A Numerical and Experimental Study of Turbulent Single and Multiphase Forced Plumes and Jets at Moderate Reynolds Numbers

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A Numerical and Experimental Study of Turbulent Single and Multiphase Forced Plumes and Jets at Moderate Reynolds Numbers
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Taub, Gordon N
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Doctorate ( Ph.D.)
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University of Florida
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Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Balachandar, Sivaramakrishnan
Committee Co-Chair:
Sherif, Sherif Ahmed
Committee Members:
Lear, William E, Jr
Narayanan, Ranganatha

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Subjects / Keywords:
cfd -- dns -- jets -- multiphase -- numerical -- plumes -- similarity -- swirl -- turbulent
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, Ph.D.
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Abstract:
Flows where a narrow body of fluid, carried by initial momentum or buoyancy, entrains an ambient fluid are common in the environment. Volcanoes, smoke exiting a chimney or rising from a forest fire, tornadoes and deep water oil spills are just a few examples.  If the main driving force is momentum we term such a flow a jet. On the other hand, if the main driving force is buoyancy we typically refer to the flow as a plume. Perhaps more common, initial momentum and buoyancy are both present in which case we can refer to the flow as either a buoyant jet or a forced plume.  Forced plumes can be either single phase, where buoyancy is caused by either a difference in temperature or some other scalar phenomenon, or multiphase where buoyancy is due to the injection of particles, bubbles or drops of material other than the ambient fluid into the flow. In this study single and multiphase jets and plumes will be examined.  A laminar study will be presented which examines the effect of the addition of a small amount of nonzero angular momentum to jets and plumes. The results of direct numerical simulations of a pure jet, a pure plume and a forced plume, in the turbulent, moderate Reynolds number regime are presented and compared.  Each simulation was run long enough so that time averaged third order statistics converged, allowing each term of the turbulent kinetic energy equation and Reynolds stress transport equations to be calculated directly.  The results of laboratory experiments of a pure jet and multiphase plume, conducted at  Ecole Nationale Superieure De Mecanique et D'Aerotechnique (ENSMA) located in Poitiers, France will be presented.  Preliminary results of multiphase plume simulations using the Equilibrium-Eulerian technique will be discussed.
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by Gordon N Taub.
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Thesis (Ph.D.)--University of Florida, 2013.
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Adviser: Balachandar, Sivaramakrishnan.
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Co-adviser: Sherif, Sherif Ahmed.
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ANUMERICALANDEXPERIMENTALSTUDYOFTURBULENTSINGLEANDMULTIPHASEFORCEDPLUMESANDJETSATMODERATEREYNOLDSNUMBERSByGORDONN.TAUBADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013GordonN.Taub 2

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Tomyparents,JackandFriedaTaub,whohavealwayssupportedmeasIchosemyownpath.EventhoughtheroadIchosewasn'talwaysthemostdirectroutetomyeventualdestination. 3

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ACKNOWLEDGMENTS IwouldliketothankthealumnioftheUniversityofFlorida,MechanicalandAerospaceEngineeringDepartmentandtheNationalScienceFoundation(NSFOISE-0968313)throughthePartnershipforInternationalResearchandEducation(PIRE)forfundingthisresearch.Iwouldalsoliketotakethisopportunitytothankmyadvisorsandcollaborators,ProfessorsS.Balachandar,S.A.SherifandF.PlourdeandDr.HyungooLee. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2SINGLEPHASELAMINARJETSANDPLUMES ................. 19 2.1Background ................................... 19 2.2Methodology .................................. 22 2.2.1ProblemDescription .......................... 22 2.2.2NumericalSimulation .......................... 25 2.3SimilaritySolutions ............................... 28 2.3.1JetswithandwithoutSwirl. ...................... 28 2.3.2PurePlumeswithandwithoutSwirl. ................. 32 2.4VortexBreakdown ............................... 39 2.4.1CriticalSwirlforJets. .......................... 40 2.4.2ComparisonofVortexBreakdowninJetsandPlumes ....... 42 2.4.3CriticalSwirlforPlumes ........................ 45 2.5BuoyantJetsandInjectedPlumes ...................... 47 2.6Summary .................................... 49 3ADNSSTUDYOFHIGHERORDERSTATISTICSINATURBULENTROUNDJET .......................................... 51 3.1Background ................................... 51 3.2Methodology .................................. 53 3.3Results ..................................... 55 3.3.1FirstandSecondOrderStatistics ................... 56 3.3.2ThirdOrderStatistics .......................... 62 3.3.3TurbulentKineticEnergyBalance ................... 63 3.3.4ReynoldsStressTransportEquations ................. 66 3.3.5VorticalStructuresandEntrainment ................. 70 3.3.6Spectra ................................. 73 3.4Summary .................................... 75 4SINGLEPHASEJETS,PLUMESANDFORCEDPLUMES ........... 78 4.1Background ................................... 78 5

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4.2Methodology .................................. 80 4.3Results ..................................... 85 4.3.1MomentumandBuoyancyFlux .................... 85 4.3.2MeanFlow ............................... 88 4.3.3SecondOrderStatistics ........................ 92 4.3.4ThirdOrderStatistics .......................... 96 4.3.5TurbulentKineticEnergyBalance ................... 100 4.3.6ReynoldsStressTransportEquations ................. 104 4.3.7VorticalStructuresandEntrainment ................. 110 4.3.8Spectra ................................. 115 4.4Summary .................................... 116 5PIVEXPERIMENTSOFAMULTIPHASEPLUME ................ 120 5.1Background ................................... 120 5.2ExperimentalSetup .............................. 122 5.3Results ..................................... 128 5.3.1JetExperiment ............................. 128 5.3.2MultiphasePlume ........................... 132 5.4Summary .................................... 138 6PRELIMINARYRESULTSTOWARDMULTIPHASEPLUMESIMULATIONS .. 140 6.1Motivation .................................... 140 6.2Methodology .................................. 143 6.3PreliminaryResults .............................. 147 6.4FutureWork ................................... 150 7CONCLUSIONS ................................... 152 REFERENCES ....................................... 159 BIOGRAPHICALSKETCH ................................ 164 6

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LISTOFTABLES Table page 2-1Domain,gridsize,andbuoyancyandangularmomentumux .......... 26 2-2Pureandbuoyantjetnumericalsimulationscompletedinthecurrentstudy,withmomentumux,@JM=@z,buoyancyux,LowerStagnationPoint(LSP)andUpperStagnationPoint(USP)indicated. ................... 27 2-3Pureandforcedplumenumericalsimulationscompletedinthecurrentstudy,withmomentumux,@JM=@z,buoyancyux,LowerStagnationPoint(LSP)andUpperStagnationPoint(USP)indicated. ................... 28 2-4Approximatevalueofcriticalswirl,Scforvariousbuoyantjetsandinjectedplumes ........................................ 49 4-1TableofDNSrunscompletedinthecurrentstudy ................. 85 7

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LISTOFFIGURES Figure page 1-1Schematicofamulti-phaseplumewithasingleintrusion. ............ 15 2-1Schematicofthebuoyantswirlingjetorplumewithinthecomputationaldomain. 23 2-2Axialvelocityprolesofswirlingjets. ........................ 30 2-3Comparisonoftangentialvelocityofthenumericalsolutionofswirlingjetswithsimilaritysolutions .................................. 31 2-4NumericalresultsofthevalueofJBandJasafunctionofzforGr=3000andS=0.0,0.3,0.45,and0.6 ........................... 37 2-5NumericalresultsofthevalueofJB,JMandJasafunctionofSforplumeswithGr=3000andjetsofRej=100 ....................... 38 2-6Comparisonofthenumericalsolutionofswirlingplumeswithsimilaritysolution. 40 2-7Streamtracesplottedonther)]TJ /F4 11.955 Tf 9.3 0 Td[(zplaneforanabovecriticalswirljet.(Re=100,Ri=0,S=0.9). ..................................... 43 2-8Streamtracesplottedonther)]TJ /F4 11.955 Tf 11.59 0 Td[(zplaneforaplume(Gr=3000)withnoswirl(left)andaswirlvalueofS=0.9(right.) ..................... 44 2-9@~ur=@ezatthelowerboundaryofaplumeforvariousvaluesofswirl. ...... 44 2-10Streamtracesplottedonther)]TJ /F4 11.955 Tf 10.69 0 Td[(zplaneforabuoyantjet(Re=100,Ri=0.05,S=0.9)andaninjectedplume(Gr=3000,uJ=0.1,S=0.9). ......... 49 3-1Schematicofthecomputationaldomain. ...................... 53 3-2Initialinletvelocity,UJ,overmeancenterlineaxialvelocity,hUz,ci,ofapurejet. 57 3-3AxialandRadialvelocityprolesofapurejetinthesimilaritycoordinate. ... 57 3-4Turbulentintensityalongthejetaxis. ........................ 58 3-5ComparisonofReynoldsstresstermsradialproleswithpreviousstudies. ... 60 3-6Contributionstothetotalmomentumintegralbyvariousterms. ......... 62 3-7Thirdorderstatisticsofcurrentstudycomparedtopreviousstudies. ...... 62 3-8Comparisonofcurrentstudy'sturbulentkineticenergybalancewithpreviousstudies. ........................................ 65 3-9ComparisonofdissipationofturbulentkineticenergybetweenthecurrentDNSstudyandvariousearlierstudies. .......................... 67 8

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3-10Reynoldsstresstransportequationsofcurrentstudycomparedtoexperimentalresults. ........................................ 69 3-113Diso-surfacesofci=0.5calculatedattwodifferentinstantsoftime. ..... 70 3-12EntrainmentcoefcientP(z)asafunctionofzasdenedbyPhametal.[ 41 ] 73 3-13Onedimensionalvelocityspectraintheazimuthaldirectionforeachvelocitycomponent,attwodifferentlocationsonther)]TJ /F4 11.955 Tf 11.95 0 Td[(zplane. ............ 74 4-1Schematicofthecomputationaldomain. ...................... 81 4-2Contributionstothetotalmomentumintegralofapurejetbyvariousterms. .. 87 4-3Conservationofbuoyancyuxforapureandforcedplume. ........... 87 4-4Scaledmeanaxialvelocityandbuoyancyprolesofthepureplume. ...... 90 4-5CenterlineAxialVelocity ............................... 91 4-6Scaledmeanaxialvelocityandbuoyancyprolesoftheforcedplume. ..... 92 4-7Comparisonofturbulentintensityalongtheaxisforthepurejet,pureplumeandforcedplumeofthecurrentDNSstudy. .................... 93 4-8ComparisonoftheradialprolesofReynoldsstresstermsforthepurejet,pureplumeandforcedplumeofthecurrentDNSstudyattwoaxialdistances. 94 4-9ComparisonoftheradialprolesofsecondorderbuoyancytermsforthepureplumeandforcedplumeofthecurrentDNSstudyattwoaxialdistances. ... 95 4-10Comparisonoftheradialprolesofthethirdordervelocitystatisticsforthepurejet,pureplumeandforcedplumeofthecurrentDNSstudyatanaxialdistanceofz=15. .................................. 97 4-11Comparisonoftheradialprolesofthethirdordervelocitystatisticsforthepurejet,pureplumeandforcedplumeofthecurrentDNSstudyatanaxialdistanceofz=30. ................................. 98 4-12ComparisonoftheradialprolesofthethirdorderbuoyancystatisticsforthepureandforcedplumeofthecurrentDNSstudyataxialdistancesofz=15andz=30. ..................................... 99 4-13Turbulentkineticenergybalanceofapurejet,pureplume,andforcedplumeatz=15andz=30. ................................ 102 4-14Prolesofthetransportofhu02ziReynoldsstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. ... 106 4-15Prolesofthetransportofhu02riReynoldsstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. ... 107 9

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4-16Prolesofthetransportofhu02iReynoldsstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. ... 108 4-17Prolesofthetransportofhu0zu0riReynoldsshearstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. .......................................... 109 4-18Comparisonofiso-surfaceofci=0.5ofapurejet,pureplumeandforcedplume. ........................................ 111 4-19ComparisonofthecoefcientofentrainmentasdenedbyPham[ 41 ],p(z).Forapurejet,pureplumeandaforcedplume. .................. 114 4-20Onedimensionalvelocityspectraintheazimuthaldirectionforeachvelocitycomponent,attwodifferentlocationsonther)]TJ /F4 11.955 Tf 11.95 0 Td[(zplane ............. 116 5-1Diagramofamultiphaseplume ........................... 121 5-2Photographoftheexperimentalsetup. ....................... 123 5-3ExperimentalWaterTank .............................. 124 5-4Waterandbuoyancyparticleinjectionsystem. .................. 125 5-5Contourplotoftheaxialvelocity,hUzioftheexperimentaljet. .......... 129 5-6Initialinletvelocity,UI,overmeancenterlineaxialvelocity,hUz,ci,ofapurejet. 130 5-7Axialvelocityprolesoftheexperimentaljetinthesimilaritycoordinate. .... 131 5-8Scaledhu02zialongthecenterline(left)andradialproleatz=D=15(right)ofajet. ......................................... 132 5-9ImagefromPIVcameraTracerParticles(left)andBuoyancyParticles(right). 133 5-10Contourplotofaxialvelocity,hUzi,ofthecontinuousphase(left)anddispersedphase(right). ..................................... 134 5-11MultiphasePlumeExperiment,meanaxialvelocity,hUzi. ............. 136 5-12Differencebetweendispersedandcontinuousmeanaxialvelocitycomparedtotheoreticalsettingvelocity. ............................ 136 5-13Scaledhu02zialongthecenterline(left)andradialproleatz=D=15(right)ofamultiphaseplume. ................................. 137 6-1Schematicofthecomputationaldomain. ...................... 143 6-2Continuousphaseaxialvelocityofasimulatedmultiphaseplumecomparedtoexperimentalresults. ............................... 148 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyANUMERICALANDEXPERIMENTALSTUDYOFTURBULENTSINGLEANDMULTIPHASEFORCEDPLUMESANDJETSATMODERATEREYNOLDSNUMBERSByGordonN.TaubMay2013Chair:S.BalachandarCochair:S.A.SherifMajor:MechanicalEngineeringFlowswhereanarrowbodyofuid,carriedbyinitialmomentumorbuoyancy,entrainsanambientuidarecommonintheenvironment.Volcanoes,smokeexitingachimneyorrisingfromaforestre,tornadoesanddeepwateroilspillsarejustafewexamples.Ifthemaindrivingforceismomentumwetermsuchaowajet.Ontheotherhand,ifthemaindrivingforceisbuoyancywetypicallyrefertotheowasaplume.Perhapsmorecommon,initialmomentumandbuoyancyarebothpresentinwhichcasewecanrefertotheowaseitherabuoyantjetoraforcedplume.Forcedplumescanbeeithersinglephase,wherebuoyancyiscausedbyeitheradifferenceintemperatureorsomeotherscalarphenomenon,ormultiphasewherebuoyancyisduetotheinjectionofparticles,bubblesordropsofmaterialotherthantheambientuidintotheow.Inthisstudysingleandmultiphasejetsandplumeswillbeexamined.Alaminarstudywillbepresentedwhichexaminestheeffectoftheadditionofasmallamountofnonzeroangularmomentumtojetsandplumes.Theresultsofdirectnumericalsimulationsofapurejet,apureplumeandaforcedplume,intheturbulent,moderateReynoldsnumberregimearepresentedandcompared.Eachsimulationwasrunlongenoughsothattimeaveragedthirdorderstatisticsconverged,allowingeachtermoftheturbulentkineticenergyequationandReynoldsstresstransportequationstobecalculateddirectly.Theresultsoflaboratoryexperimentsofapurejetandmultiphase 11

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plume,conductedatEcoleNationaleSuperieureDeMecaniqueetD'Aerotechnique(ENSMA)locatedinPoitiers,Francewillbepresented.PreliminaryresultsofmultiphaseplumesimulationsusingtheEquilibrium-Euleriantechniquewillbediscussed. 12

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CHAPTER1INTRODUCTIONFlowswhereanarrowbodyofuid,carriedbyinitialmomentumorbuoyancy,entrainanambientuidarecommonintheenvironment.Volcanoes,smokeexitingachimneyorrisingfromaforestre,tornadoesanddeepwateroilspillsarejustafewexamples.Ifthemaindrivingforceismomentumwetermsuchaowajet.Ontheotherhand,ifthemaindrivingforceisbuoyancywetypicallyrefertotheowasaplume.Perhapsmorecommon,initialmomentumandbuoyancyarebothpresentinwhichcasewecanrefertotheowaseitherabuoyantjetoraforcedplume.Jetsandplumes,whicharetypicallyturbulent,canincludevariousadditionalfeatureswhichcomplicatetheow.Asaplumerises,itoftenentrainstheambientuidunevenly,thiscausestheplumetohavenon-zeroangularmomentumux,orswirl.Consequencesofthisphenomenoncanbedramaticsuchascanbeseenin,dustdevilsrisingoffthedesertsand,tornadoesandrewhirls.Plumescaneitherbesinglephaseormultiphase.Inasinglephaseplume,buoyancyistypicallycausedbyatemperaturedifferencebetweentheambientuidandtheuidalreadyentrainedintotheplume.Inthecaseofamultiphaseplume,buoyancyiscausedbytheinjectionofsmallparticles,dropsorbubblesintoanambientuid.Sincetheinjectedparticles(bubbles,ordrops)haveadifferentdensitythantheambientuid,theybegintoriseorfall,entrainingambientuidastheydoso.Thiscausesthebulkuid,entrainedbytheparticles,tohaveadifferentbulkdensitythantheambientuidwhereparticlesarenotpresent.Theentraineduidbeginstoriseorfallenmassewiththeparticles,formingaplume.Itisinterestingtonotethattheresultingplumewillbehaveasifitistheresultofthedensitydifferencebetweentheparticle-uidmixtureandtheambientuid,asopposedtothedensitydifferencebetweentheindividualparticlesandtheambientuid.Themixtureofparticlesanduidcanoftenbetreatedasacontinuum.Assuch,thebulk 13

