ContentslistsavailableatScienceDirect
Applied
Ocean
Research
journalhomepage:www.elsevier.com/locate/apor
Experimental
and
numerical
hydrodynamic
analysis
of
a
stepped
planing
hull
Agostino
De
Marco,
Simone
Mancini,
Salvatore
Miranda,
Raffaele
Scognamiglio,
Luigi
Vitiello
∗DepartmentofIndustrialEngineering,UniversityofNapoliFedericoII,Italy
a
r
t
i
c
l
e
i
n
f
o
Articlehistory: Received16July2016
Receivedinrevisedform2January2017 Accepted13February2017
Keywords: Shiphydrodynamics Steppedhull Towingtanktest CFDsimulation Overset/chimeragrid Morphinggrid
a
b
s
t
r
a
c
t
Thisworkaddressestheexperimentalandnumericalstudyofasteppedplaninghullandtherelatedfluid dynamicsphenomenatypicallyoccurringinthesteppedhullintheunwettedaftbodyareabehindthe step.Inthelastfewyears,theinterestinhigh-speedplaningcrafts,withlowweight-to-powerratios, hasbeenincreasingsignificantly,and,insuchcontext,navalarchitectshavebeenorientingtowardthe steppedhullsolution.Steppedplaninghullsensuregooddynamicstabilityandseakeepingqualities athighspeeds.Thisismainlyduetothereductionofthewettedarea,whichiscausedbytheflow separationoccurringatthestep.Thispaperpresentstheexperimentalresultsoftowingtanktestsincalm wateronasingle-stephullmodel,whichisthefirstmodelofanewsystematicseries.Thesameflow conditionsareanalyzedviaReynoldsAveragedNavier-Stokes(RANS)andLargeEddySimulations(LES), withdifferentmovingmeshtechniques(overset/chimeraandmorphinggrid),performedatdifferent modelspeeds.Thenumericalresultsareinaccordancewithexperimentaldata,andoverset/chimeragrid isfoundtobethebestapproachbetweentheanalyzedones.Theflowpatternsobtainednumerically throughLESonarefinedgridappearsimilartotheonesobservedintowingtankinvestigationsthrough photographicacquisitions.Theseflowpatternsaredominatedbyarathercomplex3Darrangementof vorticesoriginatingfromairspillageatbothsidesofthestep.Theunderstandingofthesephenomenais importantfortheeffectivenessofsteppedhulldesigns.
©2017ElsevierLtd.Allrightsreserved.
1. Introduction
Inthelastfewyears,thedevelopmentoflightweightengines and propulsionsystems, alongwiththe developmentoflighter boatsbuiltbyshipyardswithnewtechnologyandmaterials,has forceddesignerstopayincreasingattentiontohulldesign. Out-boardenginesinparticulararecharacterizednowadaysbyavery lowweighttopowerratioandhighreliability.Thesefeaturesmake them suitable for several installation types, including military, commercial,pleasure,and racing. Thenewcompositematerials allowaboatweightreductionof30%withrespecttoatraditional hand-madelayup.Inthisscenarioinrecentyears,thehigh-speed planingcraftformilitary,commercial,andpleasureuse,withavery lowweighttopowerratiohasspreadevenfurther.
Thereductionofweighttopowerratioinvolvesanincreaseof maximumspeed;asaconsequence,navalarchitectsareoriented
∗ Correspondingauthor.
E-mailaddress:luigi.vitiello@unina.it(L.Vitiello).
evenmoreinsteppedhulldesigntoreducetheresistanceathigh speedand ensuringgooddynamicstabilityand seakeeping.The stepisasharpdiscontinuitylocatedinthebottomsurfaceofthe hull;itistransversalandusuallyV-shaped,withthevertexfacing aftward,ontheoutboardsidesofthehullthestepterminateswith largeapertures(alsonamed‘inlets’)forincomingair.
Aspointedoutby[1],thesteppedhullischaracterizedbyalow hydrodynamicdrag-to-liftratioathighspeedsandbyawettedarea reductionduetoaflowseparation,whichoccursatsteplocation andthenreattachesattheaftbody[2].Moreover,steppedplaning hullshaveasmallvariabilityoftrimangleandimprovecontrolof thelongitudinalattitudebecausetheyaresailedalwaysonan+1 wettedtriangle,wherenrepresentsthenumberofsteps.Infact, forasinglesteppedhull,anadditionalaft-liftiscreated,duetothe presenceofthereattachmentline(stagnationline)intheaft-body. Thisforcekeepstherunningtrimofthevesselalmostconstantwith Froudenumber(athighspeed).Thisfeatureisbeneficialbecause avoidstheporpoisinginstability,whichoccursinsteadtostepless planinghulls.
http://dx.doi.org/10.1016/j.apor.2017.02.004
K Constantvalue L Waterlinelength(m) LOA Lengthoverall(m) N Numberofcontrolvertices Sn Numericalsimulationresult pk Observedorderofaccuracy Pr Precisionuncertainty
rij Magnitudeofdistancebetweentwovertices rk Refinementratio
Rk Convergenceratio RTM Totalmodelresistance(N) S Wettedsurface(m2) SDevj Standarddeviationofjthrun U Uncertainty
Ur Totaluncertainty
Uk k-inputparameteruncertainty UI Iterativeuncertainty
UG Griduncertainty UTS Timestepuncertainty
USN Numericalsimulationuncertainty UV Validationuncertainty
UD Experimentaldatauncertainty V Hullspeed(m/s) Z Sinkage(m) Displacementvolume(m3) Greeksymbols A Constantvalue Displacementweight(N) E Solutionchange P Density(kg/m3) Expansioncoefficient t Timestep(s)
T Dynamictrimangle(deg) Acronyms
AIAA AmericanInstituteofAeronauticsandAstronautics AMG Algebraicmultigrid
ASME AmericanSocietyofMechanicalEngineer CF CorrectionFactor
CFD Computationalfluiddynamics CFL CourantFriedrichsLewynumber CNC Computernumericalcontrol DAQ Dataacquisitiondevice DFBI Dynamicfluidbodyinteraction DOF Degreeoffreedom
DT Downthrust
EFD Experimentalfluiddynamics FRP Fiberreinforcedplastic GCI Gridconvergenceindex
HRIC Highresolutioninterfacecapturingscheme ITTC InternationalTowingTankConference
SIMPLE Semiimplicitmethodpressurelinkedequations UA Uncertaintyanalysis
VOF Volumeoffluid
V&V Verificationandvalidation
Anotherwaytoobtainahydrodynamicresistancereductionis theapplicationoftheartificialbottomcavitieswithsideskegs.The stateoftheartoftheairlubricationtechnologiestogetherwith researchactivitiesinRussiauntil2010canbefoundinSverchkov [3].
Nowadaystherearethreeoptionsforthehydrodynamic anal-ysisofsteppedhulls:experimentaltesting,empiricalestimation methods,andnumericalsimulations.
ExperimentalFluidDynamics(EFD)tests,thatistowingtank tests,areveryexpensiveandtimeconsuming.Moreover,theonly steppedhullsystematicseriesexperimentalresultsavailableare thoseperformedattheUniversityofSouthampton[4].
Two empirical hydrodynamic prediction methods are those publishedin[1]and [5].The firstmethodexperimentally stud-iedthelongitudinalsurfacewakeprofilesaftofprismatichulls; thesecondmethodcombinestheequationsof[1]withthe equa-tionsofSavitsky’smethodforconventionalplaninghullsforpower predictionofasteppedhull.
Numericalmethods,suchasthosebasedonComputationalFluid Dynamics(CFD)simulations,canbeusednowadaystocalculatethe hydrodynamicperformanceofasteppedhullwithgoodaccuracy. Inthelastfewyears,severalstudiesinvestigatedthisresearchfield. GarlandandMaki[6]conductedanumericalstudyona2Dstepped planingsurface.Theirresultsshowedthatthelift-to-frictional-drag ratiovariesverylittlewithrespecttothesteplocation.Makasyeyev [7]developedasolutionmethodforthe2Dmathematical prob-lemofplaningofthesteppedaircavityhulls.Matveev[8]applied hydrodynamicdiscretesourcesfor2Dmodelingofstepped plan-ingsurfaces,calculatingthewatersurfacedeformations,wetted hulllengths,andpressuredistributionatgivenhullattitudeand Froudenumber(Fr).Matveev,inanotherstudy[9],presentedthe steadyhydrodynamicmodelingofsemi-planinghullswith pres-surizedandopenaircavities.Thismethodisbasedonalinearized potential-flowtheoryforsurfaceflows.BrizzolaraandFederici[10] developedanintegratedsemi-theoretical/numerical(CFD)method forthedesignofV-shapedsteppedplaninghullsthatpresented aconsiderableresistancereductionswithrespecttoconventional hullforms.Lotfi[11]usedanunsteadyRANSsolver(ANSYS-CFX) basedonaVolumeofFluid(VoF)approachforexaminingthe char-acteristicsandperformanceofaplaninghullhavingonetransverse step.SimilarresearchwasconductedbyBakhtiari[12].Moreover, anextendedoverviewofthestateofartofthesimulationsinthe airlayerdragreductionisreportedinSternetal.[13].
