Predicting the effect of biofouling on ship resistance using CFD

Contents lists available atScienceDirect

Applied

Ocean

Research

journal homepage:www.elsevier.com/locate/apor

Predicting

the

effect

of

biofouling

on

ship

resistance

using

CFD

Yigit

Kemal

Demirel

,

Osman

Turan,

Atilla

Incecik

DepartmentofNavalArchitecture,OceanandMarineEngineering,UniversityofStrathclyde,HenryDyerBuilding,100MontroseStreet,Glasgow,G40LZ,UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received27June2016

Receivedinrevisedform1December2016 Accepted6December2016

Keywords: Biofouling Shipresistance

Computationalfluiddynamics Hullroughness

a

b

s

t

r

a

c

t

ThispaperproposesaComputationalFluidDynamics(CFD)basedunsteadyRANSmodelwhichenables thepredictionoftheeffectofmarinecoatingsandbiofoulingonshipresistanceandpresentsCFD simula-tionsoftheroughnesseffectsontheresistanceandeffectivepowerofthefull-scale3DKRISOContainer Ship(KCS)hull.

Initially,aroughnessfunctionmodelrepresentingatypicalcoatinganddifferentfoulingconditionswas developedbyusingtheroughnessfunctionsgivenintheliterature.Thismodelthenwasemployedin thewall-functionoftheCFDsoftwareandtheeffectsofatypicalasappliedcoatinganddifferentfouling conditionsonthefrictionalresistanceofflatplatesrepresentingtheKCSwerepredictedforadesign speedof24knotsandaslowsteamingspeedof19knotsusingtheproposedCFDmodel.Theroughness effectsofsuchconditionsontheresistancecomponentsandeffectivepowerofthefull-scale3DKCS modelwerethenpredictedatthesamespeeds.Theresultingfrictionalresistancevaluesofthepresent studywerethencomparedwitheachotherandwithresultsobtainedusingthesimilaritylawanalysis. Theincreaseintheeffectivepowerofthefull-scaleKCShullwaspredictedtobe18.1%foradeteriorated coatingorlightslimewhereasthatduetoheavyslimewaspredictedtobe38%atashipspeedof24 knots.Inaddition,itwasobservedthatthewaveresistanceandwavesystemsaresignificantlyaffected bythehullroughnessandhenceviscosity.

©2016TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Shippinghasbeen,andstillis,oneofthemostimportant meth-odsoftransport, withmorerelianceandimportancenowbeing placedonthismodeoftransportasaconsequenceofadvancesin shippingtechnologyandtheabilityofshipstostoreandtransport increasingcapacitiesofgoods.However,theseimprovementsbring someproblemstotheindustryduetoanincreaseinfuel consump-tion,whichisdetrimentaltotheenvironmentandwhicherodes companyrevenues.Althoughotherformsoffuelpowerexist,such aswindenergyandsolarpower,carbon-basedfueliscurrentlythe onlywayforshipstoruneffectively.Forthisreason,minimising fuelconsumptioniscrucialforshippingcompanies.Such compa-nieshavethereforeattemptedtodeterminetheoptimumoperation andmaintenanceapproachestoeitherdecreasethecostof opera-tionsortoincreasetheprofitofthecompany.Thereleaseofharmful gasesduetotheuseofcarbon-basedfuelisanotherreasonthat shippingcompaniesshouldaimtoreducethefuelconsumptionof theirships.Someregulations,suchastheEnergyEfficiencyDesign

∗Correspondingauthor.

E-mailaddress:[email protected](Y.K.Demirel).

Index(EEDI)[1]andtheShipEnergyEfficiencyManagementPlan (SEEMP)[2],andrecommendedpracticessuchastheEnergy Effi-ciencyOperationalIndicator(EEOI)[3]have beenimplemented inrecenttimestolimit thequantitiesofharmfulgasesthatare releasedintotheenvironmentasaresultofthefuelconsumedby ships.

Althoughshippingismarginallymoreenvironmentallyfriendly thanotherformsoftransportation,suchasaviationandland,it wasreportedthatshipsreleased870milliontonsofCO2in2007,

whichisequivalentto2.7%ofthetotalCO2emissionsthatyear[4].

TheInternationalMaritimeOrganization(IMO)hasthereforebeen forced,dueinparttoanincreaseinpublicawareness,todevise andimplementenergyefficiencyandGHGregulations.As95%of theworld’scargoistransportedbysea[5],ameansofreducingthe frictionalresistanceofshipswoulddramaticallyreducetheirfuel consumption,leadingtoreducedcarbonemissionsworldwide.The bestmethodtoreducefrictionalresistanceistoapplyatreatment toaship’shull,tominimiseitsphysicalandbiologicalroughness. Physicalroughnesscanbeminimisedbyapplyingsome preventa-tivemeasures,butbiologicalroughness(fouling)ismoredifficult tocontrol.

Marinebiofoulingisanincreasingproblemfrombotheconomic andenvironmentalpointsofviewintermsofincreasedresistance,

http://dx.doi.org/10.1016/j.apor.2016.12.003

Nomenclature

ks Equivalentsandgrainroughnessheight

Rt50 Averagehullroughness

ı Boundarylayerthickness u frictionvelocityFrictionvelocity

U+ _{Roughnessfunction}

k+ _{RoughnessReynoldsnumber}

␬ vonKarmanconstant

y+ _{Non-dimensionalwalldistance}

B Smoothwalllog-lawintercept

␳ Density

¯

ui AveragedCartesiancomponentsofthevelocity vec-tor

u_iu_j Reynoldsstresses p Meanpressure ¯

_ij Meanviscousstresstensorcomponents

Dynamicviscosity

t Timestep V Shipspeed U Axialvelocity L Shiplength

LBP Lengthbetweentheperpendiculars

LWL Lengthofwaterline

BWL Beamatwaterline

D Depth

T Designdraft S Wettedsurfacearea

Displacement CB Blockcoefficient

1+k Formfactor Fr Froudenumber Re Reynoldsnumber RT Totalresistance RF Frictionalresistance RR Residuaryresistance

RW Waveresistance

RVP Viscouspressureresistance PE Effectivepower

CT Totalresistancecoefficient

CF Frictionalresistancecoefficient

CR Residuaryresistancecoefficient

CW Waveresistancecoefficient

CVP Viscouspressureresistancecoefficient

CT,smooth Totalresistancecoefficientinsmoothcondition

CT,rough Totalresistancecoefficientinroughcondition

CT Increaseintotalresistancecoefficientduetosurface

roughness

CF Addedresistancecoefficientduetosurface

rough-ness

PE Increaseineffectivepowerduetosurfaceroughness

D(%) Relativedifference pa Apparentorder r,r21,r32 Gridrefinementfactors

k Keyvariableonthekth_grid

21

ext Extrapolatedvalue

e21

a Napproximaterelativeerror

e21

ext Extrapolatedrelativeerror

GCI21_fine Fine-gridconvergenceindex

increasedfuelconsumption,increasedGHGemissionsand trans-portationofharmfulnon-indigenousspecies(NIS).Itshouldbekept inmindthatevenasmallamountoffoulingmayleadtoasignificant

increaseinfuelconsumption.Inparticular,hard-shelledfoulingcan causeaconsiderableriseinshipfrictionalresistance,andhencea ship’sfuelconsumption.Hard-shelledbarnaclescanalso deterio-ratethepaintandcauseotherproblemssuchascorrosion.Itshould benotedthattheimpactoffoulingonshipperformanceisgreatly dependentonthetypeandcoverageoffouling[6].

Duetoitsnegativeeffects onship efficiencyandthemarine environment,itisverydesirabletomitigatetheaccumulationof biofoulingonshiphulls.Marinecoatingsareprevalentlyusedto smoothhullsurfacesandifappliedwithapropercathodic pro-tectionsystemalsopreventscorrosion[7].Anidealmarinecoating shouldbesmoothenoughtoimprovethesurfacepropertiesofahull intheasappliedconditionandshouldbeeffectiveagainstmarine biofoulingwhichoccursovertime.