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velocityoftheplumewillbeverydifferentthantheterminalvelocityoftheindividualparticles.Iftheambientuidisstratied,sothatthedensityoftheambientuiddecreaseswithheight,itispossiblethatatsomeheighttheuidalreadyentrainedbytheplumewillhaveahigherdensitythantheambientuidatthatheight.Ifthisoccurstheentraineduidcanpealawayandwesayeitheranintrusionhasformedorthatpealinghasoccurred(Seegure 1-1 ).Theparticlesmay,insuchcasesbegintoentrainnewuidandasecondaryplume,widerthantheoriginal,mayformatthatheight.Examplesofsuchmultiphaseplumesarenumerousandagreatdealofthephysicsinvolvedinsuchowsisstillnotwellunderstood.[ 16 52 ]Indeepwateroilspillsamixtureofoilandnaturalgasunderintensepressurecanescapeintotheocean.Thereleasedoilandgasquicklybreakupintosmalldropletsformingabubblyplumewiththeentrainedseawater.Evenwhenagreatdealofdataisavailable,determiningtheinitialowrate,maximumplumeheight,andrateatwhichtheescapinggaswilldissolveintowatercanbedifculttoanswerandhavesignicanteconomicconsequences.AswasthecaseintherecentDeepwaterHorizonoilspillintheGulfofMexico.LakesandotherstandingbodiesofwatercancontaindissolvedgaseswhichcanabruptlyeruptsuchasoccurredAugust21st,1986inLakeNyosinCameroon.InthateventdissolvedCO2suddenlyeruptedfromthebottomofthelake.ThereleasedCO2rusheddownthevalleysurroundingthelakeandpassedthroughanearbyvillagekillingover1700individuals[ 58 59 ].Whyandhowsucheruptionsoccurisnotwellunderstood.IthasrecentlybeensuggestedthatinordertoreducetheeffectsofglobalwarmingCO2frompowerplantsberedirectedtotheoceanooratdepthsgreaterthen3000meters[ 1 4 ].AtthisdepththeCO2istobereleasedasliquiddroplets.ThereleasedCO2droplets,alongwithentrainedwater,willformabubblyplume.Astheplumerises,it 14

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Figure1-1. Schematicofamulti-phaseplumewithasingleintrusion. isintendedthattheCO2willdissolveintotheambientwater.TherateatwhichtheCO2dissolves,themaximumheightoftheplume,andtheaffectthishasonthepHofthesurroundingwaterareofinterest.Whenavolcanoeruptsoraforestreoccurs,hotashisinjectedintotheatmosphere.Oftenwithalargeinitialvelocity.Althoughtheashparticlesthemselvescanbeheavierthanair,theyheattheairaroundthemcausingadensitydifferencewiththeambientair.Asthehotairandashplumerisethroughtheatmosphere,thesurroundingambientairdensitydecreases.Atsomeheighttheheatedairwillnotlongerbelessdensethanthesurroundingambientairandanintrusionwilloccur.Ithasbeensuggestedthatthestrengthofavolcaniceruptionorforestrecanbedeterminedbyanalyzingtheheightandsizeofsuchanintrusion[ 58 ].Sincejetsandplumesarecommonintheenvironmentalandengineeringapplicationsandsincetheirbehaviorcanbecomequitecomplexwithvariousexternal 15

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phenomenaaddingcomplications,alargeamountofresearchhasbeendedicatedtounderstandingtheirbehavior.Itwillbehelpfultodiscusssomeofthemajorworksandseminalpapersonthesubject.Schlichting's[ 45 ]performedsomeoftheearliestworkonaxisymmetricjetsinhispioneeringworkonboundarylayertheoryrstpublishedin1933.Startingfromturbulentboundarylayertheory,Schlichtingdevelopedasimilaritysolutionforthemeanowofanantisymmetricturbulentjet.Heshowedthattheaxialcenterlinevelocitydecreasesasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1,whilethejetwidthgrowsasz1.Wherezrepresentsthedistancefromthejetinletalongthejetaxis.Schmidt[ 46 ]performedsomeoftheearliestworkonplumesin1941.Schmidtusedmixinglengthhypothesestoobtainmeanvelocityandbuoyancyprolesforanantisymmetricplume.Hefoundthatthemeanowaxialvelocityscalesasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3andthebuoyancyscalesasz)]TJ /F9 7.97 Tf 6.58 0 Td[(5=3andproposedasimilaritysolutionsforbothquantitiesbasedonGaussiandistributions.Variousresearchers,mostnotablyYih[ 62 ],Rouse,YihandHumphreys[ 44 ],renedSchmidt'ssimilaritysolutionusingexperimentalresults.Inaseminalpaper,Morton,TaylorandTurner[ 35 ]examinedplumesinastratiedenvironment.Bymakingtheassumptionthattherateofentrainmentofambientuidisproportionaltotheaxialvelocityoftheplume,theywereabletodeterminetheaxialvelocityofaplumeatagivenheight.Theyalsoexaminedthemaximumheightobtainedbyaplumeinastratiedenvironment.Morton[ 34 ]examinedforcedplumeswherebothbuoyancyuxandmomentumux,arepresentattheplumesource.Hesuggestedthatforconvenience,jetsandplumescanbegenerallyreferredtoasforcedplumes,withpurejetsandplumesbeingtheextremecases.Wewillfollowthisconventionthroughoutthisdocument.Mortonpointedoutthatsinceaxialvelocityscalesasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1forajet,butasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3foraplume,thatinthefareld,plumebehaviorwilldominateandsuggestedaparameterthatcanbeusedtodetermineatwhatheightaforcedplumewillhaveessentiallypureplumelikebehavior. 16

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Kobus[ 27 ]wasoneoftherstresearcherstoworkwithmultiphaseplumes.Usingairbubblesasabuoyancysource,heperformedlaboratoryexperimentsina4.7mtank.HepostulatedGaussianprolesandalongwithanentrainmentassumptionsimilartothatofMorton,etal.[ 35 ],proposedanintegralmodelforsuchows.Manyresearchershavebuiltonthisearlywork.Foranexcellentreviewofsomeofthisworkandadescriptionofourstateoftheartunderstandingofforcedplumesinnature,seetherecentannualreviewarticlebyWoods[ 58 ].Wewillcontinuetodiscusspastresearchonjetsandplumesinmoredetailineachchapterofthisdocument.Inthiscurrentstudyweexaminesingleandmonodispersedmultiphaseplumesinauniformstagnantambientenvironment.Whilejetsandplumesoccurringinnatureareusuallyturbulent,itisinstructivetostudysuchowsinthelaminarregimewheresomeoftheunderliningphysicscanbedescribedanalyticallywithoutresortingtocomplexexperimentsornumericalsimulations.SuchastudyhasbeencompletedandtheresultsofwhichhaverecentlybeenpublishedinPhysicsofFluids.Theresultsarereprintedhereaschapter 2 .Whilealargenumberofresearchershaveexaminedturbulentjetswithexperimentalandnumericaltechniques.ManynumericalsimulationsrelyonmodelsofReynoldsstressinthesub-griddomain.OnlyrecentlyhasitbecomepossibletoeconomicallyperformDirectNumericalSimulations(DNS)offullyturbulentjetswherenomodelingisneeded.MostDNSstudiesofjetshavefocusedonrstandsecondorderbehavioraswelltheformationofinstabilitiesandvorticalstructuresnearthesourceofthejet.InordertoexaminethebalanceofturbulentkineticenergyandthetransportofReynoldsstressthederivativesofthirdordermomentsmustbecalculated.ADNSstudyofaturbulentjethasbeencompleted.Thesimulationranlongenoughforallthirdorderstatisticstoconverge.AlltermsoftheturbulentkineticenergybalanceandReynoldsstresstransportequationswerecalculateddirectly.Theresultsofthisstudywillbepresentedinchapter 3 17

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Directnumericalsimulationshavealsobeencompletedforapureandforcedplume.Inchapter 4 theresultsoftheDNSstudiesofthepurejet,pureplumeandforcedplumewillbecompared.Morton'sassertion,whichwasbasedonasimpleboxmodel,astowhereaforcedplumecanbetreatedasapureplumewillbeexamined.Inchapter 5 ,resultsoflaboratoryexperimentsonpurejetsandmultiphaseplumeswillbepresented.TheseexperimentswereconductedatEcoleNationaleSuperieureDeMecaniqueetD'Aerotechnique(ENSMA)locatedinPoitiers,Franceinafacilityconstructedforthispurpose.Inconjunctionwithlaboratoryexperimentsonmultiphaseplumesnumericalsimulationsaretobecompleted.Inchapter 6 preliminaryresultsarepresented.Inchapter 7 conclusionsandfutureworkarediscussed. 18

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CHAPTER2SINGLEPHASELAMINARJETSANDPLUMES 2.1BackgroundSince1933whenSchlichting[ 45 ]publishedhisseminalworkonboundarylayertheory,agreatdealofresearchhasbeendoneonaxisymmetricjetsandplumes.Avarietyofresearchershaveextendedtheworktoconsidertheeffectofrotationonjetandplumeevolution.Inthelaminarregimesimpliedanalyticsimilaritysolutionshavebeenfoundwhichallowforanunderstandingoftheowdynamicswithouthavingtoperformcomplexexperimentsornumericalsimulations.Similaritysolutionshavebeendevelopedforjets,swirlingjetsandbuoyantplumes.Ananalyticsolutionforabuoyantplumewithrotationhasnotyetbeendeveloped.Thesimilaritysolutionsdiscussedinthisstudyareallderivedfromtheboundarylayerequationswhichareonlyvalidfornarrowows.Iftheamountofangularmomentumuxpresentintheowexceedssomecriticalvaluetheboundarylayerequationsarenolongervalidanditissaidthatvortexbreakdownhasoccurred.Variousresearchershaveexaminedtheupperandlowerboundsforthisthresholdvalueforvortexbreakdown,knownasthecriticalswirlvalue,Sc[ 11 43 ].UsingtheboundarylayerequationsSchlichting[ 45 ]obtainedasimilaritysolutiontothelaminar,non-buoyant,non-swirlingaxisymmetricjet.Inthiscaseheobservedthatthemomentumux,eJM,oftheowisconservedandthatthecenterlineaxialvelocitydecreasesasez)]TJ /F9 7.97 Tf 6.59 0 Td[(1.Thejetwidthgrowsproportionaltoez1,whereezistheaxialcoordinate.Herethetilderepresentsadimensionalquantity.Revueltaetal.[ 43 ]investigatedtheswirlingnon-buoyantaxisymmetriclaminarjetusingtheboundarylayerequations.Inthiscase,eJMisnolongerindependentofez, 1Thecontentsofthischapterhavepreviouslybeenpublishedinthejournal,PhysicsofFluids.[ 54 ] 19

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howevertheowforce,eM,denedbelow,isconserved.Gortler[ 24 ]andLoitsianskii[ 29 ]foundthatthesimilaritysolutionofSchlichtingcanstillbeusedfortheaxialvelocityoftheswirlingjet,ifthemomentumux,eJM,intheSchlichting'sscaling,isreplacedbytheowforce,eM.Workingindependently,Yih[ 63 ]andBrandandLahey[ 15 ]bothfoundthatforthepureplume,theboundarylayerequationscanbereducedtoasystemofordinarydifferentialequationswithoneindependentsimilarityvariable.ThissystemhasanexactsolutionforPrandtlnumbersPr=1and2.ForallotherPrandtlnumbers,thesystemofequationscanbesolvednumerically.Byanalyzingtheresultingsimilaritysolution,itisobservedthatforplumes,thecenterlineaxialvelocityremainsconstant,whiletheplumewidthincreasesasez1=2.Inthecaseofapureplume,thebuoyancyux,eJB,isconserved,ratherthanmomentumux,eJM,whichisnotconserved.Inbothjetsandplumes,whentangentialvelocityatthejetorplumeentranceisnon-zero,angularmomentumux,eJ,isconserved.ThequantitieseJM,eJB,eM,andeJaredenedbyequations( 2 )though( 2 ) eJM=2Z10eu2zerder(2) eJB=2Z10euz~g0erder(2) eJ=2Z10euzeuer2der(2) eM=2Z10)]TJ 5.54 -9.19 Td[(eu2z+eperder(2)Hereeg0isthebuoyancyorreducedgravity,denedaseg0=g(eT)]TJ /F8 11.955 Tf 14.5 3.16 Td[(eTa),whereisthethermalexpansioncoefcientandeTandeTaarethetemperatureatagivenlocationandtheambienttemperaturerespectively.Thequantities,er,ez,eu,euz,epare 20

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thedimensionalradialandaxialcoordinates,thetangentialandaxialcomponentsofvelocity,andthepressurerespectively.Thereareanumberofapplicationswheretheinowattheinletcarriesbothbuoyancyandmomentumux.Ifthemomentumuxisstronger,thenthejet-likebehaviordominates,atleastclosetotheinlet,andtheowcanbetermedaheatedorbuoyantjet.Ontheotherhand,ifbuoyancyuxisstronger,thentheplume-likebehaviordominatesandtheowcanbetermedaninjectedplume.Sincetheaxialvelocityofajetdecaysasez)]TJ /F9 7.97 Tf 6.59 0 Td[(1,whilethatofaplumeremainsconstant,sufcientlydownstreamthebuoyancydrivenplume-likebehavioralwaysdominates.Itshouldbepointedoutthateveninnaturalconvectionowssuchswirlingplumesnaturallyforminthesystem.[ 19 ]OfrelevancetothepresentworkistheinvestigationofBillantetal.[ 11 ],whostudiedtheeffectofvortexbreakdown.Swirlingjetsandplumesexperienceanadversepressuregradientintheezdirection.Thispressuregradientdeceleratestheowandifthestrengthoftheangularmomentumuxisincreasedtosomecriticalvalue,theowwillformastagnationpointalongthecenterlineandvortexbreakdownoccurs.Theamountofswirlintroducedintotheowcanbedenotedbythenon-dimensionalswirlnumberS.ThecriticalvalueatwhichvortexbreakdownoccursisdenotedasSc.Inthecurrentstudytheestablishedsimilaritysolutionsforjets,jetswithswirlandbuoyantplumeswillbereviewed.UsinganasymptoticapproachthesimilaritysolutionforaplumeestablishedindependentlybyYih[ 63 ]andBrandandLahey[ 15 ]willbeextendedtothecaseofaswirlingplume.Numericalsimulationswillbeperformedinordertoverifythesimilaritysolutionsandtofurtherexaminethebehaviorofjetsandplumeswithandwithoutswirl.ThephenomenonofvortexbreakdownwillbeexploredandtheanalyticallyderivedupperandlowerboundsonthecriticalswirlvalueScforpurejetsdescribedinRevueltaetal.[ 43 ]andBillantetal.[ 11 ]willbediscussed.Asimilar,simpleempiricalvalueforScforplumeswillbediscussed.Acomparisonwillbemadebetweenthebehaviorofabove-criticalswirlinjetsandplumes.Interestingly 21

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abovecriticalswirlinplumeswillbefoundtohavelittleeffectonthefareldbehaviorofplumes,whereasinjets,abovecriticalswirlleadstoatotalbreakdownofjet-likebehavior.Simulationresultsofbuoyantjetsandinjectedplumeswillbepresentedandthebehaviorofvortexbreakdownofthesehybridowsisdiscussedinreferencetothepurecases.Wewillbegininsection 2.2 bydeningtheproblemandintroducingthesimulationmethodologythatwillbeusedthroughouttherestofthearticle.Insubsection 2.3.1 wewillbrieyreviewtheestablishedsimilaritysolutionsforjetswithandwithoutswirl.Insubsection 2.3.2 asolutiontoplumeswithasmallamountofangularmomentumwillbederivedusinganasymptoticapproach.Totheleadingorder,thesolutionfortheaxialvelocityisthesameasthesolutionfornon-swirlingplumes.Insection 2.4 wewillcomparetheonsetofvortexbreakdowninswirlingjetsandplumesanddiscusshoweachowisaffectedbyabovecriticalswirl.Simulationresultsinhybridowswherebothinitialmomentumuxandbuoyancyarepresentwillbediscussedinsection 2.5 .Wewillpresentourconclusionsinsection 2.6 2.2Methodology 2.2.1ProblemDescriptionWeassumeajetemergingfromacircularpipeofdiameterdoraplumegeneratedbyaheatedplateofdiameterd.Fluidexitingthecircularpipeorheatedbythecircularplateentrainsambientuidabove.Figure 2-1 showsthecomputationaldomainwhichisalargecylinderofdimensionlessheightLzandradiusLr,wheredisusedasthecharacteristiclengthoftheow.Thequantities~r,ezandedenotetheradial,axialandazimuthalcoordinatesinthecylindricalcoordinatesystem.Thequantitiesex,ey,andezarethecoordinatesintheCartesiancoordinatesystem.Theoriginofthecoordinatesystemispositionedatthecenterofthejet(orplume)inowboundary.Thegravitationalacceleration,g,actingontheuidisdirectedinthenegativez-direction.WeusetheBoussinesq 22

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Figure2-1. Schematicofthebuoyantswirlingjetorplumewithinthecomputationaldomain. approximationandtheuidproperties,suchaskinematicviscosity(),thermaldiffusivity(),andthermalexpansioncoefcient()alongwiththegravitationalacceleration(g)arekeptconstantforpurposesofthisanalysis.ManyresearchershavepointedoutthattheBoussinesqapproximationisnotvalidneartheheatsourcewherelargechangesindensitycanbeexpectedandsuggestalow-Mach-numberweaklycompressibleformulationasamorevalidmodel.[ 36 41 65 ]Here,byusingtheBoussinesqapproximation,werestrictourselvestoaweakheatsource,suchascanbeexpectedinalaminarplume.Twodifferentscalingsarenecessaryinordertonondimensionalizepureplumesandpurejets.ForapurejetthereferencevelocityisdenedaseuJ,thetophataxialvelocityatthejetinlet.But,forpureplumes,wherebydenitioneuJ=0,thevelocityisscaledbyq geTd.HereeTisthedifferencebetweenthetemperatureatthejetinletandtheambienttemperature.Forcaseswherebothinitialmomentumandbuoyancyuxare 23

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nonzeroeitherscalingcanbeusedbutitislogicaltousethescalingthatcorrespondstotheuxwhichisdominant.Whenconsideringajet,thevariablesarescaledbythefollowingdimensionalquantities:jetinletdiameter(d),jetinletvelocity(euJ),andthedifferencebetweenjetexittemperature(eTJ)andsurroundinguidtemperature(eTa).Thesubscripts`J'and`a'denotetheconditionsatthejetentranceandsurroundingambientuid,respectively.Theresultingdimensionlessvariablesarexi=exi=d,ui=eui=euJ,p=ep=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(eu2J,T=eT)]TJ /F8 11.955 Tf 13.76 3.15 Td[(eTa=eTJ)]TJ /F8 11.955 Tf 13.76 3.15 Td[(eTaandt=euJet=d.Atophatinitialvelocityprole,(~uz=~uJ)isassumed.Whenbuoyancyistobeconsideredthetemperatureatz=0isassumedtobeeT=eTJfor~r
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lengthscaleasd,thepressureandtimescalescanbeaccordinglydened.Thisscalingresultsindimensionlessgoverningequationsoftheform( 2 )to( 2 )butwith, ReJ=p Gr=q geTd3 ,(2)whereGristheGrashofnumberandthelastterminthemomentumequation( 2 )becomesT^k.Howeverthedimensionlessinletvelocityforatophatprolenowbecomes1=p Ri,whileinthejetscalingitisunity.Theamountofswirlpresentattheinletoftheowisdictatedbythenon-dimensionalparameterswirlnumber,S.HerewewilldeneSas, S=1 4ed uref(2)Whereforjetsuref=euJandforplumesuref=q g4eTd.Usingthejetscalingdescribedabovethisisequivalentto S1=2 deJJ(z=0) eJMJ(z=0),(2)whichisthesameasthedenitiongiveninRevueltaetal.[ 43 ]andBillantetal.[ 11 ].ForjetsthevalueofS1remainsconstantinezasbotheJandeJMareconservedquantities.Forplumesandjetswithbuoyancythiswillnotbethecase.ForplumeswhereJM(z=0)=0,S1isundenedbutusingtheplumescalingequation 2 becomes, S2=1 4ed q geTd.(2) 2.2.2NumericalSimulationAsecondorderaccuratenitevolumeschemeisusedtonumericallysolveequations( 2 )though( 2 ).ThesimulationsuseanO-typegridsystemintheradialandazimuthaldirections.Thegridpointsfortheradialdirectionareclusteredcloseto 25

