J.A.Neufeld,R.E.GoldsteinandM.G.Worster - PDF document

J.A.Neufeld,R.E.GoldsteinandM.G.Worster Figure1.Asequenceofhigh-resolutionimagesshowingthemeltingofacylinderoficeinairoffar-“eldtemperature=21Ctakenattimes=10214and266min.Thewhitesquareforscaleis1cmonaside.shownin“gure1andmovie1.Theicecoolsthesurroundingair,which”owsradiallyoutwardsalongthetopofthecylinderanddownthecurvedsides.However,inwhatfollowswefocusourattentiononthe”owalongthetopsurfaceofthecylinderofice,highlightingthebalanceofphysicaleectsneededtodescribetheformoftheboundarylayer.Weleaveforfuturediscussiontheultimateevolutionoftheshapeoftheicicle.Buoyancy-drivenconvectionabovea“nite,horizontal,cooledplatewithoutphasechange(orequivalentlybelowaheatedplate)hasbeenthesubjectofmanypreviousstudies,yetopenquestionsremain.Thecooled”uid(air)poursovertheedgesoftheplate,andthe”owiscontrolledbyconditionsatorneartheedges,inthemannerofaweirforexample.Thereareexcellentvisualizationsofsuch”owsinthepaperbyAihara,Yamada&Endo(1972),andoftheassociatedthermalboundarylayerinthepaperbyHat“eld&Edwards(1981).Thesepapers,aswellasourownexperimentsandnumericalcalculations,showthe”owtobelaminarandprovidemeasurementsofvelocityandtemperaturepro“les,aswellaslocalandmeanratesofheattransferwhichwecomparewithourtheoreticalpredictions.Stewartson(1958)describedthedierentialboundary-layerequationsappropriatetosuch”owsandfoundasimilaritysolutiontothem,whichheincorrectlyinterpretedasthe”owaboveacooled,horizontalplate.Hissolution,whichwesummarizein2,actuallydescribesthe”owfromtheleadingedgeaboveaheated,horizontalplate(Gill,Zeh&DelCasal1965).Mostanalyticalstudiesofconvective”owadjacenttohorizontalplateshaveusedintegralboundary-layerequationswithassumedverticalshapefunctionsintroducedtoevaluatetheintegrals.ThesearenicelysummarizedbyHiguera(1993),whoemphasizestheellipticcharacteroftheproblemandtheneedforaboundaryconditionattheedgeoftheplate.Ithadvariouslybeenproposedthatthethicknessoftheboundarylayershouldbezeroattheedgeoftheplate(Wagner1956),orthatitshouldattainacriticaldepthbyanalogywithopen-channelhydraulics(Clifton&Chapman1969),orthatasingularityintheintegralequationsshouldcoincidewiththeedge(Singh&Birkebak1969),whichwaslatershowntobeequivalenttomaximizingtheheat”uxfromtheplate(Fujii,Honda&Morioka1973).Morerecently,ithasbeensuggestedthattheboundarylayershouldadjustsoastomaximizethedischargerate(mass”ux)overtheedge(Dayan,Kushnir&Ullmann2002).Higuera(1993)himselfputconsiderableeortintoananalysislocaltotheedgeoftheplateandconcludedthatitseemslikelythatbuoyancyinthecornerregioncandealwithany”owsupplied Onthemechanismsoficicleevolutionbytheboundarylayer,implyingthereforethatthemass”uxreachingtheedgemustbeaslargeaspossible.Inthispaper,weshowthatthepartialdierentialequationsgoverning”owandtransportintheboundarylayercanbeseparatedintoordinarydierentialequationsdescribingtheverticalstructuresofthevelocity,pressureandtemperature“eldsandthehorizontalvariationoftheboundary-layerthickness.Theseparationisexactatin“nitePrandtlnumber,andweshowittobeanexcellentapproximationfor“nitePrandtlnumbersaslowasthatforair(7).AsimilarapproachhasbeenusedbyHiguera&Weidman(1995)toanalysethe”owaboveacooledplateinaporousmedium.Theequationgoverningthehorizontalvariationofboundary-layerthicknesshastheformofthatdescribingasteadyviscousgravitycurrentinwhichthelocaldivergenceof”uid”uxisbalancedbydiusiveentrainment,andthedierentialsystemiscloseduniquelybyrequiringsimplythatthemass”uxattheedgeoftheplatebenon-zeroand“nite.2,wepresentourtheoreticalapproachintwodimensionsandcompareitspredictionswithexistingexperimentaldataandourownfullnumericalcalculations.3,weextendouranalysistoanaxisymmetricdiskandcomparethepredictionswithmeasurementstakeninairaboveameltingcylinderofice.In4,weapplyourresultstoourmeasurementsoftheinitialmeltingofacylinderoficeanddiscoversurprisinglylargerolesplayedbycondensationofwatervapourfromtheairandradiativeheattransferfromthesurroundings.Generalconclusionsandpossiblefutureapplicationsofthismethodoftreatingtheboundary-layerstructurearepresented2.Convectionaboveatwo-dimensionalstripTheoreticaldevelopmentWeconsiderbuoyancy-driven”owsaboveisothermal,horizontalplates,asillustratedin“gure2.The”owaboveaheatedorcooledplateisexactlyanalogoustothe”owbeneathacooledorheatedplate,respectively,andourresultscanbeappliedtothelattercasesbysimplyreversingthesignofgravity.Whentheplateisheatedthe”uidaboveformsathin,laminarboundarylayerthatdevelopsandgrowsasthe”uid”owsfromtheleadingedgeoftheplate(see“gure2).Incontrast,whenthesurfaceiscooledthe”uidaboveformsathin,laminarboundarylayerwhosethicknessdiminishesasthe”uid”owsfromthecentretowardsthetrailingedgeoftheplate.Inbothcases,weconsiderlaminar”owthatcanbewelldescribedbytheusualBoussinesqboundary-layerequationswhenthedistancealongtheplateismuchlargerthanthecharacteristicboundary-layerthickness(Schlichting&Gersten2000).Thesteadytwo-dimensionalBoussinesqboundary-layerequationsdescribingmomentumbalancesandheattransferaboveahorizontalplateare xw zŠ1  x2u ,)1  zŠ )u xw z2T ,) J.A.Neufeld,R.E.GoldsteinandM.G.Worster (Hot)(Cold) (Cold)(Warm) Figure2.Geometryofconvectionaboveisothermal,horizontalplates.In()awarmthermalboundary-layer”owsfromtheleadingedgeofaheatedplate.Thecoolthermalboundarylayeraboveacoldplateisshownin().Nearthecentreoftheplate,the”owhasthecharacterofastagnation-point”ow.