Itisclear,nowadays,thatCFDisbecomingafundamental sup-portforhydrodynamicinvestigationsinordertoperformdetailed analysisandtoreducethenumberofmoreexpensivetowingtank
tests.EFDhowever,arealwaysnecessaryalongsidevalidationof numericalresults.
Inthiswork,anintegratedapproachbetweenEFDandCFDtests hasbeenappliedwiththeaimtoinvestigatethequitecomplex hydrodynamicfieldgeneratedbythesteppedhull.All experimen-talandnumericaltestswereperformedforasteppedhullincalm water.Severalsimulationswerecarriedoutwithdifferentmeshing techniquesinordertoachievegoodnumericalconvergenceandto capturethevorticalstructuresobservedintowingtanktests.
Thephysicsofthehydrodynamicfieldgeneratedbythestepped hullarequitecomplex,morethanplaningcraft.Forthisreason, par-ticularattentionhasbeenpaidintheunsteadyRANSsimulationsof theresistancetests.Hence,morethanonemeshapproachhasbeen usedforthebodymotionsimulation,suchastheoverset/chimera gridandmorphingmeshtechnique[14].Infact,planing/stepped hullperformance isvery sensitivetohull position(sinkageand trim)[15] asopposedtothedisplacementhull.For thisreason, accuratesimulationsofthehullmotionarenecessary.Therefore, non-conventionalapproaches (i.e.: oversetmesh and morphing grid)forthesimulationofthehullmotionarerequired,asindicated in[16].
Overset/chimera meshing allows bodies to move freely throughoutthecomputationaldomainwithoutbeingartificially constrainedbythemesh.Twodifferentmeshregionswereused, i.e.,backgroundmeshandbodyfittedmesharoundthegeometry ofinterest(overlappingmeshes).Themorphingmeshtechnique allowsforcomplicatedandarbitraryrelativemotionbymorphing thenodesofthemesh.Typically,themeshmorphingcanbeused tomodelscenarioswherecomponentsdeformandchangeshape, (seeRefs.[17]and[18]asexamplesofapplicationinthemarine field).However,thistechniquecanalsobeemployedfortheRigid BodyMotion(RBM)case.
Furthermore,thereisalackofasatisfactoryinterpretationof thecomplicatedflowphenomenaintheseparatedregionbehind thestep.Thispaperpresentsadetailedstudyofthevortex struc-turesintheunwettedaftbodyareaandtheirdevelopmentintothe downstreamwaterflowbymeansofphotographicacquisitionsand LargeEddySimulations(LES)onarefinedgrid.
Theremainingpartof thisworkisorganizedasfollows:the experimentaltestsarepresentedinSection2.Thenumericalmodel ispresentedanddiscussedinSection3.Inthesamesection, numer-icalsimulationscarriedoutbyusingboththeoverset/chimeragrid andthemorphingmeshapproachesarealsopresented,including theresultsofa verificationandvalidationstudy.Thenumerical resultsarecomparedwiththeexperimentalonesanddiscussedin Section4.InAppendixA,AppendixB,AppendixC arereported, respectively, the laboratory instrumentation and measurement techniques,experimental,andnumericaluncertaintyanalysis.
2. Experimentaltests
2.1. Descriptionofthemodel
Themodelusedinthisstudyrepresentsanexampleofamodern high-speedhullforRigidInflatableBoats(RIB).Thishullcanbea representativehullfortypicalpleasureormilitaryhigh-speedcraft. ThismodelisthehullnumberC03,oneoftheeightmodelsofa systematicseriesnowunpublished.ThebodyplanoftheC03hull isreportedinFig.1.
TheparenthullforthisresearchisaRIBbuiltbyMVMarineS.r.l., typeMito31poweredwithtwooutboardengines.
Themodelisahardchinehullwithonetransversestep,located inthesamelongitudinalpositionofcenterofgravitywithaforward Vshape(Fig.1).Thisshapeofthestepisdifferenttothetransverse bottomstepandthearrow-like shapeasreportedinSverchkov
Table1
Modeldetails.
Description
Lengthoverall:LOA[m] 0.935
Breadthmax:BMAX[m] 0.335
Deadriseangleattransom[◦][◦] 23
Stepheight[mm] 6
Displacement[N] 30.705
LCG[%L] 33
Modelscale 1:10
[3].Thehullconsideredinthisstudyisatraditionalsteppedhull, withoutartificialbottomcavity,withoutartificiallyinflatedairin thecavity,withoutskegsputonsideofthecavitynearthechine. Themodelhasthesamemaindimensions(keelline,chineline, deadriseangle,displacement,LCG,stepshape,stepangle)ofRIB Mito31,with1:10scale.
DetailsofthehullmodelarereportedinTable1.
Themodelfortowingtanktestswasdesignedin3DCADand builtwithcompositematerials.Themodelhasatransparent bot-tombuiltonlywithisophtalicresintoprovideafullviewofthe waterflowunderthehull.ThesideofthemodelwasbuiltinFiber ReinforcedPlastic(FRP),andthemodelsurfacewasbuiltwith high-glossneopentilicgelcoattransparent.
Themodelwasmanufacturedinhand-madelayupthrougha mold,thatwasdesignedin3DCAD/CAMandwasbuiltinFRP.
Inaccordance to[19],themodelhulltolerancesfor breadth, drought,andlengthare±0.5mm.Themanufacturingtolerancefor lengthislessthan0.05%,andspecialattentionwaspaidintothe shapingofchinesandstep.
2.2. Laboratoryinstrumentationandmeasurements
Thecalmwaterexperimentswereconductedinthetowingtank oftheUniversityofNapoli,FedericoII(Italy).Themaindimensions ofthebasinare:length137.5m,width9m,depth4.25m.
The details on the laboratory instrumentation employed in theexperimentaltestsand onthemeasurement techniquesare reportedinAppendixA.
ApeculiarfeatureoftheproposedEFDtestsarerepresentedby thewettedsurfacemeasurements.Thishavebeenpossiblethrough avideocameraplacedontowingcarriage,inperpendicularposition withrespecttothemodel’scenterofgravity.Thevideocamerawith 50mmlensrecordedeachtestfromthestarttotheend.Insucha way,ithasbeenpossibletoreconstructthedynamicofthevortex phenomenanearthestep[20].
2.3. Testprocedure
Themodelwastestedincalmwater,inaccordanceto[21].The minimumtime betweentwo consecutiverunswas10min,and eachrunwasinitializedwiththerecordingofzerolevelsforeach transducer.Theresistancetestswereconductedforadisplacement conditionof30.705Natzerotrimandateightdifferentspeeds: 1.290,2.357,3.131,4.631,5.368,6.340,7.301,and8.050m/s,(see AppendixB,TableA5).
Thequantitiesmeasuredarespeed,resistance,sinkage,andtrim angle.Digitalphotographsandvideosforeachrunwereacquiredto determinethedynamicwettedsurfaceandthevorticalflowunder thehullnearthestep.
Theresistancedynamometerhad beenplaced onthetowing carriageandconnectedtothemodelthroughawiretomeasure theresistance.
Thecalmwaterresistanceexperimentswereconductedwith “down-thrust”(DT)methodologyproposedin[22]withatowed pointlocatedinthehullbow(seealsoFig.2).Thechoiceofthe
Fig.1.C03modelbodyplan(transversalsectionevery0.100m)andprofile(buttocklineevery0.025m).
Fig.2.Towingtanktestwithdown-thrustmethodology.
DTmeasurementsolutionisduetothehighsensitivityofthehull modeltotheexternalappliedforces,i.e.,instrumentationweight. TheDTsolutionreleasesthetestedmodelfromtheinstrumentation weightandpromoteshigheraccuracyinmeasurementsof resis-tance,sinkage,andtrim.Thissolutionhasproventoreproducethe realsystemofforcesexertedbytheoutboardengines.
2.4. EFDresultsanduncertaintyanalysis
The results of calm water resistance tests are presented in Figs.3–6,andTable2.Allvaluesareplotted againstvolumetric Froudenumber(Fr∇).Inparticular,thenondimensionaltotal resis-tance(RTM/)isshown inFig.3;thedynamic trimangle()is shown in Fig.4; thenondimensional dynamic sinkage (Z/
∇
1/3)Fig.3. NondimensionaltotalresistancecomparisonbetweenEFDandCFDsimulationswithuncertaintybars.