While improving the energy efficiency of existing ships retrofittedwithnewantifouling(AF)paints,itisequally impor-tanttoaccuratelymodelthepotentialeffectsofbiofoulingonship resistanceandtodemonstratetheimportanceofthemitigation of sucheffects by carrying out scientific research. However,at present,thereisnocompletemethodavailabletopredicttheeffect ofbiofoulingonshipfrictionalresistance.TheITTC[8]therefore rec-ommendsresearcherstodevelopnewformulaeormethods,using experimentaldata,forthepredictionoftheeffectsofcoatingsand biofoulingonshipresistance

Granville [9,10] proposeda similarity lawscaling procedure forthepredictionoftheeffectsofaparticularroughnessonthe frictionalresistanceofanyarbitrarybodycoveredwiththesame roughness,utilisingtheexperimentallyobtainedroughness func-tionsofsuchsurfaces.Theonlyrealassumptionofthemethodis thattheouterlayersimilarityholdsinthemeanvelocityprofilesfor smoothandrough-wallboundarylayers.Thatistosay,the velocity-defectprofilescollapsetoasinglecurveintheouterlayer.Some examplesoftheuseofthismethodaregivenbyLoebetal.[11], HaslbeckandBohlander[12],Schultz[6,13–15],Shapiro[16],Flack andSchultz[17]andSchultz,Bendick[18].Recently,Walkeretal. [19]conductedexperimentsusingboth antifoulingand fouling-releasehullcoatingsandscaleduptheresultstopredicttheeffects ofthesecoatingsonamid-sizednavalship.Grigson[20]proposed amethodwhichispartlyexperimentalandpartlytheoretical,just liketheonesproposedbyGranville[9,10].Somenumerical meth-odswerealsoproposedforgeneralroughsurfacesratherthanhull roughness,suchasChristophandPletcher[21],Lakehal[22]and Gre´ıgoireetal.[23].

WhenitcomestoCFD-basedmodels,therearefewerstudies investigatingtheroughnesseffectsofcoatingsandbiofoulingon shipresistance.Patel[24]mentionedthatthemostcomplex prob-lemsforCFDarefullscaleReynoldsnumberflowsandsimulating surfaceroughness.Currently,physicalmodellingoftheroughness sources,suchascoatingsorbiofouling,inCFDispractically impos-sibleduetotheircomplexgeometries.However,oncetherelation ofU+_=f(k+_{)isknown,itcanbeemployedinthewall-function}

or theturbulence modelsof theCFD software,as discussedby Patel[24].TheuseofCFD-basedunsteadyRANSmodelsisofvital importance,sincethephenomenoncanbesimulatedbymeansof afullynon-linearmethod.Forinstance,oneparticularroughness Reynoldsnumbervalue,k+_{,androughnessfunctionvalue,U}+_,are

takenintoconsiderationwhenapredictionismadeforaspecific conditioninthesimilaritylawscalingprocedureofGranville[9]. However,thek+_vs.U+_{valueisnotuniformevenonaflatplate}

duetodifferencesinthefrictionvelocity,u,distribution.Thatisto say,uvariesalongtheflatplate.Thiseffect,however,canbe sim-ulatedusingCFD-basedmodelsasu␶isdynamicallycomputedfor eachdiscretisedcell.Therefore,theresultingfrictionalresistance canbemoreaccuratelycomputedusingCFDmethods.

eitherusingwall-functions(e.g.Sugaet al.[25],Apsley[26])or usingnear-wallresolution (e.g.Krogstad[27],Aupoix[28]).Ec¸a andHoekstra[29]showedthattheeffectofuniformsand-grain roughnessonthefrictionalresistanceofflatplatesoffull-scaleship lengthsatfull-scaleshipspeedscanbeaccuratelysimulatedusing eitherwall-functionsor near-wallresolution. Dateand Turnock [30]demonstratedtherequiredtechniquestopredicttheskin fric-tionofflatplatesusing RANSsolvers andalsoshowedthatthe effectofsurfaceroughnessonskinfrictioncanbepredictedusing CFDsoftware.Theymodifiedthewall-functionsofapieceof com-mercialCFDsoftwarebymodifyingthewallfunction-coefficient (log-layerconstant).Nevertheless,thismethoddoesnotdirectly reflecttheroughnesseffectonthefrictionalresistanceanddoes notcomputetheseeffectsdynamically.Leer-Andersenand Lars-son[31],ontheotherhand,employedroughnessfunctionsin a commercialCFDcodeandpredictedtheskinfrictionoffullscale ships.However,theyusedaspecificmoduleofthesoftware,which incorporatesthin boundarylayermethodswithapotentialflow solver,andthestudydoesnotincludeunsteadyRANScalculations. Izaguirre-Alzaet al.[32] usedthe CFDsoftware package STAR-CCM+tosimulatetheirexperimentsandvalidatetheroughness featureofthesoftware.Althoughthecomparison showsa very goodagreementbetweentheexperimentaldataand the evalu-atedresults,thereisnoevidenceoftheuseofaspecificroughness functionmodel,ratherthanthebuilt-inroughnessfunction.Khor andXiao[33]investigatedtheeffectsoffoulingandtwo antifoul-ingcoatingsonthedragofafoilandasubmarinebyemployinga CFDmethod.Theyusedtheequivalentsandgrainroughnessheight andthebuilt-inwall-functionwhichconsiderstheuniform sand-grainroughnessfunctionmodelproposedbyCebeciandBradshaw [34],basedonNikuradse’sdata[35].Currently,theITTC[36]isstill questioningthevalidity oftheroughnessmodelandequivalent sandgrainroughnessusedinCFDapplicationsforhullroughness, sinceit isknownthat thebuilt-inroughness functionmodel is basedonuniform, closely packedsand roughness,whereas the roughnessfunctionsofrealengineeringsurfacesdonotshowthis behaviour.Castro,Carrica[37]carriedoutunsteadyRANSCFD sim-ulationsofafull-scaleKCSmodelwithhullcoatingroughnessusing wall-functions.However,theyusedaconstantroughnessfunction andtheroughnessallowanceformulationproposedbytheITTC [38].Theydidnot attempttoemploya newtypeof roughness functionmodelwhichismoreappropriateforrealengineering sur-faces,especiallyforfouledsurfaces.Recently,Demireletal.[39] proposedaCFDmodelforthefrictionalresistancepredictionof antifoulingcoatings.Haaseetal.[40]showedtheapplicabilityof theCFDapproachtopredictthesandgrainroughnesseffectson thefrictionalresistanceofflatplatesaswellasontheresistanceof catamarans.

Asdiscussedabove,currentnumericalmethodsforthe predic-tionoftheeffectofbiofoulingonfrictionalresistancearelimited bytheuseofboundarylayersimilaritylawanalysis.Thismethod canonlycalculatetheeffectofagivensurfaceroughnessonthe frictionalresistanceofaflatplateofshiplength.Althoughthiscan beseenasareasonableassumption,sincethesurfaceroughness isnotexpectedtosignificantlyaffectthepressuredrag,itisstill worthinvestigatingthephenomenonbymeansofafullynonlinear method,suchasCFD,toinvestigatetheroughnesseffectof biofoul-ingontheresistancecomponentsofashipindetail.Inaddition, atypicalCFDworkcouldtaketheeffectofspatialdistributionof foulingonthetotaldragofthehullintoconsideration.

Tothebestofthisauthor’sknowledge,nospecificCFDmodel existstopredicttheroughnesseffectofbiofoulingonship resis-tance.Theaimofthepresentpaperisthereforetofillthisgapby employingamodifiedwall-functionintheCFDsoftwarepackage andtoinvestigatetheroughnesseffectofbiofoulingonthe resis-tancecomponents.Theproposedapproachenablestheprediction

oftheresistancecoefficientsoffull-scale3Dshiphullsbearinga typicalcoatingandarangeoffoulingconditions.

