J.FluidMech.2006CambridgeUniversityPressdoi:10.1017/S0022112005008141P - PDF document

J.FluidMech.2006CambridgeUniversityPressdoi:10.1017/S0022112005008141PrintedintheUnitedKingdomCollidingturbulentplumesByN.B.KAYEP.F.LINDEN 1.IntroductionCollidingplumesoccurwhentherearetwoopposingsourcesofbuoyancyincloseproximity.Forexample,anairconditioningventplacedaboveelectronicequipment,suchasacomputer,willresultinplumesthatcollide.Thenegativelybuoyantplume N.B.KayeandP.F.Lindenseparationoftheplumesources,andtheratioofthebuoyancy”uxes.Themodelisthencomparedtoouroriginalexperimentaldata.OurconclusionsaregiveninVerylittleworkhasexaminedtheinteractionofcollidingplumes,theonlyrelevantpaperbeingMoses,Zocchi&Libchaber(1993).Thatpaperfocusesonthestartingcapoflaminarplumesandthemergingofco-”owinglaminarplumes,butitalsolooksatthecollidingofapproximatelyequalaxisymmetriclaminarplumes.Twocasesareexamined:theaxisymmetricproblemwhentheplumesarealigned,andthecaseofslightmisalignment.Turbulentplumesarenotdiscussed.Intheaxisymmetriccase,theplumescollideandspreadoutwithasharpinterfacebetweenthetwoplumes.Nomixingorcoalescencewasobserved,andthediskofplume”uidthatresultedfromthecollisionwasmaintainedforsometime.Allresultspresentedweredimensional,asthee
ectiveradiusofthePeltiercoolerwasundetermined.Forthecaseofslightmisalignment,theplumesde”ectpasteachotherandthenrotatearoundeachother.Theperiodofrotationwasfoundtobestableover10…20cycles,butvariedoverlongerperiodsoftime.Notheorywaspresentedtodescribeorexplainthese”ows.RotationofthetypeobservedbyMosesetal.(1993)isnotrestrictedtocollidingplumes.Konig&Fiedler(1991)observedthataturbulentjetinacounter-”owoscillatesarounditsaxisofsymmetry.Theirpaperismainlyconcernedwiththedepthofpenetrationintothecounter-”ow,andthemeanconcentrationofapassivescalarinthejet.Theauthorsalsoobservedthatthejetoscillatesarounditsaxisofsymmetryandthatthepenetrationdepth”uctuates.Theoscillationoccursbecausethejethasnopreferreddirectionwhenitisreversedbythecounter-”ow.Theoscillationwasalsoobservedwhenthejetenteredthecounter-”owatasmallangle,buttheamplitudeoftheoscillationdecreasedastheangleincreased.Yoda&Fiedler(1996)suggestthattheamplitudeoftheoscillationisafunctionofthejettocounter-”owvelocityratio,thoughthishypothesisisnotexaminedindetail.Similaroscillationsarealsoobservedintheheightofturbulentfountains(seeTurner1966).WorkhasalsobeendoneoncollidingjetsbyWitze&Dwyer(1976).Theyshowedthattwoaxiallyalignedturbulentjetswithequalmomentum”uxwillcollideandspreadoutintheformofaradialjet.2.ExperimentalinvestigationExperimentalobservationsPreliminaryexperimentswereperformedtoinvestigatequalitativelythecollisionofopposingturbulentplumes.Opposingturbulentplumeswerecreatedbyaddingpositivelyandnegativelybuoyant”uid,freshandsaltwater,respectively,atconstant”owratesintoaPerspextank.Thetankwasinitially“lledwithasalinesolutionwithdensityatthemeanofthetwoplumesource”uids.Theaxialseparationoftheplumesvariedintherange01,whereistheaxialseparation,andistheverticalseparationofthesources.Thebuoyancy”uxesoftheplumeswereapproximatelyequalinmagnitudebutoppositeinsign.Reddyewasaddedtothefreshwaterplume,andbluedyetothesaltwaterplumeforthepurposeofvisualization.Threeobservationsweremade.First,whentheplumeswerealmostverticallyaligned,theywereobservedtorotatearoundeachother,asobservedbyMosesetal.(1993)fortheslightlymisalignedlaminarcase.Therotationperiodwasnotconstant,withoneplumeoccasionallydominatingtheother.Aftersomeperiodoftimethis”owregimewouldreverseandtheplumethathadbeendominatedwouldstarttode”ectanddominatetheother.Thisoscillationwasnotobservedforvaluesof Collidingturbulentplumes Figure1.Aseriesofimagesoftwoalignedopposingturbulentplumesshowingtheplumescolliding,de”ectingpasteachotherandmixing.Notethatsomeofthereddyedlighterplume”uidhasmixedwiththeblueheavy”uidandis”owingdownwards.Alsonotethattheambientremainsclearofplume”uid.greaterthanabout0.1.Itisreasonabletoassumethatthebasicmechanismforthisrotationissimilartothatofthelaminarcase,orforajetinacounter-”ow.Theplumesde”ectpasteachotherbutdonothaveapreferreddirectionofde”ection,sotheyrotatearoundtheiraxisofsymmetry.Aseriesofimagesfromavisualizationexperimentisshownin“gure1.Second,theplumesdidnotspreadouthorizontally,aswasobservedforlaminarplumes.Rathertheyde”ectedpasteachotherandcontinuedpastthesourceoftheopposingplume.Thisimpliesthattheplumesmaintainsomeformofstructuralintegrity.Theydonotcollide,losemomentum,andmixuniformly.Insteadtheypassbyeachother,withmixingoccurringalongthesurfacebetweenthem.The“nalobservationwasthatno”uidwasejectedfromtheplumesduringthecollision.Theambient”uidremainedcompletelyclear.Thisimpliesthatatanyheightthereisonlyanin”owof”uidintothetwoplumes.Any”uidofintermediatebuoyancythatiscreatedasaresultofmixingduringthecollisionisimmediatelyentrainedintooneoftheplumes.Aftertheplumeshadpassedbythesourceoftheopposingplumetheycontinuedtorise(orfallfornegativelybuoyantplumes)intheformofaturbulentplume.Thefocusofthisexperimentalinvestigationistodeterminethebuoyancy”uxoftheplumesaftertheycollide.The“nalobservation,thatthereisnoout”owofneutral”uidintotheambient,isincontrasttoboththelaminarplumecaseofMosesetal.