(Stewartson1958).Thesearecoupledbythelinearequationofstatetate1Š(TŠT)].(2.2)Continuityisassuredbyusingatwo-dimensionalstreamfunctionx,z)suchthatthevelocity“eldu,w).Here,x,z)andx,z)arethepressureandtemperature“elds,arecoordinatesalongandperpendiculartotheplate,arethekinematicviscosityandthermaldiusivityofthe”uid,isthecoecientofthermalexpansion,isthedensity“eld,andisareferencedensityofthe”uid,measuredatthefar-“eldtemperatureAnumberofpreviousauthors(Gilletal.1965;Wagner1956;Singh&Birkebak1969,forexample)haveapproachedthisproblembyassumingaprescribedverticalstructureforthetemperature,velocityandpressure“elds.Here,weshowthatboththeverticalstructureanditslateralvariationcanbefoundindependentlybyjudiciousapproximationof(2.1)and(2.2).Tothatendwelookforself-similarsolutionsoftheseequationsoftheformx,zT x,zx,zTgh Onthemechanismsoficicleevolutionisthetemperaturedierencedriving”uidmotionandisthetemperatureoftheplate.Thesimilarityvariable whilethedimensionlessfunctions)and)arevertical-structurefunctionsfortemperature,streamfunctionandpressurerespectively,and)and)representthehorizontalvariationsofthemass”uxandthethicknessoftheboundarylayer,respectively.Notethatthespatialvariablesandthethicknesshavesofarbeenleftdimensional.Substitutionoftheseexpressionsinto(2.1)and(2.2)leadstothesetofdierentialequationsgT 3h /isthePrandtlnumber.Theseequationsstillinvolvetwoindependent,butthefunctionsinvolvedinthemareallfunctionsofjustoneindependentvariable.Theprimesdenotedierentiationwithrespecttotheargumentoftherespectivefunctions,eitherTheseequationscanbemadedimensionlessbyscalingwithahorizontalscale,suchasthelengthoftheplate,scalingandscaling,wheretheRayleighnumbergTL Wesee,therefore,thattheboundary-layerassumptionisappropriateprovidedtheRayleighnumber1.Inaddition,thesescalingsprovideanaturalde“nitionofthecharacteristicReynoldsnumber Ra =/5 Fortheexperimentsdetailedin2and4theReynoldsnumberis10…35,whichissigni“cantlylessthanthecriticalReynoldsnumber,400,forinstabilityofshear-driven”ows(Drazin&Reid1981).Withthesescalings,thegoverningequations areindependentofRayleighnumberanddependonlyonthePrandtlnumber,sothereisjustaone-parameterfamilyofsolutions.Ouraimistoseparatetheverticalandhorizontalstructuresoftheboundarylayer,resultingindecoupledsetsofordinarydierentialequationsdescribingeach.Equation(2.8)alreadyinvolvesonlytheindependentvariable.Equation(2.8)and J.A.Neufeld,R.E.GoldsteinandM.G.Worstertheviscousandbuoyancytermsof(2.8)canbeseparatedbychoosing=1and Separationof(2.8)requiresthatbeconstant,anditisnecessaryforthatconstanttobepositiveinorderthatdecaysas.Withoutlossofgenerality,theconstantcanbesetequaltounityin(2.9).Theright-handsideof(2.8)mustbenegativeinorderthattheverticalgradientofvorticitybenegativeneartheplate,somustbepositivewhenispositive(theplateisheated)andnegativewhenisnegative(theplateiscooled).Again,withoutlossofgenerality,themagnitudeoftheconstantin(2.9)canbetakentobeunity.Once(2.9)aresatis“ed,theequationsdescribingtheverticalstructure(2.8becomeTheseequationsaresubjecttotheboundaryconditions=0(=0)whichexpressthefactthattheplateisisothermalandthattheairisstationarythere,andthatinthefar“eld,thetemperaturehasadierentuniformvalue,thepressureisknownandthehorizontalvelocityvanishes.Weseethat(2.10)isnotquiteseparatedbecause-dependenceremainsinthe.Below,weexaminetwocasesinwhichtheseparationcanbecompletedexactly:”owaboveaheatedorcooledplate.Wealsoshowthat,inthecaseof”owaboveacooledplate,ismostlyverysmall,andthatgoodapproximatesolutionscanbeobtainedbyeitherneglectingitorevaluatingitlocally.TheheatedplateItisclearthatseparationof(2.10)iscompleteifisconstant.Itisstraightforwardtoshowthatforconvectionaboveaheatedplate,asillustratedin“gure2(),thisconditionwith(2.9)leadstothepower-lawsolutions 1/5x2/52 5 andthatthemomentumequation(2.10)becomes