostmorepaddles,wecancreateasocalledsnakewavemakerthroughwhichwecangeneratebothstraightandobliquewaves.Sincetherearen’twavemakersontheotherstwosides,reflectedwavescouldcreatetestingproblems.Thecornerwavetankshavewavemakersonadjacentsidesofthetank;withthissolutionsitispossibletogenerateobliquewaveswithagoodaccuracyandwecanavoidtherearrangementofthesetupduringtests.ThebasicsolutionisthedualfacesnaketypewavemakerbuttherearealsotheCandthesquaredtypeinwhichwecanbothgenerateandabsorbwavesthroughwavemakers.

Frontal wave tank

Dual face snake type wave maker C type

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Thelasttypologiesarethecurvedandcircularwavetank.Ifwedriveallwavemakerwithnophasedelaybetweenthem,wegenerateacurvedwavedirecttothecenterofthewavemakersarc;otherwise,givingaphaselagbetweenadjacentpaddles,it’spossibletogenerateastraightwave.Inthecurvedtankweneedakindofbeachontheoppositesideofwavemakerswhilethisisnotrequiredinthecirculartankbecausewecanusewavemakerstoabsorbwaves.

Squared type

Curved Type Circular Type

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1.3Collectionoftesttanks1.3.1TesttanksintheworldInthissectionwearegoingtopresentsomeofthemostimportantfacilitiesintheworldtohaveageneraloverviewaboutthestateofartofthesetestsandsomereferences.OneofthemostimportantinstitutionsoftheworldfortesttanksistheNavalSurfaceWarfareCenter,CarderockDivision“DavidWatsonTaylor”(Maryland,USA).ThisinstitutionispartoftheNavalSeaSystemsCommand,thebiggestsystemcommandoftheUnitedStatesNavy;theaimofthisorganizationistheresearch,developandmaintenanceforthefleetofUnitedStates.Thoughtherearemanytestingfacilitiesinthissite,wementiontheMASKbasin(ManeuveringandSeakeepingBasin);thisisthebiggesttankintheworldinwhichitispossibletogeneratewaves.Thebasindimensionsare106x76x6manditisequippedwith216flapwavemakers.Itispossibletogeneratewaveswithamaximumheightof0.6m,amaximumwavelengthof12.2mandthereisalsoatowingcarriagethatcanreachthemaximumvelocityof7.2m/s.Themaximummodellengthis9m.

TheKrylovStateResearchCentreisthereferenceinstitutionfortheshipbuildingindustryinRussianFederation;itwasfoundedin1894inSt.PetersburgwiththecreationofthefirstmodeltankoftheRussianEmpire.A.N.Krylov,aworldfamousRussiannavalengineerandmathematician,becamethepersoninchargeofthisinstitutionin1900.Theconstructionoftheactualfacility,locatedinthesouthpartofSt.Petersburg,wasstartedin1936andendedin1960.Theactivitiesofthisresearchcentrecoverthemostimportantfieldsofshipbuildinginbothcivilandmilitarysector.Fromallthefacilitiespresentinthisinstitute,wefocusontheLargeScaleSeakeeping/maneuveringmodelbasin.Itsdimensionsare162mx37mx5mandtherearesegmentedflaptypewavegeneratorsontwoofthefoursidesofthisrectangulartank.Themaximumwaveheightreachableis0.45mwithaperiodof2sandthereisacarriagethatcanreach6m/sinthexdirectionand4m/sintheyone.Themaximummodellengthis12m.

ThemostimportantinstitutioninChinaistheChinaShipScientificResearchCenter.CloselylinkedtotheChinaShipbuildingIndustryCorporation,CSSRChasawiderangeofresearchactivitiesinshipbuilding,offshoreandunderwaterstructuredesign.Itwasestablishedin1951inthecityofWuxy,Jiangsuprovince,notfarfromShanghai;from1978ispartofITTC.Alsointhiscase,wefocusontheSeakeepingbasin.Itsdimensionsare69mx46mx4anditisequippedwith188flaptypewavemakersontheadjacentshortandlongsideofthebasin;ontheoppositesidestherearethebeaches.Itispossibletogenerateregularwaveswithamaximumwaveheightof0.5mandwithaperiodbetween0.5–5s;thereisalsoacarriagethatcantowmodelwithamaximumvelocityof4m/s.Themaximummodellengthis5m.StillinthefarEast,wefindtheNationalMaritimeResearchInstituteofJapan.Thefirstunitofthisinstitutionwasestablishedin1916underthecontroloftheMinistryofTelecommunicationandTransportanditdevelopedwithcontinuousimprovementsalongthelastcentury.TheActualSeaModelBasinis80mx40mx4.5mandithas382flaptypeabsorbingwavemakers.Themaximumwaveheightreachableis0.35mandthewaveperiodsarebetween0.43–4s;thereis

18

alsothepossibilitytorealizetowingtestswiththemaincarriagethatcanreachamaximumvelocityof3.5m/s.Themodelsizerangeiscomprisedbetween2and4.5m.InSouthKoreawefindtheKoreaResearchInstituteofShipsandOceanEngineering.Itwasestablishedin1973anditsmainactivitiesareshippingtechnology,marinetrafficsystemtechnology,marineaccidentresponseandunderwaterrobot.Thetowingtankis221mx16mx7m(variousdepth)andithasasingleflaptypewavemakerthatcoversallthewidthofthetank;ontheoppositesidethereisthebeach.Itispossibletogenerateregularwaveswithamaximumheightof0.5m;forirregularwaves,theheightlimitis0.3m.Sinceitisatowingtank,thereisacarriagetodragmodelsthatcanreachamaximumvelocityof6m/s.Themodelsizerangeisbetween6–12mAveryimportantinstitutioninAustraliaistheAustralianMaritimeCollege.Itwasestablishedin1978inLauncestonanditcollaborateswithinternationalgovernmentandindustrialorganizations.ItstowingtankistheonlyoneactiveinAustraliaanditis100m(length)x3.5m(width)x1.5m(depth);thereisasingleflaptypethatcoversallthewidthandthroughwhichitispossibletogeneratewaveswithamaximumheightof0.7m.Thetowingcarriagecanreachthemaximumvelocityof5m/sandthemodellengthrangeis1-2.5m.TheNavalScienceandTechnologicalLaboratoryispartoftheDefenseResearchandDevelopmentOrganizationofIndia.Itwasestablishedin1969inVisahapatnamanditsresearchactivitiesarecloselylinkedtomilitaryapplications.TheSeakeepingandManeuveringbasinis135m(length)x37m(width)x5m(depth)anditisequippedwith256flaptypewavemakersontwoadjacentsides;themaximumwaveheightachievableis0.5m.Thetowingcarriagecanreachamaximumvelocityof6m/sandthemodellengthrangeis3–5m.InIranthereistheNationalIranianMarineLaboratory(NIMALA).Itwasestablishedin2012inTehrananditsresearchesarefocusedonoffshorestructuresandresistance,maneuvering,seakeepingandpropulsionofvesselsandsubmarines.Thetowingtankis400m(length)x6m(width)x4m(depth)andithasapistontypewavemakerthatcangeneratebothregularandirregularwaves;themaximumregularwaveheightreachableis0.5m.Themaximumcarriagespeedis19m/s.AveryimportantinstitutioninSouthAmericaistheBrazilianOceanTechnologyLaboratorywithitsOceanBasin.PartoftheOceanEngineeringDepartmentoftheRiodeJaneiroFederalUniversity,itworkssince2003anditwascreatedmainlyfortestingoilandgasoffshorestructures.Withitsrecorddepthof15m,thatbecomes25minaholeof5mofdiameterinthebottomofthetank,theOceanBasinisthedeepestbasinintheworld.Ithasarectangularshapewiththedimensionof40m(length)x30m(width)x15m(depth);thecentralholehasadiameterof5manditreachesadepthof25m.Ontheshortersidethereare75flaptypewavemakersthatcangenerateamaximumregularwaveheightof0.5m;thewaveperiodrangeisbetween0.5–5s.

19

Thefollowingtablesumsupthemaininformationabouttheabovecitedfacilities:StateandInstitution

Tank Tankdimensions

Wavemakertype

WaveHeight

WaveHeight(not

specified)

MaximumRegularWaveHeight

MaximumIrregularWaveHeight

USA

NavalSurfaceWarfareCenter,

CarderockDivision

Maneuveringand

SeakeepingBasin

106mx76mx6m Flap 0.6m

RussianFederation

KrylovStateResearchCentre

Maneuveringand

SeakeepingBasin


PublicRepublicof

China

ChinaShipScientificResearchCenter

SeakeepingBasin


Japan

NationalMaritimeResearchInstitute

ActualSeaModelBasin

80mx40mx4.5m Flap 0.35m

SouthKorea

KoreaResearchInstituteofShipsandOcean

Engineering

TowingTank 221mx16mx7m Flap 0.5m

20

Australia

AustralianMaritimeCollege

TowingTank 100mx3.5mx1.5m Flap 0.7m

India

NavalScienceand

TechnologicalLaboratory

Seakeepingand

Maneuveringbasin


Iran

NationalIranianMarineLaboratory

TowingTank 400mx6mx4m Piston 0.5m

Brazil

BrazilianOcean

TechnologyLaboratory

OceanBasin 40mx30mx15m Flap 0.5m

1.3.2TesttanksinEuropeNowwearegoingtofocusonEuropeanfacilities.InformationsaremainlytakenfromthewebsiteofMarinet2.ThislastoneacronymstandsforMarineRenewablesInfrastructureNetwork,thatisaEuropeanprojectthataimstodevelopwaveenergyconverterdevicescoveringthecostsofaccessingtestfacilities.OneofthebestfacilityinEuropeistheFloWaveOceanBasinoftheUniversityofEdinburgh.Itwasbuiltin2014bythealreadymentionedEdinburghDesignanditismainlyusedtotestwaveenergyconverterdevices;thankstoitscirculargeometry,itispossibletosimulatecomplexseastatecombiningwaveandcurrentgenerationcapability.TheFloWaveisacircularbasinwithadiameterof25𝑚and2𝑚deepequippedwith168active-absorbingflaptypewavemakers.Itisoptimisedforgeneratingwaveswithaperiodof2𝑠andawaveheightof0.7𝑚andthemaximumcurrentvelocityis1.6𝑚 𝑠;thetestingscalesareapproximatelybetween1: 20 − 1: 30.InIrelandwefindtheLirNationalOceanTestFacility,anexcellencecenterformarinerenewableenergyresearchandmarineengineeringingeneral.OneofthefourtesttankspresentinthisresearchcenteristheOceanEmulator,arectangularbasinwithdimensions25𝑚𝑥18𝑚𝑥1𝑚;thereisalsoatrenchof11.2𝑚𝑥10𝑚equippedwithamovablefloorplatethatmakespossiblethedepthregulationbetween1𝑚and2.5𝑚.Thereare80flaptype

21

wavemakersontwosidesofthebasinandthisallowstogeneratewaveswithadjustabledirection.Themaximumregularwaveheightis0.32𝑚andthemaximumirregularwaveheightis0.16𝑚.Thistankisusedfortestingdevicesintheearlydevelopmentphaseandthemodelsscalerangeisbetween1: 50 − 1: 100.InGermany,UniversityofHannoverhasoneofthebiggestwaveflumeoftheworld.Itis300𝑚𝑥5𝑚𝑥7𝑚anditisequippedwithpistontypewavemakerwithamaximumstrokeof4𝑚.Themaximumwaveheightreachableismorethan2𝑚.ThemostimportantwavetankinFranceistheHydrodynamicandOceanEngineeringTankoftheÉcolecentraledeNantes;theactivityofthisfacilityisfocusedonnaval,oilandgasandmarinerenewableenergysectors.Thedimensionsare50𝑚𝑥30𝑚𝑥5𝑚withacentralsquarecavityof5𝑚𝑥5𝑚𝑥5𝑚;onthe30𝑚sidethereisasnaketypewavemakermadeof48flapsindependentlycontrolledthatallowstosimulatecomplexseastate.Itispossibletogenerateregularwaveswithamaximumwaveheightof1𝑚andirregularwaveswithamaximumsignificantwavesheightof0.65𝑚.TheWaveandCurrentBasinoftheAalborgUniversity,Denmark,wasbuiltin2017anditismainlyusedfortestingcoastalandoffshorestructuremodelsandwaveenergyconverter.Itis14.6𝑚𝑥19.3𝑚𝑥1.5𝑚andthereisalsoapitof6.5𝑚𝑥2𝑚inwhichitispossibletohaveulterior6𝑚ofdepth.Inthistankthereare30pistontypewavemakersthatrealizeasnaketypewavemaker.Themaximumregularwaveheightis0.45𝑚andthemaximumirregularoneisbetween0.25 − 0.30𝑚.Thepossibilitytogeneratecurrents,allowstestswithwavesandcurrentcombination.InSpainwefindtheCantabriaCoastalandOceanBasin,afacilityoftheUniversityofCantabria;itissuitablefortestsoncoastalandoffshorestructuresandonwaveenergyconverter.Thedimensionsare44𝑚𝑥30𝑚𝑥4𝑚withthepossibilitytoadd8𝑚tothedepthusingadepthadjustablepitof6𝑚ofdiameter.Thereare64flaptypewavemakerscapableofgeneratingwaveswithamaximumheightof1𝑚.Thereisalsothepossibilitytogeneratecurrentwithaflowrateupto18𝑚4 𝑠.InPortugaltheUniversityofPortohasawavetankwithdimensionsof28𝑚𝑥12𝑚𝑥1.2𝑚;acentralpit 4.5𝑚𝑥2𝑚𝑥1.5𝑚 makespossiblethereproductionofdeepwaterenvironment.Itisequippedwithawavemakercomposedbyseveralpistonsthatallowsthereproductionofregularandirregularwaves.

22

Thefollowingtablesumsupthemaininformationabouttheabovecitedfacilities:StateandInstitution

Tank Tankdimensions

Wavemakertype

WaveHeight

WaveHeight(not

specified)



UK

FloWaveOceanBasin

OceanBasin diameter25m

2mdeep

Flap 0.7m

Ireland

LirNationalOceanTestFacility

OceanEmulator

25mx18mx1m Flap 0.32m 0.16m

Germany

UniversityofHannover

WaveFlume 300mx5mx7m Piston 2m

France

ÉcolecentraledeNantes

HydrodynamicandOceanEngineering

Tank

50mx30mx5m Flap 1m 0.65m

Denmark

AalborgUniversity

WaveandCurrentBasin

14.6mx19.3mx1.5m Piston 0.45m 0.25− 0.30m

Spain

UniversityofCantabria

CantabriaCoastalandOceanBasin

44mx30mx4m Flap 1m

Portugal

UniversityofPorto

Wavetank 28mx12mx1.2m Piston N.A. N.A. N.A.

23

1.3.3TesttanksinItalySincewehavealreadypresentedseveralfacilitiesinsidethehistoricalintroduction,hereweonlysummarizethemainfeatures.

Institution Tank Tankdimensions Wavemakertype

WaveHeight

WaveHeight(not

specified)



INSEAN TowingTank

220mx9mx3.5m Flap N.A. 0.45m N.A.

UniversitàFedericoIINapoli

TowingTank

140.2mx9mx4.25m Flap N.A N.A N.A

PolitecnicodiBariLIC

CoastalDynamicTank

90mx50mx1.2m N.A. N.A N.A N.A

OffshoreModelTank

30mx50mx3m N.A. N.A N.A N.A

WaveFlume

40mx2.4mx1.2m N.A. N.A N.A N.A

WaveFlume

40mx2.4mx1.2m N.A. N.A N.A N.A

UniversitàdeglistudidiFirenze

LABIMA

Wave-CurrentFlumeone

37mx0.8mx0.8m Piston N.A. 0.35m N.A.

Wave-CurrentFlumeTwo

52mx1.5mx2m Piston N.A. 1.1m N.A.

UniversitàdeglistudidiPadova

WaveFlume

35mx1mx1.3m Piston N.A N.A N.A

MarineTank

20.6mx17.8mx0.8m Piston N.A N.A N.A

24

Chapter2

Linearwaterwavetheory

Thischapterintroducesthelinearwaterwavetheoryuntilthetwodimensionalsolutionforstandingandprogressivewaves.Themainreferenceforthischapteris[1].

2.1LinearwaterwavetheoryAvolumeofliquidwithoutanyforceactingonitpresentsaflatinteractionsurfacebetweenairandliquiditself,exactlylikethewaterinabasinafterfewsecondsafterfilling;thisisduetotheactionofgravityandsurfacetension.Theoppositesituation,aninteractionsurfacecrossedbywaves,isanevidenceoftheactionofforcesactingontheliquidvolumeagainstgravityandsurfacetension;theseforcescanbecausedbydifferentreasons:themotionofanobjectinsidetheliquidvolume,thewind,anearthquakeandalsothegravitationalattractionofastronomicalbodies.Afterthecreationofthewaves,theactionofgravityandsurfacetension,thattendtorecreateaflatinteractionsurface,causesthepropagationofthewaves.Letobservenowthefollowingsketch:

Fromnowon,thereferencesystemhasthe𝑥directionlikeshown,the𝑦oneenteringintheplaneofthesheetandthezeroforthe𝑧directionisatthestillwaterlevel,sothatthebottomislocatedat𝑧 = −ℎ.Referringtothesketch,thefundamentalphysicalquantitiesfordescribingwavesare:

- thelength𝐿,thedistancebetweentwocrestsortroughs;- theheight𝐻,thedistancebetweenacrestandatrough;- thewaterdepthℎ,thedistancebetweenthestillwaterlevelandthebottom.