(1993),andtheturbulentjetcaseofWitze&Dwyer(1976).Inboththesecasesthe”owscollidedandspreadoutradially.Theturbulentplumesdonotexhibitthisbehaviourfortworeasons.First,theout”owwouldbeunstableasitwouldhavetheheavier”uidfromthedownward-”owingplume”owingoutoverthelighter”uidfromtheupward-”owingplume.Second,as”uidfromthetwoplumesmixesitisre-entrainedintooneorother N.B.KayeandP.F.Linden Figure2.Schematicdiagramoftheparametersgoverningthecollisionofopposingturbulentplumes.oftheplumes.Theradialout”owobservedinthelaminarplumecasemust,therefore,bestabilizedbyviscosity.Collidingaxisymmetricturbulentplumescanbecharacterizedintermsoffourparameters:thebuoyancy”uxofeachplume,theaxialseparationofthetwoplumes,andtheverticalseparationoftheplumesources.Thisleadstotwodimensionlessparameters:thebuoyancy”uxratio andtheaspectratio Notethatlengthsarescaledontheheightnottheaxialseparationtoavoidasingularitywhenthereiszeroaxialseparation.Wechosetheconventionthat,andtherefore01.WithoutlossofgeneralityforBoussinesqplumes,therisingplumeistakentohavethelargermagnitudebuoyancy”ux.Therefore,0and0.Theseparametersareshownschematicallyin“gure2.Theplumesthatemergefromthecollisionregionwilleachhaveabuoyancy”uxdenotedasforthestrongerplumeandfortheweakerplume.Wede“nethe Collidingturbulentplumesdimensionlesspost-collisionbuoyancy”uxofeachplumeas |F1|2=|F2.0| ByconservationofbuoyancywecanwriteHence,themagnitudeofthebuoyancy”uxlossineachplumeisthesame.Indimensionlessform(2.5)isTherefore,foragivenvalueofweonlyneedmeasurethebuoyancy”uxofoneofthepost-collisionplumes.Wecanalsouse(2.6)tosetlimitsonthevalueof.Ifweassumethatthereisnomixingatalltheneachplumewillretainitsoriginalbuoyancy”uxand=1and.If,ontheotherhand,theplumesmixcompletely,thentheweakerplumewillloseallofitsbuoyancy”uxand=0.Therefore,willbeintherangeAsallmeasurementswillbefocusedonthestrongerplumewewilldropthesubscript1fromwhenreferringtothestrongerplume.ReviewofprevioustechniquesVariousexperimentaltechniqueshavebeenusedtomeasurethebulkpropertiesofbuoyancy-driven”ows.Forexample,Hacker,Linden&Dalziel(1996)usedalightattenuationtechniquetoexaminethetime-varyingbuoyancypro“leinlockreleasegravitycurrents.Morton,Taylor&Turner(1956)measuredtheriseheightofplumesinastrati“edambientandinferredfromtheirmeasurementsthevalueoftheentrainmentcoecient.Lightattenuationtechniquesinvolveaddingadyetothe”uidandmeasuringhowthedyeattenuatesalightofknownintensity.Theattenuationisrelatedtotheconcentrationofthedye,whichactsasasurrogateforthebuoyancy(orpassivetracer)thatisbeingmeasured.Inordertousethistechniquewithcollidingturbulentplumes,three”uidsneedtobeconsistentlydyed,thetwoplumesandtheambient”uid.Theonlywaytodothisistodyetheambientandoneplume,whileleavingthesecondplumeun-dyed.Assumingthattheambient”uidhasadensityequaltotheaverageofthedensitiesofthetwoplumesources,theambientdyeattenuationresultsinanattenuatedlightintensityinthemiddleofthecamerarange(say=128foratypical0…255digitalscale),asthedyeconcentrationscalesonthedensity.Thesource”uidsprovideintensitiesof0and255,respectively.Oncetheplumesmixwiththeenvironmenttheirbuoyancyisclosertothatoftheambient”uid.Typicalexperimentsresultinplumebuoyancytosourcebuoyancyratiosofapproximately1/40overtheheightoftheexperiment.Thisdilutionresultsinintensitiesapproximatelyintherangeof125to131,yieldingatotalresolutionofonly7points.Itis,therefore,onlypossibletousethistechniquewithplumesofthesamesign(Kaye&Linden2004),becausetheambient”uidisnotdyed,andthesourcesolutioncanbedyedsuchthatthefullresolutionof256pointscanbeusedoncetheplumehasmixed.Baines(1983)developedatechniqueformeasuringthevolume”owrateinaplumebyplacingtheplumeinatankwithaconstantambient”owrateinthesamedirection N.B.KayeandP.F.Linden