Fig.4. TrimcomparisonbetweenEFDandCFDsimulationswithuncertaintybars.
Fig.6.NondimensionaldynamicwettedsurfacecomparisonbetweenEFDandCFDsimulationswithuncertaintybars.
Table2
Experimentalandsimulationresults.
Fr∇ RTM Z ∇1
⁄
3 S ∇2⁄
3 Exp CFD Mor-phing CFD Overset Exp CFD Mor-phing CFD Overset Exp CFD Mor-phing CFD Overset Exp CFD Mor-phing CFD Overset 1.984 0.183 0.164 0.173 3.55 3.53 3.87 0.011 0.051 0.056 6.631 6.110 7.171 3.898 0.263 0.307 0.268 3.27 2.94 3.31 0.086 0.161 0.167 3.881 5.885 5.092 5.337 0.418 0.484 0.379 2.69 2.13 2.70 0.149 0.186 0.181 3.316 5.358 4.349 6.777 0.570 0.642 0.540 2.58 2.07 2.66 0.184 0.213 0.210 2.845 4.601 3.873isshowninFig.5; thenondimensionaldynamicwetted surface (S/
∇
2/3)isshowninFig.6.These resultsare typical of a steppedhull. In particular, as expected,thetrimanglesarequitelowforhigherFroudenumbers andchangeslightlywithFr∇.
Asignificantoutcomeof thepresent experimental investiga-tionisgivenbytheobservationofsomeclearvorticalpatternsthat developintotheunwettedaftbodyareabehindthestep,andpartly continuedownstreaminthewaterwake.Thevortexinthe unwet-tedaftbodyareaappearsfortowingspeedsgreaterthan2.36m/s (Fr∇>1.97).Thesephenomenaarevisiblethankstothe transpar-entbottomofthehullmodel,whichwasexpresslydesigned.Some picturesoftheflowpatternsareshowninFig.14(upperpart),and Fig.22(upperpart).Additionally,avideorecordingofatowingtank runhasbeenreleasedinthepublicdomain[20];itdocumentsthe timedevelopmentofwellvisiblevortexstructuresunderthehull, asseenfromanobserverlookingfromthetopandthroughthe transparentbottom.
AnUncertaintyAnalysis(UA)oftheEFDresultshasbeenalso performedaccordingtoITTC[23],asreportedinAppendixB.
3. Numericaltests
Inordertoconfirmtheexperimentalresultspresentedabove, includingtheexistenceoftheobservedvortices,acampaignofCFD simulationshasbeendevised.Moreover,adetailed studyofthe flowfieldhasbeenmadetoreconstructthevortexpatternsboth intotheunwettedaftbodyareaandinwaterdownstreamthestep. Inthissection,thedetailsofthenumericalsetupusedinallthe simulationsarereported.Inparticular,thedetailedanalysisof com-monanddifferentsetupusedinthetwodifferentmeshapproaches appliedinthisstudyarepresentedanddiscussed.
Unsteady RANSsimulations were performed on four model speeds: 2.357, 4.631, 6.340, and 8.050m/s, corresponding to Fr∇=1.984,3.898,5.337,and6.777.
3.1. Physicalmodelingandcoordinatesystem
TheunsteadyReynoldsAveragedNavierStokes(RANS) equa-tionsaresolvedusinganimplicitanditerativesolverinorderto findthe field of all hydrodynamic unknownquantities in each time step. The velocity–pressure coupling and overall solution procedurewerebased ona semi-implicitmethodfor pressure-linkedequations(SIMPLE)toconjugatepressureandvelocityfields. The discretizedalgebraic equations were solved usinga point-wiseGauss–Seideliterativealgorithmandanalgebraicmultigrid (AMG)methodwasemployedtoacceleratesolutionconvergence. Furthermore,asegregatedflowsolverapproachwasusedforall simulationsandtheturbulenceclosureoftheRANSequationswas ensuredbythek-shearstresstransport(SST)turbulencemodel. TheRANSsolverisbasedonafinite-volumemethodtodiscretize thephysicaldomain.Alltheresultspresentedinthisstudywere achievedbysolvingthefullunsteadyflowequationsandmarching thenumericalsimulationsintimewithapseudo-compressibility approach.Theequationsforthemultiphasefluidarejoinedwith onemoretransportequationfortheVoFmodel.Thisequationhelps toensurethecorrectshapeofthefreesurfacebetweenwaterand air.In thesesimulations,themodified high-resolution interface capturingscheme(HRIC)wasusedasadvectionschemes.The mod-ifiedconfigurationoftheHRICschemeremovesthedependency oftheschemeonlocalCourant-Friedrichs-Lewynumber(CFL),as reportedin[24].Thiscorrectionisappliedtoavoidthecommon problemoffreesurfaceinterfacespread,alsoknownas Numeri-calVentilation(NV)[25]and[26].Thisproblemisoneofthemain
sourcesoferrorinthenumericalsimulations,inparticularfor plan-inghullresistancetestsimulations,asindicatedby[27].
Inordertosimulatethehullfreetomoveinthepitchandheave directions,theDynamicFluidBodyInteraction(DFBI)modelwas used.TheDFBImodelallowedtheRANSsolvertoevaluatetheforce andmomentsonthehullandtosolvethegoverningequationsof bodymotioninordertorelocatethebody.TheDFBImodelhasbeen appliedbyusingtwodifferentmethods:theoversetmeshmethod andthemorphinggrid.
Theright-handorthogonalcoordinatesystemO-XYZ,asusedin allsimulations,hastheoriginofthereferenceframelocatedinthe longitudinalpositionofthecenterofbuoyancy(LCB)(Xcoordinate), intheverticalpositionofthecalmfreesurface(Zcoordinate),and transversalpositioninthesymmetryplane(Ycoordinate).The pos-itivex-axisisforwardorientedandparalleltothebaselineofthe hull,thepositivey-axispointstoportside,andthepositivez-axis pointsmverticallyupward.
ACFDcodecommerciallyavailable,STAR-CCM+byCDAdapco, wasusedformeshgenerationsandcomputations.
3.1.1. Computationaldomainandtime-stepanalysis
Asforthecomputationaldomain,theboundarieswereplaced farenoughaway from theship hull inorder toavoid the well knownproblemoffluidflowreflection.Theoversetmeshtechnique requiredtwo differentregions, i.e.,thebackgroundand overset regions,asshowninFig.7(a).Itisimportanttonotethatinthe shiphydrodynamicsfieldnodefinedrecommendationsinterms ofdomaindimensionsareavailablefortheoversetregion,as indi-catedin[28].However,thedimensionsofthebackgroundregion werechosenincompliancewiththeITTC’sCFDrecommendations [29],inparticularthedimensionsweresettomorethantwiceof thevaluesrecommendedbytheITTC.
Inregardstothemorphingmeshapproach,onlyoneregionwas required,andtherelevantdimensionsofthecalculationdomains wereincompliancewiththeITTCprescriptions[29](inparticular thedimensionswerenotlessthan2.5timestheITTC recommen-dations),asshowninFig.7(b).
Thedimensionsofthecalculationdomainsandtheboundary conditionsforthetwodifferentmeshapproachesareillustrated, respectively,inFigs.8–11
Theflowaroundtheshiphullwasassumedsymmetricalwith respecttothecenterplane.Thisisa typicalboundarycondition usedintheCFDresistancetestsimulations,asindicatedintheITTC guidelines[29],andtheeffectsofthisassumptionarenegligiblein termsofsimulationresults,asindicatedinmanyworks.Thisisa reasonableassumptionthatleadstosignificantreductioninterms ofcomputationaltime(seeFig.10).
Thetime-stepusedinthesimulationsisafunctionofthehull speed,accordingtothefollowingITTCequation[29]:
t=0.01∼0.005Vl (1)
whereVisthehullspeedandlisthelengthcharacteristicvalue oftheanalysedbody.In thisstudy,lwasassumedequal tothe dynamicLWL,detectedbytowingtanktestsateachanalysedspeed. Furthermore,thetime-stepisalsoafunctionofthegriddensity when,asinthiscase,theCFLnumberhastobekeptconstant.
Asummaryofthemainpropertiesofthenumericalsolverused fortheoversetandmorphingmeshcasesarereportedinTable3. 3.2. Overset/chimeragridsetup
One of the main advantages of the overset/chimera grid technique for the hull motion is that this approach ensures a goodnumericalaccuracynearthefarfieldboundaries.Thisisa typicalproblemoccurringinplaninghullsimulationswherethere
Table3
Summaryofnumericalsimulationsetup.