Themainadvantageoftheproposedmodelisthatitenablesthe useofasimpleroughnesslengthscaletopredicttheeffectof bio-foulingonfrictionalresistanceofaflatplateofshiplength,similar tothatofDemireletal.[39]aswellasontheresistancecomponents andeffectivepowerofafull-scaleship.

Inthisstudy,theexperimentaldataofSchultzandFlack[41] wereusedtoestablisha suitable roughnessfunctionmodel for differentfoulingconditions.Correspondingroughnessheightsof Schultz[6]representingdifferentfoulingconditionswereusedto modelthedifferentsurfaces.Thisroughnessfunctionmodelwas thenemployedinthewall-functionoftheCFDsoftwarepackage STAR-CCM+.

Followingthis,atypicalcasestudywasperformedthroughCFD simulationsoftowingtestsinvolvingaflatplateoflength232.5m, representingtheKrisoContainerShip(KCS),withdifferentsurface conditionsatdifferentservicespeeds.Frictionalresistance coeffi-cientswerecomputedandtheincreasesinthefrictionalresistance oftheflatplaterepresentingKCSduetosuchfoulingconditions werepredicted.

UnsteadyRANS CFDsimulations of the roughness effects of marinecoatingsandbiofoulingonthefull-scale3DKCSwerethen performedusingthe sameCFDmodel. Afull-scale 3DKCS hull appendedwitharudderwasusedduetotheexistenceofavailable experimentaldataforcomparisonpurposes,andinordertoenable areasonablecomparison. Themodelwasfirsttowed insmooth conditionsatadesign speedof24knotsandtheresultingtotal resistancecoefficientwascomparedandvalidatedwiththetotal resistancecoefficientextrapolated usingtheexperimentaldata. Followingthis, typical parametriccase studies wereperformed atadesign speedof 24knotsandaslowsteamingspeed of19 knots.Theseinvolvedchangingthesurfaceconditionsby employ-ingtheroughnessfunctionmodeland correspondingroughness lengthscalesproposed,torepresentatypicalcoatingandarange offoulingconditions,whileholdingtheotherparametersconstant. Frictional,residuary and total resistancecoefficientsof theKCS weredirectlycomputedwhereasthewaveresistancecoefficients werecalculatedusingtheformfactoroftheKCS.Theeffectofhull roughnessonthewavesystemswerealsoinvestigated.Moreover, theincreasesintheeffectivepoweroftheKCSduetosuchsurface conditionswerepredictedusingthepresentresults.

Thepresentresults,obtainedusingflat-plateCFDsimulations and using full-scale 3D KCS model were compared with each otherandwiththoseobtainedusingthesimilaritylawanalysisof Granville[3].

Thispaperisorganisedasfollows:Theroughnessfunctionsused torepresentarangeofbiofoulingconditionsarepresentedin Sec-tion2,while a newwall-function formulation is proposedand detailsofthenumericalsetuparecoveredinSection3.InSection4, theCFDresultsobtainedusingflat-plateandfull-scale3Dapproach werepresentedtogetherwiththeresultsobtainedusingthe sim-ilaritylawanalysis.Finally,theresultsofthestudyarediscussed inSection5,alongwithrecommendationsforfutureavenuesof research.

2. Roughnessfunctions

Thevelocityprofileinthelog-lawregionoftheturbulent bound-arylayercanbedefinedby

U+= 1ln(y+)+B−U+ (1)

inwhich_␬isthevonKarmanconstant,y+_{isthenon-dimensional}

Fig.1.Roughnessfunctionvs.roughnessReynoldsnumbers[6].

canrepresentthechangeinthevelocityprofileduetoroughness usingU+_{,andthevelocityprofilecanbedefinedbysimply}

sub-tractingU+_{fromthesmoothvelocityprofile.Itshouldbeborne}

inmindthatU+_{simplyvanishesinthecaseofasmooth}

condi-tion.U+_{valuesaretypicallyobtainedexperimentally,sincethere}

isnouniversalroughnessfunctionmodelforeverykindof rough-ness.ReferencemaybemadetoJiménez[42]foracomprehensive reviewonroughwallturbulentboundarylayers.

SchultzandFlack[41]determinedtheroughnessfunctionsfor threedimensionalroughsurfacessimilartothoseusedby Shock-lingetal.[43].Schultz[6]proposedthattheroughnessfunction behaviourofarangeoffoulingconditionsfollow theroughness functionsofSchultzandFlack[41]andShocklingetal.[43],based onhispreviousworkpresentedinSchultz[15].Thisisa reason-ableassumption,sincetheroughnessfunctionsofrealsurfacesare expectedtoshowbehaviourthatisbetweenthemonotonic Cole-brookandinflectionalNikuradsetyperoughnessfunctions,suchas thosepresentedbySchultzandFlack[41]andShocklingetal.[43], asshowninFig.1.Inaddition,Schultz[6]presentedtheequivalent sandroughnessheightsforarangeofcoatingandfouling condi-tionstogetherwiththeNSTM(NavalShips’TechnicalManual)[44] ratingandaveragecoatingroughness(Rt50)basedonhisextensive

experimentsincludingSchultz[15](Table1).

Inthispaper’sstudy,theroughnessfunctionvaluesofSchultz andFlack[41]showninFig.1wereusedtodeveloparoughness functionmodeltobeemployedintheCFDsoftwaretorepresent thecoatingandfoulingconditionsgivenbySchultz[6],asshown inTable1.

Thepresentpredictionsweremadebasedontheassumptions that the given fouling conditions can be represented by these roughnessfunctionsandroughnesslengthscales.Schultz[6] vali-datedtheseassumptionsandthismethodbycomparinghisresults

Table1

Arangeofrepresentativecoatingandfoulingconditions[6]. Descriptionofcondition NSTMratinga _k

s(␮m) Rt50(␮m)

Hydraulicallysmoothsurface 0 0 0

TypicalasappliedAFcoating 0 30 150

Deterioratedcoatingorlightslime 10–20 100 300

Heavyslime 30 300 600

Smallcalcareousfoulingorweed 40–60 1000 1000

Mediumcalcareousfouling 70–80 3000 3000

Heavycalcareousfouling 90–100 10000 10000

a_NSTM[44].

withotherstudiessuchasHundleyandTate[45]andHaslbeckand Bohlander[12],documentingtheeffectsofcoatingsandbiofouling onshippoweringthroughfull-scaletrials.

An appropriate roughness function model was fitted to the roughnessfunctionvaluesofSchultzandFlack[41],giveninEq. (5).Thisroughnessfunctionmodelispresentedsuchthatitisin theformofthebuilt-inroughnessfunctionmodelofSTAR-CCM+ forapplicationconvenience.

3. Numericalmodelling

3.1. Mathematicalformulation

An Unsteady Reynolds-Averaged Navier-Stokes (URANS) methodwasusedtosolvethegoverningequationsinthisstudy. Thesemassandmomentumconservationequationsweresolved bythecommercialCFDsoftwareSTAR-CCM+.Theaveraged con-tinuityand momentumequations for incompressible flowsare givenintensornotationandCartesiancoordinatesbyEqs.(2)and (3)

∂

∂

∂

∂

∂

∂

u¯iu¯j+uiuj

=−

∂

∂

+

∂

∂

(3)

where␳isdensity, ¯uiistheaveragedCartesiancomponentsofthe velocityvector,uiujistheReynoldsstressesandpisthemean pressure. ¯ij arethemean viscousstresstensorcomponents,as showninEq.(4)

¯

ij=

∂

∂

+

∂

∂

(4)

inwhichisthedynamicviscosity.

Thesolverusesafinitevolumemethodwhichdiscretisesthe governingequations.Asecondorderconvectionschemewasused forthemomentumequationsandafirstordertemporal discreti-sationwasused.Theflowequationsweresolvedinasegregated manner.The continuity and momentumequations were linked withapredictor-correctorapproach.