Figure3.Schematicdiagramoftheexperimentalset-upformeasuringthebuoyancy”uxlossincollidingplumes.Thediagramshowsthecollidingplumes,thedenselayerofthicknessformedbelowthecollision,andthebottomopeningandresultingout”ow.astheplume”ow.Atwo-layerstrati“cationdevelops,withtheplume”owrateatthedensityinterfaceequaltotheambient”owrate.Oneprobleminapplyingthistechniquetocollidingplumesisthat”uidoftheambientdensitywouldneedtobeaddedinordertoholdthedensityinterfaceinplace,whichrequirespumpingalargevolumeofsaltwaterintothetankatameasuredrate.Typically,thiswouldinvolvevolumesoftheorderofhundredsoflitres…mainlyalogisticalproblem.Asecondproblemisthatonceaninterfacehasbeenestablished,anyinteractionbetweenthecollidingplumesbelowthefront(assumingthatthefallingplumeisbeingmeasured)wouldnotoccurunderthesameambientconditions.Notethatinthiscon“gurationthetechniquewouldnotmeasurethe”owrateofasingleplume,butratherTechniquedevelopedformeasuringbuoyancy”uxToovercometheseproblems,anewexperimentaltechniquewasdevelopedtomeasurethebuoyancy”uxofaplume.BasedontheventilationmodelofLindenetal.thetechniqueinvolvesallowingthetwoplumestocollideandinteract,andthencatchingthefallingplumeinanopen-toppedventilatedbox.Aschematicisshownin“gure3.Belowtheinteractionarea,thefallingplumehasbuoyancy”ux.Thisplume“llstheboxtoasteadystateheight,justasinthebasicventilationproblem(seeLindenetal.1990).The”owrateoftheplumethroughtheinterfaceismatchedbythedraining”owoutoftheloweropening(denotedby).Afteraninitial Collidingturbulentplumestransientperiodthebuoyancyofthelayerwillbeuniform(denotedby),andequaltothatofthemeanbuoyancyoftheplumeasitentersthelowerlayer.Thebuoyancy”uxoftheplumeisthengivenbyBysamplingthe”uidinthedenselayerandtheambient,itispossibletoestablishthebuoyancyofthelayer.Fromthisitispossibletocalculatethe”owratethroughthebox,byusingthedrainingtheoryofLindenetal.(1990),as Providedthattheopeningareaoftheinletisconsiderablylargerthantheoutletarea,thenisgivenbyThebuoyancy”uxisthereforegivenby Asthesourcebuoyancy”uxisknown,thevalueofisgivenby 2g03/2h1/2 Thedischargecoecientforasharp-edgedori“ceistypicallyintherange0.6to0.65(seeMassey1989,p.96).Toverifythisvalue,experimentswereperformedusingonlyasingleplumeofknownbuoyancy”ux(inthiscase,=1).Averagingoveraseriesofexperimentsgaveavalueof015,whichiswithintheexpectedrange.AdetailedanalysisofallerrorsrelatedtotheseexperimentsisgivenintheAppendix.Notethatthedistancebetweentheplumesourcesistakenasthedistancebetweenthenozzleoutletsplusthedistancetothevirtualoriginforeachplume.ThecorrectionsweremadeusingthemethodpresentedbyHunt&Kaye(2001)andaccountedforabout20%ofthetotalheight(10%ateachend).TransientsandtimescalesWhentwoplumescollidetheyde”ectpasteachotherandoscillatearoundeachother.Thisoscillationisobservedtotakeanumberofforms,witheitherthetwoplumesde”ectinginoppositedirections,oronlyoneplumede”ectingandtheothertendingtodominatethe”ow.Bothsituationswereobservedforequalplumes,thoughforsmallervaluesofthestrongerplumetendedtodominate.Asthe”owisunsteady,thebuoyancy”uxdrainingfromtheboxisnotconstant.Therefore,itisnecessarytotakemeasurementsofthebuoyancy”uxovertimeandaveragethem.Inordertoestablishtheaveragingtimeweconsiderthedrainingtime,whichisthetimeittakesforalayerofdense”uidtodrainoutofaventilatedboxetal.1990).Thistimescaleisgivenby 2Ao 2Cd h isthecross-sectionalareaofthetank.Substitutingtypicalexperimentalvaluesinto(2.13)gives ×5× 2×0. =103s N.B.KayeandP.F.Linden 5 10 15 20 25 30 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Figure4.Measuredvaluesoffor91and=0.Thesquaresaretheinstantaneousmeasuredvaluesandthecrossesarevaluesaveragedover5minutes.Thesolidlineistheaverageoverthelast20minutesandthedashedlinesare7%ofthatvalue.orapproximatelytwominutes.Therefore,variationsthatoccuronthistimescaleorlongerwillappearinatimeseries.Any”uctuationswithshorterperiodsaredampedoutbythecapacityofthebox,andsowillnotbemeasured.Inordertoestablishthebestaveragingprocedure,anexperimentwasrunwith91andMeasurementsweretakeneveryminutefor30minutes,fromthetimetheplumeswereturnedon.Theresultsareplottedin“gure4.Thecrossesaretheinstantaneousresults,whilethesquaresarea“veminuteaverage.Thethicklineistheaveragevalueforthelast20minutes,andthethinlinesare7%(see(A3)).Thebox“llsupoveraperiodof,butsigni“cant”uctuationsareobservedoutsidetheerrorrangepredictedearlier.The“veminuteaverage(),however,canbeseento“lteroutthese”uctuations,withallpointswithintheestimatederror.Basedonthispreliminaryexperiment,thefollowingprocedurewasadoptedforourexperiments.Theplumesareturnedon,and17minuteslater(typically)“vemeasurementsaretakenat1minuteintervals,andthetime-averagedetermined.Resultsforthecollidingplumeexperimentsarepresentedfor=1.0,0.8,0.6,0.4,and0.2,withintherange0to0.3.Theresultsareplottedin“gures5to7.The“rstpointtonoteabouttheexperimentalresultsisthatall34measuredvaluesof