Pressurelink SIMPLE
Pressure Standard
ConvectionTerm 2ndOrder TemporalDiscretization 1stOrder
Time-step(s) Functionofvelocityandgrid Iterationpertime-step 5
TurbulenceModel k-SST Table4
Oversetcase:meshpropertiessummary.
Typeofmesh Trimmed(Background.Region) Polyhedral(OversetRegion) No.ofcells 662290 1994122
Basesize(m) 0.3 0.7
isasignificantvariationofthedynamictrimangle,asreportedin [27].
However,thesimulationsperformedwiththeoverset/chimera grid methodrequireadditional settings.In particular, to estab-lish the connectivitybetween the background and the overset regions,aninterpolationschemeisrequired.Accordingto previ-ousnumericaltestsconductedonplaninghullmodels[30],ithas beenidentifiedthatthelinearinterpolationschemeisthemost suitablein termsofadequacy ofthenumericalresultswiththe experimentaldata.However,thecomputationaleffortrequiredis higherwithrespecttotheothermethods.Adetailedanalysisofthe differentinterpolationschemesavailablewiththeoverset/chimera gridmethodarereportedin[14].
The oversetmesh solutionfor this analysisis similarto the solutionadoptedin[31]withtrimmedmesh(backgroundregion) andpolyhedralmeshembeddedzone(oversetregion),asshown in Fig.7(a).The maindataof themeshsolutionchosenfor the backgroundandoversetregionsaregiveninTable4.
Thedimensionsofthecalculationdomainsandtheboundary conditionsareillustratedinFig.8andFig.10.
3.3. Morphingmeshgridsetup
Itisknownthatmorphinggridsrequirespecialtreatmentof themovingnodesin ordertocontroltheaccuracy ofthespace derivatives and of the time-stepping scheme. This is done by appropriatelyinterpolatingthefluidflowvariables.Then,the inter-polationfieldisusedtodisplacetheverticesofthemeshbasedon theRadialBasisFunctions(RBF)method.Togeneratethe inter-polationfield,a systemofequationsissolved,usingthecontrol verticesandtheirspecifieddisplacements:foreverycontrolvertex i,itsdisplacementdiisapproximatedbythecombination: di= n
j=1 j r2 ij+c2j +˛, (2)whererij=|xi–xj|isthemagnitudeofthedistancebetweentwo vertices,jistheexpansioncoefficient,xithepositionofi-vertex, nisthenumberofcontrolvertices,cjthebasisconstant,and␣a constantvalue.Moredetailsarereportedin[32].
Differently fromwhat happens for theoverset/chimera grid method,themorphinggriddoesnotrequireadditionalsettings.The dimensionsofthecalculationdomainandtheboundaryconditions areillustratedinFigs.9and11,respectively.
InthesimulationspresentedinSection4(seeFig.3–6),the mor-phingtechnique is combinedwithrigidmotions, asmentioned above. Asmorphingstrategies canlead topoorquality cells, it becomesimportanttokeepundercontrolthetopological deforma-tionstakingplaceinthesurfacemeshandinthevolumecells,by
Fig.7. (a)Oversetgridvisualizationwithdifferentmeshesforregions(polyhedral:oversetregion,trimmed:backgroundregion):(b)morphingmeshvisualization.
Fig.8. Domaindimensions:oversetgridcase.
Fig.9.Domaindimensions:morphinggridcaseandLESsimulation.
meansofspecificmeshqualitymetrics.Tominimizethe topologi-caldeformationsofthegriddomain,thehullatthestartingpoint hasaninitialtrimangleof2◦aft.
Avisualizationofthegridwithrefinementintheair/water inter-faceareaisshowninFig.7(b).
3.4. Wally+treatment
Thewallfunctionwasusedforthenearwalltreatment,andthe All-Wally+[14]isthewalltreatmentusedforallsimulations.Itis ahybridapproachthatattemptstoemulatethehighy+wall treat-mentforcoarsemeshes(fory+>30),andthelowy+walltreatment forfinemeshes(fory+≈1).Itisalsoformulatedwiththe desir-ablecharacteristicofproducingreasonableanswersformeshesof
intermediate resolution (for y+ in the buffer layer), [14]. This approach is considered a reasonable compromise among the acceptablequalityoftheboundarylayerdescriptionandthe calcu-lationtime.Thevaluesofwally+nearthehullareshowninFig.12, whereit ispossibletoobservethewally+range onthehullat maximumspeedtestedforthetwodifferentmeshtechniques.
3.5. Verificationandvalidationstudy
A verification and validation study (V&V) is conducted for RTM/,,Z/
∇
1/3,andS/∇
2/3atonespeed(4.631m/s,Fr∇=3.898). TheresultsofthisanalysisarereportedinAppendixC.Fig.10.Boundaryconditions:oversetgridcase(viewfromthebackofthedomain).
Fig.11.Boundaryconditions:morphinggridcase(viewfromthebackofthedomain).
Fig.12.Wally+visualizationonthehullatFr∇=3.898forthedifferentmeshtechnique:overset/chimeragrid(top),morphinggrid(down).
TheV&Vstudyresultsandtheuncertaintyanalysisforthetwo differentsimulationtechniques,i.e.,oversetandmorphingmesh, arelistedinTableA6andA7,respectively.
Themonotonicconvergenceisreachedinallcasesexceptfor thesinkageintheoversetmesh.Thevalidationprocessisachieved fortheresistance/displacementratioandforthetrimangleinthe oversetcase.Inspiteofwhathappenswiththeoversetmesh,forthe
Fig.13.WettedsurfacetopviewatFr∇=3.898,comparisonbetweenexperimental(topside),andRANSEoversetsimulation2.5×106cells(downside),air/waterinterface asisosurfaceatVoF=0.5.
Fig.14.Experimentalwettedsurfaceright-sideviewatFr∇=3.898.
morphingmeshcaseahigherscatterinthesolutionsisdetected. Thisisprobablyoneofthemainreasonsforwhichthevalidation isachievedonlyforthesinkageandtrimangle.Whenthe compar-isonerrorismuchhigherthanvalidationuncertainty(EUV),in particularforthewettedsurface,thevalidationisnotachieved,but itissignificant,andthemainchallengebecomestheimprovement ofthesimulationmodeling.
Theiterativeuncertainty isnot reportedin thetablesabove becauseitwasfoundtobealwaysnegligiblewithrespecttoother sourcesofuncertainty,asshowninsimilarstudies,e.g.[29].
4. Resultsanddiscussions
Inthissection,acomparisonbetweenexperimentaland numer-icalresultsispresentedanddiscussed.
Themainaimsaretoevaluatethereliabilityof thedifferent simulationtechniquesadopted,aswellastofindaconfirmation inthenumericalsimulationsofthevortexphenomenaobserved experimentallyintheaftbodywaterplaneareabehindthestep.
Secondly,thenumerical analysisisused toreconstructwith greaterdetailtherecirculationpaths,whichareseen experimen-tallyfromtop-andside-viewphotographs.
TheresultsintermsofRTM/,,Z/
∇
1/3,andS/∇
2/3versusFr∇ arereportedinFigs.3–6,respectively.4.1. Totalresistance,dynamicsinkage,andtrimangle
Thenumericalresultsareinaccordancewiththeexperimental dataandalloftheexperimentalandnumericalresultsare sum-marizedin Table2. Thebetterperformance of theoversetgrid approachisevidentbecause,ascomparedwiththeexperimental
Fig.15. Wettedsurfaceright-sideviewatFr∇=3.898,RANSEoversetsimulation (2.5×106cells),air/waterinterfaceasisosurfaceatVoF=0.5.
values,thepercentagedifferencesareintherangefrom2.2%to9.3% fortheRTM/.
Similarresultsareachievedfortrimangle,inparticularforthe velocitiesgreaterthanthehumpspeed (3.13m/s,2.61Fr∇),the oversetgridhasbetterperformancerespecttothemorphinggrid approachwhencomparedwithexperimentalvalues.Thedifference betweenthetwomethodsismainlyduetotheunderestimationof trimwithmorphinggridforvelocitiesgreaterthanthehumpspeed, asshowninFig.4.
Thevalueofthenumericalnondimensionaldynamic sinkage withthetwodifferentmeshtechniquesarecloserandthe per-centagedifferences,comparingtheoversetapproachandmorphing grid,areupto8.9%,asshowninFig.5.
However,theoversetmeshtechnique ismoreeffectivethan themorphingtechnique,butmoredemandingintermsof com-putationaltimesrequired(1800versus980sofCPUtimepertime step).
4.2. Dynamicwettedsurface
Theamountof wetted surfaceareareduction is not directly relatedtothedragreductionaspointedoutbyAmromin[33]. How-everinthissection,forthesakeofcompleteness,theprocessof dynamicwettedsurfacecalculationisdiscussed.