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Table2-1. Domain,gridsize,andbuoyancyandangularmomentumux DescriptionLrLzLNrNzNGridPointsJBJ Coarse25502120210641.61060.7340.106Standard25502170300643.31060.7250.104Fine25502220390837.11060.7220.103 thejetexitborder(r=1=2)toresolvethehighshearrates.Also,thegridpointsfortheaxialdirectionareclusteredclosetothejetentrance(z=0).Auniformspacingisemployedintheazimuthaldirection.Thebranchcuttechniqueisusedforcoveringthediscontinuityofthegridsystem.Typicaldifcultyinthejetowcomputationsassociatedwiththesingularityproblemalongthejetaxisisresolvedbyanappropriatetreatmentofcellcenterboundarycondition.Thedomainsizewaschosensothateffectsduetoconnementinthelateraldirectionareminimized.WeconsultedpreviousnumericalstudieswheresimilarorlargerReynoldsnumberswereexamined,suchasBoersmaetal.[ 12 ]andAissiaetal.[ 2 ]astwoexamples,todeterminethesizeofourcomputationaldomainandgridresolution.Threedifferentgridresolutionswerecomparedandfoundtoyieldessentiallythesameresultswithinthetolerancenecessaryforthisstudy.Thedomainsize(Lr,LzandL)andnumberofgridpointsusedinthepresentsimulationsareshowninTable 2-1 alongwiththevaluesobtainedfortheconservedquantitiesJBandJ.Attheinowboundary,atop-hatproleisusedforbothaxialvelocity(uz)andtemperature(T).Theazimuthalvelocity(u)atthejetinowboundaryisdenedbythelinearrelationu=4Sr.Weassumethenon-dimensionaltemperatureatthejetentrance(TJ)tobeunity.Thelowerboundaryoutsidethejetinow(r1=2),isconsideredasano-slipandadiabaticwall.Weallowtheowandheattofreelyenterorleavethelateralboundary(r=Lr).Similartothelateralboundaryconditions,weuseNeumannboundaryconditionsforthevelocitiesandtemperatureattheoutlet(z=Lz).Asecond-orderaccuratecentraldifferenceschemeisemployedforthespatialdiscretizationofthegoverningequationsonanon-staggeredbody-ttedgridsystem. 26

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Table2-2. Pureandbuoyantjetnumericalsimulationscompletedinthecurrentstudy,withmomentumux,@JM=@z,buoyancyux,LowerStagnationPoint(LSP)andUpperStagnationPoint(USP)indicated. RiSJM@JM=@zJBLSPUSP 0.00.00.650.00.0--0.00.30.580.00.0--0.00.60.370.00.0--0.00.70.280.00.00.971.90.00.90.0250.00.00.425.50.01.20.0250.00.00.334.2 0.050.0-0.110.73--0.050.3-0.110.70--0.050.6-0.110.69--0.050.9---0.442.180.051.2---2.57 0.10.0-0.160.73--0.10.3-0.150.73--0.10.6-0.160.73--0.10.9---0.680.760.11.2---0.351.96 0.30.0-0.290.73--0.30.6-0.300.73-Afractionalstepmethodisusedfortimeadvancement.Intheadvection-diffusionstep,thenonlineartermsaretreatedexplicitlyusingasecond-orderAdams-BashforthschemeandthediffusiontermsaretreatedimplicitlywiththeCrank-Nicholsonscheme.ThepressurePoissonequationissolvedasapressurecorrectionstep.Analdivergence-freevelocityisobtainedateachtime-stepwithapressurecorrectionstep.AllsimulationswererunwithaPrandtlnumberof0.7whichistheapproximatevalueforair.ForpurejetstheReynoldsnumberwasheldconstantatReJ=100.ForpureplumestheGrashofnumberwasheldconstantatGr=3000.Inallover37simulationswereconducted.Thesimulationscanbebrokendownintofourcategories,purejets,pureplumes,buoyantjetswheretheowwasdominatedbymomentumuxandinjectedplumeswheretheowwasdominatedbybuoyancy.Thesimulationsaredescribedintables 2-2 and 2-3 27

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Table2-3. Pureandforcedplumenumericalsimulationscompletedinthecurrentstudy,withmomentumux,@JM=@z,buoyancyux,LowerStagnationPoint(LSP)andUpperStagnationPoint(USP)indicated. ujS@JM=@zJBLSPUSP 0.00.000.190.058--0.00.100.190.058--0.00.300.190.054--0.00.450.180.049-0.100.00.600.160.042-0.270.00.900.140.030-0.530.01.200.120.022-0.710.01.600.110.017-0.920.02.000.100.015-1.08 0.050.000.230.080--0.050.450.210.069--0.050.600.200.061--0.050.700.190.0550.0730.32 0.10.000.250.100--0.10.450.240.092--0.10.600.240.084--0.10.700.220.077--0.10.800.210.700.110.370.10.900.210.650.080.48 2.3SimilaritySolutionsSinceSchlichting[ 45 ]rstpublishedthesimilaritysolutionoftheaxisymmetricjetin1933avarietyofresearchershaveextendedonhiswork.Inthissectionwewillbrieyreviewtheanalyticalsolutionsforjetswithandwithoutswirl.Wewillthenextendthesimilaritysolutionusinganasymptoticapproachtondananalyticsolutiontoplumeswithasmallamountofangularmomentumux.Wewillthencomparetheanalyticsolutionswiththenumericalresults.Allofthesimilaritysolutionsdiscussedherearederivedfromtheboundarylayerequationswhereitisassumedthat@=@z@=@r. 2.3.1JetswithandwithoutSwirl.Schlichting[ 45 ]startedwiththeboundarylayerequationsinaxisymmetricformandobtainedthefollowingself-similarvelocityproleforanon-swirlingpurejet. 28

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uz=3 81 zReJJM (1+1 42)2 (2) ur=1 4zr 3JM )]TJ /F9 7.97 Tf 13.15 4.7 Td[(1 43 (1+1 42)2 (2) wherethescaledradialcoordinateisgivenby, =ReJr 3JM 16r z(2)Foranite-sizedsourceavirtualorigin(zv)needstobedenedandthezcoordinateintheabovedenitionsneedstobeadjustedas(z+zv).Fromexperimentalmeasurementsandnumericalsimulationsithasbeenestablishedthatforatop-hatinletvelocityproleavirtualoriginofzv=0.072ReJdprovidesexcellentagreementwiththeaboveself-similartheory.Gortler[ 24 ]andLoitsianskii[ 29 ]extendedSchlichting'ssolutiontothecasewhereasmallamountofangularmomentumuxispresentintheow.Theyshowedthatintheswirlingcase,equations( 2 )through( 2 )arestillvalidforaxialandradialvelocitybutwiththemomentumuxeJMreplacedwiththeowforce,eM.Thetangentialvelocityisgivenby, u=ReJS3 5121 rz2 (1+1 42)2.(2)Figure 2-2 comparestheaxialvelocityproleoffournumericalsimulations(S=0.0,S=0.3andS=0.6andS=0.9forReJ=100)ofnon-swirlingandswirlingjetswithboththesolutionobtainedbyGortlerandLoitsianskiiandSchlichting'ssolutionforanonswirlingjet.Thecomputationalresultsofthenon-swirlingjetcomparewellwiththeSchlichtingsolution(withzv=7.2)sufcientlydownstreamoftheinlet.TheDNSresultswithaswirlvalueS=0.3comparesverywellwithGortlerandLoisianskii'ssolutionwithavirtualoriginofzv=8.55.ForthenumericalsimulationswithaswirlvalueofS=0.6 29

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Figure2-2. ComparisonofaxialvelocityofthenumericalsolutionofswirlingjetswiththesimilaritysolutionsbySchlichting[ 45 ]andGortler[ 24 ]atfourdifferentheights,z,abovejetinlet.z=1,20,30and46.ReJ=100,S=0.0(zv=7.2),S=0.3(zv=8.6),S=0.6(zv=18),andS=0.9(zv=16).Notethatz=1,isshownonadifferentscalethanz=20,30,and46. 30

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Figure2-3. ComparisonoftangentialvelocityofthenumericalsolutionofswirlingjetswiththesimilaritysolutionsbyGortler[ 24 ]andLoitsianskii[ 29 ]atfourdifferentheights,z,abovejetinlet.z=1,20,30and46.ReJ=100,S=0.3(zv=8.6),S=0.6(zv=18),andS=0.9(zv=16).Notethateachvalueofz,isshownonadifferentscale. neithersimilaritysolutioncapturestheowforheightslessthenz=30.S=0.9isabovecriticalswirlandneitherSchlichting'snorGortler'ssolutionisvalidastheowhasundergonevortexbreakdown.Thiswillbediscussedfurtherinsection 2.4 .Figure 2-3 comparesthetangentialvelocityproleofthreenumericalsimulations(S=0.3,S=0.6andS=0.9forReJ=100)ofswirlingjetswiththesolutionobtained 31

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byGortler[ 24 ]andLoisianskii[ 29 ].Hereagain,thenumericalresultsofajetwithaswirlvalueofS=0.3compareverywellwiththeGortlerandLoitsianskiisolution( 2 ).Astheamountofswirlisincreasedagreateraxialdistancefromthejetinletisrequiredbeforethesimilaritysolutionbecomesvalid.ForjetswithabovecriticalswirlsuchasS=0.9thesimilaritysolutionisnotvalidatanydistancefromtheorigin. 2.3.2PurePlumeswithandwithoutSwirl.Herewewillobtainanasymptoticanalyticsolutionforthecaseofanaxisymmetricswirlingpureplume.Westartwiththeboundarylayergoverningequations,writtenas, ~uz@~uz @~z+~ur@~uz @~r=1 ~r@ @~r~r@~uz @~r+~g0)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 @~p @~z (2) ~u2 ~r=1 @~p @~r (2) ~uz@~u @~z+~ur~u ~r+~ur@~u @~r=@ @~r1 ~r@ @~r(~r~u) (2) ~uz@~g0 @~z+~ur@~g0 @~r= Pr1 ~r@ @~r~r@~g0 @~r. (2) Theaxialandradialvelocitiesarethenwrittenintermsofthestreamfunctionas, @~ @~r=~r~uz,@~ @~z=)]TJ /F7 11.955 Tf 8.91 0 Td[(~r~ur.(2)Bysetting~uand~g0identicallyequaltozerotheaboveequationsreducetothesamesetofboundarylayerequationsusedbySchlichting[ 45 ]toderivethesimilaritysolutionforthejet,giveninsubsection 2.3.1 .Wenotethatsuchalaminarsolutionwilldependontheaxialandradialcoordinates(~z)and(~r),viscosity(),buoyancyux(~JB)andangularmomentumux(~J).Notethatinthiscaseboththebuoyancyandangularmomentumuxareconservedastheplumeevolvesalongtheaxialdirection,whilethemassandaxialmomentumuxarenotconservedandarenotconsideredascontrolparameters.Thedimensionlessgroupsto 32

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beformedoutofthesevariablesare ~r ~z,~z~J1=2B 3=2and~J1=2B~J 7=2.(2)Thedependentvariables,namelystreamfunction(~ ),buoyancyperturbation(~g0=g(~T)]TJ /F7 11.955 Tf 13.97 2.66 Td[(~Ta)),andangularvelocity(~v)canalsobecombinedwiththecontrolparameterstoformthefollowingdimensionlessgroups: ~ ~J1=2B 5=2,~g05=2 ~J3=2Band~u1=2 ~J1=2B.(2)Byappealingtointermediateasymptotics(Barenblat[ 9 ])thefunctionalformofthesolutioncanthenbeexpressedas: ~ (~r,~z)=5=2 ~J1=2B"~z~J1=2B 3=2#a1"~J1=2B~J 7=2#a2F() (2) ~g0(~r,~z)=~J3=2B 5=2"~z~J1=2B 3=2#b1"~J1=2B~J 7=2#b2G() (2) ~u(~r,~z)=~J1=2B 1=2"~z~J1=2B 3=2#c1"~J1=2B~J 7=2#c2H() (2) wheretheF,GandHarenon-dimensionalfunctionsofthescaledradialcoordinatedenedas =~r ~z"~z~J1=2B 3=2#d1"~J1=2B~J 7=2#d2.(2)Intheabove,theexponentsa1,a2,etc.,areconstantsthatcannotbedeterminedpurelyfromthedimensionlessanalysis.However,weusethehorizontallyintegratedaxialmomentum,energyandangularmomentumequationstoobtaintheadmissiblevaluesoftheseexponentsas a1=1,b1=)]TJ /F7 11.955 Tf 9.3 0 Td[(1,c1=)]TJ /F7 11.955 Tf 10.49 8.08 Td[(3 2,d1=1 2,a2=0,b2=0,c2=1,d2=0.(2)Weseekaperturbativesolutiontotheswirlcomponentofaplume.Inotherwords,wewilltakethepureplumetobetheleadingordervelocityandthermalstructureof 33

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theowandconsidertheswirltobeasmallperturbation.Theeffectoftheswirlistointroduceapressureperturbationwhoseleadingorderscalingcanbeestablishedas~p=~v2~J1=2B~J2)]TJ /F9 7.97 Tf 6.58 0 Td[(7=2~z)]TJ /F9 7.97 Tf 6.59 0 Td[(3.Thispressureperturbationinturnisresponsiblefordrivingasecondaryowandthermalvariationawayfromthepureplumesolution.Fromtheaxialmomentumequationaconsistentsmallquantity()forthisperturbationcanbeestablishedas ="~JB #)]TJ /F9 7.97 Tf 6.59 0 Td[(1~J1=2B~J2 7=2~z3=~J2 ~J1=2B5=2ez3.(2)Intermsofthesolutionforaswirlingplumecanbeexpressedasanasymptoticexpansionas e (z,r)=z[F1()+F2()+] (2) eu(z,r)=eJ1=4BeJ 7=4ez3=2[0+H2()+H3()+] (2) eg0(z,r)=eJB ez[G1()+G2()+] (2) ~p(z,r)=~J1=2B~J2 7=2~z3[0+2()+3()+] (2) Fromthedenitionofthesteamfunctionwend, euz(z,r)=1 er@e @er= eJB !1=2F01 +F02 + (2) eur(z,r)=1 er@e @ez=eJ1=4B1=4 ez1=2F01 2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(F1 +F02 2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(F2 + (2) Notethatbecomeslargeasez!0andcannotbeconsideredsmall.Butthisisconsistentwiththeintermediateasymptoticsolutionsoughthere,whichisnotvalidforverysmall,orverylargeez.TherstordertermsdonotincludepressureorangularvelocityandaresimilartotheassumptionsmadebyYih[ 63 ].Bycarryingthroughthesubstitutionoftherstorder 34

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termsintotheboundarylayerequationsweobtain, F0001)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1F001+)]TJ /F9 7.97 Tf 6.59 0 Td[(2F01+)]TJ /F9 7.97 Tf 6.59 0 Td[(1F1F001)]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(2F1F01+G1=0 (2) G01+PrF1G1=0, (2) whichisthesamesetofequationsfoundinbothYih[ 63 ]andBrandandLahey[ 15 ].Thecorrectboundaryconditionsalongtheaxisoftheplumeare: F1=@ @F01 =G01=0at=0,(2)andthefareldconditionsare F01 =G1=0at!1,(2)whichcorrespondto e (0)=@euz=@er(0)=@eT=@r(0)=euz(!1)=eT(!1)=0.(2)InadditionwerequirefromconservationofeJBthat, 1=2Z10F01G1@.(2)Yih[ 63 ]demonstratedthatforPrandtlnumbersof1or2,thesystemhasanexactsolution.ForexampleforPr=1, F1()=3 22 6p 2+2 (2) G1()=1 31+2 6p 23 (2) ForallPrandtlnumbersthesystemcanbesolvednumericallyaswasdonebyBrandandLahey[ 15 ]. 35

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Wearenowinapositiontoextendthesimilaritysolutiontothecaseofaplumewithasmallamountofswirl.TheleadingordersolutionfortheangularvelocityisobtainedfromtheeumomentumequationwrittentoO().Theresultingordinarydifferentialequationis, F01H2+F1H02+H2F1+2H002+H02)]TJ /F4 11.955 Tf 11.95 0 Td[(H2=0.(2)Theappropriateboundaryconditionsare, H2(=0)=H2(!1)=0,(2)whichcorrespondtotheangularvelocitybeingzerobothalongtheaxisofsymmetryandinthefareld.Wefurtherrequirefromconservationofangularmomentum(eJ)that 1=2Z10F01H2@.(2)Whilethesystemofordinarydifferentialequationsgivenin( 2 )and( 2 )arenonlinearandrequireaNewtoniterationschemeforconvergence,theaboveordinarydifferentialequationforH2()islinearandcanbeeasilysolved.Theradialmomentumboundarylayerequation( 2 )canthenbeappliedtosolvefor2.Wenowcomparethenumericalsolutionsfortheswirlingplumewiththeasymptoticsolutionoftheboundarylayerequations.Comparisonsarenotasstraightforwardasinthecaseofajet,sincethebuoyancyuxandangularmomentumux,althoughexpectedtobeconstant,arenotexplicitlystatedasinputparametersinthenumericalsimulations.Inthepresentcaseofapureplume,onlyaheatsourceisspeciedatez=0,withnomassormomentuminput.Onlysufcientlyabovetheheatsourcetheplumefullydevelopsandtheself-similarsolutionbecomesapplicable.Onlyinthisregime,conservationofbuoyancyuxandangularmomentumuxcanbeexpected. 36