,fallwithintherangerequiredbyconservationofbuoyancy,see(2.7).Thisaddstoourcon“denceinonlymeasuringthepost-collisionbuoyancy”uxofthestrongerplumecanbecalculatedusingthemeasurementsforand(2.6).Secondly,withonlyoneexception,thereisnomeasurablebuoyancy”uxexchangeforseparationsgreaterthan25.Forseparationslessthanthis(25)thereisasteadydeclineinreachingaminimumwhentheplumesareaxiallyaligned(notethatfor2,“gure5(),thefourdatapointswithsmallestvaluesareallessentiallyattheminimumvalueof8).Asazero-ordermodelweassumethatfor25nomixingoccursand.Foralignedplumesweassumethemixingiscomplete,thatistheweakerplumeisfullyabsorbed,and.We Collidingturbulentplumes 0.10 0.15 0.20 0.25 0.30 0 0.2 0.4 0.6 0.8 1.0(a)(c)(e)  0.2 0.4 0.6 0.8 1.0  0.2 0.4 0.6 0.8 1.0 1.2 0.050.100.150.200.250.300.050.100.150.200.250.300.050.100.150.200.250.300.050.100.150.200.250.30Figure5.Experimentalresultsfor)with(0,(8,(6,(2.Thelineisgivenby=min(4thereforeplotalinein“gure5,representingalinearinterpolationbetweenthesetwo)for0Withtheexceptionof6themajorityofpointsdonotlieonthelinegivenbythelinearmixingmodel(2.15).Forlargervaluesofthereislessmixingandbuoyancy”uxexchange(larger)astheaxialseparationisreduced.Whentheplumes N.B.KayeandP.F.Linden 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0  Figure6.Experimentalresultsfor)for=0.Theline(=1)istheminimumpossiblevaluewhilestillconservingbuoyancy.0.20.40.60.81.0 Figure7.Calculatedvaluesof)showingtheneartotalabsorptionoftheweakerplumeforarealigned(=0)thetotallossinbuoyancy”uxislessthantheminimumvalueof.Forsmaller=0theweakerplumeiscompletelyabsorbedbythestrongerplume().Astheaxialseparationisincreasedtherateofincreaseofislessthanthesimplelinearmodel,butincreasesas25.Thedi
erencebetweenthemeasuredvaluesoffor=0andlargervaluesof,areshownmoreclearlyin“gures6and7.Forthecaseofalignedplumes(=0)andsmallthevalueofisclosetotheminimum(=1).However,forlargervaluesofthisisnotthecase.Thisimpliesthatwhenoneplumeissigni“cantlystrongerthantheotheritwillcompletelyabsorbtheweakerplumewhenaxiallyaligned.Formorebalancedplumestrengthsthereisstillsubstantialmixing,butbothplumessurvivethecollisionwithreducedbuoyancyClearlythesimplelinearmodelforquantifyingthelevelofmixingbetweencollidingturbulentplumesshowspooragreementwiththeexperimentalmeasurements.Wenowlookforamoresophisticatedmodeltodescribetheexperimentalresultspresentedabove.Themodelwouldideallycapturedetailsthatarenotincludedinthelinearinterpolation.Thesemajordiscrepanciesbetweentheexperimentalresultsandthe Collidingturbulentplumes 1 Figure8.Schematicoftheentrainmentmodelforcollidingplumes.Thediagramshowstheequivalentsystemoftwoplumesthatexchangeonlybuoyant”uid.simplelinearmodelare:(i)foralignedplumes,=0)(ii)for6,therateofincreaseofislessthanthelinearmodel.3.EntrainmentmodelforplumecollisionWebeginourmodelbyrecallingthat,intheinitialvisualizationexperimentsreportedabove,theplumesdidnotmixandspreadouthorizontally,butratherpassedbyeachotherandcontinuedonasplumesbeyondthecollisionregion.Itis,therefore,reasonabletoassumethattheplumesmaintainsomeformofstructuralintegritythroughoutthecollision.Consequently,wedescribethetwoplumesseparately,butwiththemodelforeachplumecontainingmixingtermstoaccountfortheinteractionbetweenthetwoplumes.WethenwriteplumeconservationequationssimilartothoseofMortonetal.(1956)withmodi“cationsfortheentrainmentbyeachplumefromtheother.Aschematicofthismodellingapproachisshownin“gure8.ThisapproachissimilartothattakenbyBloom“eld&Kerr(2000)tomodeltheinteractionoftheupwardanddownward”owinturbulentfountains.Theconservationequationswrittenintermsoftheplumevelocityandradius,forauniformenvironmentandtop-hatpro“les,are dz
b2iwibitwi, dz
b2iw2i=
b2igi, Thesubscript=1,2referstotheplumenumber,andthesubscriptontheentrainmentcoecientmeansitisthecoecientappropriateforusewithtop-hat N.B.KayeandP.F.Lindenpro“les.Twoassumptionsarenowmaderegardingthenatureandgeometryoftheplume-to-plumeentrainment:(i)thetwoplumesinteractovertheoverlappingfractionoftheircircumferences;(ii)the”uidexchangebetweentheplumesisthedi
erencebetweentheamountof”uidthateachplumewouldentrainfromtheotherwereitpartoftheambient.Thesetwoassumptionsarediscussedindetailbelowtogetherwithtwomodelsforbuoyancy”uxexchange.AngleofinteractionItisassumedthatatanyheighteachplumecanbetakenascircularandthattheradiusgrowslinearlywithdistancefromtheplumesource.Thisisnotstrictlycorrect,astheplumeswillnotbeself-similarintheinteractionregionaseachwillbedeformedbythepresenceoftheother.Nevertheless,thecircleassumptionprovidesastartingpointforscalingtheextentoftheplumeinteraction.Theassumptionthattheplumescanbemodelledasoverlappingcircularconesmeansthatatanyheightacertainfractionofthecircumferenceofeachwilloverlaptheotherplume.Theanglesoftheoverlaparedenotedby2andvarywithheight.Thechoiceofradius)isessentiallyarbitrary,butneedscarefulconsideration,asitisassumedthatforseparationsgreaterthanthesumofthetwoplumeradiinointeractionwilloccur(notethateachplumeradiusgrowslinearlywithheightinoppositedirections,thereforethesumoftheirradiiisconstant).Wecouldchoosearadiussuchthatnointeractionoccursfor25basedonourearlierexperimentalresults.However,thereareinsucientdatatoestablishaccuratelythisvaluefromourexperiments.Themodelwepresentherealsoaccountsforthedrawingtogetheroftheplumesduetotheentrainmentvelocityinamannersimilartothatforco-”owingplumesKaye&Linden(2004).Thereforeasmallerstartingradiusisrequired.Theobviouschoicesfortheradiusarethetop-hatpro“leradius,theradiusoftheequivalentGaussianpro“le( ),andaradiusgivenbythedistanceatwhichthemeanbuoyancyisacertainfractionofthecentrelinevalue(say5%).InteractionwillstilloccurwhentheplumesareataseparationslightlygreaterthantheGaussianradiusastheturbulentedgeoftheplumeisfurtherawayfromtheaxis.Agreaterlengthisrequired,andforthisreasonthethirdchoice,the5%radius,isusedasa“rstapproximation.Theradiusofeachplumeisthengivenby 5
|=6g(1Š) 5
|,i=rig=t/ 2.Itisimportanttonotethattheradiusisindependentofandthatforseparationsgreaterthanthemaximumradius =0.19),nointeractionwilloccurinthismodel(as=0).Theanglesaregivenby 201,2=20+22Š21 Itisassumedthatwhenoneplumecompletelysurroundstheother,theangleforthelargerradiusplumeis0andtheotherangleisAplotofasafunctionofheightisshownin“gure9,forvariousvaluesofForclarity,heightisplottedvertically.Theplotindicatesthat,forlargervaluesof,theheightoverwhichthereisalargeangleofinteractionissmaller(forexample,for=0.180.8for0.1).Forsmallerseparations,theheightatwhichincreases(0.5for=0.12and0.6for=0.06).Forplumeswithnoaxialseparation(=0)for0.5and=0for0.5.Thisimpliesthattherising Collidingturbulentplumes 1.0 1.5 2.0 2.5 3.0 0 0.2 0.4 0.6 0.8 1.0 1 = 00.12 0.18 Figure9.Plotofasafunctionoffor=0,0.06,0.12and0.18.plumeiscompletelysurroundedbythefallingplumeuntilthepointhalfwaybetweenthesources,andthentherisingplumesurroundsthefallingplumeabovethatheight.Exchangeofvolume”uxAsdiscussedearlier,themainconsequenceofthecollisionoftwoplumesisareductioninthebuoyancy”uxofeachplume.Inorderforthistooccur,theplumesmustexchange”uid.Tocalculatetheamountof”uiddrawnintoeachplumeatanyheightweassumethateachplumeregardstheotheraspartoftheambient”uid.Wethencalculatetherateofentrainmentintoeachplumebasedontheentrainmentvelocity,plumeradiusandtheangleofoverlap.Theamountof”uidentrainedintothedominantplumeistakentobethedi
erencebetweenthesetwocalculatedvalues.Thedominatedplumelosesthesamevolumeof”uid.Theentrainmenttermontheright-handsideof(3.1)willcontaintheregularentrainmenttermforasectorofangle2()plusthedi
erencebetweenthecalculatedentrainmentrates.Thedi
erencebetweentheentrainmentratesisgivenbyEquation(3.1)thereforebecomes .Thevaluesofaretakenasthemagnitudesoftheplumevelocities,andarealwayspositive.Notethat(3.7)isequivalentto(3.1)withanadditionaltermforthe”uidremovedfromtheplumeduetoentrainmentbytheExchangeofbuoyancy”uxHavingestablishedanestimateofthevolume”uxexchangedatanyheight,weneedtoexaminepossiblemodelsfortheexchangeofbuoyancy”ux.Therearetwomodelstoconsider.Onepossibilityisthatwecouldassumethatentrainmentisatwo-wayprocessandthat,ateachheight,eachplumeentrains”uidfromtheother N.B.KayeandP.F.Lindenandbuoyancyexchangeisinbothdirections.Inthatcasetheappropriateexpressionforthebuoyancy”uxexchangeis Alternatively,theentrainmentcouldberegardedasanessentiallyone-wayprocessatanygivenheight.Thatis,atanygivenheight”uidexchangebetweentheplumesoccursinonlyonedirection.Thebuoyancytransferwouldthereforebethenet”uidtransferratemultipliedbythemeanbuoyancyoftheplumethatisnetlosing”uid.Hence,if0,theplumeisgaining”uid(3.6)andthebuoyancyexchangeisgiven.Ontheotherhandif0,thebuoyancyexchangeisgivenby.Inthiscasethenetbuoyancyexchangeisgivenbyi, j, �Theequivalentexpressionto(3.8)forthisentrainmentmodelis Wewillrefertothesemodelsas(A)forthebidirectionalexchangemodeland(B)fortheunidirectionalexchangemodel.Notethattherateofexchangeofvolume”uxatanyheightisthesameforbothmodels.OtherpotentialchangestotheplumeequationsTheabovemodi“cationstotheplumeentrainmentequationsaccountfortheexchangeofbuoyancybetweenthetwoplumes.Otherchangescouldalsobemade.Themodelignoresanymomentumlossduetothecollision.Thisisareasonableomissionbasedontheobservationmadeearlierthattheplumestendtopassaroundeachotherratherthanblockeachother.Thisobservationdoesnot,however,meanthattheplumeswillnotexchangemomentumduetoentrainment.Itcouldalsobearguedthateachplumewillberelativelymorebuoyantwheninteractingwiththeopposingplumeasthenetbuoyancydi
erencebetweentheplumeanditssurroundingsisgreaterduetothepresenceoftheopposingplume.Wecould,therefore,addatermfortheexchangeofmomentum)to(3.2)andasecondtermto(3.2)toaccountforthedi
