Numerical simulation

Numericalsimulationofthreedimensionalcavitationsheddingdynamicswithspecialemphasisoncavitation–vortexinteractionBinJia,XianwuLuoa,n,RogerE.A.Arndtb,YulinWuaaStateKeyLaboratoryofHydroscience&Engineering,TsinghuaUniversity,Beijing100084,ChinabSaintAnthonyFallsLaboratory,UniversityofMinnesota,Minneapolis,MN55414,USAarticleinfoArticlehistory:Received15July2013Accepted17May2014Availableonline13June2014Keywords:CavitationHomogeneousmodelHydrofoilVorticalflow3DstructureabstractRecentexperimentsshowedthatthereisaninteractionbetweenthefluidvortexformationandcavitation,butthemechanismisstillanopenproblem.Inthepresentpaper,thestructureofthecavitatingflowaroundatwistedhydrofoilwasinvestigatednumericallyusingthemasstransfercavitationmodelandthemodifiedRNGk-εmodelwithalocaldensitycorrectionforturbulenteddyviscosity.Thepredictedthreedimensionalcavitystructuresandthesheddingfrequencyagreefairlywellwithexperimentalobservations.Threetypesofflowbehavioralongthesuctionsideofthetwistedhydrofoilarediscussed.Furtheranalysisoftheflowfieldrevealsthatcavitationpromotesvortexproductionandincreasestheboundarylayerthicknesswithlocalseparationandtheflowunsteadiness.Finally,theinfluenceofcavitationonthevorticitydistributionisillustratedusingthevorticitytransportequationinavariabledensityflowandisdemonstratedbythecontributionofvortexstretching,vortexdilatationandbaroclinictorqueterms.&2014ElsevierLtd.Allrightsreserved.1.IntroductionThestudyofunsteadyfeaturesofpartialcavitationandshed-dinghasbeenreceivedgreatattentionduetoitsstrongback-groundintheengineeringapplicationinpumps,turbinebladesandshippropellers.Sheetcavitationsheddingoftenleadstocloudcavitation,whichstronglyaffectshydrodynamicperformanceandproducesvibration,noiseandcavitationerosion(Brennen,1995;FrancandMichel,2005).Inmostindustrialapplications,cavitationandturbulentflowoftenresultintheformationoflarge-scalevorticalstructures,whichwillinvolvecomplexinteractionsbetweenphase-changeandvortexstructures(Arndt,2002).Inordertocontroltheunsteadysheddingandtheformationofcloudcavitation,itisveryimportanttounderstandtheseinteractions.Inthepast,numerousexperimentshavebeenconductedtostudypartialcavitationstructuresespeciallyonhydrofoils(Astolfietal.,2000;Reismanetal.,1998;Tassinetal.,1995)orinVenturi-typesections(Barreetal.,2009;StutzandReboud,1997a,1997b).Theseexperimentsshowedthatpartialsheetcavitiesareperiodi-callybroken-upandrolledupintobubbleclouds.Althoughmanyinterestingstudieshavebeenreportedonthesephysicalmechanisms,theyarenotyetfullyunderstoodduetothecomplexfeaturesofpartialcavitatingflowssuchasbubblyflow,laminartransitionorturbulentflows,detachedandreattachedflows,shearlayersandvorticalstructures.Kubotaetal.(1989)successfullymeasuredtheunsteadystructureofcloudcavitationusinglaserdoppleranemo-metry(LDA)andmatchedthemeasurementswithunsteadycavitiesphotographedbyahigh-speedcamera.Theirresultsshowedthatthecloudcavitationobservedintheexperimenthadavorticitymaximumatitscenterandaclustercontainingmanysmallcavitationbubbles.Thestructureofthetwo-phaseflowinsidethecavitywasinvestigatedbyStutzandReboud(1997a,1997b).Theysucceededinmeasuringthelocalvoidfractionandthevelocityinsidethecavitiesandconfirmedtheexistenceofreversedtwo-phaseflowalongthewall.Leetal.