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Figure2-4. NumericalresultsofthevalueofJBandJasafunctionofzforGr=3000andS=0.0,0.3,0.45,and0.6 Fromthesimulationresultsdimensionlessbuoyancyux(JB)andangularmomentumux(J),denedasin( 2 )and( 2 )butusingthecorrespondingdimensionlessvariables,arecomputed.VerticalvariationsintheuxesforGr=3000,forfourdifferentswirlvaluesofS=0.0,0.3,0.45,and0.6areshowningure 2-4 .ByintegratingtheappropriatemomentumequationsitcanbeshownthatJBandJareindependentofz.ItcanbeseeninthegurethatbothJBandJobtaintheirasymptoticvaluewithin3diametersabovethelowerboundary.Figure 2-5 graphstheconstantvaluesofJBandJforplumesaswellasJMandJforjets.Intheplotsolidsymbolscorrespondtobeloworatcriticalswirl,whileopensymbolscorrespondtoabovecriticalswirlandthustocasesofvortexbreakdown.TheasymptoticconstantvalueofJBattainedbytheplumesufcientlyabovetheinletdecreaseswithincreasingswirl.Evenabovecriticalswirlthebuoyancyux 37

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Figure2-5. NumericalresultsofthevalueofJB,JMandJasafunctionofSforplumeswithGr=3000andjetsofRej=100 suppliedtotheplumefromthesourceremainsniteandshowsacontinuoustrend.Correspondinglyinajet,theasymptoticmomentumuxdecreaseswithincreasingswirl.But,incontrasttothepureplume,adrasticchangeincharacterabovethecriticalswirlcanbeobserved.Aswillbeseenbelow,forthecaseofajetwithabovecriticalswirl,mostoftheinjectedowexitsthedomainsidewaysandtheaxialmomentumcoupledtothedownstreamevolutionisnegligibleduetovortexbreakdown.ThiscanalsobeseenintheplotofJ,whereforthecaseofaplumetheangularmomentumuxcontinuestoincreaselinearlywiththeinletswirlmagnitudeevenacrossthecriticalswirlvalue.Whileforajet,theincreaseislinearforsubcriticalswirl,andforabovecriticalswirlvirtuallynoneoftheinjectedangularmomentumisretainedclosetotheaxisfartherdownstream.Figure 2-6 comparesaswirlingplumewithvariousswirlvalues,bothaboveandbelowcriticalswirl,tothesimilaritysolutionderivedabove.Inordertocomparetheaxialvelocity(uz),azimuthalvelocity(u),andbuoyancy(g0)withtheasymptotictheorywerstrescalethesedimensionlessquantitiesobtainedfromthesimulationsasfollows: 38

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uz J1=2BGr1=4,u(z)]TJ /F4 11.955 Tf 11.95 0 Td[(zv)3=2 J1=4BJGr7=8,andT(z)]TJ /F4 11.955 Tf 11.96 0 Td[(zv) JBp Gr,(2)andplotthemalongwithF1=,H2()andG1()obtainedfromthesolutionsoftheODEsinFigure 2-6 .AswewilldiscusslatervortexbreakdowninplumesoccursaboveacriticalswirlvalueSc0.45.Itisinterestingtonotethatthesimilaritysolutionderivedaboveappearstoremainvalidevenforabovecriticalplumes.Thiscontraststhebehaviorinjetsandsuggests,thatforplumes,theowisstillnarrowandtheboundarylayerequationsapply,evenforabovecriticalows.Thisbehaviorisconsistentwiththeobservationsreportedingure 2-5 .Anotherdifferencetonoteincomparingthesimilaritysolutionstojetsandplumesisinthevirtualorigin,zv.Injetszvwasfoundtobehighlydependentonswirl,withalargervalueofzvrequiredforgreaterswirl.Ingure 2-2 itwasfoundtobezv=7.2,8.6and18forswirlvaluesS=0.0,0.3and0.6respectively.However,forplumesthisdoesnotseemtobethecase.Usingtherstorderapproximationofg0( 2 )anapproximationofzvcanbedenedas, zvp GrJBG1(=0).(2)whichisonlyveryweaklydependentonswirl.Thevirtualoriginingure 2-6 iszv=0.7whichisofthesamemagnitudewendusing( 2 )andisindependentofS.Itisoffurtherinteresttonotethatboththeaxialvelocityandtemperatureofaswirlingplumecontinuestocompareverywelltotheleadingorderanalyticsolutions(F1=andG1).Thissuggeststhattheadditionofamoderateamountofinitialswirldoesnoteffectaplumesaxialvelocityortemperatureinthesimilarityregion. 2.4VortexBreakdownFromtheangularmomentumboundarylayerequation( 2 )itisclearthatthepressureminimumisalongtheaxisofthevortex.Sincetheswirlweakenswithincreasingz,forowswithanon-zeroangularmomentumuxanadversepressure 39

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Figure2-6. Comparisonofthenumericalsolutionofswirlingplumeswithasymptoticsimilaritysolutionforvariousvaluesofswirlandattwodifferentheightsz=20,46.(Gr=3000,S=0.3,0.6and1.6.)(a)AxialvelocityandTemperature.(b)Tangentialvelocity. gradientwillformabovetheowinletboundary.Ifthevalueofswirlisabovesomecriticalvalue,theadversepressuregradientwillbestrongenoughforastagnationpointtoformalongtheaxis.Ifthishappenstheassumptionsonwhichtheboundarylayerequationsarebasedwillnolongerbevalidandtheabovesimilaritysolutionswillbeinappropriate.Thisphenomenonisreferredtoasvortexbreakdown.ThevalueofSwherevortexbreakdownbeginstooccurisknownasthecriticalswirlvalue,Sc. 2.4.1CriticalSwirlforJets.ForapurejetRevueltaetal.[ 43 ]andBillantetal.[ 11 ]derivedupperandlowerestimatesforthevalueofSc.Theyassumedajetwithaninitialtophataxialvelocityandinitialsolidbodyrotation.Revueltaetal.[ 43 ]notedthatM,mustbepositiveinorderfortheboundarylayerequationstobeappropriate.SinceMisconservedwecanderivethevalueofMusingtheknownvaluesforuzanduatz=0.Itiseasytoshowfromequation( 2 )that, 40

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pJ(r)=S2(8r2)]TJ /F7 11.955 Tf 11.95 0 Td[(2).(2)Sowehaveforthetophatinletaxialvelocityproleandsolidbodyinlettangentialvelocityproleassumedhere, M=2Z1=20r+S2(8r3)]TJ /F7 11.955 Tf 11.95 0 Td[(2r)dr= 4(1)]TJ /F4 11.955 Tf 11.96 0 Td[(S2).(2)FromtheaboveitisclearthatSmustbelessthen1forMtobepositive.IfS>1,thenaspressureincreaseswithincreasingzandapproachestheambientpressure,theverticalvelocitymustapproachzero,indicatingstagnation.ToobtainalowerlimitforthecriticalswirlBillantetal.[ 11 ]usedtheBernoulliequation.Assumingabovecriticalswirl,astagnationpointistakentoexistonthejetaxis.TheBernoulliequationforthestreamlinealongthejetaxisfromtheinowboundarytothestagnationpointandfromthestagnationpointtothefareldyields, pJ(r=0)+1 2u2J=ps=pa(2)HerethesubscriptsJ,a,andsindicatethevaluesatthejetinlet,ambientandstagnationpointrespectively.Asdenedin( 2 )thenon-dimensionalambientpressureissettozeroandpJ(r=0)=)]TJ /F7 11.955 Tf 9.3 0 Td[(2S2.Fromwhichwegetthecondition, Sc>1 2,(2)fortheexistenceofastagnationpointalongtheaxis.Thisisanecessarybutnotsufcientconditionforastagnationpointtoform.Soforajetwithtophatinitialaxialvelocityandasolidbodyswirlwehave,1=2
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2.4.2ComparisonofVortexBreakdowninJetsandPlumesInjets,tillaswirlvalueof0.7thejetemergingfromtheinletcontinuestoowoutwithoutdisruption.Forlargerswirlthejetbreaksdownimmediatelydownstreamoftheinlet.Ithasbeenwellestablishedintheliterature[ 23 30 51 ]thatinthisregimetwodifferentbistableowpattensarepossible.Inonepossibleowpattern(gure 2-7 )twostagnationpointsformalongtheaxisofsymmetrywitharecirculatingstagnationbubblebetweenthetwostagnationpoints.Ahyperbolicxedpoint(markedH)canbeseeninthehalfplaneandtheseparatrixoriginatingfromthehyperbolicxedpointtowardtheinlet,separatesthepartofinowthatcontinuestoproceedasajetalongtheaxisfromthatpartthatspreadsradially.InthesimulationdisplayedinthegureforReJ=100andS=0.9,theseparatrixfromthehyperbolicxedpointintersectedthebottomboundaryatr=0.37,implyingthatabout45%oftheinowspreadsradially.Thestreamlinesshowningure 2-7 arenotequispacedinordertoclearlydepictthequalitativenatureoftheowandasaresultarenotquantitativelyinformativeastothestrengthofthejet.Forinstance,thestrengthofthejet,measuredintermsofeithermassormomentumux,signicantlydecreasespastthevortexbreakdownabovethestagnationbubble.Postvortexbreakdownthereexistsanotherpossibleowpatternwithonlyonestagnationpointalongtheaxisofsymmetryandwherealloftheinowspreadsradially.Theambientuidabovethestagnationpointisentrainedbytheowandobtainsaveryweaknegativeaxialvelocity.Inrealapplications,suchastornadoes,outowsofvortexchambers,abovethedeltawingsofaircraft,etc,twodifferentstatesofvortexbreakdownhavebeenobservedwithhysteretictransitionsbetweenthemthataredifculttopredict,buttheyhappenabruptlyandareoftendangerous[ 51 ].Inthecurrentnumericalstudy,whichowpatterndominatesishighlydependentoninitialconditions.ThecriticalswirlvalueforinitiationofvortexbreakdowninajethasbeenshowntobeSc,J0.7byRevueltaetal.[ 43 ]andthepresentresultsareincompleteagreement.Itisnowofinteresttoestablishasimilarcriterionforvortexbreakdowninapureplume 42

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Figure2-7. Streamtracesplottedonther)]TJ /F4 11.955 Tf 11.96 0 Td[(zplaneforanabovecriticalswirljet.(Re=100,Ri=0,S=0.9). andcompare/contrastthenatureoftheresultingow.Figure 2-8 showsthestreamlinesonaverticalhalf-planepassingthoughtheaxisoftheplume(Gr=3000)forswirlS=0.0(left)andS=0.9(right).AstagnationbubblecanbeobservedclosetothebottomboundaryneartheoriginforS=0.9.ThisbubblebeginstoformforplumeswithswirlgreaterthanS=0.45anditgrowsinsizewithincreasingswirl.Notethatforthecaseofapureplumetheuidvelocityalongtheaxialdirectionatthebottomboundaryisspeciedtobezeroandtheplumedevelopmentisentirelybyentrainmentofambientuid.Abetterindicationofonsetofbreakdowncanbeobtainedbyinvestigatingwallnormalgradientofradialvelocityplottedalongtheradialdirection.Figure 2-9 showsthisquantityplottedfordifferentvaluesofswirl.TheowclosetothebottomboundaryremainspointedinwardstowardstheoriginforS<0.45,whilearegionofradialoutwardsow(positiveshearstress)developsclosetotheaxisforS>0.45,indicatingtheformationofastagnationbubble. 43

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Figure2-8. Streamtracesplottedonther)]TJ /F4 11.955 Tf 11.96 0 Td[(zplaneforaplume(Gr=3000)withnoswirl(left)andaswirlvalueofS=0.9(right.) Figure2-9. @~ur=@ezatthelowerboundaryofaplumeforvariousvaluesofswirl.Filledinsymbolsindicatesowswithlessthencriticalswirl.Thoseplottedwithopensymbolsrepresentvortexbreakdownasindicatedbythezerocrossingoftheshear. 44

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Althoughthetopologyoftheowclosetotheswirlingheatsourcedramaticallychangesabovethecriticalswirl,unlikeinthecaseofaswirlingjet,thecharacteristicsoftheplumewellawayfromthebottomboundarycontinuestoevolvesmoothlyevenbeyondvortexbreakdown.Ascanbeseeningure 2-5 boththebuoyancyandangularmomentumuxescontinuetodecreaseandincreasesmoothly,respectively,withoutanysharpvariationwiththeinitiationofvortexbreakdownatSc,p=0.45.Forabovecriticalswirl,arecirculatingregionofswirlingmassofuidsitsontopoftheswirlingheatsource.Theplumedevelopmenthappensabovethisswirlingheatedmassofuid,insteadofdirectlyovertheheatedplate.Thus,theimpactofswirlontheverticaldevelopmentoftheplumesufcientlyabovethesourceremainscontinuous. 2.4.3CriticalSwirlforPlumesThecriticalswirlfortheplumecanbecomparedwiththatforthejet.Fromequation( 2 )thenon-dimensionalcenterlinevelocityofafullydevelopedplumecanbeexpressedas uz(r=0)=J1=2BGr1=4F01 =0.(2)Fromgure 2-5 athenon-dimensionalbuoyancyuxatcriticalswirlcanbetakentobe0.049andfromthesolutionof( 2 )-( 2 )presentedingure 2-6 weobtain[F01=]=0=0.37.ThesevaluesalongwithGr=3000yieldanon-dimensionalcenterlinevelocityof0.61forthefullydevelopedplume.Ifwenowproposetheswirltoaxialvelocityatbreakdowntobethesameforajetandaplume,thewecanuseSc,J=0.7givenbyRevueltaetal.[ 43 ]andestimatethecriticalswirlforaplumetobe0.43,whichissurprisinglyveryclosetothecriticalvalueobservedfortheplumeinthepresentsimulations.Thebehavioroftheswirlingplumecanbefurtheranalyzedbyintegratingtheaxialmomentumequationalongahorizontalplanetoobtain @ @z(JM+Jp)=2Z10rg0dr,(2) 45

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wherethenon-dimensionalpressureforceisdenedasJp(z)=2R10rp0dr.Itcanbeseenfrom( 2 )thatJM+Jp=M.Thus,intheabsenceofbuoyancyeffectMisconservedalongtheaxialdirectionandRevueltaetal.[ 43 ]andBilliant[ 11 ]establishedSc,J<1ascriterionforvortexbreakdowninajet.Incaseofaswirlingpureplumethenon-dimensionalpressureforceisnegativeatz=0,butapproacheszerowithincreasingzasthecircumferentialvelocityoftheowdecays.Forasolidbodyrotationofthecircularheatsourceatz=0,thelimitingvaluesofthepressureforceare Jp(z=0)=)]TJ /F3 11.955 Tf 10.49 8.09 Td[( 4S2andJp(z!1)!0.(2)Thustheeffectofswirlistodecreasetheowmomentumawayfromthesource.Incontrast,thebuoyancyeffectontherighthandsizeremainspositiveandcontributestoasteadyincreaseinowmomentum.Therighthandsidecanbeexpressedas 2Z10rg0dr=2J1=2B Gr1=4Z10G1d.(2)NotethatJB,andtherighthandsideof( 2 ),isidenticallyzeroatz=0duetozeromassuxboundaryconditionatthesource,butquicklyreachestheasymptoticconstantvalue.Theconditionforowseparationincaseofapureplumecanthenbeexpressedintermsofthefollowing, @Jp @zz!0>2Z10rg0dr.(2)FromthedenitionofJpandtheleadingtermoftheasymptoticpressureexpansion( 2 )itcanbeshownthat @Jp @z=)]TJ /F7 11.955 Tf 9.3 0 Td[(4z)]TJ /F9 7.97 Tf 6.59 0 Td[(3GrJ2Z102d(2)Recallingfromgure 2-5 thatJwasfoundtobeJS,whereisaconstantforagivenGrashofnumber,itiseasytoshowthatvortexbreakdownwilloccurif, 46

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S>1 )]TJ /F7 11.955 Tf 10.49 8.08 Td[(1 2R10G1d R102d1=2J1=4BGr)]TJ /F9 7.97 Tf 6.59 0 Td[(5=8z3=2(2)TheintegralsofG1and2areevaluatedtobeapproximately0.946and)]TJ /F7 11.955 Tf 9.29 0 Td[(9.2610)]TJ /F9 7.97 Tf 6.59 0 Td[(4respectively.Intheaboveequationzmustbetakentobedistancebetweentheinletandthevirtualorigin(i.e.,zv).Iftheestimateforthevirtualorigin( 2 )derivedfromthetemperatureproleisusedforzvandthebuoyancyuxisobtainedfromgure 2-5 forthecriticalswirlvaluefoundinthesimulations(Sc=0.45.)tobe0.049,areusedin( 2 )toevaluatethecritialswirl,thenperhapsfortuitously,theaboveestimateisinverygoodagreementwiththesimulationresults.SinceScinjetsisindependentofReJitissurprisingtondadependencyonGrintheaboverelation.AssuchhowchangeswithGrisofinterestanddeservesfurtherstudy. 2.5BuoyantJetsandInjectedPlumesDuetothedifferenceinthebehaviorofjetsandplumesitbecomesofinteresttostudythebehaviorofjetsperturbedwithasmallamountofbuoyancyandplumesperturbedwithasmallamountofinow.SimulationswereranforjetswithRichardsonnumbersof0.05,0.1and0.3,andofplumeswithaninletvelocityof0.05and0.1,withvariousamountsofswirl.Asummeryoftheresultsispresentedintables 2-2 and 2-3 .Sincethecenterlineaxialvelocityinthesimilarityregionoflaminarplumesisindependentofzandthatoflaminarjetsdecaysasz)]TJ /F9 7.97 Tf 6.58 0 Td[(1,plumelikebehaviordominatesinthefareld.AssuchJMisnolongerconservedinbuoyantjetsevenforverylowRichardsonnumbers.However,thequantity@JM=@zisconservedinalljetsandplumes.Thiscaneasilybeseenfrom( 2 )wheretherighthandsideisconservedandJPdecaystozerosufcientlyfarawayfromz=0,incaseofnon-zeroswirl.Asoneexpectsincreasingbuoyancyinjetsorincreasinginowinplumesincreases@JM=@z.Increasingswirlhasverylittleeffecton@JM=@zineitherbuoyantjetsorinjectedplumesuntiltheamountofswirlbecomescritical.Forabovecriticalswirlincreasedswirlleadstoaslightdecreasein@JM=@z. 47

