ThenearwallbehaviorofanimpingingjetDanielleR.S.Guerra,JianSu,AtilaP.Si - PDF document

ThenearwallbehaviorofanimpingingjetDanielleR.S.Guerra,JianSu,AtilaP.SilvaFreireMechanicalEngineeringProgram(PEM/COPPE/UFRJ),C.P.68503,21945-970„RiodeJaneiro,BrazilNuclearEngineeringProgram(PEN/COPPE/UFRJ),C.P.68509,21945-970„RiodeJaneiro,BrazilReceived2January2005;receivedinrevisedform31January2005Thepresentworkinvestigatestheapplicabilityofscalinglog-lawstotheturbulentimpingingjet.Both,thevelocityandthetemperature“eldsarestudiedunderthisassumption.Tovalidatetheproposedexpressions,adetailedexper-imentalprogramwascarriedoutbasedonthermalanemometry.Theexperimentswereconductedforonenozzle-to-platespacing(=2.0)andReynoldsnumberof35,000.Aconstantwallheat”uxconditionwasachievedbyconductingelectricitythroughthinresistorsthatwereplacedbeneathanaluminumdisk.Measurementsoflocalveloc-ityandoftemperaturedistributionsarepresentedaswellaslongitudinalturbulencepro“les.Themeantemperaturepro“lesweremeasuredthroughthermocouples.2005PublishedbyElsevierLtd.Impingingjet;Lawofthewall;Hot-wireanemometry1.IntroductionAturbulentjetimpingingontoasurfaceisaverye
ectivemeanstopromotehighratesofheatexchange.Asanobviousimplication,thisgeometricalarrangementhasbeenextensivelyusedinindustrialprocessesthataimtoachieveintenseheating,coolingordryingrates.Typ-icalapplicationsarethetemperingandshapingofglass,theannealingofplasticandmetalsheets,thedryingoftextileandpaperproducts,thedeicingofaircraftsys-temsandthecoolingofheatedcomponentsingastur- Correspondingauthor.Tel./fax:+552125627748.E-mailaddresses:daguerra@terra.com.br(D.R.S.Guerra),sujian@lmn.con.ufrj.br(J.Su),atila@mecanica.coppe.ufrj.bratilafreire@gmail.com(A.P.SilvaFreire). InternationalJournalofHeatandMassTransfer48(2005)2829…2840 www.elsevier.com/locate/ijhmt Forwalljets,wecannotpositivelysaythatthelog-lawisawellestablishedconcept.Infact,severalauthorsauthorshavereportedalargerangeofvaluesforthelog-lawconstants.Thiscertainlyraisessomeimportantquestionsastothevalidityontheuseofthelog-lawfortheestimationofsurfacefrictionandoftheheattransfercoecient.Someresearchersresearchers,however,haveshownthatforanobliqueimpingingjetthelawofthewallcanbeob-servedforboththevelocityandthetemperature“elds.Morethanthat,theseauthorsproposeafunctionalbehaviorforthelog-lawparametersthatresortstoascalingprocedurebasedonthestream-wiseevolutionofthe”owcharacterizedbyitsmaximumvelocity.Infact,Narasimhaetal.al.wereoneofthe“rsttoacknowledgethatthetraditionaluseofthenozzlediam-eterasthereferencescalingforwalljet”owswasnotappropriate.Theyproposedascalinglengththattookintoconsiderationthe”owevolution.Thepurposeofthepresentworkistocarryoutfur-therinvestigationsonthescalinglawsgoverningthemo-tionofanorthogonaljetimpingingontoasurface.Forthatmatter,thelawofthewall,forboththevelocityandthetemperature“elds,willbeinvestigatedindetail.Infact,thisisanecessary“rststepforthefutureinvesti-gationofsomegoverningparametersontheskin-frictionandontheheattransfercharacteristicsofacircularimpingingjetonaheated”atplate.Twonewexpressionswillbeadvancedforthelawofthewallforthevelocityandtemperature“eldsthatwillresorttoparametricargumentsinthelineoftheworksofNarasimhaetal.al.andofOzdemirandWhitelawWhitelaw.Measurementsoflocalpressure,velocityandtemperaturedistributionswillbepresented.Wewillalsopresentdataforthelongi-tudinalturbulentintensities.Theexperimentswerecon-ductedforonenozzle-to-platespacingandReynoldsnumberof35,000.Aconstantwallheat”uxconditionwasachievedbyconductingelectricitythroughthinresis-torsthatwereplacedbeneathanaluminumdisk.Tem-peraturewasmeasuredusingthermocouples.Atthispoint,itisimportanttomakeitcleartothereaderthatotherauthorshavespeci“callystudiedtheroleofthescalinglawsinwalljet”ows.ThatisthecaseoftheworkofWygnanskietal.al.wheretherelevanceofthewalltotheevolutionofthelargecoherentstruc-turesinthe”owisstudied.Here,intheimpingingjet,theproblemisfurthercomplicatedbyade”ectionofthestreamlinesandbythepresenceofastagnationTherecognitionthatforthewalljetthevelocitypro-“ledoesnotexhibittheconventionallawofthewallbehaviorisalsoveri“edinanarticlebyHammondHammond.Aimingatdevelopingananalyticexpressionforthecompletevelocity“eldinthefullydeveloped”owregimeofaplane,turbulentwall,Hammondproposedtocou-pleSpaldingssingleformulaformulafortheinnerlayerwithasinefunctionforthewakecomponent.Forturbulent”ows,themathematicaldescriptionofthe”ow“eldisgreatlycomplicatedbythenecessaryspeci“cationofturbulencemodelsthatcancaptureallrelevantcharacteristicsoftheproblem.Frequently,tur-bulencemodelsoftheeddyviscositytypeareusedto-getherwithsomeheattransferanalogyconsiderationforthedescriptionofthetemperature“eld(seee.g.,e.g.,.Thisleads,forexample,toseriousdicultiesatthestagnationpointwheretheReynoldsanalogybe-tweeneddy-di
usivityandeddy-viscositybreaksdown.Indeed,whentheequationsofmotionareintegratedtothewallandthehypothesisofaconstantturbulentPrandtlnumberisused,thecalculatedheattransferratesatthestagnationpointareobservedtoexceedbymuchtheiractualvalues. parametersinlawofthewalldiameterofjetnozzleheattransfercoecientdistancebetweennozzleandimpingingwallelectriccurrentacrossresistancesVonKarmanconstant(=0.41)Nusseltnumberheat”uxradialdistancefromjetcenterlinetemperature,frictiontemperature[%]turbulentintensity(=





jetbulkvelocitylocallongitudinalmeanvelocity,locallongi-tudinalvelocity”uctuationfrictionvelocityvoltageacrossresistancesCartesiancoordinatesGreeksymbolskinematicviscositythermaldi
usivitymaximum,minimumwallcondition,adiabaticwallconditionD.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 Despitethecriticsofmanyresearchers,theuseofwallfunctionstoby-passthedicultiesinvolvedwiththemodelingoflowReynoldsnumberturbulenceisstillanattractivemeanstosolveproblemsinasimpleway.Forinstance,CruzandSilvaFreireFreirehavepro-posedanalternativeapproachwherenewwallfunctionsareusedtodescribethevelocityandtemperature“eldsinthewalllogarithmicregionofaseparating”ow.Asthestagnationpointisapproached,thesefunctionsre-ducetopower-lawsolutionsrecoveringStratfordssolu-tion.ThepaperofCruzandSilvaFreireresortedtoKaplunlimitsforanasymptoticrepresentationofthevelocityandtemperature“elds.Resultswerepre-sentedfortheasymptoticstructureofthe”owandfortheskin-frictioncoecientandStantonnumberatthe2.ShortliteraturereviewForthewalljet,the“rststudieswereseverallylimitedbythelackofanysophisticatedinstrumentation.Asaresult,the“rstexperimentswerelimitedtomeasure-mentsofmeanvelocitiesinthevicinityofthenozzle(seee.g.,e.g.,).SigallaSigallawasthe“rsttotrytoevaluatetheskin-frictionandalsothe“rsttomeasurethemeanvelocityatlargedistancesfromthenozzle.Thedevelop-mentofthehot-wireanemometerinthe60smadeitpossiblethedevelopmentofmuchmoredetailedTaillandandMathieuMathieunoticedthattherateofspreadofawalljetandthedecayofitsmaximumveloc-ityweredependentontheReynoldsnumber,afeaturethatisnotobservedinafreejet.Thatraisedquestionsonthereasonforsuchdi
erence.Scalinglawsforwalljetswereparticularlystudiedbyby,OzarapogluOzarapogluandIrwinIrwin.Inrelationtotheimpingingjet,thefollowingaresomeofthenotablereferences.Intheearly90s,OzdemirandWhitelawWhitelawstudiedtheproblememphasizingthelarge-scaletransportoftemperaturebyspatiallycoherentstructures.Theseauthorsshowedthatanobliqueimpingementintroducedverticalvelocitiesthatrenderedtheboundarylayerequationinapplicableandresultedina”owstructurewithstrongazimuthallydependence.Thelargestruc-turesimprovedthetransportofheatbutledtoaninac-tivezonenearthevortexcenter.Foxetal.al.alsostudiedthein”uenceofvorticalstructuresonthetemperature“eldofjets.Dependingonthedistanceofthejetnozzletoanadiabaticwall,sec-ondaryvorticalstructureswereobservedthatcouldre-sultinaregionoflowerwalltemperature.Thus,itisthecompetitionthatisestablishedbetweenthevortexringsformedatthejetperipheryandthesecondaryvor-ticesresultingfromtheimpingementthatdeterminesthewalltemperature.Cooperetal.al.andCraftetal.al.intwocompan-ionpapersstudiedturbulentjetsimpingingorthogonallyontoaplanesurface.Cooperetal.reportedanextensivesetofmeasurementsonthe”ow“eld;dataforthemeanvelocitypro“leinthevicinityofthesurfaceandalsoforthethreeReynoldsstresscomponentslyingintheplanewerepresented.ThesedatawereusedbyCraftetal.toexaminetheperformanceoffourdi