25

Otherimportantquantitiesarethewaveamplitude𝑎,thehalfofthewaveheight𝐻,andthesurfaceelevation𝜂,thedistancebetweenstillwaterlevelandinteractionsurface;moreover,wedefinethewaveperiod𝑇asthetimespentbytwosuccessivecrestsortroughstocrossacertainpoint.Sincethewavescrossadistance𝐿inthetime𝑇,wecandefinethespeedofthewaveas𝐶 = 𝐿 𝑇;thisquantityiscalledcelerity.Itisinterestingtonotethat,despitethismotion,thewaterabovethesurfacedoesnottranslatealongthewavepropagationdirection.Inrealcases,thesurfaceelevation𝜂appeartobemuchdifferentfromtheoneshowedinthesketchabove;probably,forafixedpointoftheseasurface,wewillhavethefollowingrecordwithrespecttothetime:

ThistrendcanbeseenlikeasuperpositionofsinusoidsanditallowstheuseofFourieranalysisfordescribingwaterwaves;moreover,toincreasetheanalysisaccuracy,statisticaltechniquesarerequiredtofacethenaturerandomness.However,inthisworkisusedthelinear,orsmallamplitude,waterwavestheoryaccordingwhichawavecanbemodeled,withagoodaccuracy,byasinusoid;thisismostlytrueforlargewaves,thesocalledshallowwaterwaves,thatpresenttheratio

ℎ𝐿 <

120

Inthefollowingsection,wewillpresentthebasisofthistheory.Ingeneral,thefirsttwoassumptionsthatwehavetomakeforformulatingthelinearwaterwavestheoryare:

- irrotationalfluidmotion;- incompressiblefluid.

Referringtothefirstone,wecansaythatthemotionofavolumeoffluidisclosetobeirrotational,thatmeansthereisnovorticityinsidethemotionfield,becausetheviscouseffects,thatgiverotationtothefluid,areconfinedtothesurfaceandthebottom.Letnowseetheconsequencesofthisfirstassumption.

26

Let𝐮bethevelocityvector

𝐮 𝑥, 𝑦, 𝑧, 𝑡 = 𝑢𝐢 + 𝑣𝐣 + 𝑤𝐤Thecurlofthevelocityvectoristhevorticity𝛚

∇×𝐮 = 𝛚Sincewemadethehypothesisofirrotationalfluidmotion,wehavenovorticity

∇×𝐮 = 𝟎Thiscoincideswiththeindependenceofpathconditionfortheresultofalineintegralinathreedimensionalproblem.Weintroducenowthevector

𝑑𝐥 = 𝑑𝑥𝐢 + 𝑑𝑦𝐣 + 𝑑𝑧𝐤andwedefinetheresultofthelineintegralof𝐮alongacertainroutethatlinksthetwogeneralpoints𝑃Xand𝑃Yas

−𝜙 = 𝐮 ∙ 𝑑𝐥\]

\̂= 𝑢𝑑𝑥 + 𝑣𝑑𝑦 + 𝑤𝑑𝑧

\]

\̂

Therespectoftheabovementionedindependenceofpathconditionimpliesthatthetermsinsidetheintegralhavetobeanexactdifferential,then

𝑢 = −𝜕𝜙𝜕𝑥 𝑣 = −

𝜕𝜙𝜕𝑦 𝑤 = −

𝜕𝜙𝜕𝑧

Thisallowsustowrite

𝐮 𝑥, 𝑦, 𝑧, 𝑡 = 𝑢𝐢 + 𝑣𝐣 + 𝑤𝐤 = −𝜕𝜙𝜕𝑥 𝐢 −

𝜕𝜙𝜕𝑦 𝐣 −

𝜕𝜙𝜕𝑧 𝐤

thatis

𝐮 = −∇𝜙Summingup,sincethefluidmotionisquiteirrotational,wecandefineapotentialfunction𝜙thegradientofwhichisthevelocityvector𝐮.Letnowseetheconsequencesoftheassumptionofincompressiblefluid.Fromthetheory,wecanderivethecontinuityequationasfollows

27

1𝜌𝜕𝜌𝜕𝑡 + 𝑢

𝜕𝜌𝜕𝑥 + 𝑣

𝜕𝜌𝜕𝑦 + 𝑤

𝜕𝜌𝜕𝑧 +

𝜕𝑢𝜕𝑥 +

𝜕𝑣𝜕𝑦 +

𝜕𝑤𝜕𝑧 = 0

Thedefinitionofbulkmodulus𝐸ofthefluidis

𝐸 ≡ 𝜌𝑑𝑝𝑑𝜌

andforwateris

𝐸defgh = 2.07 ∙ 10i𝑃𝑎Rewritingthedefinitionas

𝑑𝜌 =𝑑𝑝𝐸 ∙ 𝜌

wecannotethat,forapressureincreaseof1 ∙ 10j𝑃𝑎wehavethe0.05%ofdensityincrease.Forthisreason,wecanclearlyconsiderwaterincompressibleandwith𝜌 = 1000𝑘𝑔 𝑚4constant.Thisallowstosimplifythecontinuityequationreducingitto

𝜕𝑢𝜕𝑥 +

𝜕𝑣𝜕𝑦 +

𝜕𝑤𝜕𝑧 = 0

or

∇ ∙ 𝐮 = 0Thisequationmustbetrueeverywhereinthefluidvolumeandaflowfield𝐮thatsatisfiesthisconditioniscallednondivergentflow.Summingup,sincewecanconsiderthewaterincompressible,wecanwritethecontinuityequationinaneasierwayandthelinkedphysicalmeaningisthatisnotallowedanyfluidaccumulationinsideavolumeoffluid.Then,thecontinuityequationforanincompressiblefluidstatesthat

∇ ∙ 𝐮 = 0

Substitutingfor𝐮theresultoftheirrotationalhypothesiswehave

∇ ∙ ∇𝜙 = 0

andthisyieldstotheLaplaceequation,widespreadinengineeringproblems

∇n𝜙 = 𝜕n𝜙𝜕𝑥n +

𝜕n𝜙𝜕𝑦n +

𝜕n𝜙𝜕𝑧n = 0

28

Thisisasecondorderlineardifferentialequationand,inthiscase,itdescribesthefluidmotionunderaperiodictwo-dimensionalwaterwave;thisequationmustbevalidwithinthewholefluiddomain,constitutedbyonlyonewave

∇n𝜙 = 00 < 𝑥 < 𝐿−ℎ < 𝑧 < 𝜂

Thearemanysolutionstothisequationandnowwehavetofindtheonesthatcorrectlydescribeourphenomenonsettingupaboundaryvalueproblemandsolvingit.Beforedoingthis,itisimportanttounderlinethatthisdifferentialequationislinearandallowsustousethesuperpositionproperty,inparticular:if𝜙Yand𝜙naresolutionoftheLaplaceequation,thentheirlinearcombination𝜙4 = 𝐴𝜙Y + 𝐵𝜙n,where𝐴and𝐵arearbitraryconstants,isstillasolution.Thisgiveusthepossibilitytocreatenewsolutionswithadditionsandsubtractionsbetweenothersolutions.

29

Wehavenowtosetupthefollowingboundaryvalueproblem

Forthefreesurfaceandthebottomwehavetointroducethesocalledkinematicboundaryconditions,requirementsonwaterparticlekinematics.Theconceptisthattherecannotbeflowacrossthistwoboundaries;totranslatethisideainanexpression,wehavetostartfromthemathematicaldescriptionoftheseboundaries.Ingeneral,asurfacecanbedescribedby

𝐹 𝑥, 𝑦, 𝑧, 𝑡 = 0Applingthetotalderivativetothissurfaceexpressionandevaluatingitonthesurface,wehave

𝐷𝐹 𝑥, 𝑦, 𝑧, 𝑡𝐷𝑡 =

𝜕𝐹𝜕𝑡 + 𝑢

𝜕𝐹𝜕𝑥 + 𝑣

𝜕𝐹𝜕𝑦 + 𝑤

𝜕𝐹𝜕𝑧 stu v,w,x,f yX

= 0

becauseifwemovewiththesurfacewedonotseeanychange.Thelastequationcanberewrittenas

−𝜕𝐹𝜕𝑡 = 𝐮 ∙ ∇F

and,sincethegradientofascalarfieldthatdescribesasurfacehasthedirectionofthenormaltothissurface,wecanintroducetheunitvectornormal

𝐧 =∇F∇𝐹

then

30

−𝜕𝐹𝜕𝑡 = 𝐮 ∙ 𝐧 ∇𝐹

Solvingfor𝐮 ∙ 𝐧,weobtainthegeneralexpressionforthekinematicboundarycondition

𝐮 ∙ 𝐧 =−𝜕𝐹𝜕𝑡∇𝐹 𝑜𝑛𝐹 𝑥, 𝑦, 𝑧, 𝑡 = 0

with

∇𝐹 =𝜕𝐹𝜕𝑥

n

+𝜕𝐹𝜕𝑦

n

+𝜕𝐹𝜕𝑧

n

ThisequationstatesthatthecomponentofthefluidvelocityfieldnormaltothesurfacearelinkedtothesurfacelocalvelocityNowweapplythisconditiontothebottomandfreesurfaceboundaries.BottomboundaryconditionInatwodimensionalcase,assumingnochangeintimeforthebottom,wecandescribeitssurfacelike

𝑧 = −ℎ(𝑥)or

𝐹 𝑥, 𝑧 = 𝑧 + ℎ 𝑥 = 0andthen

𝜕𝐹𝜕𝑡 = 0

Moreover,assumingthebottomimpermeable,wehave

𝐮 ∙ 𝐧 = 0where𝐧isthenormaltothebottomsurfaceandhasthefollowingexpression

𝐧 =∇𝐹∇𝐹 =

𝑑ℎ𝑑𝑥 𝐢 + 1𝐤

𝑑ℎ𝑑𝑥

n+ 1

Developingthedotproductandmultiplyingbothsidesoftheequationbythesquarerootatthedenominator,wearriveto

31

𝑤𝑢 = −

𝑑ℎ𝑑𝑥 𝑜𝑛𝑧 = −ℎ(𝑥)

Accordingtothiscondition,theflowatthebottomhastobetangenttothebottomateach𝑥;referringtothefollowingimage

−𝑤 = 𝑢𝑑ℎ𝑑𝑥 𝑜𝑛𝑧 = −ℎ(𝑥)

andsummingthetwovectors,onedirectedalong𝑥andtheotheralong𝑧,wecanclearlyobservetheresultofthebottomboundarycondition.

Asshowninthereferencesketch,inourproblemwehaveahorizontalbottomandthen

𝑑ℎ𝑑𝑥 = 0 ⇒ 𝑤 = 0

KinematicFreeSurfaceBoundaryConditionOncestudiedthebottom,wemovetothefreesurface.Usingtheexpressionforthefreesurfacedisplacement𝜂(𝑥, 𝑦, 𝑡),theinteractionsurfacebetweenairandwatercanbedescribedby

𝐹 𝑥, 𝑦, 𝑧, 𝑡 = 𝑧 − 𝜂 𝑥, 𝑦, 𝑡 = 0Substitutingthisrelationinsidethegeneralkinematicboundaryconditionwehave

32

𝐮 ∙ 𝐧 =𝜕𝜂𝜕𝑡

𝜕𝜂𝜕𝑥

n+ 𝜕𝜂

𝜕𝑦n+ 1

𝑜𝑛𝑧 = 𝜂(𝑥, 𝑦, 𝑡)

where𝐧isthenormaltothefreesurfaceandhasthefollowingexpression

𝐧 =∇𝐹∇𝐹 =

−𝜕𝜂𝜕𝑥 𝐢 −𝜕𝜂𝜕𝑦 𝐣 + 1𝐤

𝜕𝜂𝜕𝑥

n+ 𝜕𝜂

𝜕𝑦n+ 1

Developingthedotproductandmultiplyingbothsidesoftheequationbythesquarerootatthedenominator,wearriveto

𝑤 =𝜕𝜂𝜕𝑡 + 𝑢

𝜕𝜂𝜕𝑥 + 𝑣

𝜕𝜂𝜕𝑦 𝑜𝑛𝑧 = 𝜂(𝑥, 𝑦, 𝑡)

DynamicFreeSurfaceBoundaryConditionTheinteractionsurfacebetweenairandwaterisafreesurfaceand,inreverseofsurfacesfixedinthespace,itcannotsupportpressurevariations;forthisreason,weneedanotherboundaryconditiontoimposethepressuredistributiononthefreesurface.Asshownbythefollowingimage,inourproblemthereistheatmosphericpressureactingonthewater

Toimposethiscondition,wewillusetheunsteadyBernoulliequation.Stillunderthehypothesisofirrotationalmotionandincompressiblefluid,fromthetheorywecanarriveto

−𝜕𝜙𝜕𝑡 +

12 𝑢n + 𝑤n +

𝑝𝜌 + 𝑔𝑧 = 𝐶 𝑡

33

ComparedtotheBernoulliequation,theterm𝐶 𝑡 isnotaconstantanditisthecauseofthe“unsteady”.Comingbacktotheboundarycondition,weapplytheunsteadyBernoulliequationonthefreesurface

−𝜕𝜙𝜕𝑡 +

12 𝑢n + 𝑤n +

𝑝�𝜌 + 𝑔𝑧 = 𝐶 𝑡 𝑜𝑛𝑧 = 𝜂(𝑥, 𝑡)

whereusually𝑝� = 0becauseconsideredasgagepressure.Substitutingfor𝑢nand𝑤ntheresultsoftheirrotationalhypothesis,weobtain

−𝜕𝜙𝜕𝑡 +

𝑝�𝜌 +

12 −

𝜕𝜙𝜕𝑥

n

+ −𝜕𝜙𝜕𝑧

n

+ 𝑔𝑧 = 𝐶 𝑡 𝑜𝑛𝑧 = 𝜂(𝑥, 𝑡)

thatisthedynamicfreesurfaceboundarycondition.LateralboundaryconditionInordertofinishtheformulationoftheproblem,wehavetoimposetheremainingtwoconditionstothelateralboundary.Sincewearestudyingageneralvolumeofwaterwiththesurfacecrossedbywaves,weusethefollowingperiodicboundaryconditionusedbothintimeandspace

𝜙 𝑥, 𝑡 = 𝜙 𝑥 + 𝐿, 𝑡𝜙 𝑥, 𝑡 = 𝜙 𝑥, 𝑡 + 𝑇

Settheboundaryvalueproblem,wehavetosolveit.Thegoalistofindasolution𝜙 𝑥, 𝑧, 𝑡 fortheLaplaceequationandanexpressionfor𝜂(𝑥, 𝑡)bothforstandingandprogressivewaves;duringtheresolutionwillbecarriedoutalsoanotherimportantequation,thedispersionrelationship.SinceLaplaceequationisalinearpartialdifferentialequation,wecantrytosolveitwithseparationofvariables.Toapplythismethod,wefirsthavetomakethehypothesisthatthesolutionhasthefollowingstructure

𝜙 𝑥, 𝑧, 𝑡 = 𝑋 𝑥 ∙ 𝑍 𝑧 ∙ 𝑇 𝑡 aproductofterms,eachonefunctionsofoneindependentvariable.Sincethelateralboundaryconditionimposes𝜙periodicalsointime,wecanset

34

𝑇 𝑡 = sin 𝜎𝑡

where𝜎istheangularfrequencyofthewave.Tofindarelationfor𝜎,weusethetimeperiodicboundarycondition

𝜙 𝑥, 𝑡 = 𝜙 𝑥, 𝑡 + 𝑇 then

sin 𝜎𝑡 = sin 𝜎 𝑡 + 𝑇

sin 𝜎𝑡 = sin 𝜎𝑡 + 𝜎𝑇 whichistruefor𝜎𝑇 = 2𝜋andthen,asusual

𝜎 =2𝜋𝑇

Wehavetonotethat,sincewehavealinearproblem,thechoiceof𝑇 𝑡 = sin 𝜎𝑡aspartofthesolutioncanbereplacedby𝑇 𝑡 = cos 𝜎𝑡or,ingeneral,bythelinearcombination

𝑇 𝑡 = 𝐴cos 𝜎𝑡 + 𝐵 sin 𝜎𝑡However,herewechoosethefirstchoiceandthenthevelocitypotentialhasnowthefollowingexpression

𝜙 𝑥, 𝑧, 𝑡 = 𝑋 𝑥 ∙ 𝑍 𝑧 ∙ sin 𝜎𝑡IfwesubstitutethisequationinsidetheLaplaceoneandthenwedivideby𝜙,notreportingthedependences,weobtain

1𝑋𝜕n𝑋𝜕𝑥n +

1𝑍𝜕n𝑍𝜕𝑧n = 0

Consideringnow,forexample,avariationin𝑧while𝑥ismaintainedconstant,thesecondtermoftheequationcouldvarybutthefirstonenoandthentheequationisnotsatisfiedbecausewesurelyhavenotasumequaltozero.Forthisreason,lastequationissatisfiedifthetwotermsareequaltothesameconstantbutwithdifferentsign

𝜕n𝑋 𝑥𝜕𝑥n𝑋 𝑥 = −𝑘n

35

𝜕n𝑍 𝑧𝜕𝑧n𝑍 𝑧 = 𝑘n

wheretheconstant𝑘iscalledwavenumber;wehavetonotethatitisnotrelevantwhichoneofthetwotermshastheminussince𝑘canalsobeacomplexnumber.Thelastequationsareordinarydifferentialequationsandwecansolveitseparately;thefollowingtablesummarizethepossiblesolutions

SincewehavenowasetofsolutionstotheLaplaceequation,wehavetounderstandwhichofthesedescribesourphenomenonapplyingtheabovesetofboundaryconditions;thiswillgiveusalsothepossibilitytofindtheseveralconstantspresentinthesolutions.LateralperiodicityconditionThefirstboundaryconditionwearegoingtouseisthelateralperiodicitycondition.Accordingtothisrequirement,𝜙hastobeperiodicinspace;observingthesolutionsreportedintheabovetable,thisispossibleonlyif𝑘isrealandnonzero.Stillfromthetable,for𝑘n > 0wehavethefollowingsolutionsfortheabovementionedequationsrespectively

𝑋 𝑥 = 𝐴 cos 𝑘𝑥 + 𝐵 sin 𝑘𝑥

𝑍 𝑧 = 𝐶𝑒�x + 𝐷𝑒��x

36

Consequently,wehavethefollowingvelocitypotential

𝜙 𝑥, 𝑧, 𝑡 = 𝐴 cos 𝑘𝑥 + 𝐵 sin 𝑘𝑥 𝐶𝑒�x + 𝐷𝑒��x sin 𝜎𝑡Tofindarelationfor𝑘,weusethespaceperiodicboundarycondition