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AsinplumesJBisconservedinthefareldofbuoyantjets.SurprisinglysimulationsofjetswithallthreeRichardsonnumbersresultedinapproximatelythesameamountofbuoyancyux,JB0.73,whichisanorderofmagnitudelargerthanseeninpureplumes.Thissuggeststhat0.73representsamaximumbuoyancyuxwhichaplumewillasymptoticallyapproachasuJisincreasedandtheplumetakesonmorejet-likebehavior.IndeedasuJisincreasedinplumesJBincreases.IncreasingswirlhasaveryweakeffectonJBineitherbuoyantjetsorinjectedplumeswiththeeffectbeingslightlystrongerininjectedplumes.Thenatureofvortexbreakdowninabovecriticalswirlowsisgreatlyeffectedbytheadditionofbuoyancytoajetorinjectiontoaplume.Figure 2-10 displaysdiagramsofthestreamtracesplottedonther)]TJ /F4 11.955 Tf 12.08 0 Td[(zplaneforabuoyantjetandaninjectedplumewithS=0.9,andRi=0.05anduJ=0.1respectively.Theseguresshouldbecomparedtogures 2-7 and 2-8 whichsimilarlyshowthestreamtracesofapurejetandpureplumewithS=0.9respectively.ItcanbeseeninthegurethattheadditionofarelativelysmallamountofbuoyancyRi=0.05greatlyreducesthesizeoftherecirculatingstagnationbubble.Theadditionofevenasmallamountofbuoyancycauseshysteresistonolongeroccurandonlyonevalidsolutionexistsforthisow.Theabovecriticalswirljetowpatternwithnegativeaxialvelocityalongthecenterlineofthefareldisnolongerpossible.Addingbuoyancytoapurejetorinowtoapureplumeincreasestheamountofswirlnecessaryforvortexbreakdowntooccur.Inlightofthepreviousdiscussioninsection 2.4.1 regardingtheBernoulliequationthereasonforthisshouldbeclear.Inabuoyantjetoraninjectedplume,theadversepressuregradientcausedbytheswirlinguidmustovercomeboththeinitialaxialvelocityandthebuoyancyforceinorderforastagnationpointtoformalongthecenterline.Table 2-4 showstheapproximatecriticalswirlvaluesobservedfromthesimulations. 48

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Figure2-10. Streamtracesplottedonther)]TJ /F4 11.955 Tf 11.96 0 Td[(zplaneforabuoyantjet(Re=100,Ri=0.05,S=0.9)andaninjectedplume(Gr=3000,uJ=0.1,S=0.9). Table2-4. Approximatevalueofcriticalswirl,ScforvariousbuoyantjetsandinjectedplumesJetsPlumesRiScuJSc 0.00.70.00.450.050.90.050.650.10.90.10.75 Ingeneralincreasingswirlresultsinanincreaseinthesizeoftherecirculatingstagnationbubble,andtheadditionofbuoyancyorinitialmomentumuxdecreasesthesizeofthebubble.Tables 2-2 and 2-3 indicatesthelocationoftheupperandlowerstagnationpointsforvariousabovecriticaljetsandplumes,whichgivesasenseofthesizeofthestagnationbubble.InthetabletheupperandlowerstagnationpointsarelistedasUSPandLSPrespectively. 2.6SummaryTheestablishedsimilaritysolutionsforjets,swirlingjetsandnon-swirlingplumeswerediscussed.Throughanasymptoticapproachthesolutionforthebuoyantplumewasextendedtothecasewhereasmallamountofangularmomentumuxispresentin 49

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theow.Thesimilaritysolutionwascomparedtonumericalsimulationandfoundtobevalidevenforplumeswithsubstantialabovecriticalswirl.Inparticulartheaxialvelocityandtemperatureproleswereobservedtobelittleinuencedbyswirlandcomparedwellwiththeclassicalsimilaritysolution.Thiscontrastswiththesimilaritysolutionfortheswirlingjet,whereswirlhasastrongeffectonthejetow.ForpurejetsthetheoreticalupperandlowerboundsforthevalueofcriticalswirlScwereestablishedbyRevueltaetaalandthevalueofScfoundinthecurrentDNSstudyisinagreementwiththeirresults.UsingtheoreticalargumentsasimilarcriterionforanestimateofScforplumeswasderived.Thebehaviorofjetsandplumeswithabovecriticalswirlwascompared.Inpurejets,oncevortexbreakdownhasoccurredtheassumptionofanarrowowisnolongervalidandalmostalljetlikebehaviorislost.Twodifferentowstructuresarepossibleforjetswithabovecriticalswirl.Theowpatternthatresultsinasimulationdependedontheinitialcondition.Inplumesevenalargeamountofswirldoesnotgreatlyeffecttheoverallbehaviorinthefareld.Thevalueoftheconservedquantitybuoyancyux,JB,decreaseswithincreasedswirl,buttheoverallbehaviorremainsunchanged.Byaddingasmallamountofbuoyancytoajetorasmallinowtoaplumetheamountofcriticalswirlforvortexbreakdowncouldbeincreased.ThiscaneasilybeexplainedintermsoftheBernoulliequationappliedalongthecenterline.Further,jetswithevenaverysmallamountofbuoyancynolongerundergoeshysteresisandbecomestablewithonlyonepossibleowpattern.TheaboveconclusionshavebeenfoundeitheranalyticallyusingtheboundarylayerequationsornumericallyusingtheBoussinesqgoverningequations.Theinterestingndingsofthepaperwarrantfurtherexperimentalvalidation. 50

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CHAPTER3ADNSSTUDYOFHIGHERORDERSTATISTICSINATURBULENTROUNDJET 3.1BackgroundAxisymmetricturbulentfreejetsoccurofteninengineeringapplicationsandenvironmentalows.Assuch,jetsareoftenusedasanarchetypeinthestudyofaxisymmetricfreeshearlayers.Manyresearchershaveexaminedthisproblemusinganalytical,experimental,andnumericaltechniques[ 5 26 28 39 45 56 ].UntilrecentlymostnumericalstudieshaveusedLargeEddySimulations(LES)orothermodelstoapproximatetheturbulentbehaviorofthejetow.MorerecentlyithasbecomepossibletoperformDirectNumericalSimulations(DNS)ofturbulentjetswherenomodelingisused.Mostofthesestudiesfocusonthesecondorderstatistics,theformationofinstabilities,andvorticalstructuresintheneareld[ 7 12 14 31 ].VeryfewfocusonthefareldbehaviororcapturethethirdordermomentsneededtoexaminethecompletebalanceofturbulentkineticenergyandReynoldsstress.Schlichting[ 45 ]iscreditedforsomeoftheearliestworkonturbulentshearowsincludingtheaxisymmetricturbulentjet.Hederivedasimilaritysolutionfromtheboundarylayerequationsanddeterminedthattheaxialvelocityofanaxisymmetricturbulentfreejetdecaysasz)]TJ /F9 7.97 Tf 6.58 0 Td[(1awayfromthesourceWygnanskiandFiedler[ 61 ]examinedanairjetataReynoldsnumberofReJ=105usingHot-WireAnemometry,(HWA),tomeasurevariousaspectsoftherst,secondandthirdorderstatistics.Later,usingmoremoderntechniques,Husseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]performedbothHWAandLaser-DopplerAnemometry(LDA)experimentsandpresenteddetailedresultsofthesecondandthirdorderstatistics 1Thecontentsofthischapterhavebeensubmittedtothejournal,PhysicsofFluidsandarewaitingreview. 51

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ofturbulentkineticenergybalanceandothersecondordertransportequations.Bothpapersareconsideredseminalworksintheexperimentalstudyofturbulentroundjets.VariousresearchershaveexaminedturbulentjetsnumericallyusingLEStechniqueswheretheeffectofturbulenceintheunresolvedscalesismodeledinordertoapproximateitseffectontheresolvedscales.OlssonandFuchs[ 38 ]examinedthetransitionfromlaminarownearthejetsourcetoturbulentow.BogeyandBailly[ 13 ]examinedtheinuenceoftheReynoldsnumberonthedevelopmentoftheow.TheirstudysuggeststhatselfsimilarityisachievedinashorteraxialdistanceawayfromthesourceforlowerReynoldsnumberjets.AkselvollandMoin[ 3 ]performedsimulationsofco-annularjetsinaconnedspaceinordertostudycoaxialjetcombustion.Kung[ 28 ]preformedanLESusinganapproximatedeconvolutionmodelandcomparedhisresultswithbothexperimentsandDNS.HefoundthattheLESsimulationoverpredictedtheturbulentintensitiesinthetransitionalregion.Ascomputerprocessingbecomesbothmorepowerfulandeconomical,directnumericalsimulationsbecomemoretractable.Boersmaetal.[ 12 ]examinedthemeanowandsecondorderstatisticsofjetswithdifferentinletvelocityproles.Lubbersetal.[ 31 ]examinedtheself-similarityoftheconcentrationofapassivescalar.BabuandMahesh[ 7 ]comparedtheeffectofentrainmentofjetsemanatingfromanoriceinaatsurfacewiththoseemanatingfromapipeorchimney.BrancherChomazandHuerre[ 14 ]focusedontheeffectofbreakingrotationalsymmetrybysubjectingtheirsimulatedjetstostream-wiseandazimuthalperturbations.Wangetal.[ 56 ]examinedthethirdordermomentsofafreejet.MuppidiandMahesh[ 37 ]studiedaroundturbulentjetwithacrossowandreportedontheturbulentkineticenergybalance.InthiscurrentworkweperformaDNSofaturbulentjetwithaReynoldsnumberofReJ=2000.Thesimulationisrunforover1000timescales,longenoughforthethirdordervelocitymomentstoconverge.First,secondandthirdorderstatisticsarecomparedwithpreviousexperimentalstudies.Theturbulentkineticenergybalance 52

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Figure3-1. Schematicofthecomputationaldomain. andalltermsoftheReynoldsstresstransportequationsarecalculatedanddiscussed.Vorticalstructuresarevisualizedbythecimethod[ 64 ]anddiscussedalongwithentrainmentofambientuidintothejet.Theonedimensionalenergyspectraintheazimuthaldirectionwascalculateddirectlyandisalsodiscussed.Wewillbegininsection 3.2 wherewewilldiscussthenumericalmethodologyusedinthisstudy.Ourresultswillbepresentedinsection 3.3 .Finalconclusionsandfutureworkwillbediscussedinsection 3.4 3.2MethodologyWeassumeajetemergingfromacircularpipeofdiameter,D.Fluidexitingthecircularpipeentrainsambientuidabove.Figure 3-1 showsthecomputationaldomainwhichisarectangularregionwithdimensionlesslength,widthandheightofLx,LyandLz.Thequantities~r,ezandedenotetheradial,axialandazimuthaldirectionalvectorsinthecylindricalcoordinatesystem.Thequantitiesex,ey,andezarethedirectionalvectorsintheCartesiancoordinatesystem.Theoriginofthecoordinatesispositionedatthecenterofthejetinlet.Thevelocityvectoriseuandepisthepressure.The`tilde'denotesdimensionalvariables.Theuidproperties,suchaskinematicviscosity,,andtheconstantdensity,0,arekeptconstantforpurposesofthisanalysis.Thevariablesare 53

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scaledbythejetinletdiameter,D,andthejetinletvelocity,UJ.Thesubscript`J'denotestheconditionsatthejetinlet.Theresultingdimensionlessvariablesarexi=exi=D,ui=eui=UJ,p=ep=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(0U2J,andt=UJet=D.Atophatinitialvelocityprole,UJisassumed.Withappropriatenon-dimensionalization,theincompressiblegoverningequationsare, ru=0(3) @u @t+uru=rp+1 ReJr2u,(3)wheretheReynoldsnumberisdenedas, ReJ=UJD .(3)Whendiscussingsecondandthirdordertermswewillusetheconventionthataveragedrstorderquantitiesarecapitalizedandperturbationsareprimed.Thus,Reynoldsdecompositionofauctuatingvariablecanbeexpressedas, x=hXi+x0(3)ThegoverningequationsweresolvednumericallyonaCartesiangridwithgridpointsclusteredclosetothejetinlet.Thedomainsizewaschosensothateffectsduetoconnementinthelateraldirectionareminimized.WeconsultedpreviousDNSstudieswheresimilarorlargerReynoldsnumberswereexamined,suchasBoersmaetal.[ 12 ]andAissiaetal.[ 2 ]astwoexamples,todeterminethesizeofourcomputationaldomainandgridresolution.Thedomainsizeandgridresolutionusedwere252545(LxLyLz)and251251550(NxNyNz),respectively. 54

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Attheinowboundary,atop-hatproleisusedfortheaxialvelocity,uz,forr=p x2+y2<1=2.Randomuctuationsrangingbetween0.05UJwereaddedtothetophatinletvelocityproleinordertospeedupthetransitiontofullyturbulentbehavior.Thelowerboundaryoutsidethejetinlet,r1=2,isconsideredasano-slipwall.Weallowtheuidtofreelyenterorleavethelateralboundaries(x=1 2Lx,y=1 2Ly),byapplyingNeumannboundaryconditions.Aconvectiveboundaryconditionisappliedtotheaxialvelocityatthejetoutlet(z=Lz).Asecond-orderaccuratenitevolumeschemeisemployedforthespatialdiscretizationofthegoverningequationsonanon-staggeredgrid.Afractionalstepmethodisusedfortimeadvancement.Intheadvection-diffusionstep,thenonlineartermsaretreatedexplicitlyusingasecond-orderAdams-BashforthschemeandthediffusiontermsaretreatedimplicitlywiththeCrank-Nicholsonscheme.ThepressurePoissonequationissolvedasapressurecorrectionstep.Analdivergence-freevelocityisobtainedateachtime-stepwithapressurecorrectionstep.MPIarchitecturewasusedinordertoenableparallelprocessing.Thesimulationwasrunon32processorsutilizingtheHighPerformanceComputer(HPC)centerattheUniversityofFlorida. 3.3ResultsAReynoldsnumberofReJ=2000wasselectedinordertobelargeenoughtoexhibitturbulentbehaviorbutsmallenoughastostillbetractablebyDNS.Thesimulationranforover1.3105timestepsorapproximately1000timeunits,(=D=UJ)beyondthetimeittookthesimulationtobecomestatisticallysteadystate.Averagesofturbulentvariableswerecalculatedazimuthallyaswellastemporally.RunningthesimulationoverthisnumberoftimestepsallowedforconvergenceofthethirdordermomentsnecessaryformeasuringturbulentdiffusionofturbulentkineticenergyandReynoldsstress. 55

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3.3.1FirstandSecondOrderStatisticsFigure 3-2 depictsthedecayofcenterlineaxialvelocityofthejetinthecurrentstudycomparedtoexperimentalstudiesbyHusseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]andapreviousDNSstudyperformedbyKung[ 28 ].Ascanbeseeninthegurethereciprocalofthecenterlinevelocityscaleslinearlywithrespecttoz.HeretheconstantcoefcientofdecayisfoundtobeCd=5.4whereCdcanbedenedas, Uc(z)Cd (z)]TJ /F4 11.955 Tf 11.96 0 Td[(zv).(3)ThiscompareswellwithotherpastDNSstudiessuchasWangetal.[ 56 ]whofoundCd=5.5andexperimentalstudiessuchasHusseinetal.[ 26 ],Cd=5.8,WygnanskiandFiedler[ 61 ],Cd=5.4andPanchapakesanetal.[ 39 ],Cd=6.06.Theterm,zvinapproximation( 3 )isthevirtualoriginwhichwasfoundtobezv=1.3.Husseinetal.[ 26 ]thisdatawithzv=4whereasPanchapakesanandcolleagues's[ 39 ]datadidnotrequireavirtualorigin,(zv=0).Theradialprolesofthemeanaxialandradialvelocityattwodifferentaxiallocations,z=15and25,areplottedingure 3-3 andcomparedtopreviousexperimentalstudies.Heretheradialcoordinateis, =r z)]TJ /F4 11.955 Tf 11.96 0 Td[(zv.(3)TheaxialvelocitycorrespondswelltoaGaussiancurvet, hU()i hUz,ci=exp )]TJ /F4 11.955 Tf 9.3 0 Td[(ln(1 2) 21=22!,(3)where,1=2,thejethalfwidthintermsofthesimilaritycoordinate,isfoundtobe1=2=0.094.ThisisthesamevaluethatwasmeasuredbyHusseinetal.[ 26 ],andsimilartothevalueobtainedbyPanchapakesanetal.[ 39 ],1=2=0.096.ThemaximumpositiveradialvelocityintheinteriorofthejetisfoundtobeapproximatelyhUri+max=hUz,cit0.02.Thisisslightlylargerthan0.018found 56

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Figure3-2. Initialinletvelocity,UJ,overmeancenterlineaxialvelocity,hUz,ci,ofcurrentstudy(lledincircles)comparedtoHusseinetal.[ 26 ](solidline),Panchapakesanetal.[ 39 ](dashedline)andKung[ 28 ](opensquares). Figure3-3. Comparisonofcurrentstudy'stimeaveragedvelocityeldattwodifferentaxialdistances,z=15,20withcurvetsofexperimentalresultsbyHusseinetal.[ 26 ](solidlines),Panchapakesanetal.[ 39 ](dashedlines)andWygnanskiandFiedler[ 61 ](dash-dotlines). 57

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Figure3-4. Comparisonofturbulentintensityalongthejetaxisfoundinthecurrentstudy(solidlines)withthatfoundinpreviousDNSstudybyKung[ 28 ](dashedlines)andanexperimentalstudybyAmielhetal.[ 5 ](symbols). byPanchapakesanetal.[ 39 ]or0.015foundbyWygnanskiandFiedler[ 61 ].ThemaximuminwardradialvelocityofentraineduidenteringthejetisfoundtobehUri)]TJ /F5 7.97 Tf 6.59 0 Td[(max=hUz,cit)]TJ /F7 11.955 Tf 9.3 0 Td[(0.026andisfoundat=0.23slightlyfurtherawayfromthejetaxisandslightlylargerinmagnitudethanmeasuredbyPanchapakesanetal.[ 39 ]andWygananskiandFiedler[ 61 ]whoagreedonavalueofhUri)]TJ /F5 7.97 Tf 6.59 0 Td[(max=hUz,ci=0.025locatedat=0.21.Figure 3-4 comparestheevolutionoftheaxialandradialcomponentsoftheturbulentvelocityintensityalongthejetaxiswithapreviousDNSstudybyKung[ 28 ]andanexperimentalstudybyAmielhetal.[ 5 ].TheDNSstudiespredictrelativelyconstantturbulentintensityabovethetransitionalregion,whereastheexperimentalresults,wherelargerReynoldsnumberswereexamined,suggestsamoregradualapproachtosomeasymptoticvalue.ThisfurthercorroboratesthestudyofBogeyandBailly[ 13 ]whoalsosuggestedthatlowReynoldsnumberturbulentjetsdevelopmorequicklyandreachsimilarityinashorteraxialdistanceawayfromtheirsourcethanjetswithlargerReynoldsnumbers.Indeed,Amielhetal.[ 5 ]foundthatajetwithaReynoldsnumberof 58