erentturbu-lencemodels:themodelandthree-secondordermo-mentclosures.Thepredictionsobtainedthroughthemodelandonesecondordermomentclosurere-sultedinfartoohighturbulencelevelsnearthestagna-tionpoint.Assuch,theyalsoresultedintoohighheattransfercoecients.Adaptationsonthetwoothermod-elsledtomuchbetterpredictions.Noneofthemodels,however,couldsuccessfullypredicttheReynoldsnum-bere
ectsonthe”ow.Theauthorsconcludedthatthisoughttobeduetothetwo-equationeddyviscositymodelthatwasadoptedforallcasestospantothenearwallsublayer.ThenumericalsimulationofimpingingjetsusingamodelwasalsoperformedbyKnowlesKnowles.TheauthorconcludedthattheRodiandMalincorrectionscouldnotpredictwalljetgrowth(see(seefordetails).ColucciandViskantaViskantastudiedexperimentallythee
ectsofnozzlegeometryonthelocalheattransfercoef-“cientsofcon“nedimpingingjets.Lownozzle-to-plategapswereconsideredintheReynoldsnumberrangeof10,000…50,000.Theresultswerecomparedwithsimilarexperimentsforuncon“nedjets.Animportantconclu-sionwasthatthelocalheattransfercoecientsforcon-“nedjetsaremoresensitivetoReynoldsnumberandnozzle-to-plategapsthanthoseforuncon“nedjets.Dianatetal.al.usedamodelandamodi“edsecond-momentclosuretomakevelocity“eldpredic-tionsinthestagnationaswellasinthejetregion.Thesec-ond-momentclosurewasmodi“edtoaccountforthein”uenceofthewallindistortingthe”uctuatingpressure“eldawayfromit.Withthismodi“cation,thedampingofnormalvelocity”uctuationswaswellpredicted.Theheattransferinthe”owofacold,two-dimen-sional,verticalliquidjetimpingingagainstahot,hori-zontal,surfacewasgivenanapproximatesolutionforthevelocityandtemperature“eldsbyShuandWilkinsWilkins.Thesolutionisvalidforlaminar”owsandresortstothehydrodynamicsimilaritysolutionofWatson(seetheauthorsfordetails).Theresultswerecomparedwithanumericalrealizationofthe”ow.Meolaetal.al.investigatedthein”uenceofshearlayerdynamicsonimpingementheattransfer.Again,coherentstructuresand/orrecirculationcurrentswereobservedtoalterthedistributionofheattransfercoe-cients.TemperaturesweremeasuredwithaninfraredscanningradiometerwhereasheattransfercoecientsD.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 wereevaluatedbytheheatedthinfoiltechnique.ThedistributionofNusseltnumberwasdiscussedandanewexplanationgivenforthelocalpeakinthelocalNusseltnumber.LiuandSullivanSullivaninvestigatedtheheattransferand”owstructuresinanexcitedcircularimpingingjetwithasmallnozzle-to-wallgap.Enhancementorreduc-tionofthelocalheattransfercoecientinthewall-jetregionwasshowntobeachievedbyexcitingtheimping-ingjetneartoitsnaturalfrequencyorsub-harmonics,Nishinoetal.al.reporttheturbulencestatisticsinthestagnationregionofanaxisymmetricjetimpingingverticallyonawall.Theyusedparticle-trackingveloci-metrydomeasurethe”ownearthestagnationpointandfoundthattheturbulentnormalstressoftheaxialcomponentgaveasubstantialcontributiontothein-creaseinthestaticpressurenearthewall.Turbulencewasstudiedthroughaninvariantmapoftheturbulentstressanisotropy.Inthestagnationregion,turbulencewasclosetoanaxisymmetricstate.Thestandardmodeltogetherwiththelogarith-miclawofthewallwasappliedbyAshforth-Frostandandtoasemi-con“nedimpingingjet;thenozzle-to-walldistancewastwonozzlediametersandtheReynoldsnumber20,000.Laser-Doppleranemome-tryandliquidcrystalthermographywereusedtodeter-minevelocity,turbulenceandheattransferdata.Inthedevelopingwalljet,authorsshowednumericalheattransferresultstocomparetowithin20%ofexperimen-taldata.However,atthestagnationpoint,heattransferisoverpredictedbyabout300%.Theauthorsattributedthisdiscrepancytofailureofthewallfunctiontocon-formtothephysicsofthe”ow.Sanetal.al.reportedlocalmeasurementsofNusseltnumberforacon“nedimpingingjet.Therecirculationandthemixinge