TheSST(ShearStressTransport)k-␻turbulencemodelwasused inordertocompletetheRANSequations,whichblendsthek-␻

modelnearthewalland thek-␧modelinthefarfield.For the flat-platesimulations,theCourant-Frederich-Lewis(CFL)number wasalwaysheldatvalueslessthanunitytoensurethe numeri-calstability.ItisofnotethattheITTC[36]recommendtheuseof

t=0.005∼0.01L/V,whereLisshiplengthandVisshipspeed,for theselectionofthetimestep.However,thetimestepsizeofthe simulationsoftheKCShullwassetto∼0.0005LBP/V,whichisten

Fig.2.TheproposedCFDroughnessfunctionmodeltogetherwiththeroughness functions.

3.2. Proposedwall-functionapproachforfoulingconditions

Anappropriateroughnessfunctionmodelforarangeof repre-sentativecoatingandfoulingconditionsforuseinSTAR-CCM+is proposedbyEq.(5).

U+=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

0→k+<3

1

ln

sin

2

log(k+/3) log(5)

→3<k+<15

1

ln

0.26k+

(5)

ShownforcomparisoninFig.2istheproposedroughness func-tionmodelgivenby(5)and roughnessfunctionsofSchultzand Flack[41].

Theproposedmodelforarangeofrepresentativecoatingand biofoulingconditionsinthispaperisinasimilarformtothebuilt-in wallfunctionofSTAR-CCM+intermsofflowregimes.Thatistosay, theproposedroughnessfunctionmodelandthewall-lawhave3 flowregimes,namelyahydraulicallysmoothregime,a transition-allyroughregimeandafullyroughregime,whicharesimilarto thoseproposedbyCebeciandBradshaw[34]basedonNikuradse’s data[35].ItisevidentfromFig.2thatanexcellentagreementis achievedinthefullyroughregime whileareasonable matchis obtainedinthetransitionallyroughregime.Generalinformation aboutthewall-functionapproachanddetailsoftheapplicationof roughnessfunctionsthroughwall-functionscanbefoundin[39].

3.3. Geometryandboundaryconditions

ThegeometryoftheplaterepresentingtheKCSisshowninFig.3. Theboundaryconditionsofthesimulationswerechosento repre-senttheplatebeingcompletelysubmergedinaninfinite ocean, withsimilaritytothefullscalepredictionsimulationspresentedin [39].Theboundaryconditionsandthepositioningofthe bound-arieswerethereforechosentobesimilartothosegivenin[39],as showninFig.4.

TheKRISOContainerShip(KCS)appendedwitharudderwas usedinthispaper’sfull-scale3Dsimulationssinceexperimental dataforthishullispubliclyavailableandalargebodyofrelated CFDstudiesexistintheliterature(e.g.Larssonetal.[46],Zhang [47],Castroetal.[37],Carricaetal.[48]andTezdoganetal.[49]).

Fig.3.TheplatesrepresentingtheKCS.

Fig.4.a)profileviewofthedomainandb)topviewofthedomain,showingthe dimensionsandboundaryconditions[39].

Table2

PrincipalparticularsoftheKCS,adaptedfromTezdoganetal.[49]andKimetal. [50].

Lengthbetweentheperpendiculars(LBP) 230.0m

Lengthofwaterline(LWL) 232.5m

Beamatwaterline(BWL) 32.2m

Depth(D) 19.0m

Designdraft(T) 10.8m

Wettedsurfacearea 9498m2

Displacement() 52030m3

Blockcoefficient(CB) 0.6505

DesignSpeed 24knots

Froudenumber(Fr) 0.26

Additionally,theKCShasaverysimilarshapetocommercial con-tainerships,meaningtheresultswillgiveanindicationofhow foulingeffectstheperformanceofrealcommercialcontainerships. Theprincipalparticulars,bodyplansandsideprofilesofthe full-scaleKCSmodelaregiveninTable2(adaptedfromTezdoganetal. [49]andKimetal.[50]),andFig.5[50],respectively.

The boundaryconditions of thesimulations were chosento representthefull-scaleKCSmodelbeingtowedinadeepwater condition.Fig.6depictsanoverviewofthedomainwiththeKCS modelandtheselectedboundaryconditions.

Fig.5.BodyplanandsideprofilesoftheKCSmodel[50].

Fig.6. Anoverviewofthedomainwiththeselectedboundaryconditions.

shouldbekeptinmindthattheinitialflowvelocityatallinlet con-ditionswassettothevelocityoftheflatwave,i.e.ashipspeedof 24knots,inthenegativex-direction.Theselectionofthevelocity inletforthetopandsideofthedomainthereforeenablestheflow atthetopandsideofthedomaintobeparalleltotheoutlet bound-ary,whichpreventsreflectionsfromtheseboundaries.Inaddition, therepresentationofthedeepwaterandinfiniteairconditionswas facilitatedbytheuseofavelocityinletboundaryconditionforthe topandbottomboundaries.TheKCShullitselfhasano-sliprough wallconditiontorepresenttheroughnessonthehull.

Anothercriticalselectionisthepositioningoftheboundaries, especiallythedownstreamoutletboundaryandtheupstreaminlet boundary.Theinletisplacedat∼1.5LBPlengthsupstreamandthe

outletboundaryisplacedat∼2.5LBPlengthsdownstream,toensure

boundaryindependentsolutionsareproduced.Similarly,thetop islocatedat∼1.5LBPandthebottomandthesidearepositioned

at∼2.5LBPawayfromtheKCShull.Itisofnotethattheselection

oftheseboundaryconditionsand thepositioningofthe bound-ariesweremadebasedontherecommendationsandapplications reportedinCD-ADAPCO[51].Thelocationsoftheboundariesare showninFig.7.

ItshouldbenotedthattheVOFwavedampingcapabilityofthe softwarewasappliedtotheoutletandallvelocityinletboundaries, namelytheinlet,bottom,topandside,withadampinglengthequal to∼1LBP,topreventreflectionsfromtheseboundaries.

3.4. Meshgeneration

Acut-cellgridwithprismlayermeshonthewallswas gener-atedusingtheautomaticmeshgeneratorinSTAR-CCM+.Additional refinementswereappliedtogivefinergridsinthecriticalregions, suchastheareaimmediatelyaroundtheplate,theareasaround thetrailingand leadingedges,andthetopedgeof theplateas

Fig.7. Thepositionsoftheboundaries.(L:lengthoftheshipbetween perpendicu-lars).

wellastheareaimmediatelyaroundthehullandrudder,thearea wherethebowencountersthefreesurface,theareawherewater breakswiththehullstern,andtheareainthewakegeneratedbythe ship.Themeshgenerationwasachievedusingsimilartechniquesto thoseexplainedin[39].Also,convergencetestswereperformedto ensuregrid-independentmeshconfigurations,aswellastopredict theuncertaintyoftheCFDsimulations.

Table3

Totalcellnumbersforflatplate.

SurfaceCondition(ks[␮m]) Cellnumbers

ks=0,ks=30,ks=100,ks=300 5.5×106

ks=1000 5.28×106

ks=3000 4.5×106

ks=10000 4×106

Table4

Totalcellnumbersforthefull-scaleKCShull.

SurfaceCondition(ks[␮m]) Cellnumbers

ks=0,ks=30,ks=100,ks=300 4.09×106

ks=1000 4.00×106

ks=3000 3.70×106

ks=10000 3.58×106

inTable4forthefull-scalesimulationsoftheKCShull.Near-wall meshgenerationmustbeperformedwithcaresincethisisdirectly relatedtothehullroughnessduetomarinecoatingsandbiofouling. Theprismlayerthicknessandprismlayernumberswere,therefore, determinedsuchthaty+_{isalwayshigherthan30,andhigherthan}

k+_{,asperCD-ADAPCO[51]’ssuggestion.}

Fig.8showscross-sectionsofthemesheddomainwhereasFig.9 showsthevolumemeshontheKCShullandrudder.Itisofnote that,fromthispointonward,thefiguresshowthewholesectionsas ifthereisnosymmetricalboundary,owingtothevisualtransform featureofthesoftware.