Fig.6showsacomparisonbetweennumericalandexperimental valuesoftheS/
∇
2/3versusFr∇.TheexperimentalwettedsurfaceFig.16.DetailsoftheLESsimulationatFr∇=3.898withveryfinegridusedfortheflowpatternstudyintheunwettedaftbodyarea(12×106cells).ContourmapsofVoF (redinwater;blueinair).SeealsoFig.19.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
Fig.17. LESsimulation;bottomviewatFr∇=3.898withVoFisosurfaceat0.5,andwavecuts(12×106cells).
Fig.19.LESsimulation;volumefractioncontoursatdifferentcrosssections(atxfromLCB,positiveforward),atFr∇=3.898.ContourmapsofVOF(redinwater;bluein air).(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
Fig.20.LESsimulation;streamlinesinairandwateratFr∇=3.898.
valuesareestimatedthroughdigitalanalysisofvideoframes,which arereferencedtotheoriginal3DCADmodel.Ontheotherhand, thenumericalvaluesarecalculatedbythefluiddynamicssolver. Ingeneral,especiallyathighFroudeNumbers,thevaluesof wet-tedsurfacecalculated numericallyare morereliablethan those extractedfromthevideorecordings.
Asourceofuncertaintyintheestimationofwettedsurfaceis thesprayarea,whichisdifficulttoevaluateexperimentallyonthe basisofthevideofootages.Infact,asreportedinAppendixB,the biasofS/
∇
2/3 ismainlycomposedbythebiasofwettedsurface estimation(BS),whichis80.81%ofS/∇
2/3totalbias.Theissueofevaluationofthesprayareaalsoaffectsthe numer-icalsolution;indeed,thisevaluationisanontrivialtaskforRANS solversand requiresavery high-resolutiongrid,as indicatedin [34].Forthisreason,theoverestimationofthewettedsurfacewith respecttotheexperimentalresultsisduetotheuseofaVoF thresh-oldvalueequalto0.5asindicatedin[29].
Thenumericalresultsareclosertotheexperimental measure-mentswhenobtainedwiththeoversetmeshapproach,asopposed tothemorphinggridapproach.Thedifferencebetweenthetwo numericalmethodsismainlyduetotheunderestimationoftrim, asshowninFig.4.
In particular, the experimental wetted surface amount is underestimatedwithrespecttotherealwettedsurface,butthe numericallyestimatedwettedsurfaceoverestimatesthereal wet-tedsurfaceduetothechosenVoFvaluethreshold.Consequently therealwettedsurfacevaluescanbeconsideredlocatedbetween experimentalandnumerical(oversetmesh)values.
Asmentionedabove,theuseofthetransparentbottomhullwas providedtoensureafullviewofthewaterflowunderthehullanda moreeffectiveestimationofS.Duringthetowingtanktests,a par-ticularflowphenomenonwasdetectedatallFr∇.Thisphenomenon consistsintwovortexstructuresunderthehullandbehindthestep, asshowninthevideofootage[20].
However,forthisstudythespeedconditionofFr∇=3.898was chosen becausethe phenomenon is especially wellvisible (see Figs.13and14).
Thevortexstructuresarealsovisibleinthenumerical simula-tionswithanoversetmeshapproach,asshowninthetwoviews oftheflowunderthehull(Fig.13).Theupperpictureisatopview
ofthetransparenthullmodel,showingtheunwettedaftbodyarea andawell-developedvortex.Thisvortextrailsdownstreaminto thewater,departingfromtheaftregionoftheunwettedaftbody area.Thelowerpictureistakenfromthenumericalsolutionand showstheisosurfaceatVoF=0.5.Thisvisualizationemphasizesthe presenceoftheunwettedaftbodyareabehindthestep,asopposed totheforebodywettedsurface.Thetwovisualizationsareingood agreement,bothintermsofwettedsurfaceextensionandinterms ofshapeofunwettedaftbodyarea.
Similarly,Figs.14and15showtwosideviewsofthe underwa-terflowbehindthestep.Alsointhiscase,thetwovisualizations showagoodagreementbetweentheexperimentalevidenceand thenumericalresults.
4.3. Analysisofthefluid-dynamicsintheunwettedaftbodyarea Adetailedstudyoftheflowpatternsintheunwettedaftbody areabehindthestephasbeenperformedatFr∇=3.898.Starting fromtheexperimentalvaluesoftrimandsinkageinthiscondition (seeFigs.4and5),thezero(DoF)hullhasbeensimulatedatthe givenFroudenumberwithalargeeddysimulation(LES)approach onaveryfinegridofapproximately12×106cells(seeFig.16)with thesamesizeofcomputationaldomainandboundaryconditions previouslydescribedfor themorphinggridsimulation(see[14] forthedetailsoftheappliedLESformulation).Successively,the convergedsolutionhasbeenusedastheinitialconditionof fur-thersimulationswithfree heave-and-pitchdegrees-of-freedom. Thefinaltwo-DoFconvergedsolutionhasbeenpost-processedto gaininsight intothecomplexflowpatternbehindthestep.The streamlinesbothin waterand inairwereanalyzed,anda flow patternintheunwettedaftbodyareawascharacterized.Theflow patterns(in-airandin-water)areclearlyvisiblefromthebottom andsideviewsoftheLESsolutionshowninFigs.17and18,and Figs.20–24.
AbottomviewoftheLESsolutionisshowninFig.17where someselectedstreamlinesontheair/waterinterfacehavebeen cal-culatedandvisualized.ThesamestreamlinesarevisibleinFig.18. Theserepresentationsemphasize theshape oftheunwettedaft bodyareaandtheextensionintheaftregionofthestep.In partic-ular,theextensionoftheseparatedregionisevidentfromtheflow
Fig.21.LESsimulation;flowpatternsintheunwettedaftbodyarea,topviewatFr∇=3.898.
fieldcrosssectionsofFig.19atdifferentx-stations(x=0attheLCB, Fig.1).
AmoredetailedsetofvisualizationsisgivenbyFigs.20–22.The mainfeatureoftheflowpatterninsidetheunwettedaftbodyarea dependsontheairinlet(shapeandlongitudinalposition).Theair inletislocatedattheintersectionofthechinewiththestep,and facingthesideofthemainflow.Afairlycomplex3Dflowis vis-iblefromtherepresentationofthestreamlines.Theairparticles entertheunwettedareathroughtheinletandsuccessively propa-gateinsidetheairregionaccordingtodifferentpatterns.Finally,a mixofairand(mostly)waterparticlesflowsdownstreamoftheaft bodyintoavisiblevortex.Theseparationofthewater-flowinduced bythestepissimilartowhathappensinthetransom.The pres-sureoftheflowaftertheseparationistheatmosphericpressure. Thephysicsofthis3D-phenomenonismainlyregulatedbytheFr
accordingto[1].Therefore,accordingto[7]thecavitationnumber behindthestepis=0.018.
Aclose-uppictureofthetransparenthullmodeltakenfromthe experimentaltestisshowninFig.21(upperpart).Inthe photo-graph,thetwo-mainobservedcounter-rotatingflowpatternsare depictedandnamedpatternAandpatternB.Inthelowerpartof thesamefigure,asimilarvisualizationtakenfromtheLES solu-tionisreported.Twosimilarflowpatternsarevisiblebyinspection ofthenumericallycalculatedstreamlines.Thecomplexityofthe 3DstreamlinesisvisibleinFigs.22and23.ThesideviewofFig.23 showsthatstreamlinesoriginatingfromtheinletpropagate accord-ing toa recirculating pathinduced by theexternal water flow overcomingthe step.Finally, a perspective bottomview ofthe streamlinesisshowninFig.24.
Fig.22.LESsimulation;sideairinletatthestep,sideviewatFr∇=3.898.
Fig.23. LESsimulation;closeupof3Dpatternsofairflowintheunwettedaftbodyarea,sideviewatFr∇=3.898.
Theflowpatternsobtainednumericallylooksimilartotheones observedintowing tankinvestigations. Thesameflowpatterns seemtobeconfirmedbyongoingLESsimulationsonafullscale hullwithaveryfinegridof17.7×106cells.TheLESsimulationhas beenperformedforaspeedof15.5m/s,(28.5knotsinfullscale), equaltoFr=3.898.Furthermore,theresultsintermsoftrimangle, andnondimensionaltotalresistance,sinkage,wettedsurface,are comparablewithrespecttothemodelscalesimulation.
5. Conclusions
Calmwaterresistanceexperimentshavebeenconductedona single-stephullmodelwithatransparentbottominordertostudy thevorticalflowphenomenafeaturedbythistypeofplaninghulls. Anuncertaintyanalysisoftheexperimentalresultshasbeen per-formedincompliancewiththeITTCstandards.