ectontheheattransferwereinvesti-gatedbyvaryingthejetdiameter,thesurfaceheat”ux,theReynoldsnumberandthesurfaceheatingwidth.KnowlesandMyszkoMyszkocarriedoutturbulencemeasurementsinajetimpingingontoa”atwall.Di
er-entnozzle-to-wallgapswereinvestigated.Measurementswereconductedusinghot-wireanemometry.Nozzleheightwasfoundtohavealargee
ectonturbulencepeaklevelfordistancesupto=4.5;lowernozzle-to-wallratioscausedanincreaseinpeaklevelmeasuredinallturbulentstressesinthestagnationregion.Kendoushinapreviousworkhadderivedanalyticalsolutionsfortheconvectiveheatandmasstransferpre-dictionsinalaminarjetimpingingonaplanewall.However,hissolutionwasshowntohaveastrongsingu-larityatthestagnationpoint.Thesubsequentpaperofofhad,therefore,theobjectiveofremovingsuchsingularityand“ndworkablesolutionsfortheproblem.Theresultswerecomparedwithavailableexperimentalandnumericaldata.LeeandLeeLeeexperimentallystudiedtheheattransferbehaviorofaturbulentjetimpingingonawallwithspecialattentiontothestagnationregion.Fornoz-zle-to-wallratiosof=2.0thelocalNusseltnumbervariationwithhadtwopeaksandvariedaccording.For�6.0,Nusseltnumberdecreasedmonotonicallywith.ReynoldsnumberdependencewasobservedtoincreaseasCon“nedimpingingjetsatlowReynoldsnumberswereexperimentallystudiedbyBaydarBaydarforasingleandadoublejet.Theauthorconcludesthatasub-atmo-sphericregionoccursontheimpingementwallatnozzle-to-wallgapsuptotwoandthatthereisalinkagebetweenthesub-atmosphericregionandthepeakintheheattransfercoecients.Intheirnextpaper,LeeandLeeLeestudiedexperi-mentallythelocalheattransfercharacteristicsofanellipticjetimpingingonaheated”atplateforvariousnozzleaspectratios.Thetemperaturedistributionsontheheatedplateweremeasuredusingathermochromicliquidcrystalthermometrywithadigitalimageprocess-ingsystem.Asmoke-wiretechniquewasusedtovisual-izethe”ow.Forsmallnozzle-to-wallgap,astheaspectratiooftheellipticjetincreases,theheattransferratein-creasedmorethanthatforthecircularjetinthestagna-tionregion.Thiswasattributedtothelargeentrainmentandlargescalemixingoftheellipticjet.ThecontrolonthepropertiesofanimpingingjetbyvortexpairingwasinvestigatedbyHwangetal.al..Twotypesofvortexcontrolwereperformed:secondaryshear”owandacousticexcitation.Some”owvisualiza-tion,velocityandturbulencemeasurementswereper-formedtounderstandthe”owstructure.Enhancementorreductioninheattransferwasobservedbycontrollingthevortexpairing.GuoandWoodWoodperformedmeasurementsinthevicinityofthestagnationpointforanimpingingjetwithaverylowfree-streamturbulencelevel(0.35%).ThewallshearstresswasmeasuredwithPrestontubesandStan-tonprobes.Duetotheverysmallthicknessoftheboundarylayer,0.42mm,thesizesoftheprobeshadtobekepttoaminimum.Infact,theauthorsfoundthatdecreasingprobesizetheshearstressincreased.Intheend,thevaluesfoundforthe0.11mmStantonprobewerereported.A”owdirectnumericalsimulationwasperformedbyChungetal.al.toinvestigatethevelocityandtheheattransfercharacteristicsofanunsteadyimpingingjet.Theseauthorsfoundthattheunsteadinessinwallheattransferiscausedbythevorticesemanatingfromthenozzle,andthatthesevorticessigni“cantlya