(1993)studiedtheglobalbehaviorofpartialcavities,includingcavitationpatterns,cavitylength,periodicshedding,andmeanpressureinthecavityclosureregion.Theyfoundthatthecavityunsteadinessisintimatelyrelatedtocavitythicknessandare-entrantjetthatresultsinvorticityproduction.Reliableestimatesforthesheddingrateofthecirculationbythere-entrantjetmechanismforaperiodiccavitycanthenbeobtained.Kawanamietal.(1997)thoroughlyinvestigatedcloudcavitationinaseriesofdetailedexperimentsonatwo-dimensionalEllipticNoseFoilwithhigh-speedphotographyaswellaspressuremeasurementsthroughpressurepick-upsandahydrophone.Theyestablishedaclearrelationshipbetweenthere-entrantjetandthecloudcavitygenera-tionprocess.TheythenpointedoutthatasmallobstacleattachedContentslistsavailableatScienceDirectjournalhomepage:www.elsevier.com/locate/oceanengOceanEngineeringhttp://dx.doi.org/10.1016/j.oceaneng.2014.05.0050029-8018/&2014ElsevierLtd.Allrightsreserved.nCorrespondingauthor.Tel./fax:þ861062789853.E-mailaddresses:jibin@mail.tsinghua.edu.cn(B.Ji),luoxw@mail.tsinghua.edu.cn(X.Luo),arndt001@umn.edu(R.E.A.Arndt),wyl-dhh@mail.tsinghua.edu.cn(Y.Wu).OceanEngineering87(2014)64–77atthemid-spanneartheterminationofthesheetcavitycanblockthere-entrantjet,therebypreventingthegenerationofthecloudcavity.Phametal.(1999)alsoconductedanexperimentalinves-tigationofunsteadysheetcavitationusingnon-intrusivetechni-questostudythere-entrantjetdynamicsandtheinterfacialinstabilities.Theyfoundthatthefrequenciesofthere-entrantjetsurgesareequaltothecloudsheddingfrequenciesdeterminedbyunsteadypressuremeasurements,whichdemonstratedthatthecloudsheddingisactuallydrivenbythere-entrantjet.Agravityeffectanalysisshowedthatthere-entrantjetrolepredominatesovertheinterfacialinstabilitiesinthegenerationofperiodiccloudshedding.Arndtetal.(2000)usedatwo-dimensionalNACA0015hydrofoiltoinvestigatethecomplexphysicsinvolvedinthetransitionofsheetcavitationtocloudcavitationwithanintegratedexperimental/numericalapproach.Theyindicatedthattwocom-petingmechanismsarefoundfortheinducedsheddingofcloudcavitation.Athighvaluesofs/2α,re-entrantjetphysicsdominate,whileatlowvaluesofs/2α,bubblyflowshockwavephenomenadominate.Watanabeetal.(2001)usedalinearizedfreestreamlinetheoryusingasingularitymethodtoshowthatwhenthere-entrantjetisnottakenintoaccount,cavitationinstabilityorigi-natesfromthetransitionalcavityoscillationandthetransitionbetweenpartialandsupercavities.Callenaereetal.(2001)experimentallyinvestigatedtheinstabilityofapartialcavityinducedbythedevelopmentofare-entrantjetonadivergingstep.Theyarguedthatthetwoparametershavingthegreatesteffectonthere-entrantinstabilityare:theadversepressuregradientandthecavitythicknesscomparedtothere-entrantjetthickness.LaberteauxandCeccio(2001a)observedtwotypesofpartialcavitieswithclosedpartialcavitiesformedonatwodimensionalNACA0009hydrofoilandopenpartialcavitieswith-outre-entrantflowformedonaplano-convexhydrofoil.Thecomplexityofcavitatingflowisduetothestrongcouplingbetweencavitationandturbulenceaswellasthestrongcompres-sibilityeffects.Withthedevelopmentofcomputationalfluiddynamics,numericalsimulationofcavitatingflowsisbecominganimportanttoolincavitationresearch.Thecavitationmodelandturbulencemodelinthesimulationofcavitatingflowsarethekeyfactorstoobtainthereasonableresults.Ahomogeneousmodelbasedontheassumptionthatthecavitationareacanbeconsideredasonlyonefluidwiththemixtureofwaterandvaporiscommonlyusedtosimulatethecavitation.Twokindsofcavi-tationmodelsareoftenusedinthesimulationsofcavitationflow,theStateEquationModel(SEM)(Coutier-Delgoshaetal.,2003;Fig.1.ThreedimensionalDelftTwist-11hydrofoil:(a)3Dview;(b)Sideview;and(c)Attackangledistributioninspanwise;(Jietal.,2013b).Fig.2.Computationaldomainandboundaryconditions(Jietal.,2013b).Fig.3.MeshgenerationaroundtheDelftTwist-11hydrofoilsurface(α¼�21)(Jietal.,2013b).B.Jietal./OceanEngineering87(2014)64–7765GoncalvesandPatella,2009)andtheTransportEquationModel(TEM)(Kunzetal.,2000;SchnerrandSauer,2001;Singhaletal.,2002).Recentexperimentalresultshaveconfirmedthatvorti-cityproductioninthecavityclosureregionisanimportantaspectofcavitatingflowsduetothebaroclinictorqueterm(GopalanandKatz,2000;LaberteauxandCeccio,2001a).However,SEMisnotabletoproperlyreflectthistermbecausethegradientsofdensityandpressureinSEMarealwaysparallel,whichleadstozerobaroclinictorque.TEMmaybeabetterchoicetosimulatethecomplexcavitatingflow,whichintroducesanadditionalequationforthevapor(orliquid)volumefractionincludingsourcetermsforevaporationandcondensationprocesses.ComparisonstudiesofdifferentTEMmodelshavebeenshowninRefs.(Ducoinetal.,2012;Morgutetal.,2011).Forsimulationofcavitatingflowaccurately,theturbulencemodeliscrucialbecausethecavitationprocessisbasicallyunsteadyinnatureandtheremustbestronginteractionsbetweenthecavityinterfaceandtheboundarylayerduringthecavitationdevelopment.ThoughtheReynoldsAverageNavier-Stokes(RANS)equationapproachhasbeenwidelyusedtomodelturbulentflows(KarimandAhmmed,2012;Seifetal.,2010),thecapabilityoftheRANSmodelwitheddyviscosityturbulencemodelstosimulateunsteadycavitatingflowsislimitedduetoitsover-predictionofeddyviscosityattherearpartofcavitation(Coutier-Delgoshaetal.,2003).Conversely,themoreaccuratelarge-eddysimulations(LES)anddirectnumericalsimulations(DNS)arerestrictedintheirapplicationbecauseoftheirhighrequirementsincomputingpower(Luoetal.,2012;Roohietal.,2013;ZhangandKhoo,2013).RecentresearcheffortsaredirectedtowardsdevelopmentofsomehybridRANS/LESmodel(HuangandWang,2011;Jietal.,2012b;Fig.4.Timedependentliftcoefficientfors¼1.07overseveralcycles.Fig.5.Topviewofcavitysheddingpatternsfors¼1.07duringonetypicalcycle.(Left:numericalresults,Right:experimentalobservationbyFoeth(2008).)(a)–(g)Correspondstoinstant1–7inFig.4.B.Jietal./OceanEngineering87(2014)64–7766Wuetal.,2005;ZhangandChen,2013)orRANSmodelwithsomeconsiderationofthelocalcompressibilityeffectoftwo-phasemixturesonturbulentclosuremodels(DecaixandGoncalves,2013;Liuetal.,2013;Wangetal.,2012).Itisnotedthatmanyresearchershavestudiedcavitatingflowsaround3Dhydrofoilstodiscusstheinstabilityofpartialcavitation.DangandKuiper(1999)numericallystudiedare-entrantjetusingahydrofoilwithvariousanglesofattackinthespanwisedirection.Theyfoundthatthedirectionofthere-entrantjetwasstronglyinfluencedbythecavitytopologyandthechangeinthecavityshapewasdeterminednotbythesweepanglebutbytheloading.LaberteauxandCeccio(2001b)showedforaseriesofsweptwedgesthatthecavityinstabilitywasstronglyinfluencedbythespan-wisepressuregradientsandthere-entrantjetmaybedirectedawayfromthecavityinterface,allowingsheetcavitationtoformacloudcavityfardownstream.Dularetal.