𝜙 𝑥, 𝑡 = 𝜙 𝑥, 𝑡 + 𝑇 hence

𝐴 cos 𝑘𝑥 + 𝐵 sin 𝑘𝑥 = 𝐴 cos 𝑘 𝑥 + 𝐿 + 𝐵 sin 𝑘 𝑥 + 𝐿 == 𝐴 cos 𝑘𝑥 cos 𝑘𝐿 − sin 𝑘𝑥 sin 𝑘𝐿 + 𝐵 sin 𝑘𝑥 cos 𝑘𝐿 + cos 𝑘𝑥 sin 𝑘𝐿

Theexpressionholdsforcos 𝑘𝐿 = 1andsin 𝑘𝐿 = 0andthisyieldsto𝑘𝐿 = 2𝜋andthen

𝑘 =2𝜋𝐿

Sincewearesolvingalinearproblem,superpositionprincipleisvalidanditallowsustoseparate𝜙indifferentparts.Fromnowon,itisusefulforustomaintainthefollowingtermsfromthepreviouspotentialequation

𝜙 = 𝐴 cos 𝑘𝑥 𝐶𝑒�x + 𝐷𝑒��x sin 𝜎𝑡Thanktosuperposition,wecanaddbacklatertheterm𝐵 sin 𝑘𝑥.Summingup,withthespatialperiodicityconditionwehaverestrictedthepossiblesolutionsoftheequationstotheonlyonesfor𝑘n > 0andwefoundanexpressionfor𝑘.BottomboundaryconditionforhorizontalbottomProceedinginourdevelopment,wehavenowtousethebottomboundarycondition.Asalreadysaid,theconditionstatesthat

𝑤𝑢 = −

𝑑ℎ𝑑𝑥

Beingthebottomhorizontalinourproblem,wehave

𝑑ℎ𝑑𝑥 = 0

andthen

𝑤 = −𝜕𝜙𝜕𝑧 = −𝐴𝑘 cos 𝑘𝑥 𝐶𝑒�x − 𝐷𝑒��x sin 𝜎𝑡 = 0

37

Evaluatingthelastexpressionin𝑧 = −ℎ,weobtain

𝑤 = −𝜕𝜙𝜕𝑧 = −𝐴𝑘 cos 𝑘𝑥 𝐶𝑒�� − 𝐷𝑒�� sin 𝜎𝑡 = 0

Sincethisequationhastobesatisfiedforany𝑥and𝑡,thequantityinsidetheparenthesesmustbezero;then,imposingthisconditionandresolvingfor𝐶wehave

𝐶 = 𝐷𝑒n��Substitutingnowthelastrelationintheexpressionforthepotential

𝜙 = 𝐴 cos 𝑘𝑥 𝐷𝑒n��𝑒�x + 𝐷𝑒��x sin 𝜎𝑡

𝜙 = 𝐴𝐷𝑒��cos 𝑘𝑥 𝑒� ��x + 𝑒�� x sin 𝜎𝑡

𝜙 = 𝐺 cos 𝑘𝑥 cosh 𝑘 ℎ + 𝑧 sin 𝜎𝑡with𝐺 = 2𝐴𝐷𝑒��DynamicfreesurfaceboundaryconditionWeapplynowthedynamicfreesurfaceboundaryconditiontodeterminethevalueofthisnewconstant𝐺.Asalreadyseen,wearegoingtousetheunsteadyBernoulliequation

−𝜕𝜙𝜕𝑡 +

𝑝�𝜌 +

12 −

𝜕𝜙𝜕𝑥

n

+ −𝜕𝜙𝜕𝑧

n

+ 𝑔𝑧 = 𝐶 𝑡 𝑜𝑛𝑧 = 𝜂(𝑥, 𝑡)

Sincetheequationforthesurfaceelevationisstillnotknownnow,thisconditionisappliedon𝑧 = 𝜂(𝑥, 𝑡)evaluatingthisequationat𝑥 = 0andusingthetruncatedTaylorseries

𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 xy� = 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 xyX + 𝜂𝜕𝜕𝑧 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 xyX + ⋯

Substitutingthealgebraicquantities

𝑔𝑧 −𝜕𝜙𝜕𝑡 +

𝑢n + 𝑤n

2xy�

= 𝑔𝑧 −𝜕𝜙𝜕𝑡 +

𝑢n + 𝑤n

2xyX

+ 𝜂 𝑔 −𝜕n𝜙𝜕𝑧𝜕𝑡 +

12𝜕𝜕𝑧 𝑢n + 𝑤n

xyX+ ⋯ = 𝐶 𝑡

where𝑝 = 0on𝑧 = 𝜂.

38

Wehavenowtolinearize.Referringtoverysmallwaves,𝜂issmall;then,wecanassumethatalsovelocitiesandpressuresaresmallquantities.Thentheproductsofthesetermsareverysmall

𝜂 ≪ 1𝜂n ≪ 𝜂𝑢𝜂 ≪ 𝜂Neglectingthesequantities,wecanwritethelinearizedunsteadyBernoulliequation

𝑔𝜂 −𝜕𝜙𝜕𝑡 xyX

= 𝐶 𝑡

Solvingfor𝜂,weobtainthelineardynamicfreesurfaceboundarycondition

𝜂 =1𝑔𝜕𝜙𝜕𝑡 xyX

+𝐶 𝑡𝑔

andsubstitutingthevelocitypotentialwehave

𝜂 =𝐺𝜎𝑔 cos 𝑘𝑥 cosh 𝑘 ℎ + 𝑧 cos 𝜎𝑡

xyX+𝐶 𝑡𝑔 =

𝐺𝜎 cosh 𝑘ℎ𝑔 cos 𝑘𝑥 cos 𝜎𝑡 +

𝐶 𝑡𝑔

Setting𝐶 𝑡 ≡ 0itresults

𝜂 =𝐺𝜎 cosh 𝑘ℎ

𝑔 cos 𝑘𝑥 cos 𝜎𝑡

wherethetermsinthebracketsareconstant.Referringtothereferencesketch,thatdescribesourphysicalmodel,wecanwrite

𝐻2 =

𝐺𝜎 cosh 𝑘ℎ𝑔

andsolvingfor𝐺

𝐺 =𝐻𝑔

2𝜎 cosh 𝑘ℎSubstitutingthislastrelationinsidetheequationfor𝜂and𝜙,ityields

𝜂 =𝐻2 cos 𝑘𝑥 cos 𝜎𝑡

𝜙 =𝐻𝑔 cosh 𝑘(ℎ + 𝑧)

2𝜎 cosh 𝑘ℎ cos 𝑘𝑥 sin 𝜎𝑡

39

Summingup,usingthebottomboundaryconditionandthedynamicfreesurfaceboundarycondition,wehavefoundanexpressionforthefreesurfaceelevationandforthevelocitypotential.KinematicfreesurfaceboundaryconditionThereisstillanotherboundaryconditiontouse,thekinematicfreesurfaceboundarycondition

𝑤 =𝜕𝜂𝜕𝑡 + 𝑢

𝜕𝜂𝜕𝑥 + 𝑣

𝜕𝜂𝜕𝑦 𝑜𝑛𝑧 = 𝜂(𝑥, 𝑦, 𝑡)

Usingthislastcondition,wewillfindtherelationshipbetween𝜎and𝑘.SteelusingtheTaylorseriesexpansiontorelatetheboundaryconditionattheunknownelevation𝑧 = 𝜂(𝑥, 𝑡)to𝑧 = 0,wehave

𝑤 −𝜕𝜂𝜕𝑡 − 𝑢

𝜕𝜂𝜕𝑥 xy�

= 𝑤 −𝜕𝜂𝜕𝑡 − 𝑢

𝜕𝜂𝜕𝑥 xyX

+ 𝜂𝜕𝜕𝑧 𝑤 −

𝜕𝜂𝜕𝑡 − 𝑢

𝜕𝜂𝜕𝑥 xyX

+ ⋯ = 0

Withthesameassumptionsmadebefore,wecanlinearizethelastequationobtaining

𝑤 =𝜕𝜂𝜕𝑡 xyX

or,expliciting𝑤

−𝜕𝜙𝜕𝑧 xyX

=𝜕𝜂𝜕𝑡

Substitutingtheexpressionsfor𝜙and𝜂inthelastrelation

−𝐻2𝑔𝑘𝜎sinh 𝑘 ℎ + 𝑧cosh 𝑘ℎ cos 𝑘𝑥 sin 𝜎𝑡

xyX= −

𝐻2 𝜎cos 𝑘𝑥 sin 𝜎𝑡

andthen,simplifying,weobtainthedispersionrelationship

𝜎n = 𝑔𝑘 tanh 𝑘ℎ Fixedthevaluesof𝜎andℎ,wehaveonlyone𝑘thatsatisfiesthisequation;thisisshowninthegraphbelow,inwhichisplottedthefollowingequation

𝜎nℎ𝑔𝑘ℎ = tanh 𝑘ℎ

40

Rememberingthat

𝜎 =2𝜋𝑇 𝑘 =

2𝜋𝐿

aftersomealgebra,wecanobtainanexpressionforspeedofwavepropagationC

𝐶n =𝐿n

𝑇n =𝑔𝑘 tanh 𝑘ℎ

𝐶 =𝑔ℎ tanh 𝑘ℎ

andtheexpressionforthewavelength

𝐿 =𝑔2𝜋 𝑇

n tanh2𝜋ℎ𝐿

StandingwavesSumminguptheresultsobtaineduntilnow,wehave

𝜙 =𝐻2𝑔𝜎cosh 𝑘 ℎ + 𝑧cosh 𝑘ℎ cos 𝑘𝑥 sin 𝜎𝑡

𝜂 𝑥, 𝑡 =1𝑔𝜕𝜙𝜕𝑡 xyX

=𝐻2 cos 𝑘𝑥 cos 𝜎𝑡

𝜎n = 𝑔𝑘 tanh 𝑘ℎ

41

Theseresultsdescribesasocalledstandingwavebecauseitdoesnotpropagate;itcanmodelawavecompletelyreflectedbyaverticalwall.Nowwehavetofindthesameresultsforprogressivewaves.ProgressivewavesWeconsiderthefollowingvelocitypotential

𝜙 𝑥, 𝑧, 𝑡 =𝐻2𝑔𝜎cosh 𝑘 ℎ + 𝑧cosh 𝑘ℎ sin 𝑘𝑥 cos 𝜎𝑡

itdiffersfromtheothervelocitypotentialbecausethe𝑥andthe𝑡termsare90°outofphasebutthisissteelasolutiontotheLaplaceequationanditrespectsalltheboundaryconditions.ApplingthelinearizedDynamicFreeSurfaceBoundaryConditionweobtain


= −𝐻2 sin 𝑘𝑥 sin 𝜎𝑡

SinceLaplaceequationislinearandsuperpositionisvalid,wecansubtractthenewpotentialtotheprecedentonesteelobtainingasolution

𝜙 =𝐻2𝑔𝜎cosh 𝑘 ℎ + 𝑧cosh 𝑘ℎ cos 𝑘𝑥 sin 𝜎𝑡 − sin 𝑘𝑥 cos 𝜎𝑡 = −

𝐻2𝑔𝜎cosh 𝑘 ℎ + 𝑧cosh 𝑘ℎ sin 𝑘𝑥 − 𝜎𝑡

ApplingagainthelinearizedDynamicFreeSurfaceBoundaryConditionweobtain


=𝐻2 cos 𝑘𝑥 − 𝜎𝑡

fromwhichweunderstandthatwehavenowatimemovingwave.Summingup,weobtained

𝜙 =−𝐻2𝑔𝜎cosh 𝑘 ℎ + 𝑧cosh 𝑘ℎ sin 𝑘𝑥 − 𝜎𝑡


=𝐻2 cos 𝑘𝑥 − 𝜎𝑡


Theseresultsdescribesaprogressivewave.

42

2.2SimplificationsforshallowanddeepwaterwavesIntheseexpressions,oftenoccurhyperbolicfunctionsof𝑘ℎ;forlargeandsmallvaluesof𝑘ℎ,simplificationsareadmitted.Beforeintroducingthem,wehavetodefineshallow,intermediateanddeepwaterwaves.Wedefine:

- shallowwaterwaves,thatwavesthathavealengthmuchbiggerthanthebackdropdepthonwhichtheypropagate(𝐿 > 20ℎ);

- intermediatewaterwaves,thatwavesthathavealengthcomprisedbetween2ℎ < 𝐿 < 20ℎ;

- deepwaterwaves,thatwavesthathavealengthinferiorthantwicethebackdropdepthonwhichtheypropagate(2ℎ > 𝐿).

Thefollowinggraphshowsthetrendsofsinh 𝑘ℎ,cosh 𝑘ℎandtanh 𝑘ℎ

44

Thefollowingtablesumsuptheresults

45

Chapter3

Firstorderwavemakertheory

Inthischapteritispresentedthefirstorderwavemakertheory.Thechapterincludesthemainpassagestoobtainthetransferfunction𝐻 𝑆forpistonandflaptypewavemakerandtheevaluationofforcesandpowerfortheconfigurationwithwaterononlyonesideandonbothsides.Thechapterisclosedbythemainpassagestoderivethegeneraltransferfunctionforobliquewavesgeneration.Themainreferenceforthispartis[2].3.1FirstorderwavemakertheoryFromnowon,wewillrefertothefollowingsketch

ThewaveboardmotionhereisgivenbyacombinationofrotationandtranslationThen,assumingirrotationalfluid,itsbehaviorinawavetankisdescribedbyLaplaceequation

𝜕n𝜙𝜕𝑥n +

𝜕n𝜙𝜕𝑧n = 0

withtheappropriateboundaryconditions.Asalreadyseeninthesecondchapter,wehave:

46

• BottomBoundaryConditionforhorizontalbottom

𝜕𝜙𝜕𝑧 = 0𝑎𝑡𝑧 = −ℎ

• KinematicFreeSurfaceCondition

𝜕𝜂𝜕𝑡 +

𝜕𝜙𝜕𝑥

𝜕𝜂𝜕𝑥 −

𝜕𝜙𝜕𝑧 = 0𝑎𝑡𝑧 = 𝜂

• DynamicFreeSurfaceBoundaryCondition

𝜕𝜙𝜕𝑡 +

12

𝜕𝜙𝜕𝑥

n

+𝜕𝜙𝜕𝑧

n

+ 𝑔𝜂 = 0𝑎𝑡𝑧 = 𝜂

Nowwehavetoformulatethewaveboardboundarycondition;assumingthatthewaveboardis:

• flat;• solid;• impermeable

thefluidvelocitynormaltothepaddleandthepaddlehavethesamevelocity.Then,theboundaryconditionforthissidebecomes(FlickandGuza1980;Barthel,etal.1983)

𝑟𝜕𝜃𝜕𝑡 = 𝑣t

Workingonthegeometryofthereferencesketch,wecanwrite

𝑟 = 𝑋 𝑧, 𝑡 n + 𝑙 + ℎ + 𝑧 n = 𝑙 + ℎ + 𝑧 1 + tan 𝜃 n

𝑣t = 𝑢 cos 𝜃 − 𝑤 sin 𝜃

𝑋 𝑧, 𝑡 = 1 +𝑧

ℎ + 𝑙 𝑋s 𝑡

Substitutingtheserelationsinsidetheboundarycondition,weobtain

𝑙 + ℎ + 𝑧 1 + tan 𝜃 n 𝜕𝜃𝜕𝑡 = 𝑢 cos 𝜃 − 𝑤 sin 𝜃

Assuming𝜃tobesmall,itispossibletowrite:𝑢 cos 𝜃 ≈ 𝑢𝑤 sin 𝜃 ≈ 0

47

tan 𝜃 n ≪ 1𝜃 ≈ � x,f

��x

Substituting,weobtaintheGeneralWaveBoardBoundaryCondition:

𝜕𝜙𝜕𝑥 = 1 +

𝑧ℎ + 𝑙

𝜕𝑋s 𝑡𝜕𝑡 𝑎𝑡𝑥 = 𝑋 𝑧, 𝑡

• 𝑙 = 0,boundaryconditionforaflaptypewavemakerhingedatthebottom;• 𝑙 → ∞,boundaryconditionforapistontypewavemaker.