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ReJ=7000approachedaconstantvalueofaxialturbulentintensitymuchfasterthanajetwithaReynoldsnumberofReJ=21,000.InthecurrentstudyrandomperturbationswereaddedtotheinitialtophataxialvelocityinordertospeedupthetransitiontoturbulenceandtomoreaccuratelysimulatetheinletconditionsofAmielhetal.[ 5 ].Itcanbeseeningure 3-4 thatinthepresentsimulationsthisinitialrandomperturbationdiesoutratherquickly.Kung[ 28 ]addressedtheperturbationinhisDNSsimulationbyincorporatinganinowgeneratorwhichsimulatedaturbulentpipeowatthejetsource.Assuch,theturbulentintensityreportedbyKung[ 28 ]issimilartothatfoundintheexperimentalstudyofAmielhetal.[ 5 ].TherandomperturbationsinputinthecurrentstudyisnotsimilartoexperimentalorKung's[ 28 ]resultsnearthesource.However,thetwonumericaltechniqueshavesimilarresultsfurtherawayfromthesource,whichistheareaofinterestinthecurrentstudy.Figure 3-5 comparestheradialprolesoftheturbulentintensitycomponentsscaledbythemeanaxialvelocityatthecenterlinewithpreviousstudies.Theproledatafromthecurrentstudyisatadistance,z=15,awayfromthejetinlet.ThepresentresultscomparewelltheexperimentalresultsofAmielhetal.[ 5 ],Husseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]aswelltheDNSresultsofKung[ 28 ].Asnotedbypreviousresearchers,Wangetal.[ 56 ]forexample,themaximumofboththeaxialReynoldsnormalstresshu02ziandtheReynoldsshearstressterm,hu0zu0riarenotalongtheaxis,butratherataradialdistanceslightlylessthenthejet'shalfwidthwherethereisastronginteractionbetweenthejetandtheambientuid.Panchapakesanetal.[ 39 ]noticedthatthisisalsotrueofthetangentialReynoldsnormalstressterm,hu02iandthiscanalsobeseeninthecurrentstudy.Ifthestreamwisepressuregradientcanbeignoredthenitiseasytoshowthatthattheinitialmomentumux, MJ=2ZD=20u2zrdrz=0,(3) 59

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Figure3-5. ComparisonoftheradialprolesofReynoldsstresstermswithpreviousstudies.HeretheresultsofHusseinetal.[ 26 ](solidthinlines)andPanchapakesanetal.[ 39 ](thindashedlines)arerepresentedbyaCurveFits(CF)oftheirrespectiveexperimentaldata.TheresultsofAmielhetal.[ 5 ](circles)ReJ=21,000arerepresentedbysymbols.BoththecurrentDNSstudy(solidlines)andtheDNSstudyperformedbyKung[ 28 ](dashedlines)arerepresentedbythicklines. 60

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isconserved.However,Husseinetal.[ 26 ]suggestedthatthestreamwisepressuregradientshouldnotbeignoredandbyusingthecross-streammomentumequationtoeliminatethepressuretermtheyderived, MJ=M(z)+,(3)whereM(z)isreferredtoasthemomentumintegralandisdenedas, M(z)=2Z10hUzi2+hu02zi)]TJ /F7 11.955 Tf 20.45 8.09 Td[(1 2)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(hu02ri+hu02irdr,(3)and,=2)]TJ /F7 11.955 Tf 10.49 8.09 Td[(1 2@ @zZ10(hUzihUri+hu0zu0ri)r2drz0+1 2Z10hUri2rdrz0+21 2Z101 ReJr2hUrir2drz0+Zz0Z101 ReJr2hUzirdrdz0. (3)Byacombinationofscalinganalysisandusingthesimilaritysolutiontosolvefortheintegralsinequation 3 ,Husseinetal.[ 26 ]determinedthatisnegligiblecomparedtoMJ.ThereforeitwassuggestedthatM(z)MJisanimportanttesttoconrmtheaccuracyofanymodelofanaxisymmetricjetinaninniteenvironment.ThecurrentstudysupportsboththesuggestionthatthestreamwisepressuregradientshouldbeconsideredandthatisnegligiblecomparedtoMJ.InthecurrentDNSwheretheinitialtophataxialvelocitywassettouz=1withrandomperturbationsof5%,theinitialmomentumuxisMJ=4.Figure 3-6 displayseachtermofequation 3 throughouttheentiredomainandcomparesM(z)toMJ.M(z)isfoundtobewithin5%ofMJthroughoutthedomain.Aftertheowbecomesfullyturbulentforapproximatelyz>10,thecomparisonimprovessothatM(z)iswithin1%ofMJ.Themeanowmomentumuxaccountsfor93%ofthetotalmomentumintegralinthefullyturbulentregion.Thetotalmomentumuxincludingthemeanowanductuating 61

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Figure3-6. Contributionstothetotalmomentumintegralbyvariousterms. Figure3-7. Thirdorderstatisticsofcurrentstudy(SolidLines)comparedtoexperimentalstudiesperformedbyHusseinetal.[ 26 ](Exp.-squares,CF-thinsolidlines)andPanchapakesanetal.[ 39 ](circles)andaDNSstudybyWangetal.[ 56 ](dashedlines),normalizedbyhUz,ci3. velocityintheaxialdirectionisapproximately10%largerthanthemomentumintegral,M(z),inthesameregion. 3.3.2ThirdOrderStatisticsDerivativesofthirdordermomentsplayadominaterollintheturbulentdiffusionofturbulentkineticenergyandReynoldsstress.Figure 3-7 comparestheradialproles 62

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ofthethirdordermomentsofthecurrentstudywiththeexperimentalmeasurementsofHusseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]andtheDNSstudybyWangetal.[ 56 ].Termsincludingoddpowersofu0willconvergetozeroandarenotpresented.Husseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]didnotreportonthehu0ru02itermwhichplaysasignicantroleinthediffusionoftheturbulentkineticenergyaswellasthehu0icomponentofReynoldsstress.hu0zu02ri,hu0ru02zi,andhu0zu02iarenegativenearthecenterlineofthejet.AsdiscussedinPanchapakesanetal.[ 39 ],thishasasignicantimpactontheturbulenttransportofkineticenergy.hu0ru02iwhichisresponsiblefortheradialuxofthetangentialcomponentofkineticenergy,doesnothavethisfeatureandwasfoundtobenearzerobutpositivenearthejetcenterline.ThereissomedisagreementbetweentheresultsofHusseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]inthehu03ziandhu03riterms,whereinbothcasesHusseinetal.reportsasignicantlylargermagnitude.BoththecurrentstudyandtheDNSstudybyWangetal.[ 56 ]comparewellwithPanchapakesanetal.[ 39 ]inthecaseofthehu03zitermbutagreebetterwithHusseinetal.[ 26 ]inthecaseofthehu03riterm. 3.3.3TurbulentKineticEnergyBalanceByperformingReynoldsdecompositionofthemomentumequations,subtractingofftheRANSequations,takingthedotproductwiththeuctuatingvelocityvectorandtakingtheaverage,thetransportequationforturbulentkineticenergy,k,canbederivedas:0=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(hUzi@k @z+hUri@k @r)]TJ /F8 11.955 Tf 11.96 16.86 Td[(1 r@ @r(rhp0u0ri)+@ @zhp0u0zi)]TJ /F8 11.955 Tf 11.95 16.85 Td[(hu0zu0ri@hUzi @r+@hUri @z+u02z@hUzi @z+u02r@hUri @r+hUri rhu02i)]TJ /F8 11.955 Tf 11.95 16.86 Td[(1 r@ @rr 2q2u0r+1 2@ @z\012q2u0z)]TJ /F8 11.955 Tf 11.95 20.44 Td[("1 ReJh@u0i @xj2i#. (3) 63

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Equation 3 isalsoreferredtoastheturbulentkineticenergybalance.Here k=1 2u02z+u02r+u02,(3)istheturbulentkineticenergyand, hq2uii=u02zu0i+u02ru0i+u02u0i.(3)Thevetermsinbracketsrepresentthecontributionofadvection,pressurediffusion,production,turbulentdiffusionanddissipationrespectively.ThecontributiontoturbulentdiffusionfromtheviscoustermisnegligibleduetotherelativelylargeReynoldsnumberusedinthesimulationandisnotincludedinequation( 3 ).ThissimplicationisalsomadefortheReynoldsstresstransportequations( 3 through 3 ),denedbelowinsubsection 3.3.4 .Figure 3-8 comparestheturbulentkineticenergybalanceofthecurrentDNSsimulationwiththesixthorderpolynomialbestcurvetsoftheexperimentalresultsofPanchapakesanetal.[ 39 ]andShiri[ 49 ].Thebalancefromthecurrentstudywascalculatedatanaxialdistanceofz=15abovethejetinlet.TheresultsfromShiri[ 49 ]areactuallyforajetwithamoderateamountofswirl,butasdiscussedinhispaper,andascanbeseenbythegure,theadditionofamoderateamountofangularmomentumattheinletdoesnotaffectthestatisticsinthesimilarityregion.Previousexperimentalstudieswereunabletocalculateeachtermofequation 3 directlyandmadevariousassumptionsregardingthedissipationandpressuretermsinordertoclosetheturbulentkineticenergyequation.Husseinetal.[ 26 ]discussedvariousalternativesforapproximatingthedissipationtermandeventuallysuggestedthattheapproximationbasedonajetbeinglocallyaxisymmetric, dissipationt1 ReJ"5 3h@u0z @z2i+2h@u0z @r2i+2h@u0 @z2i+8 3h@u0 @r2i#(3) 64

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Figure3-8. Comparisonofcurrentstudy's(symbols)turbulentenergybalancewithexperimentalresultsofShiri[ 49 ](solidlines)andPanchapakesanetal.andLumley[ 32 ](dashedlines).Valuesarescaledbythemeancenterlineaxialvelocitycubedovertheaxialdistancefromthejetinlet. waspreferabletootherapproximationswhichassumedlocalisotropy.Husseinetal.[ 26 ]thensolvedforthepressuretermbyclosingthebalance.Panchapakesanetal.[ 39 ]assumedthepressurediffusiontermtobenegligibleandobtainedthedissipationtermbyclosingthebalance.Inhisstudytheequationsfortheselfsimilarsolutionswereusedtocalculatetheaxialandradialderivativeofvariousquantities.Shiri[ 49 ]estimatedthepressuretermfollowingthesuggestionbyHusseinetal.[ 26 ]andLumley[ 32 ]thatthestreamwisegradientofhp0u0zicanbeignoredandthat )-222(hp0u0ri1 5hq2u0ri.(3) 65

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Shiri[ 49 ]thensolvedfordissipationbyclosingthebalance.Husseinetal.[ 26 ],Panchapakesanetal.[ 39 ]andShiri[ 49 ]assumedthat, hu0ru02ithu03ri,(3)forthepurposeofcalculatingtheturbulentdiffusionterm.TheresultsofWangetal.[ 56 ]andthecurrentstudy(seeingure 3-7 )showthatthesetwoproleshavesimilarshapebutthatthehu0ru02itermonlyachievesamaximumvalueofapproximately60%ofthatofthehu03riterm,andreachesthismaximumsomewhatclosertothejetaxis.Inthecurrentstudyalltermsofequation 3 werecalculateddirectlyfromtheDNSresults.Theerrorinturbulentkineticenergybalance,inthesamescalingasthetermsseeningure 3-8 islessthan0.02forallvaluesof.Thereissomedisagreementregardingtheshapeofthedissipationproleneartheaxisofthejet.Figure 3-9 comparesthedissipationproleofthecurrentDNSstudywithpreviousexperimentalresults.Shiri[ 49 ]andTaulbeeetal.[ 55 ]bothreportedanoff-axispeakforthedissipationterm.Husseinetal.[ 26 ]whoestimatedthistermintwodifferentwaysassumingeitherisotropyorlocaleaxialsymmetryfoundamoreGaussianproleforbothassumptions.Panchapakesanetal.[ 39 ]reportedaatregionnearthecenterofthejet.Thecurrentresultswhichweremeasureddirectlywerefoundtohaveasimilaratregionnearthejetcenterline.HoweverthemaximumcenterlinevalueofthedissipationtermwasfoundtomatchmostcloselywiththatfoundbyHusseinetal.[ 26 ]whenlocalaxialsymmetrywasassumedanddissipationwascalculatedbyequation 3 .For>0.5,awayfromthejetcenterlinethecurrentdatamatchesmorecloselywiththatofTaulbeeetal.[ 55 ]whousedthesamedatasetasHusseinetal.[ 26 ]butusedequation 3 tocalculatethepressureterm. 3.3.4ReynoldsStressTransportEquationsTransportequationsfortheReynoldsstresstermscanbederivedinasimilarfashionastheturbulentkineticenergybalance.Fromsymmetryalltermsthatinvolve 66

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Figure3-9. ComparisonofdissipationofturbulentkineticenergybetweenthecurrentDNSstudy(solidline)andexperimentalstudybyPanchapakesanetal.[ 39 ](dashedline),Husseinetal.[ 26 ]withassumptionofisotropicturbulence(longdashedlines),Husseinetal.withlocallyaxisymmetricassumption(thinsolidline),Taulbeeetal.[ 55 ](dottedline)andShiri[ 49 ](dashdotline). oddpowersofu0arezeroandthereforeonlythenormalstressesandthehu0zu0rishearstresstermneedtobeconsidered.Thesefourtransportequationsare: u2ztransportequation:0=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(hUzi@ @zu02z+hUri@ @ru02z)]TJ /F8 11.955 Tf 11.95 16.86 Td[(2hu0z@p0 @zi)]TJ /F8 11.955 Tf 11.95 16.86 Td[(2hu02zi@hUzi @z+2hu0zu0ri@hUzi @r)]TJ /F8 11.955 Tf 11.96 16.86 Td[(@hu03zi @z+1 r@ @r)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(rhu02zu0ri)]TJ /F7 11.955 Tf 17.2 8.09 Td[(2 Re"h@u0z @z2i+h@u0z @r2i+1 r2h@u0z @2i# (3) 67

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u2rtransportequation:0=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(hUzi@ @zu02r+hUri@ @ru02r)]TJ /F8 11.955 Tf 11.95 16.86 Td[(2hu0r@p0 @ri)]TJ /F8 11.955 Tf 11.96 16.86 Td[(2hu02ri@hUri @r+2hu0zu0ri@hUri @z)]TJ /F8 11.955 Tf 11.95 16.86 Td[(@hu0zu02ri @z+1 r@ @r)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(rhu03ri)]TJ /F7 11.955 Tf 11.95 0 Td[(2hu0ru0i r)]TJ /F7 11.955 Tf 17.2 8.09 Td[(2 Re"h@u0r @z2i+h@u0r @r2i+1 r2h@u0r @2i# (3) u2transportequation:0=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(hUzi@ @zu02+hUri@ @ru02+2hUri ru02)]TJ /F8 11.955 Tf 11.95 16.86 Td[(2hu0@p0 @i)]TJ /F8 11.955 Tf 11.96 16.86 Td[(2hu0zu0i@hUi @z+2hu0ru0i@hUi @z)]TJ /F8 11.955 Tf 11.96 16.86 Td[(@ @zhu0zu02i+1 r@ @rrhu0ru02i+2hu0ru02i r)]TJ /F7 11.955 Tf 17.2 8.09 Td[(2 Re"h@u0 @z2i+h@u0 @r2i+1 r2h@u0 @2i# (3) uzurtransportequation:0=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(hUzi@ @zhu0zu0ri+hUri@ @rhu0zu0ri)]TJ /F8 11.955 Tf 11.96 16.86 Td[(hu0r@p0 @zi+hu0z@p0 @ri)]TJ /F8 11.955 Tf 11.96 16.85 Td[(hu0zu0ri@hUzi @z+hu02ri@hUzi @r+hu02zi@hUri @z+hu0zu0ri@hUri @r)]TJ /F8 11.955 Tf 11.96 16.85 Td[(@ @zhu02zu0ri+1 r@ @rrhu0zu02ri)]TJ 20.45 8.08 Td[(hu0zu02i r)]TJ /F7 11.955 Tf 17.21 8.09 Td[(2 Reh@u0z @z@u0r @zi+h@u0z @r@u0r @ri+1 r2h@u0z @@u0r @i (3)Similartotheturbulentkineticenergybalancethevetermsinbrackets,foreachequation 3 through 3 ,representthecontributionofadvection,pressurediffusion,production,turbulentdiffusionanddissipationrespectively.ThecontributiontoturbulentdiffusionfromtheviscoustermisnegligibleduetotherelativelylargeReynoldsnumberusedinthesimulationandisnotincludedinequations 3 through 3 .Figure 3-10 comparestheReynoldsstresstransportequationsofthecurrentstudywithpreviousexperimentalstudiesbyHusseinetal.[ 26 ]andPanchapakesanetal.[ 39 ].IncalculatingtheReynoldsstresstransportequationsthepressurediffusiontermwasfoundbyclosingthebalanceoftheothertermswhichwerecalculateddirectly. 68

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Figure3-10. Reynoldsstresstransportequationsofcurrentstudy(Symbols)comparedtoexperimentalresultsbyHusseinetal.[ 26 ](SolidLines)andPanchapakesanetal.[ 39 ](DashedLines). Thebalanceofhu02zitransportinthecurrentstudyyieldsroughlysimilarresultstoboth,Husseinetal.[ 26 ]andPanchapakesanetal.[ 39 ],however,thepreviousstudiesbothfoundthemagnitudeofthepressurediffusiontermtobegreaterthanthatofthedissipationterm.Inthecurrentstudythetwotermsarefoundtobeofapproximatelythesameshapeandmagnitude.Also,Husseinetal.[ 26 ]reportedaclearoffaxispeakofdissipation.InboththecurrentstudyandthatofPanchapakesanetal.[ 39 ]thisfeatureisnotclearandthedissipationprolenearthejetaxisappearsfairlyat.Thebalanceofthehu02riandhu02iReynoldsnormalstresstermsinthecurrentstudyaresimilartothoseofPanchapakesanetal.[ 39 ],howevertheReynoldsshearstresstermhu0zu0riagreesmorewithHusseinetal.[ 26 ]. 69

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Figure3-11. 3Diso-surfacesofci=0.5calculatedattwodifferentinstantsoftime. 3.3.5VorticalStructuresandEntrainmentPlourdeetal.[ 42 ]examinedtheeffectofindividualvorticalstructuresofanaxisymmetricshearowontheentrainmentofambientuidintotheow.Inordertovisualizethevorticalstructureofthejetwecalculateci,theimaginarypartofthecomplexeigenvaluesofthelocalvelocitygradienttensor,assuggestedbyZhouetal.[ 64 ]andChakrabortyetal.[ 18 ].Asdiscussedintheirpapers,ifatanypointthevelocitygradienttensorhasallrealeigenvalues,thentheowhaszeroswirlatthatpoint.Ifontheotherhandthelocalowisswirling,thenthevelocitygradienttensorwillhavetwocomplexconjugateeigenvalues,andthevalueoftheimaginarypart,ci,isameasureofthestrengthofthelocalswirlingmotion. 70