ectthestagnationheattransfer.A“niteelementcodethatresortstothestreamlineupwindmethodofPetrov…GalerkinandusestheturbulencemodelwasdevelopedbyParketal.al.topredictthe”owandtheheattransfercharacteristicsofD.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 two-dimensionalcon“nedimpingingjets.Validationofthecodewasmadeagainstothercodesandsomemea-suredmeanvelocitypro“les.The”ow“eldandthelocalheattransferfortransi-tionalimpingingjetswerecharacterizedbyAngiolettietal.al.thoroughparticleimagevelocimetryandthenaphthalene“lmtechnique.Localnon-uniformitiesinheattransferwereexplainedthroughthedestructiveef-fectsthatlargecoherentstructurescreatedatthenozzlehaveuponimpingementontheboundarylayer.Narayananetal.al.studiedexperimentallythe”ow“eld,surfacepressureandtheheattransferratesforasubmerged,turbulent,impingingjet.Twonozzle-to-sur-facespacingratioswereanalyzed.Meanandturbulent”owpropertiesweredeterminedthorougha1-DLDAsystem.MeanandRMSsurfacepressureswerefoundwithapiezoresistivetransducer,surfacetemperatureswerefoundusingIRthermography.Theresultsindicatethatpastimpingement,and,irrespectiveofthenozzlespacing,locationsofhighstreamwise”uctuatingveloc-ityvarianceoccurinthewalljet.Inparticular,fortheshorterofthetwonozzle-to-surfacespacingratios,theauthorsfoundagoodcorrelationbetweenthelocationofasecondarypeakintheheattransferandthenearwallstreamwise”uctuatingvelocityvariance.TheuseofmeshscreenstoenhancethetransferofheatinimpingingjetswasinvestigatedbyZhouandand.Generally,itwasobservedthatthescreensmodi“edthe”ow“eldleadingtoanincreaseinthelocalheattransfercoecients.3.ExperimentalmethodsAdrawingoftheexperimentalapparatusisshowninFig.1.Airat18.5Cispumpedthroughacentrifugalblowerconnectedtoa1350mmlongpipewith43.5mminternaldiameter.Insidethepipe,ahoney-combis“tted,constructedfromdrinkingstrawsgluedtogether;screensarealsosetinplace.Thejetissettoemergefromthecircularnozzlewithabulkvelocityof12m/s.Theimpingement”atsurfaceismadeofa3.7mmthickaluminumcircularsheet.Thissheethas840mmindiameterandislaidoveraplenumchamberasshownFig.1.Theplenumchamberis20mmheightand Fig.1.Experimentalapparatus.(a)Overallview:(1)centrifugalblower,(2)”exiblesection,(3)contraction,(4)pipe,and(5)testsection.(b)Detailoftestsectionandheatingsystem:(1)con“nementplate,(2)electricalresistance,(3)impingementplate,(4),(5)and(6)thermalisolation.D.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 815mmindiameter.Atthebottomofthechamberaseriesofelectricalresistancesareplacedtofurnishamaximumof4000W.ThewallsoftheplenumwerecompletelyinsulatedfromtheambientasshowninFig.1Thecontrolledparametersintheexperimentsarethenozzle-to-platespacing,theresistanceheat”uxandthestagnationpressure.Ateachtest,thecenterlineofthejetislinedupwiththecenteroftheimpingementsurfaceThetemperatureofthealuminumsheetwasmoni-toredthroughthermocouples.Thereadingsofthether-mocoupleswereroutedtoanAMDAthlon+2000MHzpersonalcomputerviaaPicologacquisitionsystemmodelTC-08.Meanvelocitypro“lesandturbulenceintensitylevelswereobtainedusingaDANTEChot-wiresystemseries56N.Theboundarylayerprobewasofthetype55P15.APitottube,aninclinedmanometer,andacomputercontrolledtraversegearsystemwerealsoused.Inget-tingthedata,10,000sampleswereconsidered.Thepro-“leswereconstructedfromabout100points.Themeantemperaturepro“leswereobtainedthroughachromel-constantanmicro-thermocouplemountedonthesametraversegearsystemusedforthehot-wireprobe.Thetraversegearsystemhas0.02mmsensitivity.AnuncertaintyanalysisofthedatawasperformedaccordingtotheproceduredescribedinKlineKline.Theuncertaintyassociatedwiththevelocityandtemper-aturemeasurementswere:=0.064m/sprecision,0bias(=0.95);=0.214Cprecision,0bias(=0.99).Thelocalconvectiveheattransfercoecientwascal-culatedfrom arewalltemperatureandadiabaticwalltemperatureofthestream.isthetotalheat”uximposedthroughtheelectricalresistanceandisgivenby Theconductionheat”uxwasconsiderednegligibleandtheradiationheat”uxwascalculatedandwas2%ofthetotalimposedheat”ux.Thelocalsurfacetemper-aturewasconvertedtothelocalNusseltnumberde“ned isthejetexitdiameterandisthethermalcon-ductivityofair.Toperformtheexperiments,thefollowingprocedurewasapplied.First,the”atplatewas“ttedwith27pres-suretapsarrangedinacrossformation.Thereadingsofthepressureatthesepointsweresubsequentlyusedto“ndthegeometricalcenterofthejet;onlywhenthepres-suredistributionwasfoundtobecompletelysymmetricthejetcenterlinewasconsidereddetermined.To“ndtheadiabatictemperature,theelectriccurrentwasturnedo