Summingup,thewavemakerproblemisgivenbythefollowingsetofequations:

• � ¡�v

+� ¡�x

= 0Laplaceequation

• �¡�x= 0𝑎𝑡𝑧 = −ℎBottomBoundaryConditionforhorizontalbottom

• ��

�f+ �¡

�v��v− �¡

�x= 0𝑎𝑡𝑧 = 𝜂KinematicFreeSurfaceCondition

• �¡�f+ Y

n�¡�v

n+ �¡

�x

n+ 𝑔𝜂 = 0𝑎𝑡𝑧 = 𝜂DynamicFreeSurfaceBoundaryCondition

• �¡

�v= 1 + x

��¢ f�f

𝑎𝑡𝑥 = 𝑋 𝑧, 𝑡 GeneralWaveBoardBoundaryConditionWecansolvethewavemakerproblemusingstandardperturbationtechniques.Weassumetoexpressthefollowingtermswithpowerseries:𝜙 = 𝜖t𝜙t = 𝜖𝜙Y + 𝜖n𝜙n + 𝜖4𝜙4 + 𝑂 𝜖¥¦

tyY 𝜂 = 𝜖t𝜂t = 𝜖𝜂Y + 𝜖n𝜂n + 𝜖4𝜂4 + 𝑂 𝜖¥¦

tyY 𝜃 = 𝜖t𝜃t = 𝜖𝜃Y + 𝜖n𝜃n + 𝜖4𝜃4 + 𝑂 𝜖¥¦

tyY 𝑋s = 𝜖t𝑋s§ = 𝜖𝑋s] + 𝜖

n𝑋s + 𝜖4𝑋s¨ + 𝑂 𝜖¥¦

tyY where𝜖,theperturbationparameter,isproportionaltothewavesteepness𝐻 𝐿.Inthefollowing,weretainonlytermsupto𝑂 𝜖 forhavingthefirstorderapproximation.Substitutingtheexpressionfor𝜙insideLaplaceequationandbottomboundarycondition,wehaverespectively

48

𝜕n𝜙Y𝜕𝑥n +

𝜕n𝜙Y𝜕𝑧n = 0

𝜕𝜙Y𝜕𝑧 = 0𝑎𝑡𝑧 = −ℎ

Representingthevelocitypotentialevaluatedat𝑧 = 𝜂byaTaylorseriesexpansionabout𝑧 = 0

𝜙 𝜂, 𝑥, 𝑡; 𝜖 = 𝜖𝜙Y + 𝜖n 𝜙n + 𝜂Y𝜕𝜙Y𝜕𝑧 + 𝑂 𝜖4

andsubstitutingthisexpressionandthepowerseriesrepresentationfor𝜂intheKinematicFreeSurfaceConditionandintheDynamicFreeSurfaceBoundaryCondition,weobtainKinematicFreeSurfaceCondition

𝜕𝜂Y𝜕𝑡 −

𝜕𝜙Y𝜕𝑧 = 0𝑎𝑡𝑧 = 0

DynamicFreeSurfaceBoundaryCondition

𝜕𝜙Y𝜕𝑡 + 𝑔𝜂Y = 0𝑎𝑡𝑧 = 0

Itispossibletocombinetheselasttwoexpressionsobtainingfromthemonlyoneconditionandavoidingthepartialderivativeof𝜂Y(FlickandGuza,1980)��]�f− �¡]

�x= 0𝑎𝑡𝑧 = 0

�¡]�f+ 𝑔𝜂Y = 0𝑎𝑡𝑧 = 0

�]y�

]ª«¬]« �

¡]�f

+ 𝑔 �¡]�x

= 0𝑎𝑡𝑧 = 0

Movingonthepaddle,weapproximatethevelocitypotentialevaluatedonthesurfaceofthewaveboardbythefollowingTaylorseriesexpansionincylindricalcoordinatesabout𝜃 = 0

𝜙 𝜃, 𝑧, 𝑡; 𝜖 = 𝜖𝜙Y + 𝜖n 𝜙n + 𝜃Y𝜕𝜙Y𝜕𝜃 + 𝑂 𝜖4

Referringtothemainsketch,wecanwriteitinCartesiancoordinates

𝜙 𝑥, 𝑧, 𝑡; 𝜖 = 𝜖𝜙Y + 𝜖n 𝜙n + 𝜃Y 𝑙 + ℎ + 𝑧𝜕𝜙Y𝜕𝑥 − 𝑥

𝜕𝜙Y𝜕𝑧 + 𝑂 𝜖4

Substitutingnowthisequationfor𝜙,thepowerexpressionsfor𝜃and𝑋s,and𝜃YwiththesmallangleapproximationinsidethegeneralWaveBoardBoundaryCondition

𝜕𝜙Y𝜕𝑥 = 𝑓 𝑧

𝑑𝑋s]𝑑𝑡 𝑎𝑡𝑥 = 0

49

𝑤𝑖𝑡ℎ𝑓 𝑧 = 1 +𝑧

ℎ + 𝑙

Summingup,thewavemakerproblemwiththefirstorderapproximationisgivenbythefollowingequations:

• � ¡]�v

+� ¡]�x

= 0Laplaceequation

• �¡]�x

= 0𝑎𝑡𝑧 = −ℎBottomBoundaryConditionforhorizontalbottom

• � ¡]�f

+ 𝑔 �¡]�x

= 0𝑎𝑡𝑧 = 0KinematicandDynamicFreeSurfaceBoundaryCondition

• �¡]�v

= 𝑓 𝑧 ¯�¢]¯f

𝑎𝑡𝑥 = 0GeneralWaveBoardBoundaryCondition𝑓 𝑧 = 1 + x

��

FirstorderwavegenerationTofindthesolutionforthewavemakerproblem,weassumetowrite𝜙Yinthefollowingmanner

𝜙Y 𝑥, 𝑧, 𝑡 = 𝑋 𝑥 𝑌 𝑦 𝑇 𝑡 Makingthisassumption,itispossibletodividetheLaplaceequationintoordinarydifferentialequationswithaknownsolutions.Sinceallthemhavetobecontainedinthepotentialfunction,wewriteitasfollows([1])

𝜙Y 𝑥, 𝑧, 𝑡 = 𝜙�] + 𝜙� + 𝜙�¨ wheretheelementsinthesumarethesolutionswehavewhentheseparationconstantisrespectivelyreal,zeroandimaginary.Inparticular:

• 𝑘Yn > 1

𝜙�] = 𝐴Y cos 𝑘Y𝑥 + 𝛼Y 𝐶Y𝑒�]² + 𝐷Y𝑒��]² 𝑇Y 𝑡

• 𝑘n = 0

𝜙� = 𝐴n𝑥 + 𝐵n 𝐶n𝑧 + 𝐷n 𝑇n(𝑡)

• 𝑘4n < 1

𝜙�¨ = 𝐴4𝑒 �¨ v + 𝐵4𝑒� �¨ v 𝐶4 cos 𝑘4 𝑧 + 𝐷4 sin 𝑘4 𝑧 𝑇4 𝑡 SubstitutingthesevelocitypotentialsinsidetheBottomBoundaryConditionandevaluatingitat

50

𝑧 = −ℎweobtain

𝐶Y = 𝐷Y𝑒n�]�𝐶n = 0𝐶4 = −𝐷4cos 𝑘4 ℎsin 𝑘4 ℎ

Intheexpressionof𝜙�¨ weset𝐴4 = 0becauseitmultipliesatermwhichwouldgiveanexponentialincreaseofthe𝑢velocitycomponentwithxandthisisnotrealistic.Thethreepotentialsarenow

𝜙�] = 𝐴Y cos 𝑘Y𝑥 + 𝛼Y 2𝐷Y𝑒�]� cosh 𝑘Y ℎ + 𝑧 𝑇Y 𝑡

𝜙� = 𝐴n𝑥 + 𝐵n 𝐷n𝑇n(𝑡)

𝜙�¨ = 𝐵4𝑒� �¨ v−𝐷4

𝑠𝑖𝑛 𝑘4 ℎ𝑐𝑜𝑠 𝑘4 𝑧 + ℎ 𝑇4 𝑡

Substitutingthesethreepotentialsinsidethecombinedkinematicanddynamicfreesurfaceboundaryconditionandevaluatingitat𝑧 = 0,weobtainordinarydifferentialequationswhichcanbesolvedforthethreeunknowntimefunctions.Thevelocitypotentialsbecome

𝜙�] = 2𝐴Y𝐷Y𝑒�]� cosh 𝑘Y ℎ + 𝑧 cos 𝑘Y𝑥 + 𝛼Y 𝐵Y sin 𝜎Y𝑡 + 𝛽Y

𝜙� = 𝐴n𝑥 + 𝐵n 𝐷n 𝐸n𝑡 + 𝐹n

𝜙�¨ =−𝐵4𝐷4𝑠𝑖𝑛 𝑘4 ℎ

𝑒� �¨ v 𝑐𝑜𝑠 𝑘4 𝑧 + ℎ 𝐸4 cos 𝜎4𝑡 + 𝛼4

with𝜎Y n = 𝑔𝑘Y tanh 𝑘Yℎ 𝜎4 n = −𝑔 𝑘4 tan 𝑘4 ℎ Thelastrelationwillbeelaboratedonlater.𝜙� increasesmonotonicallywithtime;thisisunrealisticandthenweput𝐸n = 0.Usingatrigonometricidentityin𝜙�]andregroupingwehave

𝜙�] = 𝐴 cosh 𝑘Y ℎ + 𝑧 sin 𝑘Y𝑥 − 𝜎Y𝑡 + 𝛾Y + sin 𝑘Y𝑥 + 𝜎Y𝑡 + 𝜖Y

𝜙� = 𝐵𝑥 + 𝐷

𝜙�¨ = 𝐶𝑒� �¨ v cos 𝑘4 𝑧 + ℎ cos 𝜎4𝑡 + 𝛼4

51

Observingtheexpressionfor𝜙�],wecannotethatitisthesumoftwoprogressivewavesmovinginoppositedirections;sincewehaveassumedthatthewaveboardisasolid,wecanrejectthewavemovingtowardthepaddle.𝜙� willgive,aftertheapplicationofthevectordifferentialoperator∇,aconstanthorizontalvelocitycomponent;asalreadysaid,sinceitisnotpossibletohaveflowthroughthepaddle,wecansetBequaltozero.TheremainingconstantDisarbitraryandsowechooseitagainequaltozero.Moreover,𝜎Y = 𝜎4 = 𝜎de¶g·e�gh = 𝜎andthen

𝜎 = 𝑔𝑘Y tanh 𝑘Yℎ = −𝑔𝑘4 tan 𝑘4ℎ Asshownbythefollowingtwographs,fixingavaluefor𝜎andℎ,wehaveonlyonesolution𝑘Yforthedispersionrelationshipandinfinitesolutions𝑘4forthelastequation.

Asalreadyseeninchaptertwo,𝑘Yisthewavenumberofaprogressivewavewhiletheinfinitevaluesof𝑘4,thatsatisfythelastequationforfixed𝜎andℎ,arewavenumbersofstandingwaves;theselastquantitiesyieldtomanyvaluesfor𝜙�¨ thatwehavetoconsiderwithasummation.Theoriginofthisstandingwavesisduethefactthatthesolidwaveboarddoesnotexactlyfollowthevelocitymotionbeneathaprogressivefirstorderwave.Furthermore,referringtotheequationsfor𝜙�] and𝜙�¨,sincethequantities𝛾Yand𝛼4§ arearbitrary,wecansetthemzero.Summingupthelastresults,wehave

𝜙Y 𝑥, 𝑧, 𝑡 = 𝜙�] + 𝜙� + 𝜙�¨

𝜙Y 𝑥, 𝑧, 𝑡 = 𝐴 cosh 𝑘Y ℎ + 𝑧 sin 𝑘Y𝑥 − 𝜎𝑡 + cos 𝜎𝑡 𝐶t𝑒��¨§v cos 𝑘4§ 𝑧 + ℎ¦

tyY

52

Thefirsttermofthepotential𝜙Yisaprogressivewavethatpropagatesinthepositivexdirectionwithitswavenumber𝑘Y;thesecondtermisaseriesofstandingwaves,eachoneofthemwithitswavenumber𝑘4§.Asitcanbeseen,thesestandingwavesexponentiallydecaywiththedistancexfromthepaddle:accordingto[1],theamplitudeofthefirststandingwave,themostimportant,willdecreaseby99%at𝑥 = 3ℎ.WehavenowtofindtheexpressionsforthecoefficientAand𝐶t.Weassumethatthewaveboardfollowsthefollowingsinusoidalmotion

𝑋s] =𝑆s2 sin 𝜎𝑡

with𝑆sthewaveboardstrokeat𝑧 = 0.Rememberingthewaveboardboundarycondition

𝜕𝜙Y𝜕𝑥 = 𝑓 𝑧

𝑑𝑋s]𝑑𝑡 𝑎𝑡𝑥 = 0

𝑤𝑖𝑡ℎ𝑓 𝑧 = 1 +𝑧

ℎ + 𝑙

wesubstituteintheexpressionfor𝜙Yand𝑋s]obtaining

𝐴𝑘Y cosh 𝑘Y ℎ + 𝑧 − 𝐶t

¦

tyY

𝑘4§ cos 𝑘4§ 𝑧 + ℎ =𝑓 𝑧𝜎𝑆s2

Multiplyingbothsidesofthelastequationbycosh 𝑘Y ℎ + 𝑧 ,integratingitbetween𝑧 = −ℎand𝑧 = 0andresolvingfor𝐴,weobtain

𝐴 = 𝜎𝑆s2𝑘Y

𝑓 𝑧 cosh 𝑘Y ℎ + 𝑧 𝑑𝑧X��

cosh 𝑘Y ℎ + 𝑧 nX�� 𝑑𝑧

wherethesummationtermshavegonetozeroduetotheorthogonalityconsiderationsgivenbytheSturm-Liouvilletheory([1]).Then,wemultiplybothsidesoftheequationthatcomesfromthewaveboardboundaryconditionbycos 𝑘4§ 𝑧 + ℎ andweintegrateitbetweentheabovementionedintegrationsextremes;thisyieldstheprogressivewavestermstogotozeroandthen,solvingfor𝐶t,weobtain

𝐶t = −𝜎𝑆s2𝑘4§

𝑓 𝑧 cos 𝑘4§ 𝑧 + ℎ 𝑑𝑧X��

cos 𝑘4§ 𝑧 + ℎn𝑑𝑧X

��

Tocompletetheanalysis,wehavetofindtheexpressionofthesurfaceelevation𝜂Yandthegeneraltransferfunctionbetweenwaveboardstroke𝑆sandwaveheight𝐻.

53

Substitutingtheexpressionfor𝜙YinsidethefirstorderDynamicFreeSurfaceBoundaryConditionandevaluatingitat𝑧 = 0,weobtainthesurfaceelevationinthewavetank

𝜂Y 𝑥, 𝑡 =𝜎𝐴𝑔 cosh 𝑘Yℎ cos 𝑘Y𝑥 − 𝜎𝑡 + sin 𝜎𝑡

𝜎𝐶t𝑔 𝑒��¨§v cos 𝑘4§ℎ

¦

tyY

Asabovesaid,startingfromadistanceof𝑥 = 3ℎfromthewavemaker,therearenotdisturbances;forthisreason,farfromthewavemaker,wecannegletthesummationtermsthatmathematicallydescribethisphenomenon.Then,thesurfaceelevationcanbedescribedbythealreadymentionedprogressivewavesolution

𝜂Y 𝑥, 𝑡 =𝐻2 cos 𝑘Y𝑥 − 𝜎𝑡

Equatingthelasttwoexpressionsfor𝜂Ywithoutconsidertheseriesterms,weobtainthegeneraltransferfunctionbetweenwaveboardstroke𝑆sandwaveheight𝐻

𝐻 =2𝜎𝐴𝑔 cosh 𝑘Yℎ

wherethestroke𝑆siscontainedin𝐴.Thisisageneralexpressionfromwhichwecanfindtherelationfordifferenttypesofwavemakersubstitutingin𝑓 𝑧 thefunctionthatdescribestheparticularwaveboardgeometry.Wehavetonotethat,sincewehaveneglectedthesummationterms,thelastequationgivesthewaveheight𝐻enoughfarfromthewavemaker.Piston-typeandFlap-typewavemakersReferringtothemainsketch,weknowthatthewaveboardgeometryisdescribedby

𝑓 𝑧 = 1 +𝑧

ℎ + 𝑙

Asalreadysaid,thisexpressioncandescribethegeometryofbothpiston-typeandflap-typewavemakers:

• 𝑙 = 0,flaptypewavemakerhingedatthebottom;• 𝑙 → ∞,pistontypewavemaker.

Developingtheexpressionsfor𝐴and𝐶t,weobtain

𝐴 =2𝜎𝑆s

𝑘 sinh 2𝑘ℎ + 2𝑘ℎ sinh 𝑘ℎ +1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙

𝐶t = −2𝜎𝑆s

𝑘t sinh 2𝑘tℎ + 2𝑘tℎsin 𝑘tℎ +

𝑐𝑜𝑠𝑘tℎ − 1𝑘t ℎ + 𝑙

54

where𝑘standsfor𝑘Yand𝑘tfor𝑘4§.Substitutingtheexpressionfor𝐴intheexpressionfor𝐻andusingthedispersionrelationshiptoexplicit𝜎n,weobtaintheGeneralFirst-OrderWavemakerSolution

𝐻𝑆s=

4 sinh 𝑘ℎsinh 2𝑘ℎ + 2𝑘ℎ sinh 𝑘ℎ +

1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙

Fromthisrelation,wecanfindthetransferfunctionforbothpistonandflaptypewavemakers:

• 𝑙 → ∞First-OrderPistonWavemakerSolution

𝐻𝑆s=2 cosh 2𝑘ℎ − 1sinh 2𝑘ℎ + 2𝑘ℎ

𝑙 = 0First-OrderFlapWavemakerSolution

𝐻𝑆s=


1 − cosh 𝑘ℎ𝑘ℎ

WaveBoardPressureAssumingwaterononlyonesideofthewaveboard,wehaveafirstorderexpressionforthepressuredistributiononthepaddlefromthelinearizedBernoulliequation

𝑝 𝑥, 𝑧, 𝑡 = −𝜌𝜕𝜙Y𝜕𝑡 − 𝜌𝑔𝑧

inwhichwecannotethedynamicandthehydrostaticterm.Substitutingtheexpressionfor𝜙Yandevaluatingtheresultat𝑥 = 0,wehavethegeneralrelationfortheinstantaneouspressureactingonthewaveboard.Wecanderiveparticularexpressionsforpistonandflaptypewavemakerusing𝑙 → ∞or𝑙 = 0respectively

𝑝s 𝑧, 𝑡 = 𝜌𝜎𝐴 cosh 𝑘 ℎ + 𝑧 cos 𝜎𝑡 + 𝜌𝜎 𝐶t cos 𝑘t ℎ + 𝑧¦

tyY

sin 𝜎𝑡 − 𝜌𝑔𝑧

Thefirstelementistheresistivepartofthedynamicpressurefluctuationanditisinphasewiththepaddlevelocity.Thesecondelementistheinertiapartofthedynamicpressurefluctuation;thistermisinphasewiththepaddleacceleration.Thethirdelementisthehydrostaticpressureactingonthepaddleanditisusuallythemostrelevantofthethreeterms.