Notethattheplussignhasbeentakenontheright-handside,whichisrequiredforthe”owaboveaheatedplate,andthattheplussignmustcorrespondinglybetakenfortheboundaryconditiononin(2.11).Stewartson(1958)showedthatsuchapower-lawsimilaritysolutiondoesnotexistwhenaminussignistakenontheright-handsideof(2.10),whichisthecasefor”owaboveacooledplate.Thesolutionto(2.13)with(2.10),(2.10)andboundaryconditions(2.11)waspresentedbyStewartson(1958). OnthemechanismsoficicleevolutionThecooledplateatin“nitePrandtlnumberForeitheracooledoraheatedplate,itisstraightforwardtoshowthatseparationoftheverticalandhorizontalstructureiscompleteinthelimit,when(2.10becomesThisequationdescribesthedynamicsoftheinnerthermalboundarylayer,forwhichtheappropriatefar-“eldboundaryconditiononthevelocity“eldisoneofzerotangentialstress,namely)(2.15)(seee.g.Kuiken1968).Thevelocityrelaxestothefar-“eldconditiongivenby)muchfartherfromtheplateinanouterboundarylayerthathasauniformtemperature“eldandabalancebetweeninertiaandviscousdissipation.Inthecaseofacooledplate,illustratedin“gure2(),solutionofthegoverningequationsforthethermalboundarylayershowsthatthedimensionlessshearstress(0)=1andthedimensionlesstemperaturegradient(0)=0Thefull,two-dimensionalsolutionforthethermalboundarylayeriscompletedbysolving(2.9)forthehorizontalvariationofitsthicknessandvolume”ux.ThoseequationscanbewrittenintheformThe“rstoftheseequationsisrecognizableastherelationshipbetweenthethicknessandthevolume”uxinatwo-dimensional,viscousgravitycurrent.Thesecondequationexpressesabalancebetweenthedivergenceofthehorizontalvolume”ux,whichgivestheverticalentrainmentvelocityintotheboundarylayer,andtheconductiveheat”uxacrosstheboundarylayer,whichisinverselyproportionaltoitsthickness.Thisisthefundamentaladvection…diusionbalancethatcon“nestheboundarylayer.Weseeherethatthethermalboundary-layer”owsexactlylikeasteadyviscousgravitycurrentwhosethicknessobeystheequation/h.Theonlydierenceisthatthedensity“eldissmearedoutbydiusion,accordingto(2.10),ratherthanformingadiscrete,two-layersystem.WethereforecallthisadiusivegravitycurrentTheboundaryconditionsfortheseequationsona“niteplateare=0(=0)and=1)wheretheconstantvolume”uxattheedgeoftheplateis“nite.The“rstoftheseisthenaturalsymmetryconditionatthecentreoftheplate.Thesecondisobviouslytruebutwouldnotseemtogivesucientadditionalinformation.However,wecouplethisstatementwiththeobservationthat(2.17)issingularwhere=0andthatthestrengthandformofthesingularitycanbechosentoensurea“nitevolume”ux.Inotherwords,the”owovermostoftheplateseesa“nitesinkneartheedgeoftheplate.Weexpectthatthelocationofthesinksingularityshouldbewithinapproximatelyoneboundary-layerthicknessoftheedgeoftheplateso,withinthesameorderofapproximationasboundary-layertheoryorlubricationtheory,weplacethesingularityattheedgeoftheplate,where=1. J.A.Neufeld,R.E.GoldsteinandM.G.Worster 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.2 h 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1.2 q Figure3.)Boundary-layerthicknessand()volume”uxasfunctionsofdistancealongtheplate.Thesolidcurvesshowthesolutionto(2.16)with023and(0)=1162.Thedashedcurvesshowtheapproximatesolution(2.19),with=(64Withtheseconditions,(2.16)canbeexpandednear=1togive)and Theseasymptoticexpressionscanbeusedtoinitializearelaxationscheme(weusedMatlabsbvp4croutine)inwhichequations(2.16)aresolvedsubjecttocondition)at=0andtheasymptoticexpressions(2.19)at=1.Thesethreeconstraintsprovidetheconditionsnecessarytodeterminethestructureoftheboundarylayerandthe”uxattheedgeoftheplate=1.Theresultsareshownbythesolidcurvesin“gure3,forwhich023and(0)=1Theasymptoticexpressionsnearthenosecanbeusedtoprovideapproximatesolutionsoverthewholedomainbychoosing=(64954sothatthevaluegivenby(2.19)isequaltozeroat=0.Theresultisshownbythedashedcurvesin“gure3.Thethicknessofthethermalboundarylayer)andtheresultantportraitofstreamlinesabovethecooledplateareshownin“gure4.Thepicturethatemergesabovethecoldplateisofasteady,cold,viscouscurrentfedbyentrainmentalongitslengthasitpropagatestowardstheedgeoftheplate.Theverticalstructureisdominantlythatofathermalboundarylayerinastagnationpoint”ow,forwhichtheboundary-layerthicknessisuniform(Worster2000).Theanalysisofthegravity-currentequation(2.17)includingthesingularedgeconditioniskeytodeterminingthestrainrateofthestagnationpoint”ow.Wecanpiecetogetherthesolutionsaboveto“ndthatthelocaldimensionalheat”uxfromtheplate kT /5  (x), Onthemechanismsoficicleevolution 0.20.40.60.81.0 Figure4.Thethincurvesshowstreamlinesasfunctionsofthehorizontalandverticalcoordinates,calculatedatin“nitePrandtlnumber.Theboldcurveindicatestheboundary-layerthicknessasafunctionofpositionalongtheplateandcorrespondstoanisthethermalconductivityoftheair.Thenon-dimensionalheat”uxtowardstheplatecanbeexpressedintermsofaNusseltnumberde“nedby kT/L Thismeasureofthelocalheat”uxcanbeintegratedto“ndthetotalheat”uxtowardstheplateexpressedintermsofaglobalNusseltnumber /5 010dx Inthelimitofin“nitePrandtlnumber,oursolutiongives NuRaThisisshownbythedashedlinein“gure6.Weseethatitgivesagoodapproximation(towithin10%)of downtoPrandtlnumbersofabout10.Approximateresultsat“nitePrandtlnumberWhenthePrandtlnumberis“nite,eectsofinertiabegintoaectthethermalboundarylayer.Weseefrom(2.10)thattherearetwoinertialterms.The“rstterminsquarebrackets,proportionalto,representstheverticaladvectionofvorticityandactstoincreasetheshearintheboundarylayerandthereforetoretardthe”ow.Thesecondterm,proportionalto,representsthedivergenceofthedownstreammomentum”uxandcanalsobethoughtofasthekineticenergythatmustbeimpartedtothe”uid.Thislattertermhasacoecientthatinvolves,whichisthesolecomponentfrustratingcompleteseparationoftheboundary-layerequationsat“nitePrandtlnumber.However,arebothequaltozeroatthecentreoftheplate,and“gure5showsthatthemagnitudeofremainssmallforabout80%oftheplate.Wecanproceedto“ndapproximatesolutionseitherbysettingtozeroorbytreatingasaparametergivenbythesolutionsto(2.16)above,andsolvingequations(2.10)subjecttoboundaryconditions(2.11)ateachdownstreamlocation.TheresultsobtainedfortheglobalNusseltnumberfromthesetwoapproachesareshownin“gure6anddierbylessthan10%forPrandtlnumbersdownto0.5.The J.A.Neufeld,R.E.GoldsteinandM.G.Worster 0.2 0.4 0.6 0.8 1.0 10Ð410Ð310Ð210Ð1110xÐqh Figure5.Themagnitudeoftheresidualinertialcoecient,,asafunctionofpositionalongtheplate. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 110 Figure6.ThesolidcurveshowstheglobalNusseltnumber scaledby,asafunctionofPrandtlnumber,computedbysettingtheresidualinertialcoecient0in(2.10Thedottedcurveshowsthesolutionobtainedbyevaluatingtheresidualinertialcoecientlocally.Theheat”uxinthelimitisshownbythedashedline.simplestapproachissimplytoneglect,sowedothishenceforwardinordertoassesstheutilityofthisapproximation.WecompareourpredictionswiththeresultsofAiharaetal.(1972),whoconductedcarefulexperimentsinairbelowaheatedplate25cmlong.Theexperimentswerecon“nedbytwoglassplates,throughwhichphotographsweretakentorecordthetrajectoriesof“neparticles,fromwhichmeasurementsofthevelocity“eldweremade.Inaddition,verticalpro“lesofthetemperaturewererecordedinvariousplanesperpendiculartotheplateusingcarefullysuspendedchromel…alumelthermocouples.Theirdataareshownin“gures7and8alongsideourtheoreticalpredictions.Weseethatthevelocity“eldiswellpredictedovermostoftheplate,thoughthereareslightdiscrepanciesneartheedge,particularlyintheouterpartoftheboundarylayer.Thereseemtobeslightlylargerdiscrepanciesinthestructureofthetemperature“eld.Importantly,however,thetemperaturegradientatthesurfaceoftheplateseemsverywellpredicted.Itshouldalsobenotedthatnotheory(oursincluded)predictsanynegativevelocitiesinthefar-“eld()andthatthisaspectofthedatamaybeprimarilyduetothereturn”owresultingfromthenecessityofdoingexperimentsina“nitecontainer.In“gure9,wecompareourresultswiththeexperimentalmeasurements(Aiharaetal.1972)andourownnumericalcomputations(seeAppendix)ofthelocalNusseltnumber,scaledbytheRayleighnumber,asafunctionofpositionalongtheplate,and“ndexcellentagreement.WenotethatthelocalNusseltnumberisconstantalong Onthemechanismsoficicleevolution 2 4 6 8 Ð0.2 0 0.2 0.4 0.6 0.8 zu Figure7.Acomparisonofthetheoreticalhorizontalvelocitystructure(solidcurves)andtheexperimentalresultsofAiharaetal.(1972).Pro“lesaretakenat2(squares),6(triangles),8(diamonds),9(invertedtriangles)and(crosses).Opensymbolscorrespondtoatemperaturedierenceof=55CwithanassociatedRayleighnumberof.Solidsymbolscorrespondtoatemperaturedierenceof=104CandaRayleighnumberof.WenotethatAiharaetal.(1972)ascribeamaximumerrorof5%inthevelocitymeasurementsanddiscussfurtherinthetextpossiblecausesfor 1 2 3 4 5 6 Ð1.0 Ð0.8 Ð0.6 Ð0.4 Ð0.2 0 Figure8.Acomparisonofthetheoreticalthermalstructure(solidcurves)andtheexperimentalresultsofAiharaetal.(1972),forwhich=52CandhencePro“lesarefrom0(plus),4(circles),6(triangles),8(diamonds),9(invertedtriangles),95(crosses)and99(sidewaysdiamonds).Experimentaltemperaturesanddistancesarereportedtohavebeenmeasuredtowithin0.6%and0.1mm,respectively(Aiharaetal.muchoftheplate,re”ectingthestructurefoundintheboundary-layerthicknessByintegratingourresults,we“ndtheglobalNusseltnumber NuRa496,whichiswithin3%oftheexperimentalestimatesof NuRa509for NuRa500forSuchgoodagreementmaywellbefortuitousgivenuncertaintiesintheexperimentalmeasurementsaswellasourownapproximations.Forexample,therearelikelytobeedgeeectsintheexperiments,associatedwiththedicultyofmaintaininganisothermalplateinthepresenceoflargeheat”uxesandwiththe“nitesizeof J.A.Neufeld,R.E.GoldsteinandM.G.Worster 0.4 0.6 0.8 1.0 0 0.51.5 1.02.0 Nu RaÐ1/5x Figure9.LocalNusseltnumberscaledbyasafunctionofpositionalongtheplatefor7.Thesolidcurveisthetheoreticalpredictionmadebyignoringtheresidualinertialcoecient.Resultsofthefullnumericalsimulationfor”owovera”atplatewith7andareshownasdiamonds(seetheAppendixforfulldetailsofthecalculation).DatafromAiharaetal.