andthetemperaturerecorded.Next,theresistorswereturnedonandthebehaviorofthewalltemperaturerecorded.Normally,6controlpointswereusedatthisstage.Onlywhentheplatewasobservedtoreachasteadystatetheblowerwasturnedon.Inthesteadystatecondition,thewalltemperaturevaria-tionwaswithin1C.Approximately2.5harerequiredtoreachsteadystateconditionsforeachtestrun.4.ResultsTheworkwillpresentresultsforthegeometryde-“nedbytheaspectratio=2.0.TheradialpressuredistributionontheimpingementsurfaceandtheNusseltnumberdistributionareshownFig.2.Pressurewasmadedimensionlesswiththedy-namicpressure,/2,whereisthedensityofairandisthejetbulkvelocity. -4-202468r/D -0.2 0246 304050 H/D = 2.0
P/(0, 5U2) (a)(b)Fig.2.(a)Radialpressuredistributionsofthejetand(b)Nusseltnumberdistribution.D.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 Forlarge,nozzle-to-platespacingrates,theNusseltnumberisnormallyobservedinliteraturetopresentamaximumatthestagnationpointdecreasingwithincreasing.Here,thistrendhasbeencon“rmed.Thelocalheattransferbeginstoincreasefromthestag-nationpointtowardsa“rstlocalpeakpositionnear0.5.Then,Nusseltnumberdecreasesreachingaweaklocalmaximumatabout2.0.Afterthispointasecondweakpeakappearsandthendecreases.Tocharacterizethewallspreadingofthejet,thera-dialcomponentsofthemeanvelocity,ofthelongitudi-nalturbulenceintensityandofthemeantemperaturewereexaminedatvariousradialpositions.Thevelocity,longitudinalturbulentintensityandtemperaturepro“lesareshowninFigs.3…5inphysicalco-ordinates.AstudybyOzdemirandWhitelawWhitelawhasshownthataWeibulldistributionrepresentswellsomeoftheglobalfeaturesofthepro“le,suchasthepositionofthemaxi-mumandoftheouterin”ectionpoints,butisnotanadequateapproximationforthenearwallregion.Forthisregion,theyshowedthatsemi-logarithmicrelationcanbeusedtomodeltheinnerequilibriumlayer,sothatonecanwrite uus¼ 1jln isthefrictionvelocityandisthevonKarmanThemaincontributionofOzdemirandWhitelawWhitelawwastoshowthat,fortheimpingingjet,theinnerlayer y[mm] 048 048 u[m/s] r=85mmr= 90mmr= 95mmr= 100mmH/D= 2.0 048 y[mm] 048 048 048u[m/s] r= 105mmr= 115mmr= 120mmr= 125mm 048 y[mm] 048 048 048u[m/s] r= 150mmFig.3.Meanvelocitypro“les.(a)Stations80…100mmm,(b)stations105…125mm,and(c)stations130…150mm.D.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 appearstoconstituteaconsiderablepartoftheinnerboundarylayer,and,iftheouteredgeoftheequilibriumlayerisattachedtothepointofmaximumvelocity,whichisveryclosetothewall,then,thismaximum,,shouldbeanappropriatevelocityscale.Theconclu-sion,therefore,isthatparameterisnotinvariantbutchangeswithadeviationfunction.Todescribe,theseauthorsproposedasimplerela-tionoftheform denotesthemaximumvelocityforagivenvelocitypro“le.Wygnanskietal.al.remarkedthat,foraturbulentwalljet,thevelocitypro“lecannotbeuniversallyrepre-sentedinwallcoordinates,asitcanintheboundarylayer.Thatisduetolargevariationsintheadditivecon-stantinthelawofthewall.Infact,dependingonthejetReynoldsnumber,logarithmic“tscanbefoundtotheirdatainregionsde“nedbyspeci“climits.These“ttedstraightlineshavelevelsvaryingfrom5.5to9.5.Theexistenceofawellde“nedlogarithmicregionisparticu-larlyimportantforthedeterminationoftheskin-fric-tion.Wygnanskietal.furtherremarkthatinpreviousexperimentstheskin-frictionwaseitherdirectlyassessedthrough”oatingdragbalancesorindirectlybywallheattransferdevicesorbyimpactprobeslikeStantonprobes 204060 y[mm] 204060 204060 204060 204060Tu[%] r= 80mmr= 90mmr= 95mm 204060 y[mm] 204060 204060 204060 204060Tu[%] r= 110mmr= 125mm 204060 y[mm] 204060 204060 204060 204060Tu[%] r= 130mmr= 135mmr= 140mmr= 150mmFig.4.Longitudinalturbulentintensitypro“les.(a)Stations80…100mm,(b)stations105…125mm,and(c)stations130…150mm.