55

Ifinourproblemthereiswaterofthesamedepthonbothsidesoftheboard,wehavetoaddthecontributionthatcomesfromawavemakervelocitypotentialforwavesthatpropagatesinthenegativex-direction,stillevaluatedat𝑥 = 0.Thisyieldstothedoublingofthedynamicpressuretermandityieldsalsothedeletingofthehydrostaticterm.Thenwehave

𝑝ss 𝑧, 𝑡 = 2 𝜌𝜎𝐴 cosh 𝑘 ℎ + 𝑧 cos 𝜎𝑡 + 2 𝜌𝜎 𝐶t cos 𝑘t ℎ + 𝑧¦

tyY

sin 𝜎𝑡

Inthiscase,waterofthesamedepthonbothsides,intherealapplicationitwillbenecessarytodissipatewavesgeneratedbehindthepaddleinordertoavoidthereturnofwallreflectedwaves.WaveBoardForceAssumingagainwaterononlyoneside,thetotalistantaneousforceactingperunitwidthofpaddleis

𝐹 ¢̧ 𝑡 = 𝑝s 𝑧, 𝑡X

��𝑑𝑧

andthisyieldsto

𝐹 ¢̧ 𝑡 =𝜌𝜎𝐴 sinh 𝑘ℎ

𝑘 cos 𝜎𝑡 + 𝜌𝜎𝐶t𝑘tsin 𝑘tℎ

¦

tyY

sin 𝜎𝑡 +𝜌𝑔ℎn

2

Ifthereiswaterofequaldepthonbothsidesoftheboard,wewrite

𝐹 ¢̧¢ 𝑡 = 𝑝ss 𝑧, 𝑡X

��𝑑𝑧

andthen

𝐹 ¢̧¢ 𝑡 = 2𝜌𝜎𝐴 sinh 𝑘ℎ

𝑘 cos 𝜎𝑡 + 2 𝜌𝜎𝐶t𝑘tsin 𝑘tℎ

¦

tyY

sin 𝜎𝑡

Fromthegeneralequations,wecanobtaintheparticularrelationsforflapandpistontypewavemakersubstitutingtherespectivevaluesfor𝑙.WaveboardpowerNeglectingfrictionandmechanicallosses,startingagainwithwaterononlyonesideofthepaddle,theinstantaneouspowerperunitwidthofboardisgivenby

56

𝑃s 𝑡 = 𝑝s 𝑧, 𝑡X

��𝑢s 𝑧, 𝑡 𝑑𝑧

with𝑢sthehorizontalwaveboardvelocityat𝑥 = 0.ThisquantityisfoundfromtheFirst-OrderWaveBoardBoundaryConditionas

𝑢s =𝜕𝜙Y𝜕𝑡 vyX

= 𝑓 𝑧𝑑𝑋sY𝑑𝑡

Substitutingthentheexpressionsfor𝑓 𝑧 and¯�¢]

¯frespectively,weobtain

𝑢s =𝜎𝑆s2 1 +

𝑧ℎ + 𝑙 cos 𝜎𝑡

Makingnowtheappropriatesubstitutions,wecansolvetheabovementionedintegral

𝑃s 𝑡 =𝜌𝜎n𝑆s𝐴2𝑘 sinh 𝑘ℎ +

1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙 cos 𝜎𝑡 n

+𝜌𝜎n𝑆s2

𝐶t𝑘t

sin 𝑘tℎ +cos 𝑘tℎ − 1𝑘t ℎ + 𝑙

sin 𝜎𝑡 cos 𝜎𝑡¦

tyY

+𝜌𝑔𝜎𝑆s2

ℎn

2 −ℎ4

3 ℎ + 𝑙 cos 𝜎𝑡

Ifwehavewateronbothsidesofthewavesboard,wewrite

𝑃ss 𝑡 = 𝑝ss 𝑧, 𝑡X


andmakingnowtheappropriatesubstitutions,wecansolvetheintegral

𝑃ss 𝑡 = 2𝜌𝜎n𝑆s𝐴2𝑘 sinh 𝑘ℎ +

1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙 cos 𝜎𝑡 n

+ 2𝜌𝜎n𝑆s2

𝐶t𝑘t



tyY

Itisalsopossibletoobtaintheexpressionforthemeanwaveboardpoweroverawavecyclewriting

𝑃s =1𝑇 𝑃s 𝑡

�¸ n

�¸ n𝑑𝑡

andthen,substitutingthedispersionrelationship

57

𝑃s =𝜋𝜌𝑔𝑆sn

𝑘𝑇tanh 𝑘ℎ

sinh 2𝑘ℎ + 2𝑘ℎ sinh 𝑘ℎ +1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙

n

Forwateronbothsidesofthepaddlewehave

𝑃ss = 2𝜋𝜌𝑔𝑆sn



n

Again,wecanobtaintheparticularrelationsforflapandpistontypewavemakersubstitutingtherespectivevaluesfor𝑙insidetheabovementionedequations.Accordingto[1],thepistontypewavemakersaremoreefficientinshallowwaterbecausetheirmotionapproximatebetterthenearlyuniformverticaldistributionofthehorizontalfluidparticles.Instead,theflaptypewavemakersaremoreefficientforgeneratingdeepwaterwaves.

3.2FirstorderobliqueregularwavegenerationUntilnowwehavereferredtoawavemakerwithonlyonepaddle;ifweuseinourtankmorewaveboardsindependentlycontrolled,wecancreateasocalledsnakewavemakerandthengenerateobliquewaves.Thesedevicesareparticularlyusefulwhenwewanttostudythemodelreactiontowavesincomingfromdifferentdirections:infact,thepossibilitytovarythewaveangleavoidscontinuousmodelsetupchangesduringthetests,savingtimeandthenmoney.Moreover,someofthesewavemakerscanalsogeneratearandomdirectionalseaandthenprovidingaveryrealisticsimulationenvironment.Asalreadyseeninchapter1,wecanhaveseveralconfigurations.Theeasiestprovidesasnakewavemakerononlyonesideofthetank;then,sincewewanttoavoidwallreflectedwavesandalsoextendthetestingcapabilities,oftentherearesnakewavemakersalsoonothersides.Fromnowon,wewillreferalsotothefollowingsketch

58

Thereferencesystemistheoneshownintheabovesketch,withthezaxisthatcomesoutfromthesheetplaneandhasitszeroatthestillwaterlevel.ThegoverningequationisthethreedimensionalLaplaceequation

𝜕n𝜙Y𝜕𝑥n +

𝜕n𝜙Y𝜕𝑦n +

𝜕n𝜙Y𝜕𝑧n = 0

andthefluiddomaininwhichitstandsis0 ≤ 𝑥 < ∞−∞ < 𝑦 <−ℎ ≤ 𝑧 ≤ 0

+∞

Regardingtheboundaryconditions,westillusetheabovementionedequations.AlsotheWaveBoardBoundaryConditionisthesamebutherewehavetochangetheequationthatdescribesthewavemakermotion.Assumingasinusoidalmotionintheydirectionforthepaddlesofthesnakewavemaker,wecanwrite

𝑋s] =𝑆s2 sin 𝜎𝑡 − 𝜆𝑦

where𝜆isthewavenumberofthesnakewavemakersinusoidalmotion.Thisimpliesthatthewavemakerismodeledlikeacontinuouslineandthisisasignificantdifferencefromthereality.Derivingrespecttothetime,weobtainthewaveboardvelocity

𝑑𝑋s]𝑑𝑡 =

𝜎𝑆s2 cos 𝜆𝑦 − 𝜎𝑡

Asalreadymadewiththetwodimensionalproblem,wecanfoundasolutionforthefirstordervelocitypotential𝜙Ywiththeseparationofvariables([1])

59

𝜙Y 𝑥, 𝑦, 𝑧, 𝑡 = 𝐴 cosh 𝑘Y ℎ + 𝑧 sin 𝑘Yn − 𝜆n 𝑥 + 𝜆𝑦 − 𝜎𝑡 + cos 𝜆𝑦 − 𝜎𝑡 𝐶t𝑒

� �¨§ �» v

cos 𝑘4§ 𝑧 + ℎ¦

tyY

Inthelastequation,𝑘Yisthewavenumberofthegeneratedprogressivewaveinthepropagationdirectionanditcomesfromthedispersionrelationship;𝑘4§ arethewavenumbersofthealreadyseenstandingwaves.Studyingtheprogressivepartofthelastequation,wenotethatitmustbe𝑘Y ≥ 𝜆;referringtothereferencesketch,thismeansthatitisnotpossibletogeneratewaveswith𝜃biggerthan90°.Stillstudyingtherelationbetweenwavenumbers,wecanwrite

𝜆 = 𝑘Y sin 𝜃

𝑘Yn − 𝜆n = 𝑘Y cos 𝜃

where

𝑘Y =2𝜋𝐿 𝜆 =

2𝜋𝑁𝐵

Substitutingnowtheequationfor𝜙Yand𝑑𝑋s] 𝑑𝑡intheWaveBoardBoundaryCondition,weobtain

𝐴 𝑘Yn − 𝜆n cosh 𝑘Y ℎ + 𝑧 − 𝐶t 𝑘4§

n + 𝜆n cos 𝑘4§ 𝑧 + ℎ = 𝑓 𝑧𝜎𝑆s2

¦

tyY

Multiplyingbothsidesofthelastequationbycosh 𝑘Y ℎ + 𝑧 andintegratingoverthedepth,allthesummationtermsaresenttozeroaccordingtotheorthogonallyconditions;then,wecansolveforA

𝐴 =𝜎𝑆s

2 𝑘Yn − 𝜆n

𝑓 𝑧 cosh 𝑘Y ℎ + 𝑧 𝑑𝑧X��

cosh 𝑘Y ℎ + 𝑧 n𝑑𝑧X��

Asalreadydoneforthetwodimensionalcase,wecanfindanexpressionfor𝐶t.Developingthelastequationandusingtheequationfor𝑓(𝑧),weobtaintheexpressionfor𝐴

𝐴 =2𝜎𝑆s

𝑘 cos 𝜃 sinh 2𝑘ℎ + 2𝑘ℎ sinh 𝑘ℎ +1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙

inwhich𝑘standsfor𝑘Y.

60

Theexpressionforthesurfaceelevationis

𝜂Y 𝑥, 𝑦, 𝑡 = −1𝑔𝜕𝜙Y𝜕𝑡 𝑎𝑡𝑧 = 0

𝜂Y 𝑥, 𝑦, 𝑡 =𝜎𝐴𝑔 cosh 𝑘Yℎ cos 𝑘Y cos 𝜃 𝑥 + 𝑘Y sin 𝜃 𝑦 − 𝜎𝑡

+ sin 𝜆𝑦 − 𝜎𝑡𝜎𝐶t𝑔 𝑒� �¨§

�» v cos 𝑘4§ℎ¦

tyY

Ifweareenoughfarfromthewavemaker,atleast𝑥 = 3ℎ,theeffectsofthestandingwavesarenegligibleandthen

𝜂Y 𝑥, 𝑦, 𝑡 =𝜎𝐴𝑔 cosh 𝑘ℎ cos 𝑘 cos 𝜃 𝑥 + 𝑘 sin 𝜃 𝑦 − 𝜎𝑡

inwhich,again,𝑘standsfor𝑘Yand

𝐻2 =

𝜎𝐴𝑔 cosh 𝑘ℎ

isthewaveamplitude.Substitutingtheexpressionfor𝐴inthelastequation,weobtainthetransferfunctionbetweenthewaveheightandtheboardstroke

𝐻𝑆s=

4 sinh 𝑘ℎcos 𝜃 sinh 2𝑘ℎ + 2𝑘ℎ sinh 𝑘ℎ +

1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙

Thisequationdiffersfromtheotheroneforthecos 𝜃atthedenominator.Aswecannote,thisallowstouselessstrokethan2Dcaseforgeneratingthesamewaveheight𝐻.Wehavenowtointroducethedifferencesbetweentheoryandreality.Aswecanseefromthefollowingsketch,adirectionalwavemakerhasafinitelengthanditiscomposedbyseveralpaddleseachoneofwhichwithafinitewidth

61

Thesepracticalsolutionsdonotmatchthehypothesisofinfinitelylongwavemakerandpaddleswithaninfinitesimalwidth.Fromtheabovesketch,inadditiontothequantitiesalreadydefined,weintroduce:𝐿,thewavelengthinthepropagationdirection;𝐵,thepaddlewidth;𝑁,thenumberofpaddlesincludedinawavelengthofthesinusoidmadebythesnakewavemaker.Then,theexpressionforthewavenumberintheydirectionis

𝜆 =2𝜋𝑁𝐵

Substitutingthelastrelationin𝜆 = 𝑘Y sin 𝜃andexplicitingtheexpressionfor𝑘Y,wecanfindtheexpressionforthewaveangle

sin 𝜃 =𝐿𝑁𝐵

Thephaseshiftbetweenadjacentpaddlesis

𝜙 =2𝜋𝑁

whichcanbesubstitutedinthepreviousequation.Thisyieldstoarelationthatlinksthewaveangletothephaseshift

sin 𝜃 =𝐿𝜙2𝜋𝐵

62

Rememberingthat0° ≤ 𝜃 < 90°,ifwewanttoincreasethewaveanglewehavetoincreasethephaseshift.Ingeneratingobliquewaves,wehavealsotoconsidertheundesiredpresenceofspuriouswaves;theseproblemistheconsequenceofthepaddlefinitewidth.TherelationfoundbySandandMynett(1987)forminimizingthisproblemstatesthat

𝐿𝐵 > 2

with𝐿theshortestwavelengththatwewanttogenerate.Obliquewavesgenerationleadsalsotoseveraleffectsthatinfluencethetests,stillstudiedbySandandMynett(1987).Thefollowingsketchshowsinwhichareasofthetankthesephenomenaoccur

Diffractionreducesthewaveheightandsoalsotheusefultestingarea;toavoidthisproblemwecanusewaveguides,solidboundariesthatcrossthetankperpendiculartothewavecreststartingfromthewavemakersidesbutwecanhavereflectionproblems.Anotherstrategyistocompensatethediffractioneffectsbygeneratingbiggerwavesthantheonestheoreticallyrequiredinordertohavetherequiredwaveheightwheredesired.Aswecanseefromthesketch,obliquewavesimpactingonthetankboundariescancreatereflectedwavesthatdisturbthetestingarea.Itispossibletosolvethisproblemusingotherswavemakersonthelateralsidesofthetank,usingthemtoabsorbtheimpactingwaves.Wehavealreadytalkedaboutlocaldisturbancescreatedbythemismatchingbetweenwaveboardmotionandwatermotion.Thisphenomenacreatesadeformedwaveshapebutitseffectsendatadistanceofthreetimesthewaterdepthfromthewavemaker.

63

Chapter4WavemakerdesignwithfirstorderwaterwavestheoryInthischapteritispresentedthewavemakerdesignmethodfollowed,basedonthelinearwaterwavestheory.Wefirstanalyzedthewavegenerationpotentialitiesofourtank:givenaperiodrangeofinterest,itwasevaluatedthemaximumstrokereachableforeveryperiodandthecorrespondingmaximumwaveheight.Severalothervaluesforstrokesandwaveheightsaregiventohaveacompleteoverviewofthesystemfeatures;theplotsofthewavelengthandcelerityrespecttotheperiod,completethisfirstanalysis.Togeneratewaves,wehavetomakeafirstdesignofthepaddleandafirstdimensioningoftheactuationsystem.Forthesereasons,weanalyzedvelocitiesandforcesactingonthepaddletogeneratethedesiredwaves.Theresultsofthisanalysiswereusedtomakeafirstdimensioningoftheactuationsystem:inparticular,itwaschosenanelectriccylinderdrivenbyabrushlessmotorforitssimplicityandflexibilitybut,aswewillseelater,thischoiceimpliesanewlimitationinwavegeneration.Subsequently,wefocusedontheconsequencesofthechoicemadeontheactuationsystem:inparticular,wedeterminedthenewfeaturesofthereallygenerablewaves.Theworkisclosedbytheanalysisoftransmissionandsuitablebrushlessmotors.

4.1ProblemdefinitionThewatertankthathastobeequippedwiththewavemakeris30𝑚𝑥2.2𝑚𝑥0.88𝑚.

Thewaveperiodrangeofinterestisbetween0.5𝑠and2.5𝑠.Sinceweareinterestedingeneratingdeepwaterwaves,wehaveconsideredonlytheflaptypewavemaker:infact,itsmotionmorecloselyresemblesthefieldofmotionofdeepwaterwaves.

64

Fromnowon,wewillrefertothefollowingtwosketches:

inwhichℎ = 0.7𝑚andℎ¿ = 1.1𝑚;thislastvaluewasassumedinordertoleaveacertainsafetydistancebetweenthesloshingwaterandtheactuatorwiththeelectricengine.Boththeconfigurationswillhaveanactuationsystemmadeofanelectriccylinderdrivenbyabrushlessmotor;inthisway,therotationalmotionofthemotorisconvertedinatranslationthatdrivestheflapmotion.Boththeconfigurations,waterononlyonesideandwateronbothsides,willbediscussedduringtheevaluationoftheforcesduetothewaterbut,furtheron,wewillfocusonlyontheconfigurationwithwateronbothsides. ThefollowingargumentsweredevelopedusingthesoftwareMatlab.

4.2Analysisofwavegenerationpotentialitiesandfirstdimensioningoftheactuationsystem4.2.1WavesideallygenerableinourtankSummingup,theinputdataare:

• Tankgeometry:30𝑚𝑥2.2𝑚𝑥0.88𝑚• Waveperiodrangeofinterest:𝑇 = 0.5𝑠 ÷ 2.5𝑠• Waterdepth:ℎ = 0.7𝑚• Heightofthelinkbetweenflapandelectrocylinder:ℎ¿ = 1.1𝑚• Flaptypewavemaker

First,weevaluatethewavegenerationcapabilityofthetank.Foreachwaveperiodoftheabovementionedrange,wewillhaveamaximumwaveheightexpectedaccordingtothelinearwaterwavestheory;tofindtheseinformation,wehavetointroducethefollowinglimitsthatconstrainoursystem:

• Wavebreaklimit ÁÂ< Y

Ã,

Ifweovercomethisratio,wewillnothaveregularshapedwaveanditprobablybreaks.