erenceinperceivedambientbuoyancy).Wehavechosennottoaddthesetermsfortworeasons.First,thetermsareofoppositesignandwilltendtocanceleachotherout.Moreimportantly,however,fora“rst-ordercorrectionthetermsofgreatestsigni“cancewillbethosethataltertheplumedrivingforces,thatis,theirbuoyancy”uxes.Hence,wefocusontheexchangeofbuoyancyandignoreotherpossiblecorrections.Fullnon-dimensionalequationsTheseequationscanberewrittenintermsofthebulkpropertiesoftheplumetogive dz=t2M1/2iŠ2j
M1/2j,Mi dz=FiQi MiFi dz=2t
iM1/2iFj QjŠjM1/2jFi Qi) Collidingturbulentplumes dz=2t
iM1/2iŠjM1/2jFk ,andthesubscriptisthesameasusedin(3.9).Forthepurposeofsolvingtheseequationsitisconvenienttonon-dimensionalizethemintermsofthe”uxesofapureplume.Thisresultsinthedimensionlessvariablesde“nedby 8t1/36tH 55/3,i=mi5F1 8t2/36tH Thenequations(3.12)…(3.14)become d=5 62m1/2iŠ2j
m1/2j,mi d=4 3fiqi mifi d=5 3
im1/2ifj qjŠjm1/2jfi qi)fi d=5 3
im1/2iŠjm1/2jfk Theboundaryconditionsareassumedtobethoseforpureplumesattheirasymptotic(virtual)originandaregivenby=1atEquations(3.17)…(3.19),subjectto(3.20)and(3.21),weresolvednumerically.Thetechniqueinvolvedsolvingtheequationsforplume1asifplume2wereapureplumewithnointeraction.Theresultsforplume1werethenusedtocalculatetheright-handsideoftheequationsforplume2.Thisprocesswasrepeated,alternatingbetweenplumes,untilthesolutionsforeachplumeconverged.Theconvergencecriterionrequiredthedi
erencebetweeniterationstobelessthan0.001forallthree”uxes.ComparisonofmodelsolutionsThemodeldevelopedabovecanbeusedtosolveforthe”uxesofbuoyancy,momentumandvolumeforeachplumeatanyheight.However,thismodelseekstoestablishthenetlossofbuoyancy”uxasaresultofacollisionbetweenplumes.Therefore,wedonotreportmomentumandvolume”uxpredictionsandinsteadfocusonbuoyancy”uxes.2wepresentedmeasurementsofthetotallossofbuoyancy”uxafteracollision,butdidnotconsiderthelossasafunctionofheight.Thelossinbuoyancy”uxcan N.B.KayeandP.F.Linden 0.4 0.6 0.8 1.0 0 0.2(a)(b) 0.4 0.6 0.8 1.0 0.18 0.060.12  = 0.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.2 0.4 0.6 0.8 1.0 0.060.12 Figure10.Plotofthebuoyancy”uxlossasafunctionofdistancefromtheplumesourcefor()thebidirectionalbuoyancyexchangemodel(A)and()theunidirectionalbuoyancyexchangemodel(B),andaspectratiosof=0,0.06,0.12and0.18.alsobedescribedbytheparameterde“nedhereas,, Plotsof,)for=1.0areshownin“gure10()formodel(A)and“gure)formodel(B).Theplotsshowhowthebuoyancy”uxdecreaseswithheightforvariousvaluesoftheaspectratioforeachofthetwoproposedentrainmentmodels.Forthecaseofthebidirectionalentrainmentmodel(A),shownin“gure10(wenotethatthetotalbuoyancylossincreaseswithseparation.Thisismoreclearlyseenin“gure11whichshowsaplotof)with=1forbothbuoyancyexchange Collidingturbulentplumes 0.08 0.12 0.16 0.20 0 0.2 0.4 0.6 0.8 1.0 Figure11.Plotof)for=1.Thesolidlinerepresentstheunidirectionalbuoyancyexchangemodel(B)whilethedottedlinerepresentsmodel(A).models.Forcloselyalignedplumesthebulkofthebuoyancy”uxlossisinthecentreofthe”owwithonlyminimallossesneareachplumesource.However,forlargerseparationsthebidirectionalmodelshowssigni“cantbuoyancy”uxexchangeneareachplumesource.Thisresultsfromthefactthatnearthesourcetheplumebuoyancyisverylarge(infactitissingularattheorigin)so,iftheopposingplumeextractsevenasmallamountof”uidfromtheplumenearthesource,itwillresultinasigni“cantexchangeofbuoyancy”ux.Thisdoesnotoccurformorecloselyalignedplumesastheopposingplumecompletelysurroundstheplumenearthesourceand,therefore,haszeroangleofinteraction(whiletheotherplumehasanangleofinteractionofFortheunidirectionalmodel(B)thislarge-scalenear-sourcebuoyancy”uxexchangedoesnotoccurasthenet”uidtransferisintotheplumethatisnearestitsorigin.Whentheplumesarealignedvertically(=0)mostofthemixingoccursbetween0.7wheredropsfromaround0.9to0.3,implyingalossof60%oftheinitialbuoyancy”uxoftheplume.Thisbehaviouristhesameasformodel(A)astheangleofinteractionassumptionof3.1impliesthat,forthe“rsthalfofitstravelfromthesource,theplumecanonlygain”uid(asitissurroundedbytheotherplume),whereasforthesecondhalfitcanonlylose”uidsinceitcompletelysurroundstheotherplume.Forlargerseparations,thisconcentrationofmixinginthemiddlethirdofthe”owislesspronounced.Wenowcomparemodels(A)and(B)for=1asafunctionoftheplumeseparation.Aplotofthenetbuoyancy”uxparameter)isgivenin“gure11,whichshowsthatmodel(A)predictsanincreaseinmixingastheplumeseparationincreasesuntiltheplumesarenolongertouching,atwhichpointthereisnobuoyancy”uxloss.Model(B),ontheotherhand,predictsadecreaseinbuoyancy”uxexchangeastheseparationincreases.Althoughwehavenodirectmeasurementsofhowbuoyancyexchangeoccursatanygivenheight,“gure11suggeststhattheunidirectionalbuoyancyexchangemodelmoreaccuratelytrackstheexperimentalresultspresentedin2.Forthatreasonwewillcontinuetodevelopourmodelusing N.B.KayeandP.F.Linden 0.10 0.15 0.20 0.25 0.30 0 0.2 0.4 0.6 0.8 1.0 Figure12.asafunctionoffor=1.0,0.8,0.6,0.4and0.2.Thelowestlineisfor=1.0andthehighestisfor=0.2. 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 Figure13.asafunctionof.Thethinstraightline(=1)showsthelowestvaluethatcanbeachievedwhileconservingbuoyancy,andthethicklineshowsthevaluepredictedbythecollidingplumesentrainmentmodel.Detailsofentrainmentmodel(B)solutionAplotof,