Fig.8demonstratesthecross-sectionfromthecentrelineofthe hullandthefreesurfaceandshowsonlyaportionofthe

cross-Table5

CFresultsatdifferentmeshconfigurationsfortheflat-plateKCScase.

Meshconfiguration TotalNo.ofCells CF(CFD)

Coarse 2.2×106 _0.0020086

Medium 3.3×106 _0.0020145

Fine 5.5×106 _0.0020222

sectionsforvisualconvenience,sincethedomainisratherlarge. TherefinementstocapturethefreesurfaceandKelvinwakeare clearlyvisibleinFig.8.

Fig.9clearlyshowstheeffectsofadditionalrefinementsonthe KCShullandrudder,especiallytheonesappliedtothefreesurface, bowandsternregions.

4. Results

4.1. Gridsensitivitystudy

Systematicstudieswereperformedusingtheflatplatescovered withheavyslimeandusingtheKCShullappendedwitharudder withasmoothsurfacecondition,inordertocarryoutagrid sensi-tivitystudyandtopredicttheCFDuncertainties.Inordertoobserve theeffectofcellnumbersonthekeyvariable,(CFinflat-platecase

andCTinfull-scaleKCShullcase),thedomainwasdiscretisedin

threedifferentresolutionsandthesimulationswererunforeach configuration.Thegridrefinementfactor,r,waschosentobe√2as usedbyTezdoganetal.[49].

Fig.9.Volumemeshesonthea)bow,b)sternoftheKCShullandrudder.

Table6

CTresultsatdifferentmeshconfigurationsforthefull-scaleKCSat24knots(Relative

Difference,D(%),isbasedontheCTvalueextrapolatedusingtheexperimentaldata

ofKimetal.[50]).

Meshconfiguration TotalNo.ofCells CT(CFD) D(%)

Coarse 1.07×106 _{0.002120 2.30}

Medium 2.04×106 _{0.002113 1.94}

Fine 4.09×106 _{0.002097 1.17}

Thefrictionalresistancecoefficientsforeachmesh configura-tionwerecomputedandaregiveninTable5fortheflat-plateKCS case.

Similarly,thetotalresistancecoefficientsforeachmesh config-urationwerecomputedatadesignspeedof24knotsandaregiven in

Table6forfull-scalesimulationofKCShull.

FromTables5and6itisevidentthatthevariationindragwith a√2refinementratioarebelow∼0.8%andthatthetotalresistance coefficientwasover-predictedby1.17%.Therefore,thefinemesh configurationwasselectedinallsubsequentcomputations.

4.2. Verificationstudy

Averificationstudyshouldbecarriedouttoshowthecapability oftheproposedmodelandthesoftwareforparticularcalculations. TheGridConvergenceIndex (GCI)MethodbasedonRichardson extrapolation[52,53]wasusedinthispaper’sworkfor discretisa-tionerrorestimationasdescribedbyCelik,Ghia[54].

Theapparentorderofthemethod,pa,iscalculatedby

pa= 1 ln(r21)|

ln|ε32/ε21|+q(pa)| (6)

q(pa)=ln

r₂₁pa−s r₃₂pa₋s

(7)

s=1·sign

(8)

wherer21andr32arerefinementfactors,i.e.√2inthisstudy,and

␧32=␾3−2,ε21=␾2−1,kisthekeyvariable,i.e.CFandCTin

thiscase,onthekth_grid.

Theextrapolatedvaluesareobtainedby

21_ext=

/

Table7

CalculationofthediscretisationerrorforCFvaluesoftheflatplate.

CF

r21,r32 √2

1 0.0020222

2 0.0020145

3 0.0020086

pa 0.76829

ext21 0.0020474

ea21 0.38077%

eext21 1.2327%

GCIfine21 1.5601%

Table8

CalculationofthediscretisationerrorforCTvaluesofthefull-scaleKCS.

CT

r21,r32 √2

1 0.002097

2 0.002113

3 0.002120

pa 2.3853

ext21 0.0020846

ea21 0.76299%

eext21 0.59698%

GCIfine21 0.7418%

Theapproximateandextrapolatedrelativeerrorsarecalculated usingthefollowingequations,respectively.

e21_a =|1−2

1 |

(10)

e21

ext=| 12

ext−1

12

ext

| (11)

Thefine-gridconvergenceindexiscalculatedby

GCI_fine21 =1.25e21a

r₂₁p −1 (12)

TherequiredparameterswerecalculatedforCFandCTvalues

andarepresentedinTable7fortheflatplateandinTable8forthe full-scaleKCS.

AscanbeseenfromTables7and8,numericaluncertaintiesof 1.56%,0.74%werecalculatedforthecomputedCFandCT values

Fig.10. WaveprofilealongKCSat24knots.

4.3. Validationstudy

TheavailableexperimentaldatafortheKCSwasusedto val-idatetheCFDapproachusingthesmoothcondition.Duringthe towingtanktestsconductedbyKimetal.[50],theresiduary resis-tancecoefficientfora1/31.6scalemodeloftheKCSwasfoundtobe 7.250×10−4_{,atthecorrespondingmodelspeedforthefull-scale}

speedof24knots(Fr=0.26).Giventhattheresiduaryresistance is a function of Froude number, the residuary resistance coef-ficient of the full-scale KCS model is assumed tobe the same (CR=7.250×10−4)andthefull-scale frictionalresistance

coeffi-cient,CF,is calculatedtobe1.347×10−3,usingthe“ITTC1957

model-shipcorrelationline”atthecorrespondingReynolds num-ber.Thetotalresistancecoefficientofthefull-scaleKCSmodelis thereforepredictedtobe2.0725×10−3_{.Itisofnotethatform}

fac-tor,(1+k),wasnottakenintoaccountwhileextrapolating. Table6demonstratesthetotalresistancecoefficientscomputed byCFDandextrapolatedusingtheexperimentaldataofKimetal. [50],atashipspeedof24knots.Ascanbeseenfrom

Table6,thecomputedtotalresistancecoefficient,CT,isin

excel-lentagreementwiththis extrapolateddata,withadifferenceof only∼1.17%.ThisCFDapproachcanthereforebeclaimedtobe val-idatedandcanbeusedforfurtherinvestigations.Thismodelwas thereforeusedthroughoutallthecases.

ShownforcomparisoninFig.10isthecomputedwaveprofile alongthehullsurfaceoftheKCStogetherwiththewaveprofile measuredduringthetowingtanktestsconductedbyKim etal. [50].Fig.16showstheglobalwavepatternaroundthehullsurface oftheKCSwhereasFig.17showsthewaveprofilealongalinewith constanty=0.1509.TheKelvinwakegeneratedbytheshipisclearly visibleinFig.16.

ItisevidentfromFigs.10,16(smoothcondition)andFig.17 (smooth condition) that a very good agreement is achieved betweenthecurrentCFDmodelandtheexperimentaldataofKim etal.[50],aswellastheotherCFDsimulationsperformedbyother researchers(e.g.Carricaetal.[55]andCastroetal.[37]).

4.4. Effectsofhullroughness

Beforeinvestigatingtheeffectsofhullroughnessonthe resis-tancecomponentsofaship,itwouldbetimelytodescribethese componentsindetail.Thetotalresistance(drag)ofaship,RT,is mainlycomposedoftwocomponents;thefrictionalresistance,RF,

andtheresiduaryresistance,RR,asgivenby(13).