(2007)numericallyandexperimentallyinvestigatedre-entrantjetreflec-tionataninclinedcavityclosurelinearoundahydrofoilwithanasymmetricleadingedge.Saitoetal.(2007)investigatedcavitatingflowsaroundathreedimensionalhydrofoilwithuniformprofilesanduniformattackanglesalongthespanwisedirectionandpointedoutthatthesidewalleffectisthemainreasonforgenerationoftheU-shapedcavitation.Foeth(2008)andFoethetal.(2006,2008)usedtime-resolvedPIVandahighspeedcameratostudyfullydevelopedsheetcavitationonahydrofoilwithaspanwisevaryingangleofattackandclarifiedthatthesheddingofasheetcavitywasgovernedbythedirectionandmomentumofthere-entrantandside-entrantjetsandtheirimpingementonthefreesurfaceofthecavity.ThecavitatingflowaroundtheDelfttwistedhydrofoilwasusedasbenchmarkdataintwoworkshops,VIRTUREWP4andSMP11(Hoekstraetal.,2011),becauseitresemblespropellercavitationwithwelldefinedexperimentaldatathatiseasilystudied.ThiscasewasselectedtoevaluateandvalidatecurrentCFDcodestopredictthecompli-catedcavitatingflows(Bensow,2011;Jietal.,2013a,2013b;Lietal.,2010;ParkandRhee,2013;Schnerretal.,2008).Tillnow,mostresearchofpartialcavitieshasfocusedonthestructureofthecavitatingflowanditssheddingdynamics.However,therehasbeenlittleattentiongiventotheinteractionbetweenthevorticesandthecavitiesinturbulentcavitatingflow.RecentlyFig.6.Threedimensionalviewofcavitysheddinginonetypicalcyclefors¼1.07withplottingthespanwisevorticitycontourlineatz¼0,z¼0.2C,z¼0.4C,z¼0.6Candz¼0.8C.(a)–(g)Correspondstoinstant1–7inFig.4.B.Jietal./OceanEngineering87(2014)64–7767Foeth(2008)andFoethetal.(2006,2008)foundfromtheirexperi-mentthatthesheddingof3Dsheetcavitationisinfactamixinglayer.Itscharacteristicvorticalstructureisclearlyvisibleontheimagespresentedforalargescalecavity.However,itisdifficulttoobtainthequantitativedataofthesecavitatingvorticesusingPIVduetovariouslimitationsinthemeasurementtechniques.Inspiredbytheirwork,thepresentpaperwillanalyzethesheddingvaporcloudsoverathreedimensionaltwistedhydrofoil,withspecialemphasisplacedonthethreedimensionalvortexcavitationcausedbyvorticitysheddingintotheflowfield.Thecavitatingflowaroundthetwistedhydrofoilissimulatedusingtherenormalization-group(RNG)k-εmodelwithlocaldensitycorrectiontocapturetheevolutionofvorticalflowinducedbycavitationdevelopment.2.Governingequationsandnumericalmethod2.1.GoverningequationsIntheuniformityassumptionofthemixtureofwaterandvaporinthecavityflow,themultiphasefluidcomponentsareassumedtosharethesamevelocityandpressure.Thecontinuityandmomentumequationsforthemixturefloware∂ρ∂tþ∂∂xjðρujÞ¼0ð1Þ∂ðρuiÞ∂tþ∂ðρuiujÞ∂xj¼�∂p∂xiþ∂∂xjðμmþμtÞ∂ui∂xjþ∂uj∂xi�23∂uk∂xkδij����ð2Þwhereuiisthevelocitycomponentintheidirection,pisthemixturepressure,μmisthelaminarviscosityandμtistheturbulentviscositywhichisclosedbytherenormalization-group(RNG)k-εmodel(Orszagetal.,1993).Themixturedensity,i.e.