Duetothehull model’slowweightandsensitivity to exter-nalappliedforces,the“down-thrust”methodologywasappliedin
ordertoreleasethemodelfromtheinstrumentationweight.This approachallowedhighprecisionofdataacquisition,i.e.resistance, sinkage,trim,anddynamicwettedsurfacemeasurements.The pro-posedsolutionhasproventoreproducewithgoodaccuracythereal systemofforcesexertedbytheoutboardenginesonthetransom ofthehullinfullscale.
Thetransparentbottomofthehullmodelallowedthe obser-vationofsomeuniquevorticalflowphenomenaintheunwetted aftbody areabehindthestep.Thevortexin theunwetted area appearsfortowingspeedsgreaterthan2.36m/s(Fr∇>1.97). Tow-ingtanktestsrevealedthattheflowintheunwettedaftbodyarea isdominatedbyarathercomplex3Darrangementofvortices origi-natingfromairspillageatbothsidesofthestep.Oneachsideofthe unwettedaftbodyarea,oneofthevorticesinsidetheairbubble terminatesattheair/waterinterfaceanddevelopsintoatrailing vortexinthewaterwake.
Toconfirmthis experimentalevidence,thesametest condi-tionswerereproduced withCFD simulationsbyusing different
Fig.24. LESsimulationatFr∇=3.898;perspectiveviewofstreamlinesintheunwettedaftbodyarea.
unconventionaldynamicmeshtechniques,i.e.oversetmeshand morphinggrid.Theresultsoftheoversetmeshapproach,inspite ofbeingobtainedwithagreatercomputationaleffort,showeda bettercomparisonerrorwithrespecttothoseofthemorphinggrid technique.Theoversetmeshensureshigh-qualityofthe simula-tionresultsandshowshigh-adaptabilitytothewidevariationsof trimandheaveofthesteppedhull.Asconsequence,the numeri-calresults,obtainedbytheoversetmeshapproach,arecloserto theexperimentalmeasurementsasopposedtothemorphinggrid approach,butmorecomprehensivestudiesareneededinorderto improvethesimulationsandobtainhigh–fidelityresults.In par-ticular,amore in-depthevaluationof thegridapproach, ofthe turbulencemodeleffects,ofthethree-dimensionalinstabilityand surfacetensioneffectsisdesirable.
Thesimulationuncertainty analysishasshown thatthegrid represents the main source of simulation numerical error, as already identified in another works. The uncertainty in the nondimensionalvariables(i.e.,totalresistance/displacementratio, nondimensionaldynamicsinkage,dynamictrimangle,and nondi-mensionaldynamic wetted surface) is, for the hull considered in this work,relatively greater when using the morphing grid technique.Thevalidationprocesshighlightsthat,inparticularfor thedimensionlesswettedsurface,thecomparisonerrorismuch greaterthanthevalidationuncertainty, whichimplies thatit is necessarytoreducethemodelingerror.However,byusingLES approachonarefinedgrid,adetailedanalysisoftheflowinthe unwettedaftbodyareabehindthestepshowsthattheresults qual-itativelymatchedtheexperimentsandconfirmtheexistenceofthe samevorticalstructuresobservedexperimentally.
Insummary,itcanbesaidthataboutthesimulationofstepped hull,theuseoftheoversetmeshtechniqueissuitableforthe detec-tion,inasufficientlyaccurateway,ofthetotalresistanceandofthe hullrunningattitude.InsteadtheLESapproachismoresuitablefor theevaluationofthedetailsofwaterflowphenomenaaroundthe hullandwettedsurface.Moreover,inthedesignofplaning/stepped hull,andintheabsenceoftowingtanktests,asuitableapproach istocoupleRANSsimulations,performedwiththeoversetmesh technique,withLESsimulations.TheRANScanbeused forthe determinationoftheglobalparameters,i.e.resistance, trimand sinkagewhiletheLESsimulationscanbereservedforanydetailed investigations.
Furthermore,forthefirsttimeaparticularvortexpatterninthe unwettedaftbodyareahasbeenobservedforascaledsingle-step hullmodelintowingtanktests.Atthesametime,similarflow pat-ternshavebeenfoundinthenumericalsimulationsonaveryfine mesh,inwhichadditionaldetailsonthe3Dflowwereexploredin thisarea.
Acknowledgments
Theauthorsaredeeplygratefultotheanonymousrefereesand totheeditorsfortheirsuggestionandhelpinsignificantly improv-ingthemanuscript.TheyalsodeeplythanktheHPCCentreSCoPE ofUniversityofNapoli“FedericoII”,SCoPEacademicstaffforthe givensupport,andtowingtankstaffasMrAndreaBove,Antonio Alfano,BiagioD’Abbusco,LucioIadicicco,andVitaleEsposito.
Thisworkhasbeensupportedby:SIRenaprojectgrant(CUP C69F140000500008−PONRC2007/2013),ECO-RIBprojectgrant (D.M.01/06/2016−Horizon2020−PON2014/2020),andMr. Vin-cenzoNappooftheMVMarineS.r.l.
AppendixA. Laboratoryinstrumentationand measurementtechniques
ThetowcarriageusedinthetowingtankoftheDepartmentof IndustrialEngineering,SectionofNavalArchitectureandMarine Engineering,UniversityFedericoIIofNaplesisabletodevelopa maximumspeedof10m/swithamaximumaccelerationand decel-erationof1m/s2.
Thecarriagewasinstrumentedonboardwithasensornetwork andaData-AcQuisition(DAQ)device.Thesensorsusedinthese testsaretheencoderforspeedcarriage,loadcellforresistance mea-sure,balanceformodelandballastweights,thermometerforwater temperature,accelerometerfortrim,andlaserforsinkage.
Thethermometerusedduringthetestsallowsarange−5◦Cand 40◦C,withaccuracyof0.1◦Candresolutionof0.1◦C.
The speed of the carriage was measured by a high-quality encoderandacounter/timingcard.Thehigh-qualityencoderrolls withoutrollingresistancedrivenbythecarriage;itisnotfixedto theanywheeldriveofcarriage,anditgives1000pulsesperone round(1pulseforeachmm).Theencodersensorhasanaccuracyof 1mm/mandaresolutionof1mm.Theperiodbetweentwopulses wasmeasuredbyacounter/timingcardat32bitswithaclockof
80MHz.Thecardhasarangefrom1.25×10−8sto53.69s;theclock at80MHzhasanaccuracyof±4×10−3MHzandaresolutionof 1.25×10−8s.
Theresistancemeasurewasperformedbyahigh-qualityload cell(precisionclass0.003)andconditioning-acquisitioncard.The loadcellhasarangeupto50N,anaccuracyof0.003%,anda reso-lutionof0.005N.Theconditioning-acquisitioncardhasasoftware programmedrangeof50N,accuracyof0.08%,16-bitresolution, andsamplingrateupto200kSamples/s.Forthesemeasurements, therowdatawereoversampledata rateof 10kSamples/s, and compressedatarateof500Samples/sforulteriorreductionofthe noise.
Runningtrimmeasurewasperformedbyanaccelerometerand aconditioning-acquisition card.Theaccelerometersensorhasa rangeof40m/s2,accuracyof±0.1%,andresolutionvirtually infi-nite.Theconditioning-acquisitioncardhasasoftwareprogrammed rangeof40m/s2,accuracyof0.1%,and16-bitresolution.
Sinkagewasmeasuredbytwohigh-qualitylasersensorsand aconditioning-acquisitioncard.Thetwolasershavearangefrom 0.2to1m,accuracyof0.5mm,andresolutionof0.05mm.These laserdeviceswereplacedperpendicularlywithrespecttothewater surface,attwopositions,oneatforesectionandsecondataft sec-tion.Theconditioning-acquisitioncardhasasoftwareprogrammed rangeupto1m,accuracy0.1%,and16-bitresolution.
Weightofthemodelandballastaremeasuredwithabalance witharangeof600N,accuracyof±0.1N,andresolutionof0.1N.
Thecalmwaterresistanceexperimentswereconductedwith “down-thrust”(DT)methodologyproposedin[22]withatowed pointlocatedinthehullbow.Thismethodologyconsidersthat, inahorizontalpositionandinatrimangleatrestequaltozero, thedirectionofthetowingforceisappliedinapointthatisthe intersectionbetweenthedirectionofthrustofenginesandkeelline atthebow.InthecaseofDTmethodology,themodelhas4Degrees ofFreedom(DoF),withyawanddriftrestricted.Infact,toavoidthe instabilityphenomena,themodelhasbeenbuiltwithtwoguide modelmasts,onelocatedinthebowandtheotheratthestern, whichengagesintwoforks.