(1972)areshownfortwoexperimentsinwhichcircles)and(opencircles).theboundingcontainer,aswellasedgeeectsinthetheoryassociatedwiththebreakdownoftheboundary-layerhypothesis.3.ConvectionaboveacooledcirculardiskTheBoussinesqboundary-layerequationsgoverning”owaboveanaxisymmetricdiskareidenticaltothoseforthe”atplate(2.1)…(2.2),withreplacedbytheradialcoordinate.ContinuityisnowassuredbyintroducingaStokesstreamfunctionsuchu,wAgain,welookforself-similarsolutionsoftheequationsoftheformr,zT r,zr,zTghandthesimilarityvariablez/h).Wescalebytheradiusofthedisk,thethicknessoftheboundarylayerandthe”ux,wherenowgTR Thedimensionlessfunctions)and),specifyingtheverticalstructureofthethermal“eld,thestreamfunctionandthepressurerespectively,obeytheunseparateddierentialequations r(Š)f2 =h3h q(pŠ),)p=) +h r() Onthemechanismsoficicleevolution 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 1.2 10Ð110Ð210Ð310Ð4110102rh,q Ð hq/r Ð qh12 Figure10.Radialboundary-layerpro“le)(solid),”ux)(dashed)andtheresidual hq/r(dotted)asfunctionsofradialposition.Followingthesameproceduretoseparatetheseequationsasweusedinthetwo-dimensionalcase,wechoose =1andWeidentifytheleading-ordercontributiontothehorizontalinertia(proportionalto)bynotingfrom(3.4)that 1 andwritingtheequationsgoverningtheverticalstructureintheform 2f2++ Š1 Notethatthethree(non-separated)termsenclosedinroundbracketsmultiplyingtheinertialtermsumtozeroat=0,andwe“ndbelowthattheirsumremainssmallovermostoftheplate.Equations(3.4)weresolvednumericallyusingtheboundaryconditions=0(=0)and=1)andnotingthattheasymptoticformof)near=1isidentical(totheordershown)tothatgivenby(2.19),withreplacedby.Theresultantboundary-layerstructureisshownin“gure10.We“ndtheboundary-layerdepthattheorigin(0)=1018andthe”uxattheedgeofthedisk643.Thesesolutionsareusedtocalculatethenon-separatedinertialcoecient(intheroundbracketsin(3.6whichisshownbythedottedcurvein“gure10.Weseeagainthatthistermisverysmallovermostoftheplate,andwethereforeneglectitinouranalysisforsimplicity. J.A.Neufeld,R.E.GoldsteinandM.G.Worster 0.2 0.4 0.6 0.8 Nu RaÐ1/510Ð210Ð11011021031 Figure11.ThesolidcurveshowstheglobalNusseltnumberscaledbytheRayleighnumber, NuRa,asafunctionofPrandtlnumber.Thelimit,inwhichtheseparationofhorizontalandverticalstructuresisexact,isshownbythedashedline.Intheradialgeometry,thelocalNusseltnumberisde“nedby kT/R wherethelocaldimensionalheat”ux ThelocalNusseltnumberdeterminesthemeltrateasafunctionofwhichisshownbelowin“gure13,whereitiscomparedwithresultsobtainedfromtheexperimentonthemeltingofacylinderoficedescribedinAglobalNusseltnumbercanbede“nedas R0rdr R2/5 10rdr WeshowthevariationinthescaledglobalNusseltnumberwithPrandtlnumberin“gure11.Forairat0C,forwhich7,we“ndthat NuRa615,avaluewhichcompareswellwiththeexperimentsdescribedbelow.4.ThemeltingoficeOurstudyofconvectiveboundarylayersabovecooled,horizontalsurfacesismotivatedbytheshapesoficiclesandspurredbytheexperimentalresultsdescribedbelow.Westartedwithablockofcylindricalicewitha”at,horizontaltop(see“gure1).Theseicecylinderswereplacedinanenclosureatroomtemperaturetoprotectthemfromaircurrents.Digitalimageswereacquiredevery15switha4kbhigh-resolutionHasselbladcamera.Illuminationwasprovidedbya”ashunittriggeredsimultaneouslywiththecameratominimizeheatingfromthelightsource.Usingapairof6,f/4parabolicmirrors,wealsoacquiredaseriesofSchlierenimages,fromwhichameasureofthethermal“eldintheairabovetheiciclescouldbederived.Theintensityx,z)ofaSchlierenimageisproportionaltotheintegralofverticaltemperaturegradientsalongraypaths.Invertingtheimagetoreconstruct Onthemechanismsoficicleevolution 1.0 1.5 2.0 2.5 3.0 0 5 10 15 20 25 (cm)T (°C) Figure12.Thethermalpro“leaboveanicicleofradius7cm,asmeasuredbytheSchlierenapparatus(circles),iscomparedwiththepredictedthermalpro“leatthecentreofthedisk(solidcurve)usingtheparametersgivenintables1and2.theaxisymmetrictemperature“eldisacomplexinverseproblem.However,giventhattheboundarylayerhasaroughlyconstantthickness,wecanestimatethetemperaturepro“leatthecentreofthedisk)by