RT=RF+RR (13)

Thefrictionalresistancearisesduetoshearstressesonthehull surfacewhiletheresiduaryresistanceisthepressurerelateddrag whichconsistsofthewaveresistance,RW,andviscouspressure

Table9

ComparisonofthecomputedCFvaluesusingdifferentmethodsatfullscaleat24

knots(Re=2.89×109_{)(RelativeDifference,D(%)isbasedonGranvillemethod).}

SurfaceCondition(ks[␮m]) CF×103

CFD-KCShull Granville CFD-Flatplate

Result D(%) Result D(%)

0 1.421 5.52 1.347 1.351 0.3

30 1.577 7.32 1.469 1.496 1.84

100 1.840 5.01 1.752 1.750 −0.11

300 2.120 3.62 2.046 2.022 −1.17

1000 2.514 2.41 2.455 2.401 −2.20

3000 3.014 2.49 2.941 2.886 −1.87

10000 3.741 2.47 3.651 3.571 −2.20

resistance,RVP,oftheship.Thisrelationcanbeshowninamore explicitwayby(14),

RT=RF+RVP+RW (14) whereitisassumedRVP=kRF.Eq.(14)canbedefinedasfollows:

RT=RF+kRF+RW=(1+k)RF+RW (15) If these resistance components are non-dimensionalised by dividingeachtermbythedynamicpressureandwettedsurface areaoftheshiphull,theresistancecoefficientscanbedefinedas follows:

CT=CF+CR (16)

CT=CF+CVP+CW (17)

CT=(1+k)CF+CW (18) whereCTisthetotalresistancecoefficient,CFisthefrictional

resis-tance coefficient, CR is theresiduary resistance coefficient,CVP

viscouspressureresistancecoefficient,CWisthewaveresistance

coefficient.

4.4.1. Frictionalresistance

Havingperformedpredictionstudiesusingbothflatplates rep-resentingtheKCSandthefull-scaleKCSmodelitself,itwouldbe interestingtoalsocomparethesedifferentmethods.Therefore,the followingsectionaimstocompareanddiscusstheresultsobtained usingthedifferenttechniques.Thesetechniquesarefull-scale3D CFDsimulationsof theKCShull (referredtoas‘CFD-KCShull’), flatplateCFDsimulations(referredtoas‘CFD-Flatplate’)andthe similaritylaw scalingprocedureofGranville [9] (referredtoas ‘Granville’).

Table10

ComparisonofthecomputedCFvaluesusingdifferentmethodsatfullscaleat19

knots(Re=2.29×109_{)(RelativeDifference,D(%)isbasedonGranvillemethod).}

Surfacecondition(ks[␮m]) CF×103

CFD-KCShull Granville CFD-Flatplate

Result D(%) Result D(%)

0 1.452 4.82 1.385 1.386 0.07

30 1.559 5.86 1.473 1.485 0.81

100 1.834 4.57 1.754 1.750 −0.23

300 2.115 3.30 2.047 2.022 −1.22

1000 2.509 2.10 2.457 2.401 −2.28

3000 3.006 2.11 2.944 2.886 −1.97

10000 3.733 2.14 3.655 3.578 −2.10

resultsobtainedusingGranvillemethodarelistedin thetables. SincetheexperimentalCFvaluesarenotavailable,onlytheresults

fromCFDandthesimilaritylawanalysisaregivenforthefrictional dragcoefficientsinthetable.

AsTable9and10jointlyshow,theresultsobtainedusingboth ofthepresentCFDmethodsusedinthispaperagreedwiththe resultsobtainedusingGranville’smethod,withdifferencesofless than∼7%.ThisindicatesthatthepresentCFDmodelstandsasa suitabletechniquewithwhichtopredictroughnesseffectsonthe frictionalresistanceofflatplatesofmodel-scaleandfull-scaleship lengths,andoffull-scale3Dshiphulls.Thephysicaladequacyof theCFDapproach wasthereforedemonstrated. Theresultsand comparisonsofHaaseetal.[40]alsosupporttheapplicabilityof thegeneralCFDapproachtoaccuratelyaccountfortheroughness effectsonthefrictionalresistanceofflatplatesandships.

ItisinterestingtonotethattheallCFvaluesobtainedatship

speedsof 24 knotsand 19 knotsusing “CFD-KCShull”method arehigherthanthoseobtainedusing“CFD-Flatplate”methodand thoseobtainedusing“Granville”method.

TheincreaseinthefrictionalresistanceoftheKCSdueto dif-ferentsurfaceconditionswithrespecttothoseofahydraulically smooth,predictedusingthedifferenttechniques,aredemonstrated inTable11andFig.11for24knotsandinTable12andFig.12for 19knots.

Table11

Comparisonofthecomputed%CFvaluesusingdifferentmethodsatfullscaleat

24knots(Re=2.89×109_).

Descriptionofcondition %CF

CFD-KCShull CFD-Flatplate Granville

Hydraulicallysmoothsurface – – –

TypicalasappliedAFcoating 10.9 10.7 9 Deterioratedcoatingorlightslime 29.4 29.5 30

Heavyslime 49.2 49.7 51.8

Smallcalcareousfoulingorweed 76.9 77.7 82.2 Mediumcalcareousfouling 112.1 113.6 118.3 Heavycalcareousfouling 163.2 164.3 171.0

Table12

Comparisonofthecomputed%CFvaluesusingdifferentmethodsatfullscaleat

19knots(Re=2.29×109_).

Descriptionofcondition %CF

CFD-KCShull CFD-Flatplate Granville

Hydraulicallysmoothsurface – – –

TypicalasappliedAFcoating 7.4 7.1 6.3 Deterioratedcoatingorlightslime 26.3 26.2 26.6

Heavyslime 45.6 45.9 47.8

Smallcalcareousfoulingorweed 72.8 73.3 77.4 Mediumcalcareousfouling 107.1 108.2 118.3 Heavycalcareousfouling 157.1 158.2 163.9

TheincreaseintheCFvaluesoftheKCSduetoaheavyslime

con-ditionatashipspeedof24knotswaspredictedtobe∼49%,∼50% and∼52%,byCFD-KCShull,CFD-FlatplateandGranville’smethods respectively,whereasthesevaluesalteredto∼163%,∼164%and ∼171%respectivelyforaheavycalcareousfoulingcondition,ascan beseeninTable11andFig.11.

TheresultspresentedinTable12andFig.12indicatethatthe increaseinCFoftheKCSduetoheavyslimeataslowsteaming

shipspeedof19knotswaspredictedtobe∼46%,∼46%and∼48%, byCFD-KCShull,CFD-FlatplateandGranville’smethods respec-tively,whereasthesevaluesalteredto∼157%,∼158%and∼164% respectivelyforaheavycalcareousfoulingcondition.

Fig.12.EstimationofthepercentageincreaseinthefrictionalresistanceoftheKCSduetodifferentsurfaceconditionsat19knots(Re=2.29×109_).

Theresultsobtainedusing“CFD-KCShull”methodpresentedin Tables11and12indicatethattheincreaseinCFduetothehull

roughnessofatypicalantifouling(AF)coatingis10.9%at24knots and7.4%at19knots,whereastheincreaseinCFduetobiofoulingis

predictedtobedramatic,whichwouldleadtoadrasticincreasein thefuelconsumptionandhenceCO2emissions.Theincreaseinthe

frictionalresistanceoftheKCSduetoadeterioratedcoatingorlight slimesurfaceconditionwaspredictedtobe29.4%atashipspeedof 24knotsandtobe26.3%atashipspeedof19knots.Thesevalues became49.2%and45.6%whencalculatingtheincreaseinCFdue

toaheavyslimecondition.Calcareousfoulingcausessignificant

increaseinCFvalues,rangingfrom∼77%to∼163%at24knotsand

∼73%to∼157%at19knots,dependingonthetypeofcalcareous foulingandshipspeed.

The resultspresented are in accordance with theresults of Schultz[15].Itshouldbeborneinmindthattheincreasedueto roughnessofdifferentmarinecoatingsarestillofimportancewhen consideringthefuelconsumptionofaship.

4.4.2. Residuaryandwaveresistance

Residuaryresistancecoefficient,CR,valuesofthefull-scaleKCS model weredirectly predictedby thepresent CFDsimulations.

Fig.14.CWvaluesofthefullscaleKCSfordifferentsurfaceconditionsatshipspeedsof19(Re=2.29×109)and24knots(Re=2.89×109).

Table13

Computed%CWvaluesatfullscaleat24knots(Re=2.89×109)andat19knots

(Re=2.29×109_).

