• Non ci sono risultati.

Experimental and numerical hydrodynamic analysis of a stepped planing hull

N/A
N/A
Protected

Academic year: 2021

Condividi "Experimental and numerical hydrodynamic analysis of a stepped planing hull"

Copied!
20
0
0

Testo completo

(1)

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

(2)

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

(3)

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

(4)

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)

(5)

Fig.3. NondimensionaltotalresistancecomparisonbetweenEFDandCFDsimulationswithuncertaintybars.

Fig.4. TrimcomparisonbetweenEFDandCFDsimulationswithuncertaintybars.

(6)

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 3.873

isshowninFig.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

(7)

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

(8)

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.

(9)

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

(10)

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.

(11)

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.Theexperimentalwettedsurface

Fig.16.DetailsoftheLESsimulationatFr∇=3.898withveryfinegridusedfortheflowpatternstudyintheunwettedaftbodyarea(12×106cells).ContourmapsofVoF (redinwater;blueinair).SeealsoFig.19.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig.17. LESsimulation;bottomviewatFr∇=3.898withVoFisosurfaceat0.5,andwavecuts(12×106cells).

(12)

Fig.19.LESsimulation;volumefractioncontoursatdifferentcrosssections(atxfromLCB,positiveforward),atFr∇=3.898.ContourmapsofVOF(redinwater;bluein air).(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

(13)

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

(14)

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.

(15)

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

(16)

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

(17)

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)

(18)

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% %ofB␶2 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% %ofB␶2 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% %ofB␶2 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% %ofB␶2 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% %ofB␶2 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% %ofB␳2 Br2 0.07 [kg/m3] 1.12% %ofB␳2 Br3 0.655 [kg/m3]

(19)

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% %ofB␳2 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.

(20)

Riferimenti

Documenti correlati

In fact, a layered architecture, able to efficiently implement an interleaved planning and execution strategy, appears to be a suitable solution for solving those problems that

Pierre Mellarède fu uno dei più importanti funzionari del governo sabaudo tra Sei e Settecento, membro autorevole di quella schiera di «avvocati burocrati» di cui si servì

A partire da questa prima banale forma di interazione, che di volta in volta può assumere la concreta fisionomia della relazione fra testo e contesto o fra

Quanto ricevuto da chi detiene una carica pubblica è dunque percepito anche come un dono deviato e sottratto alla comunità che però può essere purificato con la restituzione

Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China, associated to 3. 66 Departamento de Fisica, Universidad Nacional de Colombia, Bogota,

Un sistema di rapporti solidali di cittadinanza (precisato quanto sopra) viene delineato anche in rapporto al superamento degli squilibri territoriali interni ad

Below the melting temperature the motion of the molecules is not sufficient to permit long range rearrangements; in particular, below the glass transition temperature the chain

La sonda elettrodinamica è un altro strumento atto alla misurazione della concentrazione delle polveri in camini il cui principio di funzionamento sta nella capacità di