AppendixB. Experimentaluncertaintyanalysis
UncertaintyAnalysis(UA)inExperimentalFluidDynamics(EFD) hasbeenalsoperformedaccording toITTC [23].Asa firststep, UAwasperformedfor each individualvariabler(model geom-etry,displacement,speed,resistance,density,runningtrim,and sinkage).Second,UAisperformedfornondimensionalcoefficients (RTM/,,Z/
∇
1/3,andS/∇
2/3).TheUAprocedurewasbasedon themethodologyproposedby[35]with95%confidenceinterval, hence,considersnormaldistributionswithalargesamplesizewith estimatesof:• systematicuncertainty,alsocalledbias(Br)iscalculatedasRoot SumofSquare(RSS)ofeachelementaryerrorsource(i.e., calibra-tion,dataacquisition,datareduction,andconceptualbias)group ofbiaserror.Theelementaryerrorsourceshavebeendividedand separatelyestimated;
• randomuncertainty,alsocalledprecisionuncertainty,(Pr),has beencalculatedforeachrun,accordingtoPj(S)=KSDevjwhere SDevj represents the standard deviation of jth run, and K=2 accordingtotheabove-mentionedmethodology;
• totaluncertaintyUrisaRSSofBrandPr.
Thebias,precision,andtheuncertaintiesfornondimensional coefficients(RTM/,,Z/V1/3,SW/
∇
2/3andFr∇)aresummarizedin TableA5belowreported.
AppendixC. Numericaluncertaintyanalysis
Thevalidationuncertainty(UV)isgivenby
U2V=U2D+USN2 , (3)
whereUDistheuncertaintyoftheexperimentaldata.
Thecomparisonerror(E)isdefinedasthedifferencebetween theexperimentaldata(D)andnumericalsimulationresult(Sn),as reportedbelow:
E=D−Sn. (4)
AccordingtoITTCprocedures[36],numericalsimulation uncer-tainty USN is mainly composedof iterative (UI), grid (UG), and time-step(UTS)uncertainty,asfollows:
U2
SN=UI2+U2G+UTS2. (5)
AccordingtoWilsonetal.[37],themostimportantsourceof uncertaintyofthenumericalresultsisthegrid.
Thesourcesofuncertaintywereevaluatedforeachofthe sim-ulationtechnique,i.e.,oversetgridandmorphinggrid.
Thenumericaluncertaintyevaluationwasperformedbyusing twodifferentmethods:theGridConvergenceIndex(GCI)method andtheCorrectionFactor(CF)method.
Thegeneralformoftheuncertaintyevaluation,basedonthe generalizedRichardsonExtrapolation(RE)method,canbewritten asfollows: Uk=FS| ε21k rpk k −1 |, (6)
where ε21k is the solution changes for the k-input parameter between thesolutions fine (Sn1k)tomedium (Sn2k) and coarse (Sn3k),rkistheconstantrefinementratio,pkistheobservedorder ofaccuracy,andFSisthesafetyfactor.
Furthermore,anotherparameteristheconvergenceratio(Rk), whichprovidesinformationabouttheconvergence/divergenceof asolution.TheRkvaluewasdeterminedbythefollowingratio:
Rk=ε21k/ε32k. (7)
Therearefourkindsofpossibleconvergenceconditions: mono-tonicconvergence(0<Rk<1),oscillatoryconvergence(Rk<0and |Rk|<1),monotonicdivergence(Rk>1),andoscillatorydivergence (Rk<0and|Rk|>1).Since0<Rk<1,themonotonicconvergenceis satisfied.
The twodifferent solutionverification methods usedin this studydifferinthechoiceofthesafetyfactor(FS).
TheGCImethodinEq.(7),proposedby[38,39],isused exten-sivelyandisrecommended,forexample,bytheAmericanSociety ofMechanicalEngineer(ASME)[40]andtheAmericanInstituteof AeronauticsandAstronautics(AIAA)[41].Roacherecommended forthreeormoregridsanalyzed1.25asFSvalue.
TheothermethodusedistheCFinEq.(8),asdiscussedin[42], whichusesavariablevalueofFS,calledcorrectionfactor(Ck).In theCFmethod,theuncertaintyoftheerrordependsonhowmuch thesolutionsareclosetotheasymptoticrange.Theexpressions definedtheuncertaintieswerereportedby[43]:
Uk=1.25·| ε21k rpk k −1 |, (8) Uk=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
9.6(1−Ck)2+1.1| ε21k rpk k −1 |, |1−Ck|<0.125 [2|1−Ck|+1]| ε21k rpk k −1 |, |1−Ck|≥0.125 . (9)
∇
0.43% 0.27% 0.23% 0.30% 0.33% 0.29% 0.36% 0.39% %ofFr∇
ModelResistance Ratio
RTM/ 0.100 0.178 0.204 0.256 0.312 0.407 0.491 0.555 [N/N]
BR 6.62E-04 6.68E-04 6.71E-04 6.78E-04 6.86E-04 7.05E-04 7.25E-04 7.42E-04 [N/N]
97.49% 97.54% 97.56% 97.61% 97.67% 97.78% 97.90% 98.00% %ofB2 RT/
B 1.06E-04 1.06E-04 1.06E-04 1.06E-04 1.06E-04 1.06E-04 1.06E-04 1.06E-04 [N/N]
2.51% 2.46% 2.44% 2.39% 2.33% 2.22% 2.10% 2.00% %ofB2 RT/
PR 4.08E-05 1.54E-07 1.54E-07 3.45E-07 2.25E-06 2.25E-06 2.25E-06 1.53E-06 [N/N]
URTM/ 6.72E-04 6.76E-04 6.79E-04 6.86E-04 6.95E-04 7.13E-04 7.33E-04 7.50E-04 [N/N]
0.67% 0.38% 0.33% 0.27% 0.22% 0.18% 0.15% 0.14% %ofRT/ TrimAngle 2.23 3.55 3.27 3.27 2.87 2.69 2.52 2.58 [deg] B-cw 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001 [deg] 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% 0.01% %ofB2 B−ix 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 [deg] 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% %ofB2 B−iy 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 [deg] 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% %ofB2 B−iz 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 [deg] 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% 33.33% %ofB2 B 0.087 0.087 0.087 0.087 0.087 0.087 0.087 0.087 [deg] 3.88% 2.44% 2.65% 2.65% 3.02% 3.22% 3.44% 3.36% %ofB2 P 0.004 0.003 0.003 0.004 0.012 0.014 0.012 0.012 [deg] U 0.087 0.087 0.087 0.087 0.087 0.088 0.087 0.087 [deg] 3.89% 2.44% 2.65% 2.65% 3.04% 3.26% 3.47% 3.39% %of Sinkage Z/∇1/3 −0.080 0.010 0.048 0.085 0.139 0.147 0.151 0.181 [mm/mm] BZCG-cw 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 [mm]
3.3E-03 3.3E-03 3.3E-03 3.3E-03 3.3E-03 3.3E-03 3.3E-03 3.3E-03 %ofBZCG2
BZCG-lf 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 [mm] 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 %ofBZCG2 BZCG-la 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 [mm] 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 %ofBZCG2 BZCG-lb 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 [mm] 0.332 0.332 0.332 0.332 0.332 0.332 0.332 0.332 %ofBZCG2 BZCG 1.735 1.735 1.735 1.735 1.735 1.735 1.735 1.735 [mm] 14.85% 114.14% 24.96% 14.03% 8.51% 8.07% 7.83% 6.56% %ofBZCG2 PZCG 6.365 7.120 6.846 4.046 8.755 5.870 6.556 4.864 [mm] BZ/∇1/3 0.016 0.012 0.014 0.017 0.023 0.024 0.025 0.028 [mm/mm] 20.64% 115.04% 28.79% 20.06% 16.67% 16.45% 16.34% 15.77% %ofZ/∇1/3 PZ/∇1/3 0.025 0.024 0.024 0.019 0.036 0.030 0.032 0.032 [mm/mm] UZ/∇1/3 0.030 0.027 0.028 0.025 0.043 0.038 0.040 0.043 [mm/mm] −37.26% 261.37% 58.95% 29.82% 30.96% 26.01% 26.56% 23.52% %ofZ/∇1/3 Modelgeometry S/∇2/3 6.730 BS 0.284 [m2/m2] 80.81% %ofB2 S/∇2/3 B∇2/3 0.138 [m2/m2] 19.19% %ofB2 S/∇2/3 PS/∇2/3 0.032 [m/m] US/∇2/3 0.317 [m2/m2] 4.72% %ofS/∇2/3 Density 1000 [kg/m3] Br1 0.071 [kg/m3] 1.15% %ofB2 Br2 0.07 [kg/m3] 1.12% %ofB2 Br3 0.655 [kg/m3]
TableA5(Continued)
Description Term Speed Units
1.290 2.357 3.131 4.629 5.368 6.336 7.300 8.054 [m/s] 97.74% %ofB2 Br 0.663 [kg/m3] 0.066% %ofBr Pr 1.00 [kg/m3] U 1.20 [kg/m3] 0.12% %ofr TableA6
Oversetgridcase:uncertaintyanalysis.