Descriptionofcondition 24knots 19knots

Hydraulicallysmoothsurface – –

TypicalasappliedAFcoating −4.4 −5

Deterioratedcoatingorlightslime −15.2 −17.9

Heavyslime −23.2 −30.1

Smallcalcareousfoulingorweed −32.2 −43.8

Mediumcalcareousfouling −43 −57.9

Heavycalcareousfouling −55.8 −72.3

Fig.13showsthecomputedresiduaryresistancecoefficientsofthe KCShullobtainedfor7differentsurfaceconditionsatshipspeeds of19knots(Re=2.29×109_{)and24knots(Re=2.89×10}9_).

Surprisingly, theresiduaryresistancecoefficients showedan increasingtrendwithincreasingfoulingratesat19knotswhereas ittendedtodecreasewithincreasingfoulingratesat24knots.This isduetothefactthatresiduaryresistancecomprisesviscous pres-sureresistanceandwaveresistanceasshownbyEq.(14).These differenttrendscanbeattributedtothefactthatthecontribution oftheviscouspressureresistancebecomesmoreimportantthan waveresistanceatlowerspeeds.Inotherwords,athigherspeeds, thewave-makingresistancebecomesdominantduetowave gen-eration.

Sinceviscouspressureresistanceisafunctionoffrictional resis-tance,itisappropriatetodecomposetheresiduaryresistanceand investigatetheeffectofhullroughnessonthewaveresistance,RW,

usingEq.(15)andtakingtheformfactor 1+k=1.1[37].Fig.14 showsthecalculatedwaveresistancecoefficients,CW,oftheKCS hullobtainedfor7differentsurfaceconditionsatshipspeedsof19 knots(Re=2.29×109_{)and24knots(Re=2.89×10}9_).

AscanbeseenfromFig.14,thewaveresistancecontinuously decreased withincreasing fouling rates. Table 13 and Fig. 15 demonstratethechangeinthewaveresistanceoftheKCSdueto differentsurfaceconditionswithrespecttothesmoothcondition atadesignspeedof24knotsandataslowsteamingspeedof19 knots,respectively.

TheresultspresentedinTable13andFig.15indicatethatthe reductionintheCWoftheKCSduetoatypical,asappliedAFcoating

werepredictedtobe4.4%and5%whereasthoseduetoa deterio-ratedcoatingorlightslimewerecomputedtobe15.2%and17.9% atshipspeedsof24knotsand19knots,respectively.Itwasshown thattheeffectofheavyslimeontheKCShullcausedareductionin theCWof23.2%at24knotsand30.1%at19knots.Thecalcareous

foulingwoulddecreaseCWbyupto55.8%at24knotsand72.3%

at19knots.Aninterestingpointtonoteisthattheeffectofa par-ticularfoulingconditiononthewaveresistanceoftheKCSismore dominantatlowerspeeds.Thiscanbeattributedtothefactthat thecontributionoftheviscouseffectsbecomesmoreimportantat lowerspeeds.

Fig.16comparestheglobalwavepatternsaroundthehull sur-faceoftheKCSinsmoothandheavycalcareousfoulingconditions at24knots,whileFig.17showsthewaveprofilealongalinewith constanty=0.1509.

It isseenfromthecomparisonin Figs.16 and17 thatwave amplitudesappeartobereducedbyroughnesseffects.Thisisan indicationoftheeffectofviscosityonthewavesystems.The result-ingfreesurfaceelevationaroundtheKCShullwasrecordedtorange from −1.406mto3.357mfor smooth condition,and −1.345m to 2.266 for heavy calcareous fouling condition (Fig. 16). This reductionofthewavesystemisinagreementwiththecomputed reductioninwaveresistancecoefficientsshowninFigs.14and15 andTable13.

AscanbeseenfromFig.17thatthebowwaveprofilescomputed forthesmoothandheavycalcareousfoulingconditionsareontop ofeachotherwhereasthewaveprofilesdeviatefromeachotherin thewakeregion.Thisdrasticreductionofthesternwavesystem andtheobviousviscouseffectsonthewaveresistanceandwave systemsisconsistentwiththefindingsofRaven,VanderPloeg[56].

4.4.3. Totalresistanceandeffectivepower

Inordertorevealtheeffectofbiofoulingonthefuel consump-tion,theincreasein thetotalresistanceandhencetheeffective poweroftheKCSwerecalculated.Anincreaseinthetotal resis-tancewouldincreasetheeffectivepower,PE,ofaship,whichisthe necessarypowertomoveashipthroughwater.PEisrelatedtothe totalresistance,RT,andshipspeed,V,whichisdefinedbyEq.(19).

Fig.15.EstimationofthepercentagechangeinthewaveresistanceoftheKCSduetodifferentsurfaceconditionsat24knots(Re=2.89×109_).

Table14

Computed%CT,PEvaluesatfullscaleat24knots(Re=2.89×109)andat19

knots(Re=2.29×109_).

Descriptionofcondition 24knots 19knots

Hydraulicallysmoothsurface – –

TypicalasappliedAFcoating 7.1 5.9

Deterioratedcoatingorlightslime 18.1 21.2

Heavyslime 30.8 37.0

Smallcalcareousfoulingorweed 49.1 59.5

Mediumcalcareousfouling 72.6 88.2

Heavycalcareousfouling 107.5 130.9

where

RT= 1 2SCTV

2 ₍₂₀₎

where␳isthedensityofwater,Sisthewettedsurfacearea,CTis thetotalresistancecoefficient.

Wecanthenre-writeEq.(19)as

PE=1 2SCTV

3 ₍₂₁₎

TheincreaseinPEduetotheeffectoffoulingcanbeexpressed by

%PE=

CT,rough−CT,smooth

CT,smooth ×

100 (22)

similartothatusedbyTezdoganetal.[49].

Totaldragcoefficientvaluesofthefull-scaleKCSmodelwere directlypredictedbythepresentCFDsimulations.Fig.18shows thepredictedtotalresistancecoefficientsoftheKCShullobtained for 7 different surface conditions at ship speeds of 19 knots (Re=2.29×109_{)and24knots(Re=2.89×10}9_).

AscanbeseenfromFig.18,theeffectivepowercontinuously increasedwithincreasingfoulingrates.Table14andFig.19 demon-stratetheincreaseinthetotalresistanceandhenceintheeffective poweroftheKCSduetodifferentsurfaceconditionswithrespect

tothesmoothconditionatadesignspeedof24knotsandataslow steamingspeedof19knots,respectively.

TheresultspresentedinTable14andFig.19indicatethatthe increaseintheCT andPE oftheKCSduetoatypical,asapplied

antifouling(AF)coatingwerepredictedtobe7.1%and5.9%whereas thoseduetoadeterioratedcoatingorlightslimemayincreaseto 18.1%and21.2%atshipspeedsof24knotsand19knots, respec-tively.TheeffectofheavyslimeontheKCShullwascalculatedto causeanincreaseintheCTandPEof30.8%at24knotsand37%at

19knots.ThecalcareousfoulingwouldincreasePEbyupto107.5%

at24knotsand130.9%at19knots.

Aninterestingpointtonoteisthattheeffectofaparticular foul-ingconditionontheeffectivepoweroftheKCSismoredominantat lowerspeeds.Thiscanbeattributedtothefactthatthecontribution ofthefrictionalresistancebecomesmoreimportantthanresiduary resistanceatlowerspeeds.Inotherwords,athigherspeeds,the wave-makingresistancebecomesdominantduetowave genera-tion.Therefore,theeffectofagivenfoulingconditiononthetotal resistanceofashipisgreateratlowtomoderatespeedsthanat higherspeeds[6].

4.4.4. Velocityandturbulentkineticenergydistribution

Velocityandturbulentkineticenergycontoursinsmoothand heavycalcareousfoulingconditionsareshowninFig.20fortheKCS shiphullandinFig.21fortheflatplate.Fig.22demonstratescross sectionsofaxialvelocitycontoursatx/L=0.25,0.5,0.75,depicting theboundarylayeroftheKCShullinsmoothandheavycalcareous foulingconditions.

Fig.16.WavepatternaroundtheKCSforsmoothandheavycalcareousfoulingconditions(V=24knots).