D.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 orPrestontubes.Sincethesedevicesarecalibratedtak-ingasreferencetheuniversallawofthewall,theycan-notbereliablyusedinregionswheretheexistenceofthelawofthewallcanbequestioned.Wygnanskietal.estimatedtheskin-frictionthroughthreedi
erenttechniques:amomentumintegralmethod,themeanvelocitygradientintheviscoussub-layer,andbyuseofaPrestontube.Theestablishmentoftheaboveconceptsforthevelocity“eldclearlyraisessomequestionsforthetem-perature“eld.Animmediatequestionconcernstheexis-tenceofanappropriatetemperaturescaleattheouteredgeoftheequilibriumlayer.AtthepointofvelocityFig.5showsthatthetemperaturepro“lesreachaminimum.Thus,drawingananalogytothevelocityanalysesofNarasimhaetal.al.andofOandWhitelawWhitelaw,onewouldexpecttheappropriatescalingtemperatureparametertobethisminimumThelawofthewallforthetemperaturepro“lecanbewrittenas Tw
Tts¼ 1ktln isthefrictiontemperatureandisthevonKarmanconstantforthetemperature“eld.Theexpectedparametricbehaviorofisthentoberepresentedby representsthewalltemperature,themini-mumtemperatureinagivenpro“leandisthefrictionTo“ndthevaluesofandof,thegraphicalmethodofColesColeswasused.Here,wemustpointoutthatthethicknessoftheinnerturbulentregionforanimpingingjetisverythin,sothatthe“ttingofastraightlinetothe 26303438 y[mm] 26303438 26303438 26303438 26303438T[ºC] r= 80mmr= 85mmr=90mmr= 95mmr= 100mm 26303438 y[mm] 26303438 26303438 26303438 26303438T[ºC] r= 105mmr= 110mmr= 115mmr= 120mmr= 130mm 26303438 y[mm] 26303438 26303438 26303438 26303438T[ºC] r= 130mmr= 135mmr= 140mmr= 145mmr=150mmFig.5.Meantemperaturepro“les.(a)Stations80…100mm,(b)stations105…125mm,and(c)stations130…150mm.D.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 fullyturbulentregionisadiculta
air.Foraplanewalljet,thefullyturbulentregionisratherarbitraryarbitrary,nor-mallybeinglocatedwithintheinterval70170.SincetheanalysisofWygnanskietal.al.suggeststhatvonKarmansparametercanbeconsideredconstantandthatvariesfrom5.5to9.5,the“ttingofastraightlinetothevelocityandtothetemperaturedatainsemi-logplotsintheregion70170canthenbeusedto“ndinEqs..ThegraphicalmethodusedforthedeterminationofparametersisillustratedinFig.6TheresultinglinearbehaviorofparametersisshowninFig.7.This“gureindicatesthatbothincreaseasthemaximumjetvelocityincreasesandtheminimumjettemperaturedecreases,respectively.Specif-ically,thefollowingequationsresult: uMus27.538;ð8ÞB¼1.031 Thus,thetrendobservedbyOzdemirandWhitelawWhitelawiscon“rmedhere.Furthermore,thepresentanalysisgivesusastronghintthatapossiblelinearbehaviorofandofasafunctionofthemaximumjetvelocityandoftheminimumjettemperaturewouldbeinorder.Whenthepro“le-shiftparametersaresub-tractedfromthevelocityandthetemperaturepro“les,theresultingcurvesexhibitthebehaviorofequilibriumlayersthatextendstothelocationsofthevelocitymax-imumandthetemperatureminimum,respectively.ThisisshowninFig.8Despiteourbriefaccountoftheproblemofanorthogonaljetimpingingonawall,thefollowing“nd-ingsareremarkable:(1)thevariationofbothiswellde“nedandisinaccordancewiththeaccountofotherauthors,(2)thelevelinthelogarithmicexpres-sionsforthelawsofthewallhaveatendencytoin- -0.8-0.400.40.81.2ln(y) u[m/s] H/D= 2.0 -1012345ln(y) 8911w (a)(b)Fig.6.Graphicalmethodforthedeterminationofparameters.