65

• Maximumwaveheightinthetank,Referringtotheabovesketches,themaximumwaveheightreachableinoursystemissetto𝐻·ev = 0.4𝑚

• Maximumactuatorstroke,𝑆¿ = 0.5𝑚Thislimitarisesfromanotherlimitonthecontrolsystem.Ifwewanttocontrolthepositionoftheflap,themaximumrecommendedangleexcursionissetto+/−12°;controllingtheforce,itispossibletoreachamaximumangleof+/−18°[EdinburghDesign].Sinceweareinterestedincontrollingtheposition,weconsider:𝜃 = 12°ℎ¿ = 1.1𝑚Ån= ℎ¿ ∙ tan 𝜃 = 0.23𝑚 ⇒ 𝑆 = 0.46𝑚

that,forpracticalconvenience,wascarriedto

𝑆¿ = 0.5𝑚

• Ballscrewmaximumacceleration,𝑎·ev = 1𝑔Thisvaluewasassumedinfirstinstance,asmadein[5]

Toapplythislimitationstooursystem,weusetheFirstorderFlapTypeWavemakerSolution

𝐻𝑆s=


1 − cosh 𝑘ℎ𝑘ℎ

Itisimportanttounderlinethat,aswecanseefromthetwosketches,thequantity𝑆sisthewaveboardstrokeat𝑧 = 0𝑚;sincethelinkbetweentheactuatorandtheflapislocatedat𝑧 = 0.4𝑚,wehavetoreport𝑆satthesameleveloftheactuatortomakeaconsistentcomparisonbetweentheintroducedfourlimits.Ingeneral,raisinguptheconnectionpointbetweenactuatorandflaprespecttothestillwaterlevel,wewillhavealostinwaveheight:connectingactuatorandflapat𝑧 = 0.4𝑚fromthestillwaterlevel,wehavelostthe20%ofthemaximumwaveheightthatwewouldhavemakingtheconnectionat𝑧 = 0𝑚.Then,wehave:

• Wavebreaklimit

66

𝐻𝐿<17ÆÇÈÆfÉfÇfÉtÊÉt¸u

𝑆ËÌÍdÈ = 𝑆X =𝜋ℎ

14 sinh 𝑘ℎ∙

sinh 2𝑘ℎ + 2𝑘ℎ𝑘ℎ ∙ sinh 𝑘ℎ − cosh 𝑘ℎ + 1

𝑎𝑡𝑧 = 0

tan 𝜃 =𝑆X2ℎ

𝑆ËÌÍdÈ∗ = 𝑆¿ = 2 ∙ ℎ¿ ∙ tan 𝜃

• Maximumwaveheigthinthetank

𝐻·ev = 0.4𝑚ÆÇÈÆfÉfÇfÉtÊÉt¸u

𝑆ËÌÍd� = 𝑆X = 0.4 ∙ ÏÐÑÒ n��n��

¥∙ÏÐÑÒ ��∙ ÏÐÑÒ �� ]ÓÔÕÖ×ØÙØÙ

𝑎𝑡𝑧 = 0

tan 𝜃 =𝑆X2ℎ

𝑆ËÌÍd�∗ = 𝑆¿ = 2 ∙ ℎ¿ ∙ tan 𝜃

• Ballscrewmaximumacceleration

𝑆 𝑡 =𝑆¿2 sin𝜔𝑡

ÛghÉ¶ÉtÊfdÉ¿ghgÆÜg¿ffsf�gfÉ·g𝑎 𝑡 = −

𝑆¿2 𝜔

n sin𝜔𝑡

𝑆¿2 𝜔

n ≤ 9.81 → 𝑆¿ =𝑔2𝜋n 𝑇

n

• Maximumactuatorstroke

𝑆¿ = 0.5𝑚

Toplotthefirsttwoequations,weneedtosolvethedispersionrelationforeachperiodoftherange

𝜎n = 𝑔𝑘 tanh 𝑘ℎAsalreadyseeninchaptertwo,ifwehaveafixedperiodandaconstantwaterdepth,thereisonlyone𝑘thatsatisfiesthisequation;thisvalueisobtainedthroughaniterationprocedure,solvingthelastequationforeachperiodinwhichtheperiodrangeofinterestwasdivided.Fromthecomparisonbetweentheabovelimitsexpressedintermsofactuatorstrokes,weobtain,foreachperiod,themaximumactuatorstrokes𝑆¿·evthatwecanuse

67

Aswecanseefromthegraph,themaximumactuatorstrokes𝑆¿·evislimitedbythewavebreaklimituntil𝑇 = 1.23𝑠andthenbytheactuatorphysicallimit.Nowwebringbackthequantity𝑆¿·evatthestillwaterlevel,𝑧 = 0:

tan 𝜃 =𝑆¿·ev2ℎ¿

𝑆X·ev = 2 ℎ tan 𝜃

68

Thefollowinggraphshowsvariousstrokesat𝑧 = 0

Theseveralvaluesfor𝑆Xareusedintheflaptypetransferfunctiontoobtainthemaximumwaveheightforeachperiod;thefollowingwaveheightsvaluesareexpectedatacertaindistanceawayfromthewavemaker,minimumthreetimesthewaterdepth,inordertoletextinguishthelocaldisturbancesintroducedbytheflapmotion

69

Tocompletethepotentialitiesoverviewofourtank,wehavetoanalyzethewavelengthandtheceleritythatisthepropagationspeedofasinglewave.Theexpressionforthesetwoquantitiesisthealreadymentioneddispersionrelationshipopportunelyrearranged


𝐿 =𝑔2𝜋 𝑇

n tanh 𝑘ℎ

𝐶 =𝑔𝑘 tanh 𝑘ℎ

Wereportnowtheobtainedvaluesforthewavelengthsandcelerityforeachperiodoftherange

Summingup,giventheperiodrangeofinterest,weevaluatedthemaximumactuatorstrokeandthecorrespondingmaximumwaveheightexpectedforeachperiod;tohaveanoverviewofthewavegenerationcapabilitiesofthesystem,weplottedalsoseveralvaluesofstrokeat𝑧 = 0andthecorrespondingwaveheightvalues.Tocompletetheanalysis,wereportedforeachperiodoftherangetheexpectedwavelengthandcelerity.

70

4.2.2Velocities,forcesandpowerrequirementOnceunderstoodthepotentialitiesofourtank,wehavetodesigntheactuationsystemthatbetterapproachestheidealcapabilities.Todothis,wehavetodeterminethevelocityfield,theforcesactingontheflapandthenthepowerrequirement.Inthefirsttwofollowingsections,wewilldeepentheactionofthewateronthepaddleforthetwoabovementionedconfigurations;thisbecause,aswewillshowlater,thisforcegivesthelargercontributionintheequilibriumoftheflapandsoitisveryclosetotheforcenecessarytodrivethepaddleduringaworkcycle.Stillinthesetwofirstsections,itismadeapreliminaryanalysisofthepowerrequiredbyoursystem.Then,wewillfocusonthecasewithwateronbothsides:hereitwillbemadeamoredetailedanalysisoftheforcesandvelocitiesduringaworkcycle.Waterononeside

Theinstantaneouspressureactingonthepaddle,isgivenbythealreadymentionedexpression

𝑝s 𝑧, 𝑡 = 𝜌𝜎𝐴 cosh 𝑘 ℎ + 𝑧 cos 𝜎𝑡 + 𝜌𝜎 𝐶t cos 𝑘t ℎ + 𝑧¦

tyY

sin 𝜎𝑡 − 𝜌𝑔𝑧

with

𝐴 =2𝜎𝑆s





inwhichweset𝑙 = 0tostudytheflaptypewavemaker.

71

Asalreadyseen,thetotalistantaneousforceactingperunitwidthofpaddleisgivenbythefollowingintegral

𝐹 ¢̧ 𝑡 = 𝑝s 𝑧, 𝑡X

��𝑑𝑧

Thisyieldsto

𝐹 ¢̧ 𝑡 =𝜌𝜎𝐴 sinh 𝑘ℎ

𝑘 cos 𝜎𝑡 + 𝜌𝜎𝐶t𝑘tsin 𝑘tℎ

¦

tyY

sin 𝜎𝑡 +𝜌𝑔ℎn

2

whereitispossibletofindinorderthedynamicresistiveterm,thedynamicinertiatermandthehydrostaticterm.Thesubscript“𝑇”in𝐹 ¢̧ standsfortotalforce;fromnowonwewilluseonlyforceforconvenience.Toplottheaboveequation,weneedtosolve

𝜎 = −𝑔𝑘4 tan 𝑘4ℎ foreachperiodoftherangeandforthefixedwaterdepth.Inthisway,weobtainthe𝑘ttermsintheexpressionsfor𝐶tand𝐹 ¢̧.Infact,wearenowstudyingtheinteractionbetweenwaterandflap,thatisweareveryclosetothewavemaker;asseen,inthissituationwecannotneglectthealreadymentioneddisturbancesterms.Thelastequationhasaninfinitenumberofroots;itispossibletosolveitusingtheMatlabfunctionfzeroandstoppingthesolutionsresearchtothethirdone.Infact,goingoninfindingmoresolutionsyieldstonegligiblevaluesinthesummationin𝐹 ¢̧ expression:𝑘tisatthedenominatorinboththe𝐶tandintheinertiatermandthesolutions𝑘tareevenmorebiggerasshownbelow

Saidthis,itispossibletoseethegraphsmadeinthepresentwork.Thefollowinggraphshowsthemaximumandminimumforcerequiredtofacethewateractionforeachperiod;itwasobtainedrunningexpressionfor𝐹 ¢̧ periodbyperiodandsavingthemaximumandminimumvalues

72

Themaximumforceisreachedfor𝑇 = 1.23𝑠.Thegraphbelowreportstheforcecomponentstrendsrespecttothetimeforthisparticularperiodandforthemaximumpossiblestroke.Itisimportanttonotethatfor𝑡 = 0𝑠theflapisverticalanditismoving.

Asitispossibletosee,thehydrostatictermismuchbiggerthantheotherstwocomponents;theoscillatingtrendisgivenbydynamicresistivetermwhiletheinertiaoneisalmostnegligibleinthisperiod.

73

Tohaveanoverviewoftheforcetrendevolutionthroughtheperiodrangeofinterest,wereportthefollowinggraphsthatshowtheforcetrendfortheextremeperiodrangevaluesandfortheperiodinwhichthemaximumforceisreached

Fromthegraphswenotethattheforceisalwayspositiveforeachoneofthethreeparticularperiodsanalyzed;thisbecausethereiswaterononlyonesideandthenwehavetoapplyalwaysapositiveforce,respecttothechosenreferencesystem,tomovethepaddleortocontaintheactionofthehydrostaticcomponent.Furthermore,asguessed,forcevaluesslackoffreducingthestroke.Itisalsointerestingtocomparetheforcevaluewiththerelativepositionoftheconnectionpointbetweenactuatorandpaddleoveracycle

Observingthelasttwographsfor𝑇 = 0.5𝑠,itispossibletonotethatthemaximumforcedoesnotoccurattheperiodbeginning,thatiswhentheflapisvertical,butlater;thisislinkedtotheinertiaeffectsthatarenotnegligibleforsmallperiods,likeshownbytheseriesofthefollowingthreegraphs

74

Passingby𝑇 = 0.5𝑠to𝑇 = 0.9𝑠,itispossibletonotethatthedynamicinertiatermlosesimportancewhilethedynamicresistivetermbecomesrelevant.Inparticular,for𝑇 = 0.5𝑠wehavesmallerforcevaluesbutthetrendisgovernedbytheinertiaterm;onthecontrary,for𝑇 =0.9𝑠wehavebiggerforcevaluesandthetrendisgivenbytheresistiveterm.Thismeansthat,forverysmallperiods,theinertiatermisnotnegligiblebecauseitgivesthesinusoidaltrendtothetotalforcewhileitbecomesirrelevantforbiggerperiods.Weevaluatenowthewavemakerpowerrequirement;followingthefirstorderwavemakerproblem,frictionandmechanicallossesareneglected.Asalreadyseeninchapterthree,theinstantaneouspowerexpressionisobtainedcarryingoutthefollowingintegral

𝑃s 𝑡 = 𝑝s 𝑧, 𝑡X


thatyieldsto

𝑃s 𝑡 =𝜌𝜎n𝑆s𝐴2𝑘

sinh 𝑘ℎ +1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙

cos 𝜎𝑡 n +𝜌𝜎n𝑆s2

𝐶t𝑘t


sin 𝜎𝑡 cos 𝜎𝑡 +𝜌𝑔𝜎𝑆s2

ℎn

2−

ℎ4

3 ℎ + 𝑙cos 𝜎𝑡

¦

tyY

wheretheconsiderationsabovemadefortheforceabout𝐴,𝐶tand𝑘tarestillvalid;again,itispossibletofindinorderthedynamicresistivepower,thedynamicinertiapowerandthehydrostaticpower.Itisusefultoremembertheexpressionforthemeanpoweroveracyclethatarisesfromthefollowingintegral

𝑃s =1𝑇 𝑃s 𝑡

�¸ n

�¸ n𝑑𝑡

Itsresolutionyieldsto

75

𝑃s =𝜋𝜌𝑔𝑆sn



n

Thelastexpressionisusedtohaveanoverviewofoursystemneedsperiodbyperiod

Themaximummeanpowervalueoccursforthesameperiodinwhichthereisthemaximumforce.Thefollowinggraphletusunderstandwhicharetherealpowerrequirementforeachperiod,thatisvaluesnotaffectedbymediaoperation;thegraphwasobtainedevaluatingtheinstantaneouspowerexpressionforeveryperiodoftheanalyzedrangeandsavingthemaximumandtheminimumpowervaluesforeachperiod

76

Itispossibletonotethatthepowerreachesalsonegativevalues;thisbecausetheforcegivenbytheactuatortothepaddleactsalwaysalongthepositive𝑥directionwhilethepaddlefollowsasinusoidalmotion,thatisthevelocityisnotalwayspositive.Themaximumpowerrequirementisreachedstillfor𝑇 = 1.23𝑠andthefollowinggraphshowstheinstantaneouspowercomponenttrendforthisperiodandforthemaximumstroke

77

Waterofthesamedepthonbothsides

Ifinourproblemthereiswaterofthesamedepthonbothsidesoftheboard,theinstantaneouspressureactingonthepaddleisgivenbythefollowingexpression

𝑝ss 𝑧, 𝑡 = 2 𝜌𝜎𝐴 cosh 𝑘 ℎ + 𝑧 cos 𝜎𝑡 + 2 𝜌𝜎 𝐶t cos 𝑘t ℎ + 𝑧¦

tyY

sin 𝜎𝑡

withagain

𝐴 =2𝜎𝑆s





inwhichweset𝑙 = 0sincewearestudyingaflaptypewavemaker.Toevaluatetheforcetoapplytofacethewateraction,wehavetodevelopthefollowingintegral

𝐹 ¢̧¢ 𝑡 = 𝑝ss 𝑧, 𝑡X

��𝑑𝑧

𝐹 ¢̧¢ 𝑡 = 2𝜌𝜎𝐴 sinh 𝑘ℎ

𝑘 cos 𝜎𝑡 + 2 𝜌𝜎𝐶t𝑘tsin 𝑘tℎ

¦

tyY

sin 𝜎𝑡

Aswecansee,wehaveadoublingofthedynamicresistiveandinertiatermswhilethehydrostatictermdisappears.Whatsaidabouttheresolutionofequation

78

𝜎 = −𝑔𝑘4 tan 𝑘4ℎ forwaterononlyoneside,isstillvalidintheactualcase.Followingtheschemealreadyusedforwaterononesidecase,wepresentnowthegraphsaboutforcesandpower.Thefollowinggraphshowsthemaximumandminimumforceforeachperiod;itwasobtainedrunningtheexpressionfor𝐹 ¢̧¢ periodbyperiodandsavingthemaximumandminimumvalues

79

Themaximumforceisstillreachedfor𝑇 = 1.23𝑠butthevaluesarelower.Comparingtotheothercase,wecanalsostarttonotetheforcesymmetrictrendrespecttozero;thefollowinggraph,inwhicharereportedtheforcecomponentstrendsrespecttothetimefortheperiodinwhichthemaximumvalueisreached,clarifiesthisaspect

Thistrendiscausedbythepresenceofwaterofthesamedepthonbothsidesofthepaddle:infact,waterispushedbythepaddleinbothdirectionsandtodothiswehavetoapplyapositiveandnegativeforcethatgenerate,respectively,positiveandnegativevelocityoftheflap.Alsointhiscasetheinertiatermisnegligible,coherentlywithitslowinfluenceforperiodsbiggerthanabout1𝑠.Tohaveanoverviewoftheforcetrendevolutionthroughtheperiodrangeofinterest,wereportthefollowinggraphsthatshowtheforcetrendfortheextremeperiodrangevaluesandfortheperiodinwhichthemaximumforceisreached

80

Again,weproposeacomparisonbetweentheforcevalueandtherelativepositionoftheconnectionpointbetweenactuatorandpaddleoveracycle

Alsointhiscase,wecannotetheeffectoftheinertiatermonthetotalforceobservingthetrendsforthreedifferenttimeperiodsandforthehalfofthemaximumstrokeavailableforthisperiod

Sinceinthiscasethereisnotthehydrostaticcomponent,theinertiatermforsmallperiodismorerelevantrespecttotheothercase.Inparticular,forverysmallperiod,thetotalforcetrendisveryclosetotheinertiacomponent;increasingtheperiod,theresistivetermrisesupuntilitbecomespredominantforperiodlessthan1𝑠.Passingtothepoweranalysis,theinstantaneouspowerrequiredisgivenbythefollowingintegral

81

𝑃ss 𝑡 = 𝑝ss 𝑧, 𝑡X


thatdevelopedbecomes

𝑃ss 𝑡 = 2𝜌𝜎n𝑆s𝐴2𝑘

sinh 𝑘ℎ +1 − cosh 𝑘ℎ𝑘 ℎ + 𝑙

cos 𝜎𝑡 n + 2𝜌𝜎n𝑆s2

𝐶t𝑘t



tyY

wheretheconsiderationspreviouslymadeabout𝐴,𝐶tand𝑘tarestillvalid.Furthermore,likefortheforceexpression,itispossibletonotethedoublingofthedynamicresistiveandinertiatermsandthedisappearingofthehydrostaticterm.Theexpressionforthemeanpoweroveracycleisagaingivenbythefollowingintegral

𝑃ss =1𝑇 𝑃ss 𝑡

�¸ n

�¸ n𝑑𝑡

thatyieldsto

𝑃ss = 2𝜋𝜌𝑔𝑆sn



n

Tohaveanoverviewofoursystempowerrequirementperiodbyperiod,itispossibletousethelastexpression;thefollowinggraphreports𝑃ssrespecttoperiod

Alsointhiscase,theperiodthatonaveragerequiresmorepoweristhesameinwhichthemaximumforceoccurs.