Fig.18.CTvaluesofthefullscaleKCSfordifferentsurfaceconditionsatshipspeedsof19(Re=2.29×109)and24knots(Re=2.89×109).

fromFig.22,theexistenceofheavycalcareousfoulingontheKCS hullcausedincreasesintheboundarylayerthickness,␦,whichis definedasthedistancebetweenthewallandthepointwherethe axialvelocitymagnitudeoftheflowreachestheproportionof0.99 ofthefree-streamvelocity,i.e.U=0.99V,comparedtoanotherwise identicalfouling-freeKCS.Thepresentfindingsareconsistentwith theexperimentaldataofotherresearchers(e.g.SchultzandFlack [41],SchultzandFlack[59],Flacketal.[60],Flacketal.[61],Schultz [62]).

Itisinterestingtonotethatthevelocityandturbulentkinetic energycontoursshowninFigs.20and22areunexpectedlyshowing aspikeatthecentreplaneduetotheimplementationofsymmetry conditionsatthesymmetryplane(seeFig.6).

5. Discussionandconclusions

ACFDmodelforthepredictionoftheeffectofbiofouling on ship resistancehas been proposedand the effect of biofouling onshipresistancewasinvestigatedusingCFD.Anewroughness functionmodel,whichwasdevelopedbasedontheroughness func-tionvaluesofSchultzandFlack[41],wasproposedandemployed inthewall-functionofthesolverandaseriesofunsteadyRANS simulationswerecarriedouttopredicttheeffectofarangeof repre-sentativecoatingandbiofoulingconditionsontheresistancesofflat platesrepresentingtheKCSandthefull-scaleKCSmodelappended witharudder.Firstly,thetotalresistancecoefficientofthefull-scale KCSmodelwasobtainedatashipspeedof24knotsandcompared

Fig.20.a)Velocityandb)turbulentkineticenergycontoursatthemidshipoftheKCShullforsmoothandheavycalcareousfoulingconditions(V=24knots).

withthetotalresistancecoefficientextrapolatedusingthe exper-imentaldataofKimetal.[50]forvalidation.Itwasshownthat thetotalresistancecoefficientwasover-predictedby1.17%witha numericaluncertaintyof∼0.74%.Followingthis,systematic stud-ieswereperformedusingtheflatplatescoveredwithheavyslime andusingtheKCShullappendedwitharudderwithasmooth sur-facecondition,inordertocarryoutagridsensitivitystudyandto predicttheCFDuncertainties.

FullynonlinearunsteadyRANSsimulationstopredicttheeffect ofarangeofrepresentativecoatingandbiofoulingconditionson thefrictionalresistancesofaflatplate,representingtheKCS,and onthefrictional,residuary,waveandtotalresistanceandeffective poweroftheKCS,havebeencarriedoutattwospeeds, correspond-ingtoserviceandslowsteamingspeeds.

TheresultingCFvaluesobtainedusingflatplateCFD

simula-tionsandusing3Dfull-scaleCFDsimulationswerecomparedwith eachotherandwiththoseobtainedusingthesimilaritylaw proce-dureofGranville[9]toexaminetheapplicabilityoftheproposed CFDmodel,sincetheliteraturedoesnotofferanyfull-scale exper-imentalresults.ItwasshownthatthepresentCFDmodelcanbe usedforsimulatingroughnesseffectsonthefrictionalresistance offlatplatesandontheresistanceoffull-scale3Dshiphullsand thatdifferenttypesofroughnesscanbedefinedbymodifyingthe wall-functionofthesoftware.ThismeansthattheCFDmethodcan beusedtopredicttheeffectsofsuchroughnessontheresistance componentsofanyarbitrarybodywithoutbeingobligedtoconduct furtherexperiments, oncethe relationshipbetween the rough-nessfunctionsandroughnessReynoldsnumbersofeachsurface isknown.

Theincreaseintheeffectivepowerofthefull-scaleKCShull werepredictedtobe7.1%atashipspeedof24knotsand5.9%at ashipspeedof19knotsforatypicalasappliedantifouling(AF) coating,18.1% at24 knotsand21.2% at19 knotsfora deterio-ratedcoatingorlightslimeconditionand30.8%at24knotsand

37%at19knotsforaheavyslimecondition.Thesevaluesalteredto 49.1%,72.6%and107.5%at24knotsand59.5%,88.2%and130.9%at 19knotsforsmallcalcareousfoulingorweed,mediumcalcareous foulingandheavycalcareousfouling,respectively.

Animportantfindingofthestudyisthatthewaveresistanceand wavesystemsaresignificantlyaffectedbythehullroughnessand henceviscouseffects,whichiscontrarytothemajorassumption whichproposesthatthewaveresistanceisnotmarkedlyaffected bysurfaceroughnessandviscosity.Thereductioninthewave resis-tanceoftheKCShullinheavycalcareousfoulingconditionwas foundtobe55.8%at24knotsand72.3%at19knots.

Itshouldbeborneinmindthatthisstudy’saimwastopropose arobustCFDmodeltopredictthefoulingimpactonship resis-tance. For this reason,an appropriate representativeroughness function model wasemployed in spite of theslight discrepan-cies betweentheindividual roughness functionvalues and the model, especially inthetransitionally roughregime.Without a doubt,theseconditionsandtheroughnessfunctionsusedinthis papermaynotnecessarilyrepresentalltypesoffoulingconditions, sincetheassumptionsmadearebasedontheobservationsmadein [6,15].Futurepiecesofworkmaybetheinvestigationofthe rough-nessfunctionbehavioursofheterogeneousfoulingaccumulation, asseenonhulls,andaninvestigationintotherangeofapplicability oftheselectedroughnesslengthscaleforthepresentconditions.

Havingshowntheapplicabilityofthewall-functionapproach toaccountfortheroughnesseffectsofAFcoatingsandbiofouling onfull-scale3Dshiphulls,thisapproachcanbeusedtosimulate thiseffectonmorecomplexstructuressuchasonself-propelled shipswitharotatingpropeller.Anotherinterestingfutureplanis toinvestigatetheroughnesseffectsonthetotaldragandeffective powerofshipsofamorerealisticspatialdistributionoffoulingon shiphulls.

Fig.22.CrosssectionscolouredwithaxialvelocitylimitedtoU=0.99Vdepicting theboundarylayer(V=24knots).

simulation.Thatistosay,therun-timeofCFDsimulationsofany arbitrarybodywithanysurfaceroughnesswouldbeliterally identi-caltothoseofCFDsimulationsofthesamebodywithanotherwise smoothsurfacecondition.

Themainadvantageoftheproposedapproachisthatitenables thepredictionoftheeffectofatypicalcoatinganddifferent bio-foulingconditionsontheresistanceofashipundertheeffectof arotatingpropellerorundertheeffectofadynamicfluid-body interaction,whichisnotpossibleusingthesimilaritylawscaling procedure.Therefore,thisapproachstandsasapracticalprediction methodforbothacademiaandindustry.

Acknowledgements

TheauthorsaregratefulfortheEPSRCsupportfortheprojecton ‘ShippinginChangingClimates’(EPSRCGrantNo.EP/K039253/1) whichenabledthemtocarryouttheresearchreportedinthispaper. Theauthorsgratefullyacknowledgethattheresearchpresented inthispaperwaspartiallygeneratedaspartoftheEUfundedFP7 projectFOUL-X-SPEL(EnvironmentallyFriendlyAntifouling Tech-nologytoOptimisetheEnergyEfficiencyofShips,Projectnumber 285552,FP7-SST-2011-RTD-1).

ItshouldbenotedthattheresultswereobtainedusingtheEPSRC fundedARCHIE-WeStHighPerformanceComputer (www.archie-west.ac.uk).EPSRCgrantno.EP/K000586/1.Theunderlyingdata in this paperis openly availablefromthe Universityof Strath-clyde data repository at: http://dx.doi.org/10.15129/adeb0db8-fec9-46ce-bbe3-9b828d0b9d35.

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