)formodel(B)isshownin“gure12.Anumberofpointsareworthnoting.First,theminimumvalueof(thatis,themaximumlossinbuoyancy”ux)occurswhentheplumesarealignedvertically,withincreasingwithseparation.Second,thevalueof=1)isnotalwayszero;evenwhentheplumesarealigned,theydonotcanceleachotherout.Infact,onlyfor0.3istheweakerplumecompletelyentrainedintothestrongerplume.Thisresultisshownmoreclearlyin“gure13,whichplotsthepredictedvalueoffor=0.Thisplotalsoshows Collidingturbulentplumes 2 Figure14.Velocityvectordiagramforunequalcollidingplumes.theminimumpossiblevalueofthatwouldoccuriftheweakerplumewerecompletelyabsorbed.Thepredictedvalueofishigherthanfor0.3,andequaltoforThemodelpresentedaboveassumesthattheplumesentrainfromeachother,butthateachisotherwiseuna
ectedbythepresenceoftheother.However,twoplumesincloseproximitytoeachotherwilltendtodrawtogetherduetotheambientvelocity“eldcausedbyentrainmentintotheplumes(Kaye&Linden2004).Wenowapplya“rst-ordercorrectiontoourmodeltoaccountforthistendencytodrawtogether.Plumede”ectionBasedonexperimentalresults(forexample,Rouse,Yih&Humphreys1952)itisreasonabletoapproximatetheambientvelocity“eldoutsideasingleplume,createdbyentrainmentintoaplume,asahorizontal”ow.Theradialvelocityintotheplumeisthereforegivenbywristheverticalvelocityintheplume,theentrainmentcoecientandtheradialdistancefromtheplumecentreline.Gaskin,Papps&Wood(1995)showedexperimentallythat,forambientvelocitiesofthesameorderastheentrainmentvelocity,thetwovelocity“eldscanbeadded,andthisresultwillbeusedbelow.Furthermore,theentrainment”owintoeachplumeisirrotationalandgovernedbyLaplacesequation.Thelinearityofthisequationjusti“estheadditionofthetwo“elds.Itis,therefore,reasonabletoassumethateachplumeispassivelyadvectedbytheentrainment“eldoftheother.Thevelocityvectordiagramshowinghoweachplumeisde”ectedbythepresenceoftheotherisshownin“gure14.Thevaluesforthevelocitiescanbecalculatedfromthesolutionto(3.17),(3.18)and(3.19),butasa“rst-orderapproximationitisassumedthatthepure-plumevelocitiescanbeused.Thevaluesofaregivenby =cF1/31zŠ1/3b1 . N.B.KayeandP.F.LindenTreatingeachplumeseparatelyleadsto dz=Šu1 w1,d2 dz=Šu2 Takingtheplumesasconical,withradiigivenby =6(HŠz) equation(3.24)becomes dz=62t 51
1/3(HŠz)2/3z1/3,d2 dz=62t Dividingbothsidesbyandintegratingleadsto 221Š20=Š62 5
1/3 0(1Š)2/31/3d, 222Š20=Š62 Approximating(1 3Š1 92Š4 3Š7 means(3.27)thatcanbeevaluatedas 20Š
1/32t )(3.31) 34/3Š2 77/3Š1 3Š4 Thede”ectionofplume2isgivenby 20Š
Š1/32t ThetotalreductioninaxialseparationofthetwoplumesisthereforegivenbyTheaverageseparation overtheheightwasthencalculatedandacoordinatetransformationfunction )evaluated.Aplotofthistransformationisshownin“gure15.Twopointsareworthnoting.First,forinitialseparationsof0.08themeanseparationiszero.Second,thetransformationisinsensitivetovariationsinovertherange0.21.0,becausethemeanvelocityoftheplumeisonlyaweakfunctionofthebuoyancy”ux.CorrectedsolutionAquadraticcurvewas“ttedtothedataplottedin“gure15(08+0  whichallowedacorrectionto“gure12forthedrawingtogetheroftheplumes.Thecorrectedplotof)isshownin“gure16. Collidingturbulentplumes 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0 Figure15.Plotofthecoordinatetransformation)whereisplottedvertically.Thefunctionisshownforvaluesof=1.0,0.8,0.6,0.4,and0.2.Thethickestlineisfor=1.0andthedot-dashedlineisfor=0.2.Notethatmostlinesoverlapandarenotclearly0.050.100.150.200.250.30 00.20.40.6 = 1.0  = 0.2 Figure16.,