82

Thenextgraphitisusefultounderstandwhataretherealpowerrequirementforeachperiod;likeinthecaseforwaterononeside,itwasobtainedevaluatingtheexpressionfortheinstantaneouspowerforeachperiodandsavingthemaximumandminimumvalues

Itispossibletonotethatthemaximumvaluesreachedinthiscasearelowerthantheoneforwaterononlyoneside;thisbecause,beinginthiscasewateronbothsides,thehydrostaticcomponentisbalanced.Furthermore,wecannotethatthepowerispracticallyalwayspositivesincetheforcetoapplytopushthewaterandthepaddlevelocityhavethesamedirection.Again,for𝑇 = 1.23𝑠wereachthemaximumpowerrequirementandthefollowinggraphoffersadetailofthepowercomponenttrends

83

Afterthisanalysisofthewaterforceandpowerrequirementaccordingtothefirstorderwavemakertheory,weintroducenowsomepracticalconsiderations.Asitispossibletoseefromthegraphsabove,forcesandpowervaluesaresmallerinthecasewithwateronbothside:thisisduetothebalancingofthehydrostaticterm.Theconfigurationwithwaterononlyonesiderequirestheuseofsealstoavoidthepassageofwaterinthedrybackoftheflap.Forpracticalreasons,wedecidedfortheoptionwithwateronbothsides.Thischoiceimpliesthenecessityofasortofbeachalsointheareabacktotheflaptoreducetheeffectsofwavereflection.Itispossibletobalancethehydrostatictermalsowithwaterononlyonesidedesigningadevicetoholditbut,againforsimplicity,wedecidedfortheotheroption.Then,fromnowon,wewillconsideronlythecasewithwaterofthesamedepthonbothsides.Sincewewanttohaveamoredetailedanalysisoftheforcesactingduringaworkcycletomakeamorereliablechoiceoftheactuationsystem,wedeepentheproblemconsideringtheotherforcesactingontheflap.First,theflaphastomoveaccordingtothefollowingequations𝜃 = 𝜃·ev sin 2𝜋 f

¸

𝜃 = 𝜃·ev

nÝ¸cos 2𝜋 f

¸

𝜃 = −𝜃·ev

¥Ý

¸ sin 2𝜋 f

¸

84

where𝜃·evishalfoftheanglesubtendedbytheparticularstroke.Thefollowingimagerepresentsthefreebodydiagramoftheflap

Takingtheclockwiseaspositive,therotationalequilibriumaroundthebottomhinge𝐶is𝐹Þ¸¸ cos 𝜃 − 𝛽 ℎß�eÜ +

𝑚·𝑔2

sin 𝜃 ℎß�eÜ − 𝐹Á àℎ·ev − 𝐼â𝜃 − 𝑚fsfℎâ𝜃ℎâ + 𝑚fsf𝑔 sin 𝜃 ℎâ − 𝐹Þ sin 𝜃 ℎâã = 0𝐹Þ¸¸ cos 𝜃 − 𝛽 ,isthecomponentoftheforceexertedbytheactuatorwhichgivesmomentrespectto𝐶;sinceweconsiderthelinkingpointbetweenflapandactuatoratthetopoftheflap,theforcecomponentismultipliedbytheflapheightℎß�eÜ.Theangle𝛽istheoneformedbythehorizontaldirectionandtheaxisofthelinearactuator;thevaluesoftheseangleareverysmallandfurtheronwewillneglectit.Thepositionofthesecondhingewaschosenconsideringthatthemaximumwaterforceoccurswhentheflapisvertical:inthisway,theactuatorcandirectlyfacethisforcewithoutusingcomponents.Sincewewereorientedtomounttheactuatorwithadoublehinge,weconsideredtheactionofhalfoftheweightofmotorandactuator;alsothisforceactsattheconnectionpointbetweenflapandactuatorand𝑚·wasestimatedtobe𝑚· ≅ 44𝑘𝑔alreadyoversizedforsafety.𝐹Á àistheforceexertedbythewater.Asmadein[],weconsideredthisforceappliedatthemaximumheightreachablebythewater:inthisway,theimportanceofthisforceintherotationalequilibriumisoverestimatedbutweareinsafety.

85

𝑚fsfisthemassoftheflapandofthesteelstructureusedtosupportit.Sincewewanttominimizetheinertia,wedesignedtheflapwithDivinycell,apolymerfoamcharacterizedbylowdensity(𝜌 ≅ 400 𝑘𝑔 𝑚4)andwidelyusedinmarineapplications.Knowingthedimensionsofthepaddle,2.2𝑚𝑥1.1𝑚𝑥0.03𝑚,itispossibletocalculatethemassoftheflap𝜌 = 400 𝑘𝑔 𝑚4𝑉ß�eÜ = 2.2 ∙ 1.1 ∙ 0.03 = 0.0726𝑚4𝑚ß�eÜ = 𝜌 ∙ 𝑉ß�eÜ = 29𝑘𝑔Asalreadysaid,theflapwillbesupportedbyastructureofstainlesssteelandthenweconsideralsothisweightestimatedtobearound5𝑘𝑔𝑚fsf = 𝑚ß�eÜ + 𝑚Æfgg� = 34𝑘𝑔𝐹Þisthebouncyforce.Infact,theflapwillbepartiallysubmergedbywaterandthenweconsiderthebouncyforceactingonit.Theminimumsubmergedvolumeconsideringthemaximumidealwaveheight,isestimatedtobe𝑉ÆÇÈ·ghÊg¯ = 0.0356𝑚4andthecenterofgravityofthissubmergedpartisℎæÊ = 0.27𝑚Now,assumingnegligibletheangle𝛽,wehave

𝐹Þ¸¸ =1

cos 𝜃 ℎß�eÜ𝐹Á àℎ·ev −

𝑚·𝑔2

sin 𝜃 ℎß�eÜ + 𝐼â𝜃 + 𝑚fsfℎâ𝜃ℎâ − 𝑚fsf𝑔 sin 𝜃 ℎâ + 𝐹Þ sin 𝜃 ℎâã

thatistheforceexertedbytheactuatoralongitsaxis.Thefollowinggraphshowsthetrendsof𝐹Þ¸¸ andalloftheothercomponentsinthelastequation

86

Itispossibletoseethatthemostimportantcomponentistheoneofthewaterforceandalltheothersarequitenegligible.Usingothersmaterialsfortheflap,likealuminum,wewouldprobablyhaveanincreaseoftheinertiaandthenanincreaseofthecorrespondingforces.Tochoosethelinearactuatorforoursystem,wehavetoevaluatealsothevelocityofthestemoftheactuator.Tofindthisexpression,wedefineanewmobilereferencesystemlocatedonthecylinderandinrotationwithit

Writingtheequationformotioncompositionforthesuperiorextremityoftheflap,wehave

87

𝑣 = 𝑣h + 𝑣f

𝑣 = 𝑣Þ¸¸ + 𝑣fwhere𝑣isthevelocityofthesuperiorextremityoftheflap,𝑣Þ¸¸ isthevelocityofthecylinderstemand𝑣fisthedragvelocity;fromthesketchaboveweobtain

𝑣Þ¸¸ = 𝑣cos 𝜃 − 𝛽 Alsointhisexpression,weneglecttheangle𝛽sinceitisassumedtobeverysmallandthenwehave

𝑣Þ¸¸ = 𝜃ℎß�eÜ cos 𝜃Summingup,weevaluatedtheforceexertedbythewaterontheflapduringitsmotionandthepowerrequirementaccordingtothefirstorderwavemakertheory.Practicalreasonsbroughtustotheconfigurationwithwateronbothsidesandthenthiscasewasstudiedevaluatingalltheforcesactingduringaworkcycle.Therotationalequilibriumoftheflapshowedthatinourcasethemostimportantforceistheonegivenbythewaterconfirmingtheapproachofthelinearwavemakertheory.Sincewewanttochoosealinearactuatortomoveourflap,itisimportanttoknowtheactuatorstemvelocity;thiswasobtainedusingmotioncomposition.4.2.3ActuatordimensioningWehavenowtomakeafirstdimensioningoftheactuationsystem.Severalsolutionswereconsideredbutquitesoonwemovedontheelectriccylinderforitssimplicityandflexibility.Then,weanalyzedcataloguesofseveralcompaniestofindagoodsolutionforourproblem;attheend,theelectrocylindersofPamocoappearstobeveryinterestingforourapplicationsincetheyintroducelowreductionsinwavegenerationandtheyhaveanIP65thatallowstheseactuatortoworkinproximityofsloshingwater.Aftersomeevaluations,wedecidedforaconfigurationwithanelectriccylinderwithdoublehingehasshowninthesketchabove

88

Then,westudiedtheworkingconditionsoftheactuatorthroughtheperiodrangeofinterest.Thefollowinggraphicsshowthemaximumandminimumcylinderstemvelocityandforceforeachperiod

Fromthegraphsaboveitresultsthatthemostintensecyclesintermsofvelocitiesandforcesaretheonesmadewiththemaximumstrokepossible.Forthisreason,weconcentrateonlyontheseworkingcycles.

89

ThefollowinggraphrepresentstheF-vcharacteristicthatwewouldliketorealizewhenweusethemaximumpossiblestrokeforeachperiodandinoverloadconditions.Fromwhatsaid,thisisthelimitworkingareaandalltheothersworkingareascorrespondingtosmallerstrokesarecontainedinsidethisone.Then,togenerateallouridealwavesabovementioned,weneedanactuatorthatcoverswithitscharacteristicthislimitworkingarea

Forcompleteness,wereportthefollowingthreegraphsthatshowthetrendofvelocity,overloadforceandnominalforceintheperiodcorrespondingtothemostintensecycle;thedifferencebetweenoverloadandnominalisasafetyfactorof1.3ontheforcesvalues,asmadein[6]

90

TheelectrocylindersofferedbyPamocoappeartobethemostsuitable,betweentheanalyzedproviders,forthisapplicationsincetheybettercovertheworkingareofinterestandhaveIP65.Fromafirstanalysisofthecatalogue,wefocusedonthefollowingtwoactuators:

• PNCE5020𝑥20Maximumaxialload𝐹·ev = 3900𝑁Maximumtravelspeed𝑣·ev = 1.1𝑚 𝑠

91

• PNCE6325𝑥25Maximumaxialload𝐹·ev = 7940𝑁Maximumtravelspeed𝑣·ev = 1.13𝑚 𝑠

OurF-vcharacteristicsandtheoneofthetwoactuatorsfoundinthecataloguearereportedinthefollowinggraphtohaveanoverview.

Ascanbeguessed,inthecataloguetheareseveralothervaluessuchasmaximumandminimumstroke,maximumaccelerationandsoon;allthesevaluesandlimitsarenotreportedherebecauseourapplicationlargelypassestheseverifiesortheyarenottakenintoaccountinthisfirstdimensioning.Boththerealvaluesof𝐹·evand𝑣·evdependontheparticularapplication;so,tochoosebetweenthetwoactuators,wefollowedtheprocedureprovidedinthecatalogue.Todothis,wefirsthavetodefinemorequantities.

92

Referringtothefollowingimage

Inourapplication,forboththeactuators,wehave

• 𝐸 = 0𝑚𝑚

• 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑠𝑡𝑟𝑜𝑘𝑒 = 600𝑚𝑚thatis𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑠𝑡𝑟𝑜𝑘𝑒 = 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝑠𝑡𝑟𝑜𝑘𝑒 + 2𝑥𝑆𝑎𝑓𝑒𝑡𝑦𝑠𝑡𝑟𝑜𝑘𝑒 = 500𝑚𝑚 + 2𝑥50𝑚𝑚 = 600𝑚𝑚

• Mountingcase:simple-simplemountForboththeactuators,weneedthefollowingmountingattachments

Theseaccessoriesintroducelimitsonthemaximumaxialload.

93

Forboththeactuatorswestartedtoevaluatetheuseofaspeedreducer,stillprovidedbyPamocoanddesignedfortheseparticularselectrocylinder,toreducethedimensionsofthemotor:

• MSDT2 𝑖 = 2 ,forPNCE5020𝑥20

• MSDT1 𝑖 = 2 ,forPNCE6325𝑥25

Nowitispossibletofollowthedimensioningprocedurethataimstoevaluatethemaximumaxialforcethattheactuatorcansupportinthisparticularapplication.PNCE5020𝑥20Enteringdata:

• 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑠𝑡𝑟𝑜𝑘𝑒 = 600𝑚𝑚

• 𝐸 = 0𝑚𝑚

• Travelspeed𝑣·ev = 1.1𝑚 𝑠,thatinourcaseisthemaximumallowedbytheelectrocylinder.

• UsingmotorsidedriveMSD-PNCE50-T2withgearratio𝑖 = 2

• Mounting:simple-simplemount

𝐹·ev functionofabsolutestrokeandmountingcase

𝐹·ev = 1300𝑁

94

𝐹·ev functionoftravelspeed,ballscrewleadandabsolutestroke

𝐹·ev = 3900𝑁𝐹·ev limitfrommaximumdrivetorque𝑀Ü,éÅÛonmotorsidedriveFromthecatalogues,wehavetheexpressionoftheloadtorque𝑀�se¯ =

uêëìêí∙�nXXX∙Ý∙�∙É

FromtechnicaldataofMSD-PNCE50-T2wehavethemaximumdrivetorque𝑀Ü,éÅÛ = 4.5𝑁𝑚Substitutingthisvaluein𝑀�se¯ andsolvingfor𝐹evÉe� wehave

𝐹evÉe� =𝑀�se¯ ∙ 2000 ∙ 𝜋 ∙ 𝜂 ∙ 𝑖

𝑙 = 2545𝑁𝐹·ev = 2545𝑁Maximumaxialloadformountingattachments𝐹·ev = 𝐹\îïð = 3900𝑁Finally,themaximumaxialloadisgivenbythelowestvalueof𝐹·ev

95

𝐹·ev = 1300𝑁 < 𝐹Þ¸¸,·evSurely,wehavetodiscardthePNCE5020𝑥20.PNCE6325𝑥25Asalreadydonefortheothercylinder,wehave:

• 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑠𝑡𝑟𝑜𝑘𝑒 = 600𝑚𝑚

• 𝐸 = 0𝑚𝑚

• Travelspeed𝑣·ev = 1.13𝑚 𝑠,thatinourcaseisthemaximumallowedbytheelectrocylinder.

• UsingmotorsidedriveMSD-PNCE63-T1withgearratio𝑖 = 2

• Mounting:simple-simplemount

𝐹·ev functionofabsolutestrokeandmountingcase

𝐹·ev = 3000𝑁

96

𝐹·ev functionoftravelspeed,ballscrewleadandabsolutestroke

𝐹·ev = 7940𝑁𝐹·ev limitfrommaximumdrivetorque𝑀Ü,éÅÛonmotorsidedrive𝑀�se¯ =


FromtechnicaldataofMSD-PNCE63-T1wehavethemaximumdrivetorque𝑀Ü,éÅÛ = 8.9𝑁𝑚Substitutingthisvaluein𝑀�se¯ andsolvingfor𝐹evÉe� wehave

𝐹evÉe� =𝑀�se¯ ∙ 2000 ∙ 𝜋 ∙ 𝜂 ∙ 𝑖

𝑙 = 4026𝑁𝐹·ev = 4026𝑁Maximumaxialloadformountingattachments𝐹·ev = 𝐹\îïð = 7940𝑁Finally,themaximumaxialloadisgivenbythelowestvalueof𝐹·ev

𝐹·ev = 3000𝑁 > 𝐹Þ¸¸,·ev

97

PNCE6325𝑥25couldbeusedinourapplication.AsalreadyseeninthefirstF-vcharacteristic,itisnotpossibletoreachourmaximumvelocitieswiththisactuator;thisyieldstoacutonthewavegenerationcapabilitiesthatwillbeanalyzedinthefollowingsection.Sincewehavetheactuator,itispossibletoevaluatethelengthofthecylinderinthetwoextremepositions.Consideringalltheaccessoriesandthespeedreducer,wehave𝐿fsfñî = 1066𝑚𝑚,totallengthatrest𝐿fsfàò¸ = 1666𝑚𝑚,maximumlengthUsingthisdataandworkingonthesketchgeometryitispossibletoevaluatetheangle𝛽;inparticular,itresults𝛽 = 6.08°,whenthecylinderisatrest𝛽 = 3.56°,whenthecylinderiscompletelyout.Thisshowsthat𝛽issmalland,whenitshouldinteractwith𝜃insideacosinefortheprojection,itdoesnotchangetoomuchthevalueofthecosineandofthecorrespondingphysicalquantity.Stillfromthecatalogue,itispossibletoevaluatethemassoftheactuator,accessoriesandspeedreducer;wehave𝑚 = 14.3𝑘𝑔Thelastvaluedoesnottakeintoaccounttheweightofthemotor.Summingup,onceanalyzedthevelocitiesandforcesoftheactuator,wefocusedonthecycleswithmaximumstroke.Foreachperiodoftherange,wereportedthecharacteristicF-vwhenthemaximumstrokeisused:theresultisthelargestworkingareaoftheactuatorintheperiodrangeofinterest.Thisinformationallowsustosearchactuators;wemovedontheelectrocylinderofPamocobecausetheycoverquitewelltheworkingareaandhasanIP65.Afterfollowingthedimensioningprocedureprovidedinthecatalogue,wechosetheelectriccylinderPNCE6325𝑥25withMSD-PNCE63-T1.Asshowninthefollowingchapter,thischoiceyieldsanywaytoareductionofwavegenerationcapabilities.Atlast,onceknowntheactuator,itisalsopossibletoevaluatethemassandthelengthsandthisallowsustoevaluatetheangle𝛽toverifythatisverysmall.