Gridratio RG pG %UG %UG %USN %UD %UV %|E|
GCI CF
RTM/ √2 0.47 1.44 9.06 16.62 16.62 0.27 16.6 3.8
√2 0.43 −2.45 3.23 0.37 3.23 2.65 4.2 3.7
Z/∇1/3 √2 1.65 −2.16 N.A. N.A. N.A. 29.82 N.A. 42.1
S/∇2/3 √2 0.89 −0.34 0.96 0.60 0.96 4.72 4.8 31.0
UGwasexpressedasapercentagevalueofthesimulationsolutionforthefinestgrid.
TableA7
Morphinggridcase:uncertaintyanalysis.
Gridratio RG pG %UG %UG %USN %UD %UV %|E|
GCI CF
RTM/ √2 0.46 −2.27 6.92 0.49 6.92 0.27 6.9 18.3
√2 0.78 −0.71 25.69 11.59 25.69 2.65 25.8 9.9
Z/∇1/3 √2 0.90 −0.30 43.77 28.14 43.77 29.82 52.9 43.1
S/∇2/3 √2 0.95 −0.14 13.65 11.95 13.65 4.72 14.4 53.2
UGwasexpressedasapercentagevalueofthesimulationsolutionforthefinestgrid.
References
[1]D.Savitsky,M.Morabito,Surfacewavecontoursassociatedwiththeforebody wakeofsteppedplaninghulls,Mar.Technol.47(January)(2010)1–16.
[2]O.M.Faltinsen,HydrodynamicsofHigh-SpeedMarineVehicles,Cambridge UniveristyPress,NewYork,2006.
[3]A.Sverchkov,Applicationofaircavitiesonhigh-speedshipsinRussia,in: InternationalConferenceonShipDragReduction(SMOOTH-Ships),Istanbul, 2010.
[4]D.Tauton,D.Hudson,R.Shenoi,Characteristicsofseriesofhighspeedhard chineplaninghulls-part1:performancincalmwater,Int.SmallCraftTechnol. 152(2010)55–75.
[5]D.Svahn,PerformancePredictionofHullswithTransverseSteps(M.Sc. Thesis),KTHCentreforNavalArchitecture,Stockolm,Sweden,2009.
[6]W.R.Garland,K.J.Maki,Anumericalstudyofatwo-dimensionalstepped planingsurface,Jo.ShipProd.28(2)(2012)60–72.
[7]M.V.Makasyeyev,Numericalmodelingofcavityflowonbottomofastepped planinghull,in:Proceedingsofthe7thInternationalSymposiumon Cavitation(CAV2009),AnnArbour,2009.
[8]K.I.Matveev,Two-dimensionalmodelingofsteppedplaninghullswithopen andpressurizedaircavities,Int.J.NavalArchit.OceanEng.4(2)(2012) 162–171.
[9]K.I.Matveev,Hydrodynamicmodelingofsemi-planinghullswithaircavities, Int.J.NavalArchit.OceanEng.3(3)(2015)500–508.
[10]S.Brizzolara,A.Federici,Designingofplaninghullswithlongitudinalsteps: CFDinsupportoftraditionalsemi-empiricalmethods,in:Proceedingsof DesignandConstructionofSuper&MegaYachts,Genoa,2013.
[11]P.Lotfi,M.Ashrafizaadeh,R.K.Esfahan,Numericalinvestigationofastepped planinghullincalmwater,OceanEng.94(2015)103–110.
[12]M.Bakhtiari,S.Veysi,H.Ghassemi,Numericalmodelingofthestepped planinghullincalmwater,Int.J.Eng.Trans.B29(2)(2016).
[13]F.Stern,J.Yang,Z.Wang,H.Sadat-Hosseini,M.Mousaviraad,S.Bhushan,T. Xing,Computationalshiphydrodynamics:nowadaysandwayforward,in: SymposiumonNavalHydrodynamics,Gothenburg,2012.
[14]CD-adapco,STARCCM+User’sGuideVersion9.06,2014.
[15]S.Brizzolara,D.Villa,CFDsimulationofplaninghulls,in:7thInternational ConferenceOnHigh-PerformanceMarineVehicles,MelbourneOctober 11–15,2010.
[16]F.DeLuca,S.Mancini,S.Miranda,C.Pensa,Anextendedverificationand validationstudyofCFDsimulationsforplaninghulls,J.ShipRes.60(2)(2016) 101–118.
[17]J.Y.Kang,B.S.Lee,Mesh-Basedmorphingmethodforrapidhullform generation,Comput.-AidedDes.42(2010)970–976.
[18]M.E.Biancolini,I.M.Viola,SailstrimoptimisationusingCFDandRBFmesh morphing,Computers&Fluids93(2014)46–60.
[19]ITTC,RecommendedProceduresandGuidelines7.5-01-01-01,2002. [20]L.Vitiello,S.Miranda,TowingTankTestofSteppedHullC03,Universityof
NaplesFedericoII,Italy,2017[Online]Available:https://doi.org/10.6084/m9. figshare.4707697.v2(Accessed2017).
[21]ITTC,RecommendedProceduresandGuidelines7.5-02-02-01,2011. [22]L.Vitiello,S.Miranda,Propulsiveperformanceanalysisofasteppedhullby
modeltestresultsandseatrialdata,in:HighSpeedMarineVehicles Symposium,Napoli,2014,ISBN9788890611216.
[23]ITTC,RecommendedProceduresandGuidelines7.5-02-02-02,2002. [24]C.Bohm,AVelocityPredictionProcedureforSailingYachtsBasedon
IntegratedFullyCoupledRANSE-Free-SurfaceSimulations,DelftUniversity, Delft,2014.
[25]J.Ferziger,M.Peric,ComputationalMethodforFluidDynamics, Springer-Verlaag,Berlin,2003.
[26]V.Andrillion,B.Alessandrini,A2D+TVOFfullycoupledformulationfor calculationofbreakingfreesurfaceflow,Proceedingsofthe24thSymposium onNavalHydrodynamics(2003).
[27]I.M.Viola,R.G.Y.Flay,R.Ponzini,CFDanalysisofthehydrodynamic performanceoftwocandidateamerica’scupAC33hulls,Int.J.SmallCraft Technol.154(B1–B12)(2012).
[28]T.Tezdogan,Y.K.Demirel,P.Kellet,M.Khorasanchi,A.Incecik,Full-Scale unsteadyRANS-CFDsimulationsofshipbehaviourandperformanceinhead seasduetoslowsteaming,OceanEng.97(2015)186–206.
[29]ITTC,RecommendedProceduresandGuidelinesPracticalGuidelinesforShip CFDApplications7.5-03-02-03,2011.
[30]S.Mancini,TheProblemoftheVerificationandValidationProcessesofCFD SimulationsofPlaningHulls–PhDThesis,UniversityofNaples,FedericoII, Naples,2016.
[31]C.Bertorello,E.Begovic,S.Mancini,Hydrodynamicperformancesofsmallsize swathcraft,Brodogr.Shipbuild.66(4)(2015).
[32]A.deBoer,M.S.Schoot,H.Bijl,Meshmorphingbasedonradialfunction interpolation,Comput.Struct.85(2007)784–795.
[33]E.L.Amromin,Analysisofinteractionbetweenshipbottomaircavityand boundarylayer,Appl.OceanRes.59(2016)451–458.
[34]S.M.Mousaviraad,Z.Wang,F.Stern,URANSstudiesofhydrodynamic performanceandslammingloadsonhigh-Speedplaninghullsincalmwater andwavesfordeepandshallowconditions,Appl.OceanRes.51(2015) 222–240.
[35]H.Coleman,W.Steele,ExperimentalandUncertaintyAnalysisforEngineers, 2nded.,NewYork,NY,JohnWiley&Sons,1999.
[36]ITTC,UncertaintyAnalysisinCFDVerficationanValidation7.5-03-01-01, 2008.
[37]R.V.Wilson,F.Stern,H.W.Coleman,E.G.Paterson,Comprehensiveapproach toverificationandvalidationofCFDsimulations−part2:applicationfor RANSsimulationofaCargo/Containership,J.FluidsEng.123(2001)803–810.