)correctedforthedrawingtogetheroftheplumes,plottedforvaluesof=1.0,0.8,0.6,0.4and0.2.Thelowestlineisfor=1.0andthehighestisfor=0.2.Thisisonlya“rst-ordercorrection,usedtoestablishascaleforthede”ectionoftheplumes.Fromtheminimumvalueofforwhichnointeractionoccurs,weget187uncorrectedand209asthecorrectedvalue(thatis,if=0.209 187).Thisisa10%increaseintheaspectratiooverwhichwecanexpectinteractionbetweentheplumes.Further,thechoiceoftheinitiallengthscaleforcalculatingtheangleofinteractionwasfairlyarbitrary.However,thisvalueisclosetotheexperimentallyevaluatedvalueof25.Themoresigni“cantresultofthecorrectionisthat,foraspectratioslessthan=0.08,weexpectmoremixingthantheuncorrectedcalculation(“gure12)suggests. N.B.KayeandP.F.Linden 0.10 0.15 0.20 0.25 0.30 0 0.2 0.4 0.6 0.8 1.0 Figure17.Plotofasafunctionoffor=1.0.Thethicklineistheuncorrectedmodelpredictionfrom“gure11.Thethinlineisthepredictioncorrectedforthedrawingtogetheroftheplumes(“gure14).Thedashedlineisgivenby=min(4 0.10 0.15 0.20 0.25 0.30 0 0.2 0.4 0.6 0.8 1.0(a)  0.050.100.150.200.250.30Figure18.Plotofasafunctionoffor(=0.8,(=0.6.Thetheoreticallinesarethesameas“gure17.ComparisonwithexperimentalresultsWenowcomparetheresultsofourmodelwiththeearlierexperiments.Theseareshownin“gures17…20.Werecallthetwomajordiscrepanciesbetweenthelinearapproximationandtheexperimentalresults.First,forequalaxiallyalignedplumeseachplumeretainssomeofitsbuoyancy”uxafterthecollision,thatis,
.Thisfeatureisclearlycapturedin“gure17.Second,for6,therateofincreaseofislessthanthelinearinterpolation.Thisfeatureisalsocapturedbythemodel(see“gure19)whichpredictsonlyaslightincreaseinwithseparation.Figures17…20showtheexperimentalpointsfrom“gures5()withthemodelpredictionsof“gures12and13.Thelinearapproximationshownin“gures5(isalsoincludedinthese“gures.Thereisreasonableagreementbetweenthefullmodelpredictionsandtheexperimentalresults,andthemodelprovidesasigni“cantlybetter Collidingturbulentplumes0.050.100.150.200.250.300.050.100.150.200.250.30 Figure19.Plotofasafunctionoffor(=0.4,(=0.2.Thetheoreticallinesarethesameas“gure17.0.20.40.60.81.0 0.20.40.60.81.0 Figure20.(a)Plotof(and(asafunctionof.Thethinlineistheminimumpossiblevaluethatconservesbuoyancy(2.6).Thethicklineisthemodelprediction.descriptionofourmeasurementsthanthesimplelinearinterpolation(2.15).Thereisparticularlygoodagreementbetweentheentrainmentmodelandourexperimentsfor=0(“gure20).4.ConclusionsAnexperimentaltechniquewasdevelopedtomeasurethebuoyancy”uxofaplume,basedontheventilationmodelofLindenetal.(1990).Thistechniquewasusedtomeasurethebuoyancy”uxlossthatoccurswhentwoplumeswithbuoyancy”uxesofoppositesigncollide.Measurementsweremadeoverawiderangeofhorizontaltoverticalaspectratiosandbuoyancy”uxratios.Theexperimentalresultsshowthattwocollidingplumeswillexchangebuoyancy”uxastheypassbyeachotherprovidedthattheirhorizontaltoverticalseparationratioislessthan25.Forsmallerhorizontalseparationstheextentofthebuoyancy”uxexchangeincreasestoamaximumwhentheplumesareaxiallyaligned.Whenoneplumeissigni“cantlystrongerthantheother,theweakerplumecanbecompletelyabsorbedbythestrongerplume.However,forbuoyancy”uxratiosgreaterthan6,bothplumesemerge N.B.KayeandP.F.Linden ParameterTypicalvalue()Typicalerror( x/xsource1.1gcm1.05gcmgcm2cmssource50cms1.0cm5source1.0cm0.630.020…3cm0.08cm10cm0.02Table1.Typicalmeasurementsandrelativeerrorsforbuoyancy”uxmeasurements.Notethattheerrorontheaxialseparationisanabsoluteerror. fromthecollisionregionwithsomeoftheirinitialbuoyancy”ux.Foraxiallyalignedplumesofequalbuoyancy”ux(=1)eachplumeretainsaboutaquarterofitsoriginalbuoyancy”ux.Basedontheseexperimentalresultsamodelhasbeendevelopedtodescribethebuoyancy”uxlossineachplumeduetoplume…plumecollision.Theplumeconservationequations(Mortonetal.1956)weremodi“edtoaccountfortheinteractionsthatoccurduringcollision.Twomodelswereconsideredtodescribetheexchangeofbuoyancy”ux.Model(A)assumedabidirectionalbuoyancyexchangewhilemodel(B)aunidirectionalexchange.Basedonourexperimentalresultsweconcludedthattheunidirectionalentrainmentmodelbetterdescribesourresults.Themodelwasusedtopredictthetotallossinbuoyancy”uxofthetwoplumesintermsoftheirsourcestrengthsandseparation.A“rst-ordercorrectiontothistheorywasmadetoaccountforthedrawingtogetherofthetwoplumesduetotheentrainment“eld.This“rst-ordermodelshowssigni“cantlybetteragreementwiththeexperimentalresultsthanasimplelinearinterpolationthatassumescompletemixingfor=0andnomixingforN.B.K.wouldliketothanktheBritishCouncilandtheAssociationofCommon-wealthUniversitiesfortheir“nancialsupportforthisresearch,andwethankDrS.B.Dalzielforhisassistancewiththeexperiments.Appendix.ErrorestimateforexperimentsHerewereviewthepossiblesourcesoferrorinourexperiments.Inordertoestablishavalueofforanygivenexperimentthefollowingmeasurementsneedtobemade:.Itisalsonecessarytomeasuretheverticalandaxialseparation,andthebuoyancy”uxratio.Asummaryoftypicalvaluesandrelativeerrorsforthesemeasurementsispresentedintable1.Analysingtheimpactoftheerrorslistedintable1on(2.12),we“nd A+ C+ 00+ h (A1)Dividingbothsidesbyweobtain  A0+ Cd+3 20 g0+1 2 + (A2) CollidingturbulentplumesSubstitutingthevaluesfromtable1gives 02+ 2×5+1 (A3)Errorsinevaluatingthevariablesaregivenby  F1+ 04(A4)  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