istheintensityofthebackground.Theresultisshownin“gure12,whereweseethatthedata“tthetheoreticalpredictionfortheshape(inparticulartheboundary-layerthickness)ofthethermalpro“lereasonablywell.Thisgivesussomecon“dencethatourtheorygivesthecorrectrateofheattransferfromtheairwhenwetrytopredictthemeltrateoftheicebelow.Theopticalimageswereanalysedusingathresholdingproceduretoproducemeasurementsofthepositionofthesurfaceoftheiceasitevolvedintime.Fromthesepro“leswemeasuredtherateofmelting)alongthetopoftheice.Thelocalmeltrateatthetopoftheblockoficecanbequantitativelyexplainedintermsofthreeheat-transferprocessesasdetailedbelow:heattransferfromtheair,thelatentheatofcondensationofwatervapourandthenetradiativeheattransferfromthesurroundings.HeattransferfromtheairConservationofheatattheice…airinterfaceisdescribedbytheStefancondition isthedensityofice,andisthelatentheatofmeltingperunitmass.Thisequationcanberearrangedandevaluatedusingthetheoreticalmodelfrom3toobtainthemeltrateatthecentreofthetopsurface scp  R/5  recallingthat,whereisthespeci“cheatcapacityoftheair. J.A.Neufeld,R.E.GoldsteinandM.G.Worster ParameterSymbolValueAir(0DensitygcmViscosity133cmCoecientofthermalexpansionThermaldiusivity187cmSpeci“cheat005JgLatentheatofvaporizationofwater500JgSaturatedvapourdensitygcmMolarmassofwater18gmolGasconstant31JKIce(0Density9167gcmLatentheat334JgTable1.Physicalpropertiesforairandice. ParameterSymbolValue7cmRelativehumidity55%Far-“eldtemperatureIcicletemperatureTable2.Parametersspeci“ctotheiceanalysedintheexperimentdetailedinthetext. Usingthephysicalparametersgivenintable1andtheexperimentalparametersgivenintable2,inwhichwehaveassumedthatthesurfaceoftheiceisisothermalatthemeltingtemperature0C,wecalculatetheRayleighnumber61000andthemeltrate(0)cms.AtaPrandtlnumberof7,oursolutions483,whichgivesameltratecms.Thispredictioncontrastswithourmeasurementofcms,whichindicatesthatthethermalboundarylayerisnotsolelyresponsiblefortheobservedmelting.Themeltwater“lmManyprevioustheoreticalstudiesofthegrowthoficicleshavefocusedattentiononthethin“lmofwatercoatingtheirsurfaces.Inthepresentcontext,itisstraightforwardtotakeaccountofthe“lmwithinthesameself-similarframeworkusedin3toevaluatethethermalboundarylayerintheair.Thin-“lm(lubrication)theoryshowsthatthethicknessofthemeltwater“lmsatis“es 3w1 r w arethedensityandkinematicviscosityofwater.Thisequationcanbecombinedwith(4.3)toshowthatsatis“esexactlythesameformofequationandtherefore,givenalsothattheysatisfythesameboundaryconditions,that,whereT ww cp Theparametervaluesintables1and2give025,sothemeltwater“lmhasathicknessofabout100m,comparedwiththethermalboundary-layerthicknessof Onthemechanismsoficicleevolutionabout6mm.Thebalanceofheat”uxacrossthe“lmrequiresthat hw k whichshowsthatthetemperaturedierenceacrossthe“lmislessthanapproximatelyonehundredthofadegree(Short2006).Themeltwater“lmisthereforeentirelynegligibleand,inparticular,cannotaccountforthediscrepancybetweenobservedandpredictedmeltrates.CondensationofwatervapourHavingestimatedandeliminatedseveralotherhypothesesforthisdiscrepancy,wecametowonderabouttheroleofwatervapourintheair,recallingtheadage,Itsnottheheatthatllkillyou,itsthehumidity!ŽThewatervapourhaslittlein”uenceonthethermalconductivityoftheairatthetemperaturesinvolvedinourexperiments(Tsilingiris2008).However,theair”owresultingfromthermalconvectioncarrieswatervapourtothesurfaceoftheice,whereitcondensesandreleaseslatentheat.Thewatervapourcanbetreatedasapassivescalarbecauseithasnegligiblein”uenceonthedensityofaircomparedwithtemperature(seebelow).Thevapourdensitysatis“es(2.1)withthethermaldiusivityreplacedbythediusivityofwatervapour.Itisreadilyshownbyscalingthatthegradientofvapourdensityisthereforeequalto(/D)timesthethermalgradient,andthatthe”uxofwaterfromtheairtotheicesurfaceistherefore R/5  independentof,whereisthedierenceinthepartialdensityofwatervapourbetweenthefar“eldandtheicesurface.Thesaturatedpartialwater…vapourdensityisgivenasafunctionofabsolutetemperature TŠMLv R1 TŠ1 (Wood&Battino1990)whereisareferencetemperature(takenheretobethefreezingtemperature=273K),isthemolarmassofwater,isthelatentheatofvaporization,andisthegasconstant.Weassumethattheairissaturatedattheicesurface,inwhichcaseistherelativehumidityoftheairinthelaboratory.Giventhevaluesintables1and2,gcm.Thisisabout100timessmallerthanthedensitydierenceTassociatedwithtemperaturevariationsintheair,soitisappropriatetotreatthewatervapourasapassivescalar.TheStefanconditionismodi“edbytheadditionallatentheatassociatedwithcondensationofwatervapourtobecome whichcanberearrangedtodeterminethemeltrate scp  R Lv cp/5  0. J.A.Neufeld,R.E.GoldsteinandM.G.Worster 0.4 0.6 0.8 1.0 v (cm sÐ1)00.5 × 10Ð41.0 × 10Ð41.5 × 10Ð42 × 10Ð4 Figure13.Thepredicted(solidcurve)andmeasured(circleswitherrorbars)meltrateasfunctionsofradialpositionalongthetopofanicicle.Thesecondterminthesquarebracketshasavalueclosetounity,whichshowsthatthelatentheatassociatedwithcondensationofwatervapourcontributesroughlythesameamountofheatasissuppliedbyconduction.Ournewpredictionforthemeltratebasedon(4.11)iscms,whichisstillonlyabout55%ofthemeasuredvalue.Radiativeheat”uxThe“nalpieceofthejigsawcomesfromarealizationthattheradiativeheattransferplaysaroleofsimilarmagnitudetothesensibleandlatentheat”uxes.Conservationofheatattheice…airinterfaceisthereforegivenby