ρ,isdefinedasρ¼ρvαvþρlð1�αvÞð3Þwherethesubscripts,landv,representliquidandvaporphaserespectively,αisthevolumefractionofonecomponent.2.2.CavitationmodelandlocalcompressibilitycorrectionforturbulentviscosityThecavitationprocessisgovernedbythemasstransferequationfortheconservationofthevaporvolumefraction,whichisdefinedas∂ðρvαvÞ∂tþ∂∂xjðρvαvujÞ¼_m¼_mþ�_m�ð4ÞThesourcetermsforthespecificmasstransferratecorrespondingtothevaporization(_mþ)andcondensation(_m�)aregivenby_mþ¼Fevap3αnucð1�αvÞρvRbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23Maxðpv�p;0Þρlsð5Þ_m�¼Fcond3αvρvRbffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23Maxðp�pv;0Þρlsð6ÞwhereFevapandFcondareempiricalcoefficientsforthevaporiza-tionandthecondensationprocesseswithrecommendedvaluesof50and0.01(Jietal.,2012a;Zwartetal.,2004).αnucHasthevalueof5�10�4andRbistheradiusoftypicalbubblesizewiththevalueof1�10�6.Fig.7.Timeevolutionofcavitationdevelopmentatdifferentplanes:(a)Numericalresultsofcavitationdevelopmentintimeatplaneswithz¼0,z¼0.2C,z¼0.4Candz¼0.8Cofthetwistedhydrofoil;and(b)Experimentalresultsofcavitationdevelopmentintimeatmiddleplane(z¼0)ofthetwistedhydrofoil(Foeth,2008).B.Jietal./OceanEngineering87(2014)64–7768ThestandardRNGk-εmodelisnotabletopredictthecavita-tionsheddingdynamicsaccuratelyasitwasdevelopedforthefullyincompressiblesingle-phaseflow,andthecavitatingflowisactuallycompressibletwo-phasefluids.AsshownbyReboudetal.(1998),thestandardRNGk-εmodelfailstosimulateunsteadycavitationduetotheover-predictionofeddyviscosityattherearpartofthecavity.Toimprovethesimulationaccuracybycon-sideringthecompressibilityofcavitatingtwo-phaseflow,somemodificationisneededanddefinedas(Reboudetal.,1998)μt¼fðρÞcμk2εð7ÞfðρÞ¼ρvþð1�αvÞnðρl�ρvÞð8Þwherecμ¼0.085andn¼10.Thismodificationhasbeenvalidatedformanycases,suchascavitatingflowaroundVenture-typesections(Coutier-Delgoshaetal.,2003;DecaixandGoncalves,2013)andhydrofoils(Dularetal.,2007;Lietal.,2010).2.3.Hydrofoilgeometry,computationaldomainandmeshgenerationAtwistedhydrofoilfromDelftUniversityofTechnologydenotedasFig.1isselectedinthepresentresearch.ThehydrofoilhasaNACA0009sectionwithvaryingattackanglei.e.αfrom01atthesidesectionto111atthemid-section,whichissymmetrywithrespecttoitsmid-spanplane.ThechordlengthoffoilCis0.15mandthespanlengthzis0.3m.Theattackangleoftheentirehydrofoilis�21andthesameastheexperimentalhydrofoilfromFoeth(2008),whichisreferredtoasDelftTwist-11hereinafter.ThecomputationaldomainisshowninFig.2.Itisnotedthatfornumericalflowsimulationonlyhalfofthetestsectionandhydrofoilareconsideredbecauseofgeometricsymmetry.Thehydrofoilislocatedinachannelwithheight2C,alengthof2Cupstreamoftheleadingedge,alengthof5CdownstreamoftheleadingedgeandawidthofC.Theboundaryconditions,imposedvelocityatinletandstaticpressureatoutlet,symmetryboundaryatz¼0,andwallconditionsatotherwalls.ThestructuredmeshesshowninFig.3areadoptedinthecalculation.AnO–Htypegridwasgeneratedforthedomainwithsufficientrefinement(30ryþr100)towardsthefoilsurface.Themeshalonghydrofoilisfineenoughtocapturethedetailstructureofthecavitation.3.ResultsanddiscussionTheflowaroundtheDelftTwist-11hydrofoilhasbeeninves-tigatedbyanumericalmethoddescribedabovewithaninletvelocityofV1¼6.97m/s.Thestaticpressureattheoutletplaneofthedomain,pout,wasassignedaccordingtothecavitationnumber,s¼ðpout