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Figure 3-11 presentstheiso-surfaceofci=0.5calculatedattwodifferentinstantsintime,t1andt2.Thevalueofci=0.5waschosensoastoclearlyseetheimportantvorticalstructures.Botht1andt2occurafterthejethadbecomestatisticallystationaryandareinstantsincludedinthetemporalaverageddata.t1occursapproximately350timescalesafterthesimulationwasdeterminedtobestatisticallystationaryandt2wasatthecompletionofaveragingover1000timescales.Whilethedetailsoftheindividualvorticalstructuresatt1andt2appearquitedifferent,thelargescalebehaviorofturbulenceindifferentregionsofthejetaresimilar.Instabilitiesareseenformingveryclosetotheoriginintheformofmostlyvaricosevortexrings.Acloseexaminationofthesevortexringsrevealsthattheinstabilitiesareslightlyhelicalinnature.Byz=5,secondaryinstabilitiesbegintoform.Intheregionbetween510,theregionofvorticalstructureswidensfarmoreslowly.Beyondanaxialdistanceofapproximatelyz>17,thestrengthofthelocalswirlingmotionbeginstovisibilityweaken,andthestructureswhereci=0.5appearlesscloselypacked.Intheirseminalpaperonplumes,Morton,TaylorandTurner[ 35 ]suggestedthatthemeanradialvelocityattheedgeofanaxisymmetricboundarylayersuchasajetoraplumeisproportionaltotheaxialvelocity.Thatis,hUri=hUzi.Theysuggestedthatthecoefcientofentrainment,M,couldbeevaluatedas, M=d dzR10hUzirdr 2R10hUzi2rdr1=2.(3)Husseinetal.[ 26 ]showedthatforaselfsimilarjetequation 3 isequivalentto, M=I1=2I1=22,(3)where, I1=2Z10hUzi hUz,cid,(3) 71

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and, I2=2Z10hUzi hUz,ci2d.(3)OnecansolveforI1andI2byusingequation 3 asanapproximationforhUzi=hUz,ci.BoththecurrentDNSstudyandtheearlierexperimentalstudyofHusseinetal.[ 26 ]agreeonajethalfwidthof1=2=0.094.Assuch,bothstudiesagreeononthevalueoftheentrainmentcoefcient,M=0.081.Phametal.[ 41 ]suggestedanalternateapproximationfortheentrainmentcoefcient, P(z)=d dzR10hUz(z)irdr (z)hUz,ci,(3)whereisthemomentumbalancelengthscaledenedas, 2(z)=2)]TJ 5.48 -.06 Td[(R10hUz(z)irdr2 R10hUz(z)i2rdr.(3)ForaselfsimilarjetPisequivalentto P=1 2p I2,(3)which,usingthedatafromthecurrentstudy,canbeevaluatedasP=0.044.P(z)canalsobeevaluatednumerically,whichenablesustoexaminethechangesinentrainmentnearthesourceofthejetwherePisnotexpectedtobeconstant.Figure 3-12 plotsPasafunctionofz.Heretheindependentvariable,z,isplottedastheordinate,sothebehaviorofentrainmentingure 3-12 canbemorereadilycomparedtothebehaviorofthevorticalstructuresingure 3-11 .Between0
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Figure3-12. EntrainmentcoefcientP(z)asafunctionofzasdenedbyPhametal.[ 41 ](equation 3 ). expectedabovez>15,wherethejethasachievedselfsimilarity,Pisapproximatelyconstant(P0.044)andtheregionofvorticalstructureswidensmoreslowly.Thecomputedvalueisinexcellentagreementwiththatpredictedfromtheself-similarsolution. 3.3.6SpectraVariousresearchers[ 13 21 41 42 56 ]haveobtainedinsightsintothebehavioroftheuctuatingvelocitycomponentsbyexaminingthetemporalspectraandusingtheTaylorhypothesistorelatetheresultstoonedimensionalspectraintheaxialdirection.Inthecurrentstudyonedimensionalspectraintheazimuthaldirectionwasobtaineddirectlybycalculatingthefouriertransform, ^ui(r,k,z,t)=1 p 2Z20ui(r,,z,t)e)]TJ /F5 7.97 Tf 6.58 0 Td[(ikd,(3)Numericalevaluationofthefouriertransformwaswith256evenlyspacedpointsinthetangentialdirectionatagivenlocationinther)]TJ /F4 11.955 Tf 12.54 0 Td[(zplane.Instantaneousspectrasare 73

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Figure3-13. Onedimensionalvelocityspectraintheazimuthaldirectionforeachvelocitycomponent,attwodifferentlocationsonther)]TJ /F4 11.955 Tf 11.96 0 Td[(zplane. thenaveragedovertime.Inall,theenergyspectrawascalculatedateightdifferentpointslocatedatr=0.6,1.6andz=20,25,30,and35.However,verylittlevariationwasseenbetweenspectraexaminedatthedifferentaxialpositions.Figure 3-13 plotstheenergyspectraofthetheaxial,radialandtangentialcomponentsofvelocityattwodifferentradialpositionsr=0.6,1.6atz=30.Inthegurethefrequency,k,hasbeendividedbytheradialdistancefromthecenterlinesothatthefrequenciesatbothradialpositionsrepresentthesamewavelengthsintheazimuthaldirection.Thestraightlinesinthegurerepresentk)]TJ /F9 7.97 Tf 6.59 0 Td[(5=3andk)]TJ /F9 7.97 Tf 6.59 0 Td[(7.The)]TJ /F7 11.955 Tf 9.3 0 Td[(5=3slopeisrelevantintheinertialrangeandhereitcanbeseenthatthisbehaviordominatesonlyforverylowfrequencies.ThisissimilartotheresultsofBogeyandBailly[ 13 ]whocomparedtheonedimensionalspectrumoftheaxialvelocityalongthecenterlineofseveraljetswithvaryingReynoldsnumbers.Aslopesteeperthan)]TJ /F7 11.955 Tf 9.3 0 Td[(5=3isanindicationthatdissipationisplayingalargerroleintheenergycascade.Asdissipationbecomesdominantathigherfrequenciesaowdependentsteeperslopeiscanbeexpected.Forexample,inbuoyantjetsandplumesa)]TJ /F7 11.955 Tf 9.3 0 Td[(3slopeisobservedinthe 74

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dissipativeregime[ 41 42 ].Thevalueof)]TJ /F7 11.955 Tf 9.3 0 Td[(7observedinthedissipativeregimeofthecurrentstudyisidenticaltothatfoundbyFellouahetal.[ 21 ].Atbothlocationsthespectraofallthreevelocitycomponentswerefoundtohavesimilarbehavior.Thissuggeststhatthesmallscaleisotropicassumptionisvalidatthesepoints. 3.4SummaryADNSstudyofanaxisymmetricturbulentjetwascompletedandthesimulationwasrunoveraverylongtimeinorderforthethirdorderstatisticstoconverge.Firstandsecondorderstatisticswerefoundtoagreewellwithpreviousstudiesinthesimilarityregionofthejet.ThemoderateReynoldsnumberjetsimulatedinthecurrentstudywasfoundtoachieveconstantcenterlineturbulentintensitiesclosertotheinlet,thanpreviousexperimentalstudieswherejetswithlargerReynoldsnumberswereexamined.ThisagreeswithanassertionbyBogeyandBailly[ 13 ]thatmoderateReynoldsnumberturbulentjetsachievesimilarityclosertothesourcethanlargerReynoldsnumberjets.Indeed,thejetinthecurrentstudy(ReJ=2000)achievedselfsimilaritywithin15inletdiametersfromthejetsource,whereasthejetstudiedbyPanchapakesanetal.[ 39 ](ReJ=11,000)didnotachieveselfsimilarityuntilz=70.ThetotalmomentumintegralwascalculatedandM(z)wasfoundtobewithin1%ofMJforallzvaluesabovez>10.Themomentumuxcontributedbythemeanowwasfoundtoaccountforapproximately93%ofthetotalmomentumintegral.Thesixnon-zerocomponentsofthethirdorderstatisticswerepresentedandcomparedtopreviousstudies.Previousexperimentalstudiesfoundthatthecrossterms,hu0zu02ri,hu0ru02zi,andhu0zu02iarenegativenearthecenterlineofthejetbutdidnotreportonthefourthcrosstermhu0ru02i.Thecurrentstudyagreedwiththepreviousstudiesandalsoreportednegativevaluesnearthecenterlineforthethreepreviouslycalculatedcrossterms.Theforthcrosstermhu0ru02iwasalsocalculatedbutwasfoundtobepositiveforallvaluesofr=z.Previousstudieshadassumedthatthehu0ru02iterm 75

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wasequaltothehu03riinordertocalculatetheturbulentkineticenergybalance.Whilethecurrentstudyfoundhu0ru02itoberoughlythesameshapeashu03ri,themagnitudeofhu0ru02iwasevaluatedtobeabout40%smallerthanthatofhu03ri.ThereissomedisagreementbetweentheresultsofHusseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]inthehu03ziandhu03riterms,thecurrentstudycompareswellwithPanchapakesanetal.[ 39 ]inthecaseofhu03zibutagreesbetterwithHusseinetal.[ 26 ]inthecaseofhu03ri.Wherepreviousstudiesmadeuseofvariousassumptionsinordertoapproximatethepressurediffusionanddissipationtermsintheturbulentkineticenergybalance,inthecurrentstudyeachtermwascalculateddirectly.Therehasbeensomedisagreementastotheshapeofthedissipationprolenearthecenterline.WithsomeresearchersreportingaGaussianprole,otherssuggestinganoffaxismaximumvalue,andstillotherssuggestingaratheratprolenearthecenterline.ThecurrentstudyagreedwithPanchapakesanetal.[ 39 ]inndingtheproleatnearthecenterline,butfoundthemagnitudenearthecenterlineclosertothatreportedbyHusseinetal.[ 26 ].FurtherawayfromthecenterlinetheshapeandmagnitudeofthedissipationprolewasclosesttothatreportedbyTaulbeeetal.[ 55 ].Thebalanceofhu02zitransportinthecurrentstudyyieldsroughlysimilarresultstoboth,Husseinetal.[ 26 ]andPanchapakesanetal.[ 39 ],however,inthecurrentstudythepressureandturbulentdiffusiontermsarefoundtobeofapproximatelythesameshapeandmagnitude,bothpreviousstudiesreportedalargerpressuretermandasmallerturbulentdiffusionterm.Also,Husseinetal.[ 26 ]reportedaclearoffaxispeakfordissipation.InboththecurrentstudyandthatofPanchapakesanetal.[ 39 ]thisfeatureisnotclearandthedissipationprolenearthejetaxisappearsfairlyat.Thebalanceofthehu02riandhu02iReynoldsnormalstresstermsinthecurrentstudyaresimilartothatofPanchapakesanetal.[ 39 ],buttheReynoldsshearstresstermhu0zu0riagreesmorewithHusseinetal.[ 26 ].Hopefullyabetterunderstandingofthetransport 76

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ofallsixReynoldsstresstermscanbeusedtoimproveLESandRANSsimulationswhichrelyonmodelingtheseterms.Vorticalstructuresandentrainmentwerediscussedanditwassuggestedthatthebehaviorofthelargescalevorticalstructuresaffectedtherateofentrainment.Theonedimensionalvelocityspectraintheazimuthaldirectionwascalculateddirectly.The)]TJ /F7 11.955 Tf 9.3 0 Td[(5=3slopeindicativeoftheinertialregimeofturbulentowswasobservedaswellasahigherwavenumberregionof)]TJ /F7 11.955 Tf 9.3 0 Td[(7slopewhichwasalsoobservedbyFellouahetal.[ 21 ]tobepresentinthedissipativeregimeofapureaxisymmetricjet. 77

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CHAPTER4SINGLEPHASEJETS,PLUMESANDFORCEDPLUMES 4.1BackgroundTurbulent,axisymmetricshearows,suchasjetsandplumes,oftenariseinindustrialapplicationsandenvironmentalstudies.Examplesincludesmokerisingfromachimney,volcaniceruptionsanddeepwateroilspills.Ifthemaindrivingforceoftheowiscausedbyinitialmomentumwetermsuchaowajet.Ontheotherhandifthemaindrivingforceoftheowisbuoyancywetypicallyrefertotheowasaplume.Ifinitialmomentumandbuoyancyuxarebothpresentwecanrefertotheowaseitheraforcedplumeorabuoyantjet.SomeoftheearliestworkonaxisymmetricboundarylayerswasperformedbySchlichting[ 45 ]whoin1933publishedhisseminalworkonboundarylayertheory.Avarietyofresearchershaveextendedonthisearlywork.Similaritysolutionsforapurejetsandplumehavebeenderived.Thenatureoftheturbulentstatisticsofbothowtypeshavebeenexploredusingavarietyofresearchtechniques.Ithasbeensuggestedthatinthefareldbuoyantjetstakeonpureplumelikebehavior.Thissuppositionhasbeenexaminedbothanalyticallyandexperimentally.Verylittlework,however,hasbeendonetoexploretheturbulentstatisticsofaforcedplumeorbuoyantjetintheregionwheretheowisinuencedbyboththeinitialmomentumuxandbuoyancy.Startingfromturbulentboundarylayertheory,Schlichting[ 45 ]developedasimilaritysolutionforthemeanowofanaxisymmetricturbulentjet.Heshowedthattheaxialcenterlinevelocitydecreasesasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1,whilethejetwidthgrowsasz1.Wherezrepresentsthedistancefromthejetinletalongthejetaxis.Schmidt[ 46 ]performedsomeoftheearliestworkonplumesin1941.Heusedmixinglengthhypothesestoobtainthegeneralformofthemeanvelocityandbuoyancyprolesforanaxisymmetricplume.Hedeterminedthatanaxisymmetricturbulentplume'saxialvelocitydecaysasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3,whileit'sbuoyancydecaysasz)]TJ /F9 7.97 Tf 6.58 0 Td[(5=3. 78

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VariousresearchersincludingYih[ 62 ]andRouse,YihandHumphreys[ 44 ]usedexperimentswithgasburnersinordertodeterminevariousparametersoftheow.In1954,usingthescalingssuggestedbySchmidt,Batchelor[ 10 ]proposedasimilaritysolutionforanaxisymmetricturbulentplume.Inaseminalpaper,Morton,TaylorandTurner[ 35 ]examinedplumesinastratiedenvironment.Bymakingtheassumptionthattherateofentrainmentofambientuidisproportionaltotheaxialvelocityoftheplume,theywereabletodeterminetheaxialvelocityofaplumeatagivenheight.Theyalsoexaminedthemaximumheightobtainedbyaplumeinastratiedenvironment.Morton[ 34 ]examinedforcedplumeswherebothbuoyancyux,eJB,andmomentumux,eJM,arepresentattheplumesource.(Herethemarkindicatesadimensionalquantity.)Hepointedoutthatsinceaxialvelocityscalesasz)]TJ /F9 7.97 Tf 6.58 0 Td[(1forajet,butasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3foraplume,inthefareld,plumebehaviorwilldominate.Inthiscontexthegenerallyreferredtoalljetsandplumesasforcedplumes,wherepurejets,withnobuoyancy,andpureplumes,withnoinitialmomentum,aretheextremecases.Wewillusethisconventionthroughtherestofthisdocument.BasedonasimpleboxmodelMorton[ 34 ]suggestedthatindeterminingthelengthrequiredforaforcedplumetotakeonpureplumebehaviortheimportantparameteris eLM,I=eJ3=4M,I eJ1=2B,I,(4)whichhasthedimensionsofalength.HerethesubscriptIindicatesconditionsattheforcedplumeinlet.ScaledusingLM,I,hesuggestedanon-dimensionalheightabovewhichaforcedplumecanbeconsideredtohavetakenonthebehaviorofapureplume.Sincethisearlywork,jetsandplumeshavereceivedagreatdealofattention.Manyresearchershaveusedmodernlaboratoryandnumericaltechniquestosheadlightontheturbulentnatureoftheow.Husseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]performedbothHWAandLaser-DopplerAnemometry(LDA)experimentsonjetsand 79

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presenteddetailedresultsofthesecondandthirdorderstatisticsaswellasthatoftheturbulentkineticenergybalanceandothersecondordertransportequations.ShabbirandGeorge[ 48 ]performedsimilar(HWA)experimentsonforcedplumesandreportedonthebehaviorinthesimilarityregionwherepureplumebehaviordominated.Phametal.[ 41 ]performedaLarge-EddySimulations(LES)onapurethermalplumeandZhouetal.[ 65 ]performedanLESonaforcedplume.Plourdeetal.[ 42 ]completedaDNSsimulationonapurethermalplumeandreportedonthedevelopmentofturbulenceandvorticalstructuresnearthesourceofbuoyancy.Mostpastresearchhasfocusedoneitherpurejetsorplumes.Surprisinglylittleattentionhasbeengiventoexploringforcedplumesintheregionwherebothinitialmomentumandbuoyancyplayaroleinthedevelopmentoftheow.Inthiscurrentstudywewillexaminepurejets,pureplumesandforcedplumesusingDirectNumericalSimulation.Acomparisonwillbemadeoftheturbulentstatisticsofthethreedifferentows.Specialattentionwillbepaidtotheturbulentkineticenergybalance,Reynoldsstresstransportequationsandentrainment.Wewillbegininsection 4.2 bydeningtheproblemandintroducingthesimulationmethodologythatwillbeusedthroughouttherestofthearticle.Insection 4.3 theresultsofthecurrentnumericalsimulationswillbecomparedwiththepreviouslydiscussedsimilaritysolutionsandpastresults.Wewillthencomparetheturbulentbehavioroftheforcedplumeswiththatofpurejetsandplumes.Finally,ourconclusionswillbesummarizedinsection 4.4 4.2MethodologyWeassumeaforcedplumeemergingfromacircularpipeofdiameterDorapureplumegeneratedperhapsbyaheatedplatealsoofdiameterD.Fluidexitingthecircularpipeorheatedbythecircularplateentrainsambientuidabove.Figure 4-1 showsthecomputationaldomainwhichisarectangularregionwithdimensionlesslength,widthandheight,ofLxandradiusLyandLz. 80

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Figure4-1. Schematicofthecomputationaldomain. Thequantities~r,ez,andedenotetheradial,axialandazimuthaldirectionalvectorsinthecylindricalcoordinatesystem.Thequantitiesex,ey,andezarethedirectionalvectorsintheCartesiancoordinatesystem.Theoriginofthecoordinatesispositionedatthecenteroftheforcedplumeinowboundary.Thegravitationalacceleration,g,actingontheuidisdirectedinthenegativez-direction.Fortime-dependentincompressibleowthegoverningequations(continuity,momentum,andenergy)inthestandardofformaregivenby, ereu=0(4) @eu @et+euer=1 aer~p+er2eu+a)]TJ /F7 11.955 Tf 12.81 0 Td[(~ ag^k(4) @ @eta)]TJ /F7 11.955 Tf 12.81 0 Td[(~ ag+eura)]TJ /F7 11.955 Tf 12.82 0 Td[(~ ag=er2a)]TJ /F7 11.955 Tf 12.81 0 Td[(~ ag(4)whereeuisthevelocityvector,episthepressureand~isthedensity.Thesubscript,a,referstouidpropertiesoftheambientuidfarfromtheplume,r!1,whichwillbeconsideredtobeconstantinthisstudy.The`tilde'inthegoverningequations 81