wheretheradiative”uxinwhichistheStefan…Boltzmannconstant,andthetemperaturesmustbeexpressedinKelvin.Thisestimateoftheradiation”uxisbasedonassumingthattheexperimentalsurroundingsradiateasablackbody.Iceitselfisalmostablackbodyintheinfra-redspectrum:ithasare”ectivityofonlyabout1%andanopticaldeptharound10matwavelengthsaround10m(Warren1984),whichisthepeakoftheblack-bodyspectrumattheroomtemperatureof295K.Waterhassimilaropticalpropertiestoiceintheseconditions,soanywater“lmpresentmayalsoservetoabsorbincomingradiation.Therefore,nearlyalltheincidentradiationisabsorbedatorneartheicesurface,whereitcontributestomelting.Themeltrateatthecentreoftheicebasedon(4.12)iscmswhichiswithin3%ofthemeasuredvalueofcmsandthereforewellwithintheaccuracyofourexperiment.Themeasuredandcomputedverticalmeltratesareshownasfunctionsofradialpositionin“gure13.Notethatthecalculationsassumeaperfectlyhorizontaltopsurface,whereastheicequicklygainsaslightlyroundedsurface.Oncethathappens,thecomponentofgravityparalleltothe”owshouldacceleratetheair,creatingadditionalhorizontaldivergenceandanassociatedthinningoftheboundarylayerandenhancementoftheheattransfer.Thiseectislikelytobemostimportantneartheroundededgeoftheiceandmayaccountforthediscrepancyin“gure13for Onthemechanismsoficicleevolution7.Predictionofthesubsequentevolutionoftheshapeoficiclesawaitsfutureevaluation.5.ConclusionsInthispaper,wehaverevisitedthenatural,buoyancy-driven”owaboveacooled,“nite,horizontalplate,whichisequivalenttothe”owbelowaheatedplate.Wehaveshownthat,toaverygoodapproximation,thethermalboundary-layerequationscanbesatis“edwithself-similarsolutionsinwhichthehorizontalvariationofthescaleheightoftheboundarylayerobeysthethin-“lm(lubrication)equationsforaviscousgravitycurrent.ThesesolutionsmakepredictionsthatareinexcellentagreementwithpreviousexperimentalmeasurementsandwithanumericalsimulationbasedonthefullNavier…Stokesequations.However,whilethisvalidatestheapproach,oursolutionsarenotsigni“cantlymoreaccuratethanpreviousapproximationsmadeusingintegralapproaches.Theimportanceofourworkliesratherintheidenti“cationoftherelationshipbetweentheboundary-layerequationsandthosegoverningviscousgravitycurrents,andthesimplicityoftheapproach:thesolutionconsistsinsolvingonesystemofordinarydierentialequationsfortheself-similarverticalstructureoftheboundarylayerandanotherordinarydierentialequationforthehorizontalvariation.Thisapproachmightproveusefulinanalysingothertypesofdiusivegravitycurrents.Aparticularfeatureofourapproachistherecognitionthat,inthecaseofahorizontalplate,theinterioroftheboundarylayerseestheedgeoftheplate,wherethedense”uidspillsover,asapointsinkwhosestrengthisuniquelydeterminedbytheupstreamconditions.Itislikelythereforethatthevariousconditionsthathavebeenproposedattheedgeoftheplateonlyaectconditionslocallyanddonothavealeading-ordereectontheglobalheattransfer.Wehaveappliedouranalysisofthethermalboundarylayertotheair”owaboveacylinderoficeplacedinwarm,stillairinordertocalculatethecorrespondinginitialrateofmelting,whiletheiceblockhasahorizontaltopsurface.Wehaveshownthatheattransferfromtheairisinsucienttoaccountfortheobservedrateofmeltingbutthatboththelatentheatofcondensationofwatervapourandthenetradiativeheat”uxfromthesurroundingstotheicemustbeaccountedfor.Fortypicaliciclesinair,thesethreemechanismsprovideroughlyequalcontributionstothemeltWehaveshownthatthe“lmofmeltwaterisentirelynegligible.Givenitsverynarrowwidthandhighthermalconductivityrelativetothethermalboundarylayerintheair,itisalmostisothermalandtheheat”uxacrossitisdeterminedbyheattransferintheair.Thiscanbededucedbyscalinganalysisandwespeculateherethatthe“lmofwateronagrowingicicleissimilarlynegligibleexceptthat,inthecaseofagrowingicicle,itmustexistinorderfortheicicletocontinuetogrow:adistinctioncanbemadebetweeniciclesthataredripping,whichthereforehaveacontinuous“lmofwatercoveringthemandwhoselengthcanthereforeincrease,andthoseforwhichthe“lmfreezesbeforereachingthetipandsimplyfatten.Anintriguingfeatureofgrowingiciclesisthattheirsurfacesbecomerippled,whichisthoughttobeaconsequenceofsomeformofmorphologicalinstability.Theresultsofthispapersuggestthatanalysesofsuchinstabilitiesshouldincorporatetheconvectionintheair,thecondensationorevaporationofwatervapour,andradiation,oneormoreofwhichmightaccountfortheobservedfeatures,andshouldperhapsdisregardthewater“lmasakeycomponentofthesystem. J.A.Neufeld,R.E.GoldsteinandM.G.Worster 