98

4.2.4ConsequencesoftheactuatorchoiceAsalreadysaidintheprevioussection,evenifthisactuatorrepresentsasuitablechoice,itintroducescutsinwavegenerationsinceitisnotpossibletoreachthevelocitiesrequiredtogeneratethedesiredwaves.Inparticular,thislimitonthevelocitiesyieldstoalimitonthestrokesandthenconsequencesontheforces;forthesereasons,itisnecessarytoevaluateagainthepossiblestrokesforeachperiodandallthequantitiesthatdependfromthem.Inparticular,weevaluatedagainthewaveheightsandtheforcesexertedbytheactuatortoquantifythereductionthatcouldbeinterestingforthemotorchoice.ThemaximumvelocityofactuatorPNCE63issetto1.13𝑚 𝑠;forsafety,thislimitwasreducedto1𝑚 𝑠.Thevelocityofthestemoftheactuatoris,neglectingtheangle𝛽𝑣ðï = 𝜃ℎß�eÜ cos 𝜃Theangularvelocityoftheflapisgivenby𝜃 = 𝜃·ev

nÝ¸cos 2𝜋 f

¸

where𝜃·evishalfoftheanglesubtendedbyageneralstrokeSubstitutingthelastequationintheexpressionfor𝑣ðï wehave𝑣ðï = 𝜃·ev

nÝ¸cos 2𝜋 f

¸ℎß�eÜ cos 𝜃

Forafixedperiodandafixedstroke,𝑣ðï ismaxwhenthevalueofthetwocosinesisonethatisfor𝑡 = 0𝑠being𝜃 = 𝜃·ev sin 2𝜋 f

¸

Then,foraparticularperiodandstroke,themaximumvelocityofthestemoftheactuatorisgivenby𝑣ðïéÞ� = 𝜃·ev

nÝ¸ℎß�eÜ

Thevelocityoftheactuatorwassetto1𝑚 𝑠;imposingthisvalueinthelastexpressionandevaluatingitforeveryperiodoftherange,weobtain,foreachperiod,theangle𝜃·evthatallowstherespectoftheconditiononthemaximumstemvelocityforthegivenperiod𝑇𝜃·ev =

¶óôõö÷∙¸nÝ∙�øíêù

Itispossiblenowtousethelastexpressiontointroduceanewlimitonthestrokes.Thegeneralexpressionforastrokeisgivenby

99

𝑆 = 2 ∙ ℎß�eÜ ∙ tan 𝜃

𝑆 = 2 ∙ ℎß�eÜ ∙ tan¶óôõö÷∙¸nÝ∙�øíêù

Inthisway,thelimitontheangleistransformedintoalimitonthestroke.Thefollowinggraphreportsthisnewlimittogetherwiththeothersalreadymentionedinthefirstsection

100

Asitispossibletosee,thenewlimitintervenesbetweenthelimitsonthewavebreakingandonthemaximumactuatorstrokedeterminingacutofourmaximumwaveheightgenerable.Thevalueofthemaximumaccelerationoftheballscrewisnow20𝑚 𝑠,accordingtotheinformationinthecatalogue.Thecorrespondingnewstrokesandwaveheightsaresummarizedinthisgraph

Asitispossibletosee,thenewmaximumwaveheightis𝐻éev = 0.265𝑚;respecttotheidealcase,inwhichthemaximumwaveheightwas0.323𝑚,thechoiceoftheactuatorcausedareductionofthe18%ofthemaximumwaveheight.Itisinterestingtoseethatthewavelengthandcelerityarenotaffectedbythisnewconstrain:infact,theybothdependonthedispersionrelationshipthatdescribeshowwavesdisperse.

101

Thenewlimitontheactuatorvelocityintroducesalimitonthestrokesandthishasconsequencesalsoontheforces.Then,itcouldbeinterestingtocalculateagaintheforcesinactionduringallthecyclesoftheperiodrangemadewiththemaximumpossiblestrokeforthatperiod.Theequilibriumistheonealreadyseenintheprevioussection.Theoretically,wehavetoupdateintheequationthevalueofthemaximumwaveheightreachablebut,respectto𝑧axis,thereisaverysmallchange;moreover,consideringthefirstvalue,weareinsafetyandsowedecidedtonotupdate.ThefollowinggraphisthenewF-vcharacteristicswhicharisesfromthisnewcalculation

Comparingwiththepreviouscharacteristic,wehavealostintheforcesvaluesandthevelocityislimitedto1𝑚 𝑠.Asbefore,wereportedalsothegeneralactuatorcharacteristicthatcouldseemtoomuchlargebutitdoesnottakeintoaccountthemountinglimitations.Summingup,thechoiceoftheactuatorintroducesalimitonthevelocitiesthatyieldstoalimitonthemaximumstrokereachableineachperiod.Then,weanalyzedtheconsequencesofthisnewconstrainonwavegenerationcapabilitiesandontheactuatorforcesobtaininganewF-vcharacteristicthattakesintoaccountthechosenactuator;thislastcharacteristicisthestartingpointforthemotorchoice.4.2.5MotorchoiceNowthatwehavetheactuatorandthenewF-vcharacteristic,wehavetofindasuitablemotortodriveit;todothis,weneedthetorqueandangularvelocityonthemotorshaft.Toevaluatethetorqueonthemotorshaftitispossibletousetheequationprovidedbythecatalogue𝑀�se¯ =


102

Fortheangularvelocityitispossibletousetherelationforthescrewtransmissionwithrectangularthread𝑣æ = 𝑣 tan𝛼𝑣æ = 𝜔¶Éfg

¯úìûntan 𝛼 ⇒ 𝜔¶Éfg =

¶ãüúìû ýÌÑþ

Toobtainthevelocityonthemotorshaftweintroducethegearratio𝑖

𝜔�se¯ = 𝑖 ¶ãüúìû ýÌÑþ

ThefollowinggraphrepresentsthecharacteristicC-nthatourmotorhastobeabletoprovidetodrivetheactuator

Forcompleteness,wereportalsothemotorangularvelocity,themotortorqueandpowerfortheperiodinwhichthereisthemostintensecyclebothfortheoverloadandnominalcondition.Thisperiod,afterthenewlimitontheactuatorvelocity,passesfrom𝑇 = 1.23𝑠to𝑇 = 1.55𝑠

104

4.2.5.1Analysiswithgearratio𝑖 = 1Themaximumvalueoftorqueimposesusachoiceofamotorwithatorquebiggerthen10𝑁𝑚.Searchingoncatalogues,weselectedthefollowingthreebrushlessmotor

• ESTUNEMG10• LSISXML-HE15A• SVTMA02-3.18-57

Allofthemappear,infirstinstances,tobesuitableforourapplication;thebestofthethreeitseemstobetheESTUNbecauseitscharacteristicisclosertoourmaximumworkingarea.HerewereportthecharacteristicC-nwithgearratio𝑖 = 1withtheotherthreecharacteristicsofthesuitablemotors

Theuseofaspeedreducer,asalreadysaidabove,givesusthepossibilitytouseamotorwithlesstorqueandmorevelocity.Thisstudywillbedoneinthenextparagraph.4.2.5.2Analysiswithgearratio𝑖 = 2Pamocoprovidesastandardspeedreducer,withgearratio𝑖 = 2,forourelectrocylinder;inparticular,wefocusedonMSD-PNCE63-T1.Thisallowsustohalvethetorqueandtodoubletheangularvelocityonthemotorshaft.ThefollowingisthenewcharacteristicC-nthatourmotorhastobeabletoprovidetodrivetheactuatorwhenweusetheabovementionedspeedreducer

105

Inwhichthecyanarearepresentsthenominalworkingconditionswhiletheblueonetheoverloadworkingcondition.Forcompleteness,wereportalsothemotorangularvelocity,themotortorqueandpowerfortheperiodinwhichthereisthemostintensecyclebothfortheoverloadandnominalcondition.Thisperiod,afterthenewlimitontheactuatorvelocity,passesfrom𝑇 = 1.23𝑠to𝑇 = 1.55𝑠

106

Inthiscase,wehavehalfoftheprevioustorqueandthedoubleoftheangularvelocity;accordingtothesevalues,weselectedthefollowingthreebrushlessmotors:

• LSISXML-SC08A• LSISXML-FC10A• LSISXML-FEP09A

Allofthemappear,infirstinstances,tobesuitableforourapplication;thebestofthethreeitseemstobegreenbecauseitscharacteristicisclosertoourmaximumworkingarea.

107

HerewereportthecharacteristicC-nwithgearratio𝑖 = 2withtheotherthreecharacteristicsofthesuitablemotors

Summingup,oncechosentheactuatorwehaveevaluatedthenewF-vcharacteristicthatisourinputdataforthemotorchoice.Infact,usingequationsfoundinthecatalogueandtheonesforthescrewtransmission,itispossibletogeneratethecorrespondingC-ncharacteristicforthemotor.Then,westudiedboththeconfigurationwithoutspeedreducerandwithgearratio𝑖 = 2obtainingforeachcasethecorrespondingC-ncharacteristicforthemotorandthedetailofthetrendofangularvelocity,overloadtorqueandnominaltorquefortheperiodinwhichthereisthemostintensecycle.Asusual,theuseofthespeedreducerhelpsusinthechoiceofthemotoranditisthesolutionchosen.

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Chapter5ConclusionsandfutureworksInthefollowingchapteritissummerizedtheworkdevelopedinthisthesisandthemainresultsobtainedarepointedout.Then,weintroducethefuturepossibleworksonthisargument.

5.1ConclusionsThegoalofthisthesisistheevaluationofthepotentialitiesoftheexistingtankintermsofwavegenerationtotestwaveenergyconverterdevicesandafirstdimensioningoftheactuationsystemthathastogeneratewaves.Thistwogoalsareachievedthroughthestudyofthelinearwaterwavestheoryandresearchingthenafirstactuationsystemsuitableforcoveringourworkingarea.Tounderstandtheproblem,itwasfundamentalthestudyofthelinearwaterwavestheorymadeonthebookWaterwavesmechanicsforengineersandscientists(DeanDalrymple-1984);forthewavemakertheory,themainreferencefollowedisPhysicalmodelsandlaboratorytechniquesincoastalengineering(StevenA.Hughes-1993).Asalreadysaid,thefirstthingdoneistheanalysisofthewavegenerationpotentialitiesoftheexistingtank;forthispart,ourreferenceis[5].Summingup,giventheperiodrangeofinterest,thatis𝑇 = 0.5𝑠 ÷ 2.5𝑠,weevaluatedthemaximumactuatorstrokeandthecorrespondingmaximumwaveheightexpectedforeachperiod;allthesevaluesareidealsincetheydonottakeintoaccountanylimitationintroducedbytheactuationsystem.Tocompletetheanalysis,wereportedforeachperiodoftherangetheexpectedwavelengthandcelerity.Tophysicallyrealizethesewaves,wehavetomakeafirstdimensioningofanactuationsystem.Todothiswestudiedvelocities,forcesandthepowerrequirementofthesystem;forthispart,ourreferenceis[6].Wefirstconcentrateontheforcesexertedbythewaterthatare,asshown,themostimportantandwealsodoneafirstanalysisofthepowerrequirement;thisfirstanalysiswascarriedonbothforwaterononesideandwateronbothsides.Then,practicalreasonsbroughtustotheconfigurationwithwateronbothsidesandthenthiscasewasstudiedindeepevaluatingalltheforcesactingduringaworkcycle.Therotationalequilibriumoftheflapshowedthatinourcasethemostimportantforceistheonegivenbythewaterconfirmingtheapproachofthelinearwavemakertheory.Sincewewanttochoosealinearactuatortomoveourflap,itisimportanttoknowtheactuatorstemvelocity;thiswasobtainedusingmotioncomposition.Forthefirstdimensioningoftheactuator,wequitesoonmovedontheelectrocylinderforthesimplicityandflexibility.Then,weanalyzedcataloguesofseveralcompaniestofindagoodsolutiontoourproblem;attheend,theelectrocylindersofPamocoappearstobeveryinterestingforourapplicationsincetheyintroducelowreductionsinwavegenerationandtheyhaveandIP65thatallowstheseactuatorstoworkinproximityofsloshingwater.Evenifthisactuatorrepresentsasuitablechoice,itintroducescutsinwavegenerationsinceitisnotpossibletoreachthevelocitiesrequiredtogeneratethedesiredwaves.Inparticular,thislimitonthevelocitiesyieldstoalimitonthestrokesandthentoconsequencesontheforces;forthesereasons,itis

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necessarytoevaluateagainthepossiblestrokesforeachperiodandallthequantitiesthatdependfromthem.Inparticular,weevaluatedagaintheforcesexertedbytheactuatortoquantifythereductionthatcouldbeinterestingforthemotorchoice;fromthiscalculationitresultsthenewF-vcharacteristicoftheactuator.Afterthechoiceoftheparticularelectrocylinder,weanalyzedthemechanicaltransmissionandseveralbrushlessmotorsthatcouldbesuitableforourapplication.ThenewF-vcharacteristicisourinputdataforthemotorchoice.Infact,usingequationsfoundinthecatalogueandtheonesforthescrewtransmission,itispossibletogeneratethecorrespondingC-ncharacteristicforthemotor.Then,westudiedboththeconfigurationwithoutspeedreducerandwithgearratio𝑖 = 2obtainingforeachcasethecorrespondingC-ncharacteristicforthemotorandthedetailofthetrendofangularvelocity,overloadtorqueandnominaltorquefortheperiodinwhichthereisthemostintensecycle.Asusual,theuseofthespeedreducerhelpsusinthechoiceofthemotoranditisfinallythesolutionchosen.Inconclusion

• thewavegenerationcapabilitiesoftheexistingtank,pointedoutinthiswork,areconsideredinterestingfortestinghulls,marinestructureandwaveenergyconverterdevicesalsowiththereductionintroducedbythechosenactuator;

• thefirstdimensioningoftheactuationsystemrepresentsasuitablestartingpointforamoredetailedanalysisthatcouldbedonewiththetechnicalofficeofPamocoorinafuturework.Clearly,itisalsopossibletoenlargetheresearchtomoreproviders.

5.2FutureworksThisworkcouldbethebasisforseveralotherthesisindifferentworkingarea;inthefollowingsection,themainideasaresummarized.First,sincethereisastronginterestinequippingourtankwithawavemaker,wehavetostudyasuitablestructuretoholdthisdevice.Theideaistohaveastructureindependentfromthetankwhichcanbeeasymoved;forthisreason,wesurelyneedabasesufficientheavytoremainfixedtothebottomofthetankduringtheworkcyclesbutalsonottoomuchheavytonotruinthebottomofthetank.Anironsteelstructurewillbefixedonthebasisandwillholdtheactuator.Moreover,weneedtothinktoasuitableconfigurationforthestructureinironsteelthatwillholdtheflapinDivinycell;thenwehavetostudyhowtorealizethelinkbetweenthisstructureandthebaseandalsostudyingindeeptheresistanceoftheflaptotheappliedloadtonothavefailure.Sincetherearecyclingloads,astudytoavoidfatiguefailuresinallthestructurehastobemade.Tousethisdevice,itisalsonecessarytochoosethebrushlessmotorfortheactuationsystem.Thisisstrictlylinkedtothetypeofcontrolwewouldliketorealize.Afterthis,itisnecessarytoimplementthecontrolarchitectureofthesystemtorealizethemotiondescribedinthecurrentwork:itcouldbestudiedthepositioncontrolandtheforcefeedbackcontrol,analyzingadvantagesanddisadvantagesofthem,andthenphysicallyimplementtheminoursystem.Asshowninthepreviouschapters,thereareseveralothertypesofwavemaker.Forourtestingneeds,wefocusedontheflaptypewavemakerhingedtothebottombutitispossibletorepeat

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thisanalysisforthevariabledraftflaptypewavemakerandstudyingadvantagesanddisadvantagesofthem.Finally,itisalsopossibletousethisworkasthebasisforthecreationofanumericalwavetankthroughCFDcodessuchasStarCCM+toincreasethetestingpotentialitiesandthepossibilitytodeepenthedesignofthetesteddevice.

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Bibliography[1]DeanR.,DalrympleR.,WaterWaveMechanicsforEngineersandScientists,Singapore,PrenticeHall,1984.[2]HughesS.A.,Physicalmodelsandlaboratorytechniquesincoastalengineering,Singapore,WorldScientificPublishing,1993.[3]BieselF.,SuquetF.etungrouped’ingénieursauLaboratoireDauphinoisd’Hydraulique(Neyrpic,Grenoble),Lesappareilsgénérateursdehouleenlaboratoire,inLaHouilleBlanche1951.[4]PilchM.,BieselF.,SuquetF.andagroupofengineersattheLaboratoireDauphinoisd’Hydraulique(Neyrpic),Laboratorywavegeneratingapparatus,UniversityofMinnesota,ProjectReportNo.39,October1953.[5]DeBarrosPassosR.,FernandesA.C.,Designofawave-makerfacilitythroughlinearwavetheory,inProceedingsoftheXXXVIIberianLatin-AmericanCongressonComputationalMethodsinEngineeringNeyAugustoDumont(Editor),ABMEC,RiodeJaneiro,RJ,Brazil,November22-25,2015.[6]LiuY.,CavalierG.,PastorJ.,VieraR.J.,GuilloryC.,JudiceK.,GuiberteauK.,KozmanT.A., DesignandconstructionofawavegenerationsystemtomodeloceanconditionsintheGulfofMexico,inInternationalJournalofEnergy&Technology4(31)(2012)1–7[7]KrvavicaN.,RužićI.,OžanićN,Newapproachtoflap-typewavemakerequationwithwavebreakinglimit, inCoastalEngineeringJournal.[8]GuillouzouicB.,MarinetD2.12CollationofWaveSimulationMethods.[9]S.A.Sannasiraj,ModuleII-LaboratoryWaveGeneration.[10]A.Mancini,Originiesviluppodellevaschenavali,inRivistaMarittimaGiugno2000[11]ChappellE.R.,TheoryandDesignofawavegeneratorforashortwaveflume,UniversityofBritishColumbia,April1969.[12]H.E.Saunders,ManeuveringandSeakeeping(MASK)FacilityNewDirectionalWavemaker.[13]A.Mancini,Originiesviluppodellevaschenavali,inRivistaMarittimaGiugno2000.

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WebsitesITTC:https://ittc.infoMaRINET2:http://www.marinet2.euEdinburghDesign:http://www4.edesign.co.ukImperialWarMuseum:https://www.iwm.org.ukPamoco:https://www.pamoco.itWikipedia:https://it.wikipedia.org

POLITECNICO DI TORINO Design of a wavemaker for the ...

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