�pvÞ=ð0:5ρlV21Þ¼1:07.Notethatthesteadysolutionwithnon-cavitatingflowwascalculatedandusedasinitialconditiontosimulatetheunsteadysituation.Fromtheresultsofunsteadycavitatingflow,theoscillationperiodofcavityandlargevaporcloudsheddingareobtained.Theoscillationfrequencyduetocavitysheddingisabout30.2Hz,whichisclosetotheexperi-mentalresulti.e.32.2Hz(Foeth,2008).Oncecavitysheddingoccurs,theliftforceactingonthehydrofoilvariesperiodicallyduetoapressuredistributionchangebycavitation.Fig.4showsthecalculatedliftcoefficientsi.e.CLoverseveralcycles.Thetimeaveragedliftcoefficientis0.463,whichisalsoveryneartheexperimentallymeasuredvalueof0.5167(Foeth,2008).Itisnotedthattheevolutionoftime-dependentliftcoefficientascavitysheddingisverycomplicated.ItisbelievedthatthenumericalsimulationwiththemodifiedRNGk-εmodelcanresolvemoreoftheturbulentkineticenergyandovercometheover-predictionsoftheturbulentviscosityattherearpartofthecavity.Fig.9.ComparisonofdifferentterminvorticitytransportequationatplaneZ¼0.2C.(a)3Dviewofvapourvolumefraction.(b)2Dviewofvaporvolumefraction.(c)2Dviewofvortexstretchingcontourline.(d)2Dviewofvortexdilationcontourline.(e)2Dviewofbaroclinictorquecontourline.(f)2Dviewofviscousdiffusioncontourline.Fig.8.Threedimensionalviewofspanwisevorticitycontourlineatz¼0,z¼0.2C,z¼0.4C,z¼0.6Candz¼0.8Cfornon-cavitatingcase.B.Jietal./OceanEngineering87(2014)64–7769Tovisualizethesheddingbehaviorofcavitationindetail,sevennumericalsnapshotshavinganintervalofT/7areshowninFig.5(a)–(g),wherethepredictedcavityshapeisobtainedfromthevaporvolumeiso-surfacevalueof0.1.Forcomparison,thecorre-spondingexperimentalpicture(Foeth,2008)isshownintherightcolumnofFig.5.InFig.5(a),thesheetcavityhasreachedmaximumlengthandhasaconvexshapeattherear.Fromthepressuredistributionitisfoundthatastrongadversepressuregradientonthehydrofoilsurfaceispresentattheendofthecavityandforcesthere-entrantflowintothevaporregion.Thenthisre-entrantflowmovesupstreamalongthesuctionsurfaceofthehydrofoilandcausestheprimarysheddingofthecavityinFig.5(b).Duringthisprocess,thesheddingcavityquicklychangesfromasmoothpocketvaporintoahighlyturbulentvaporcloud.Frominstant3toinstant5asshowninFig.5(c)–(e),anewsheetcavitygrowsfromtheleadingedgeofthehydrofoilwithaconcaveshapeoftheclosureline,whilethesheddingvaporbecomesmoreturbulentandisentraineddownstreambythemainflow.Afterthatthesheetcavitygrowsslowly(Fig.5(f)–(g))andthesheddingvaporcloudquicklyshrinks(Fig.5(f)–(g))andfinallycollapses(Fig.5(d)).Itshouldbenotedthatattheinstant6thereisasecondarysheddingofbothdownstreamlobesofthesheetcavityasthesheetcavitydevelopedasindicatedinFig.5(f).Thus,thepresentsimulationclearlyreproducesthecavitationpatternsandtheirevolutionaroundthetwistedhydrofoilwithprimaryandsecondarysheddingvaporcloudsandisconsistentwithexperi-mentalobservationbyFoeth(2008).AccordingtotheexperimentalobservationbyFoeth(2008),sheetcavitationandthetransitiontocloudcavitationonthetwistedhydrofoilresultedinahighlyunstableflowthatcouldFig.10.Comparisonofvaporvolumefraction(left)andvorticity(right)contoursfors¼1.07attheplanewithz¼0.(a)–(g)Correspondstoinstant1–7inFig.4.B.Jietal./OceanEngineering87(2014)64–7770inducesignificantvorticitysheddingintotheflowfi