(a)Determinationofand(b)determinationof 202428323640 -44816 B = 1.031* ((T)-25.869 10152025303540 -15-10 A= 1.124*(Utau)-27.538(a)(b)Fig.7.Deviationfunctionforthe(a)velocityandthe(b)temperaturepro“les.D.R.S.Guerraetal./InternationalJournalofHeatandMassTransfer48(2005)2829…2840 creasedwithincreasingmaximumjetvelocityandwithdecreasingminimumtemperature.Therelationsforandforderivedherearepartic-ularlyimportantforthedeterminationoftheskin-fric-tioncoecientandoftheheattransfercoecient.Thisissuewillbedealtwithinseparatearticle.5.ConclusionThepresentworkhasdescribedthebehaviorofasemi-con“nedimpingingjetoveraheated”atplate.Experimentaldataforthepressuredistribution,velocityandtemperature“eldswereobtained.Theheattransferdatacon“rmedtheexistenceofaminimumintempera-turepro“leawayfromthewall.Theexistenceofaveloc-ityandatemperatureequilibriumlayerwasalsoinvestigated.Theresultsfoundatthisinvestigationindi-catethatthelevelofthelogarithmicportionofthevelocityandthetemperaturelawsofthewallincreaseswithincreasingmaximumjetvelocityanddecreasingminimumtemperature.Thisfact,forthetemperaturepro“les,hasbeenobservedforthe“rsttimeinthecourseofthepresentresearch.Thepresentresearchisparticularlyrelevantduetoitsapplicationforthedevelopmentofmethodsthatcanbeusedforthedeterminationofthelocalskin-frictionandofthelocalheattransfercoecient.DRSGisgratefultoCAPES(BrazilianMinistryofEducation)fortheawardofaD.Sc.scholarshipinthecourseoftheresearch.APSFisgratefultotheBrazilianNationalResearchCouncil(CNPq)fortheawardofaresearchfellowship(GrantNo.304919/2003-9).Theworkwas“nanciallysupportedbyCNPqthroughGrantNo.472215/2003-5andbyFAPERJthroughGrantsE-26/171.198/2003andE-26/152.368/2002.JShasalsobene“tedfromaCNPqresearchfellowship(GrantNo.[1]R.P.Patel,Selfpreservingtwodimensionalturbulentjetsandwalljetsinamovingstream,M.Sc.Thesis,McGillUniversity,Montreal,1962.[2]A.Tailland,J.Mathieu,Jetparietal,J.Mecanique6(1967)[3]V.Ozarapoglu,Measurementsinincompressibleturbulent”ows.D.ScThesis,LavalUniversity,Quebec,1973.[4]H.P.A.H.Irwin,Measurementsinaself-preservingplanewalljetinapositivepressuregradient,J.FluidMech.61(1973)33…63.[5]I.B.Ozdemir,J.H.Whitelaw,Impingementofanaxisym-metricjetonunheatedandheated”atplates,J.FluidMech.240(1992)503…532.[6]R.Narasimha,K.Y.Narayan,S.P.Pathasarathy,Para-metricanalysisofturbulentwalljetsinstillair,Aeronaut.J.77(1973)335.[7]I.Wygnanski,Y.Katz,Horev,Ontheapplicabilityofvariousscalinglawstotheturbulentwalljet,J.FluidMech.234(1992)669…690.[8]G.P.Hammond,Completevelocitypro“leandoptimumskin-frictionformulasfortheplanewall-jet,J.FluidsEng.104(1982)59…66.[9]D.B.Spalding,Asingleformulaforthelawofthewall,ASMEJ.Appl.Mech.28(1961)455…458.[10]M.Behnia,S.Parneix,P.A.Durbin,Predictionofheattransferinanaxisymmetricturbulentjetimpingingona”atplate,Int.J.HeatMassTransfer41(1998)1845…1855.[11]M.Behnia,S.Parneix,Y.Shabany,P.A.Durbin,Numer-icalstudyofturbulentheattransferincon“nedanduncon“nedimpingingjets,Int.J.HeatFluidFlow20(1999)1…9. 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