4 6 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7(a)(b)  0 1 2 3 4 5 6 Ð1.0 Ð0.8 Ð0.6 Ð0.4 Ð0.2 0 Figure14.NumericalsimulationsofthefullNavier…Stokesequationsarecomparedwiththetheoreticalpredictionsoutlinedin2.Verticalpro“lesofthevelocityareshownin(athorizontalpositions8and09.Thethermalpro“lesareshownin()athorizontalpositions9.Theinsetshowsthethermal“eldaroundthebluntwedgewithlength(centretoedge)=125cm.Thetemperaturedierenceis=10parametersrepresentativeofairwereused,thus7andWeareverygratefultoRobStylefordiscussionsaboutcondensation.ThisworkwassupportedinpartbytheSchlumbergerChairFund.J.A.N.issupportedbyfellowshipsfromLloydsTercentenaryFoundationandtheLeverhulmeTrust.M.G.W.wasprivilegedtobeacolleagueofSteveDavisforthreeyearsintheDepartmentofEngineeringSciencesandAppliedMathematics,NorthwesternUniversity.Atthattime,around1990,Steverantwobrown-bag-lunchseminarseries,oneonthin“lm”owsandtheotheronsolidi“cation,whichwerewonderfullystimulating.Wearepleasedtooerthispaper,combiningaspectsofbothofthoseinterests,inhonourofSteves70thbirthday.Appendix.NumericalsolutionoffullNavier…StokesequationsAsacomplementtotheboundary-layerapproximationdevelopedinthebodyofthepresentwork,wehaveperformeddirectnumericalsimulationsoftheBoussinesqequa-tionsofmotionforthethermalandvelocity“eldsaroundasolidbodyheldat“xedtemperature.Thenumericalcalculationsweredonewithacommercial“nite-elementcode(Comsol)whichallowedforanon-uniformgridnearthecornersofthebody.Theuppersurfaceofthebodywaschosentohavetheshapeofarectangularslab,thebottomofwhichwasdeformedintoadownward-facingbluntedpointinordertoshedthedescendingboundarylayerinasmoothmanner(seetheinsetin“gure14 OnthemechanismsoficicleevolutionAllcomputationsstartedwithuniformplateandfar-“eldtemperaturesandwereconductedwith=10C.Parametersrepresentativeoftheexperimentsofetal.(1972)werechosen,namely187cm133cm=125cm,whichcorrespondtogoverningnon-dimensional7and.Thethermalboundary-layerstructurewasresolvedwithapproximately10elementsandwasexaminedoncethesystemhadreachedasteadystate,characterizedbythetransporttimeacrosstheplate.Theresultantcomparisonbetweenthenumericallycalculatedvelocityandthermalstructureshownin“gure14showsexcellentagreementovermostoftheplatewiththeboundary-layermodelof2.Indeed,theexcellentcomparisonbetweentheboundary-layermodel,experimentalresultsandfullnumericalcalculationslendscredencetoboththeboundary-layerapproximationsandtheexplicitseparationofverticalandhorizontalstructure.Thus,we“ndthatthecontributionofthethermalboundarylayertothemeltingoficeasexempli“edbythevariationofthelocalNusseltnumbershownin“gure9iswellcharacterizedbytheboundary-layermethoddevelopedthroughoutthispaper.Aihara,T.,Yamada,Y.&Endo,S.1972Freeconvectionalongthedownward-facingsurfaceofaheatedhorizontalplate.IntlJ.HeatMassTrans.,2535…2549.Clifton,J.V.&Chapman,A.J.1969Natural-convectionona“nite-sizehorizontalplate.IntlJ.HeatMassTrans.,1573…1584.Dayan,A.,Kushnir,R.&Ullmann,A.2002Laminarfreeconvectionunderneathahothorizontalin“nite”atstrip.IntlJ.HeatMassTrans.,4021…4031.Drazin,P.G.&Reid,W.H.HydrodynamicStability.CambridgeUniversityPress.Fujii,T.,Honda,H.&Morioka,I.1973Atheoreticalstudyofnaturalconvectionheattransferfromdownward-facinghorizontalsurfaceswithuniformheat”ux.IntlJ.HeatMassTrans.,611…627.Gill,W.N.,Zeh,D.W.&DelCasal,E.1965Freeconvectiononahorizontalplate.Zeit.Ang.Math.Phys.,539…541.Hatfield,D.W.&Edwards,D.K.1981Edgeandaspectratioeectsonnaturalconvectionfromthehorizontalheatedplatefacingdownwards.IntlJ.HeatMassTrans.(6),1019…Higuera,F.J.1993Naturalconvectionbelowadownwardfacinghorizontalplate.Eur.J.Mech..Fluids(3),289…311.Higuera,F.J.&Weidman,P.D.1995Natural-convectionbeneathadownwardfacingheatedplateinaporous-medium.Eur.J.Mech.B.Fluids(1),29…40.Kuiken,H.K.1968AnasymptoticsolutionforlargePrandtlnumberfreeconvection.J.EngngMath.(2),355…371.Ogawa,N.&Furukawa,Y.2002Surfaceinstabilityoficicles.Phys.Rev.,041202.Schlichting,H.&Gersten,K.BoundaryLayerTheory.Springer.Short,M.B.,Baygents,J.C.&Goldstein,R.E.2006Afree-boundarytheoryfortheshapeoftheidealdrippingicicle.Phy.Fluids,083101.Singh,S.N.&Birkebak,R.C.1969Laminarfreeconvectionfromahorizontalin“nitestripfacingdownwards.Zeit.Ang.Math.Phys.(4),454…461.Stewartson,K.1958Onthefreeconvectionfromahorizontalplate.Zeit.Ang.Math.Phys.Tsilingiris,P.T.2008Thermophysicalandtransportpropertiesofhumidairattemperaturerangebetween0and100EnergyConvers.Manage.,1098…1110.Ueno,K.2003Patternformationincrystalgrowthunderparabolicshear”ow.Phys.Rev.Ueno,K.2004Patternformationincrystalgrowthunderparabolicshear”ow.Part2.Phys.Rev.,051604. 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