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denotesdimensionalvariablesand^kisaunitvectorintheverticaldirection,ez.Welimitourselvestoplumeswherethedensityvariation,~a)]TJ /F7 11.955 Tf 13.75 0 Td[(~,issufcientlysmalltobeignoredexceptforwhenconsideredwithgravitationalacceleration(i.e.weusetheBoussinesqapproximation).Theuidproperties,suchaskinematicviscosity,,andthermaldiffusivity,,alongwiththegravitationalacceleration,g,arekeptconstantforthepurposeofthisanalysis.Twodifferentscalingsarenecessaryinordertonon-dimensionalizepureplumesandpurejets.ForpurejetsthereferencevelocityisdenedasUI,thetophataxialvelocityatthejetinlet.Hereandbelowthesubscript(I)indicatesconditionsattheinlet.But,forpureplumes,wherebydenitionUI=0,thevelocityisscaledbyp gD(a)]TJ /F3 11.955 Tf 11.95 0 Td[(I)=a.Forforcedplumes,withbothnon-zerobuoyancyandinitialmomentum,eitherscalingcanbeused,butitislogicaltousethescalingthatcorrespondstotheuxwhichisdominant.Whenconsideringajet,thevariablesarescaledbythefollowingdimensionalquantities:jetinletdiameter,D,jetinletvelocity,UI,andthedensitydecitatthesourceofbuoyancy,a)]TJ /F3 11.955 Tf 12.41 0 Td[(I.Theresultingdimensionlessvariablesarexi=exi=D,ui=eui=UI,p=ep=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(aU2I,=[(a)]TJ /F7 11.955 Tf 12.81 0 Td[(~)=(a)]TJ /F3 11.955 Tf 11.96 0 Td[(I)]gandt=UIet=D.Atophatinitialvelocityprole,UIandbuoyancyproleI=g(a)]TJ /F3 11.955 Tf 11.95 0 Td[(I)=aisassumed.Withappropriatenon-dimensionalization,weobtainthedimensionlessgoverningequations(continuity,momentumandenergy)as, ru=0(4) @u @t+uru=rp+1 ReIr2u+Ri^k(4) @ @t+ur=1 ReIPrr2(4) 82

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Thedimensionlessparametersappearingintheaboveequation,theReynoldsnumber(ReI),thePrandtlnumber(Pr),andtheRichardsonnumber(Ri),aredenedas: ReI=UID ,Ri=(a)]TJ /F3 11.955 Tf 11.95 0 Td[(I)gD aU2I,Pr= .(4)Whenconsideringapureplumeorwhenbuoyancyuxattheinletdominatesthemomentumuxamoreappropriatechoiceforvelocityscaleisp gD(a)]TJ /F3 11.955 Tf 11.95 0 Td[(I)=a.KeepingthelengthscaleasDthepressureandtimescalescanbeaccordinglydened.Thisscalingresultsindimensionlessgoverningequationsoftheform 4 to 4 butwith, ReI=p Gr=p (a)]TJ /F3 11.955 Tf 11.95 0 Td[(I)gD3 (4)WhereGristheGrashofnumberandthelastterminthemomentumequation( 4 )becomes1^k.Whendiscussingsecondandthirdorderstatisticswewillusetheconvectionthataveragedrstorderquantitiesarecapitalizedandperturbationsareprimed.Thus,Reynoldsdecompositionofauctuatingvariable,x,canbeexpressedas, x=hXi+x0.(4)Threedifferentsimulationswereruncorrespondingtoapurejet,apureplumeandaforcedplume.Eachsimulationwasrunlongenoughforthirdorderstatisticstoconverge.AllsimulationswererunwithaPrandtlnumberof0.7whichistheapproximatevalueforair.ForthepurejetsimulationtheReynoldsnumberReI=2000wasused.ForthepureplumecaseaGrashofnumberofGrI=4.0x106wasused,thisisthesameasajetsimulationwithaReynoldsnumberofReI=2000,aRichardsonnumberofRi=1.0andsettingthevelocityattheinletUI=0.Inthecaseofaforcedplume,ReynoldsnumberwassettoReI=1684withaRichardsonnumberofRi=0.025.Thesevalueswerechoseninordertobelargeenoughfortherelatedows 83

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toexhibitturbulencebutsmallenoughastostillbetractablebyDNS.Theparametersoftheforcedplumewereselectedsuchthattheaxialdistancez=5LM,I,thedistancesuggestedbyMorton[ 34 ]beyondwhichaforcedplumecanbetreatedasapureplumeiswellwithinthecomputationaldomainofthesimulation.ThegoverningequationsweresolvednumericallyonaCartesiangridwithgridpointsclusteredclosetothesourceofmomentumandbuoyancy.Asecond-orderaccuratecentraldifferenceschemeisemployedforthespatialdiscretizationofthegoverningequationsonanon-staggeredgridsystem.Afractionalstepmethodisusedfortimeadvancement.Intheadvection-diffusionstep,thenonlineartermsaretreatedexplicitlyusingasecond-orderAdams-BashforthschemeandthediffusiontermsaretreatedimplicitlywiththeCrank-Nicholsonscheme.ThepressurePoissonequationissolvedasapressurecorrectionstep.Analdivergence-freevelocityisobtainedateachtime-stepwithapressurecorrectionstep.Attheinowboundary,atop-hatproleisusedforbothaxialvelocity,uz,andbuoyancy,,forr=p x2+y2<1=2.Randomuctuationsrangingbetween0.05UIwereaddedtothetophatinletvelocityproleofthepurejetandforcedplumeandtothebuoyancyofthepureplumeinordertospeedupthetransitiontofullyturbulentbehavior.Thelowerboundaryoutsidethejetinlet,r>1=2,isconsideredasano-slipwall.PreviousDNSstudies[ 2 12 41 ]wereconsultedinordertodeterminethedomainsizesothateffectsduetoconnementinthelateraldirectionwereminimized.Weallowtheuidtofreelyenterorleavethelateralboundaries,x=1 2Lx,y=1 2Ly,byapplyingNeumannboundaryconditions.Thestabilityofthesimulationswasfoundtobedependentontheboundaryconditionusedatthetopofthedomain,z=Lz.Duetotheturbulentnatureoftheowapureoutowboundaryconditioncouldnotbeused.Forboththepurejetandthepureplumeconvectiveboundaryconditionswereusedfortheboundaryatz=Lz.TheforcedplumesimulationwasfoundtobemorestablewhenNeumannboundaryconditionsweresetatthez=Lzboundary.Thenondimensional 84

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Table4-1. TableofDNSrunscompletedinthecurrentstudy ParameterPureJetPurePlumeForcedPlume UI1.0-1.0ReI2000-1684Ri0-0.025Gr-20002-LxLyLz252545252545252550NxNyNz251251550251251700251251750OutowB.C.ConvectiveConvectiveNeumann parameters,domainsize,gridsizeandoutowboundaryconditionsusedforthethreesimulationsaresummarizedintable 4-1 4.3ResultsAllthreesimulationsranforover1.3105timestepsorapproximately1000timescales,=D=UI,beyondthetimeittookthesimulationstobecomestatisticallysteadystate.Averagesofturbulentvariableswerecalculatedazimuthallyaswellastemporally.ThesimulationofthepurejetwithReI=2000,isthesamesimulationthatwasdiscussedindetailinChapter 3 .Inordertomoreclearlycomparethevariousaxisymmetricowssomeofthemoresignicantresultswillberepeatedhere. 4.3.1MomentumandBuoyancyFluxAsdiscussedatlengthinchapter 3 ,integrationoftheaxialvelocitymomentumequationacrossther)]TJ /F3 11.955 Tf 12.5 0 Td[(planeofapurejetyieldsarelationshipbetweeninitialaxialmomentumux,JM,I,atz=0andthethemomentumuxatagivendistancefromthejetsource,JM(z).Forlaminarowsthepressuregradientcanbeignoredanditisfoundthat, JM(z)=2ZD=20u2zrdrz=0=JM,I.(4)However,Husseinetal.[ 26 ]suggestedthatthestreamwisepressuregradientshouldnotbeignoredandbyusingthecross-streammomentumequationtoeliminatethepressuretermtheyderived, JM,I=M(z)+,(4) 85

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whereM(z)isreferredtoasthemomentumintegralandisdenedas, M(z)=2Z10hUzi2+hu02zi)]TJ /F7 11.955 Tf 20.45 8.09 Td[(1 2)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(hu02ri+hu02irdr,(4)and,whichisdenedinequation 3 inchapter 3 ,isfoundtobeverysmall.InthecurrentDNSwheretheinitialtophataxialvelocitywassettouz=1withrandomperturbationsof5%,theinitialmomentumuxisJM,I=4.Figure 4-2 ,whichisthesameasgure 3-6 fromchapter 3 ,displayseachtermofequation 4 throughouttheentiredomainandcomparesM(z)toJM,I.M(z)isfoundtobewithin5%ofJM,Ithroughoutthedomain.Aftertheowbecomesfullyturbulentforapproximatelyz>10,thecomparisonimprovessothatM(z)iswithin1%ofJM,I.Themeanowmomentumuxaccountsfor93%ofthetotalmomentumintegralinthefullyturbulentregion.Thetotalmomentumuxincludingthemeanowanductuatingvelocityintheaxialdirectionisapproximately10%largerthanthemomentumintegral,M(z),inthesameregion.Duetothenonzerobuoyancytermintheaxialvelocitymomentumequation,momentumuxwillnotbeconservedinplumes.However,integrationofthebuoyancyequationacrossther)]TJ /F3 11.955 Tf 11.96 0 Td[(planeyields, JB(z)=2Z10hUzihi+hu0z0irdr=JB,I.(4)Thatis,thetotalbuoyancyux,JB,includingboththemeanowandthetheuctuatingtermisconserved.Sinceinapureplume,theaxialvelocityatthesourceis,bydenition,uz(z=0)=0,itisnotpossibletoknowthebuoyancyuxapriori.Figure 4-3 displaystherelativecontributionsofthemeananductuatingcontributionstothetotalbuoyancyuxforthepureandforcedplumesimulatedinthecurrentstudy.Itisclearinthegurethatthebuoyancyuxofthepureplumenonethelessbecomesnonzeroandrelativelyconstantashortdistanceawayformthesourceoftheplume.Thetotalbuoyancyuxisfoundto 86

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Figure4-2. Contributionstothetotalmomentumintegralofapurejetbyvariousterms. Figure4-3. Conservationofbuoyancyuxforapureandforcedplume. beapproximatelyJB0.01158.Nearthesourcethecontributionfromtheuctuating,hu0z0itermisapproximately17%ofthetotalmomentumux,whichisapproximatelythesamepercentageasfoundfarfromthesource.ThisissimilartotheresultsofShabbirandGeorge[ 48 ]whomeasuredauctuatingcontributionofbetween15%and18%fortheirexperimentalplume.Between0
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Iftheconditionsatthesourceofaforcedplumeareknown,thenthetotalbuoyancyuxcanbedeterminedapriori.Withthenon-dimensionalizationusedinthecurrentstudythebuoyancyuxoftheforcedplumeatthesourceisJB,I=Ri=4.Whilerandomperturbationsofapproximately5%wereaddedtothetophataxialvelocitythese,uctuationsquicklydieout,andthecontributionfromthehu0z0itermisnegligiblefor05. 4.3.2MeanFlowBothjetsandplumesbecomeselfsimilarinthefareld.Themeanowaxialvelocityofajet,andboththemeanowaxialvelocityandbuoyancyofaplumetaketheformof, hi=A(z+zv))]TJ /F6 7.97 Tf 6.58 0 Td[(exp"ln(0.5) 21=22#,(4)where,hicanbeeitherhUziorhi.Here,isthesimilaritycoordinatedenedas, =r z+zv.(4)1=2isthehalfwidthoftheGaussiancurveandzvisthevirtualorigin.Theexponentsigniestherateofdecayofthevariableinquestionwithrespecttoz,andthecoefcientAistheslopeofthelinearizedcurvet, hi=A(z+zv))]TJ /F6 7.97 Tf 6.59 0 Td[(.(4)AswasdiscussedinChapter 3 ,forjetsjet=1.IncomparingthecurrentDNSjetwithpreviousstudiesavirtualoriginofzv=1.3wasused,whereasHusseinetal.[ 26 ]ttheirdatawithzv=4andPanchapakesanandcolleagues's[ 39 ]datadidnotrequireavirtualorigin,(zv=0).ThesimilaritysolutionforapurejetisuniversalifAisstatedin 88

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termsofthesquarerootofthemomentumux, Ajet=Bjetp JM.(4)InthecurrentstudyBwasfoundtohavethevalueBjet=6.1whichcompareswellwithotherpastDNSstudiessuchasWangetal.[ 56 ]whofoundBjet=6.2andexperimentalstudiessuchasHusseinetal.[ 26 ],B=6.5,WygnanskiandFiedler[ 61 ],Bjet=6.1andPanchapakesanetal.[ 39 ],Bjet=6.8Schmidt[ 46 ]performedsomeoftheearliestworkonplumesin1941.Heusedmixinglengthhypothesestoobtainmeanvelocityandbuoyancyprolesforanaxisymmetricplume.Hedeterminedthatfortheforaplumeplume,U=1=3intheequationforaxialvelocityandplume,=5=3forthebuoyancyequation.Aplume,UandAplume,canbeexpresseduniversallyas, Aplume,U=Bplume,UJ1=3B,(4)and Aplume,=Bplume,J2=3B,(4)respectively.Figure 4-4 comparesthemeanowaxialvelocityandbuoyancyatvariousheightsabovethepureplumesourcewiththeexperimentalstudyperformedbyShabbirandGeorge[ 48 ]andanLESstudybyZhouetal.[ 66 ].ThegureisscaledsothatthevalueofBplume,UandBplume,canbedeterminedfromthevaluesatthecenterline,=0.FrombesttcurvesoftheirdataShabbirandGeorge[ 48 ]determinedthatBplume,U=3.4andBplume,=9.4bothofwhicharelessthanfoundinboththecurrentDNSstudyandtheLESstudyperformedbyZhouetal.[ 66 ]whichbothmoreorlessagreeonvaluesofBplume,U=3.8andBplume,=10.1.ShabbirandGeorge[ 48 ]foundthehalfwidthofthesimilarityequationstobe1=2,U=0.109and1=2,=0.101fortheaxialvelocityandbuoyancyrespectively.Zhou 89

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Figure4-4. ScaledmeanaxialvelocityandbuoyancyprolesofthepureplumeofthecurrentstudyatvariousaxialdistancesfromthesourcecomparedwithapreviousexperimentalstudybyShabbirandGeorge[ 48 ]andanLESstudybyZhouetal.[ 66 ] etal.[ 66 ]determined1=2,U=0.106and1=2,=0.109fortheirdataatanaxialdistanceofz=D=14whichisthefurthestdistancefromtheoriginthattheyreportedon.TheplumeinthecurrentDNSstudyisnarrowerthanthoseexaminedintheearlierstudies,withboththeaxialvelocityandbuoyancycurveshavingroughlythesamehalfwidth,1=2,U=1=2,=0.102.Inttingdatatothesecurves,ShabbirandGeorge[ 48 ]didnotuseavirtualorigin,zv=0,whereasZhouetal.[ 66 ]usedzv=0.5Dandzv=)]TJ /F7 11.955 Tf 9.3 0 Td[(4Dwasusedinthecurrentstudy.Morton[ 34 ]examinedforcedplumeswherebothbuoyancyux,JB,andmomentumux,JM,arepresentattheplumesource.Hepointedoutthatsincejet=1forajetbut,plume=1=3foraplume,thatinthefareld,plumebehaviorwilldominate.Hesuggestedthatindeterminingthelengthrequiredforaforcedplumetotakeonpureplumebehaviortheimportantparameteris, LM,I=J3=4M,I J1=2B,I.(4) 90

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Figure4-5. CenterlineAxialVelocity IftheaxialcoordinateisnondimensionalizedbyLM,Ithenaforcedplumetakeonmostlyplumelikebehaviorforz>~z=5eLM,I.Figure 4-5 comparesthecenterlineaxialvelocityofthethreeows.Inthecurrentsimulation,thecenterlineaxialvelocityofthepurejetisdeterminedtodecreaseatit'scharateristicdecayrateofz)]TJ /F9 7.97 Tf 6.58 0 Td[(1forz>8.BogeyandBailly[ 13 ]suggestedhoweverthatlowReynoldsnumberjetsacheiveselfsimilarityfasterthanhighReynoldsnumberjets.Theaxialvelocityofthepureplumeinitiallyacceleratesuntilapproximatelyz=5,andthenbeginstodecreaserapidlyatarateofz)]TJ /F9 7.97 Tf 6.59 0 Td[(2.Theaxialdecayrateofz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3isachievedaboveanaxialdistanceofz>13.Thecenterlineaxialvelocityoftheforcedplumeremainsfairlyconstantforz.7.Atanaxialdistancebetween7.z.15theforcedplumedeceleratesatarateofroughlyz)]TJ /F9 7.97 Tf 6.59 0 Td[(1similartothatofapurejet.Abovez&15,buoyancydominatesandtheforcedplumetakesonepureplumebehaviorwithanaxialdecayrateofz)]TJ /F9 7.97 Tf 6.58 0 Td[(1=3.Theaxialdistance5LMisnotedinthegure.TheplumetakesontheaxialdecayrateofapureplumemuchsoonerthanthevelengthscalespredictedbyMorton[ 34 ].Infactbasedonthegureitappearsthatboththepureplumeandtheforcedplumeachieveanaxialvelocitydecayratepredictedbytheoryatapproximatelythesameaxialdistanceabovethesource. 91

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Figure4-6. ScaledmeanaxialvelocityandbuoyancyprolesoftheforcedplumeofthecurrentstudyatvariousaxialdistancesfromthesourcecomparedwithapreviousexperimentalstudybyShabbirandGeorge[ 48 ]andanLESstudybyZhouetal.[ 66 ]. Foranotherpointofviewgure 4-6 comparesthemeanowaxialvelocityandbuoyancyoftheforcedplumetothesamedatasetsthatthepureplumewascomparedtoingure 4-4 .Hereitisclearthatthedatafromthetwoaxialdistanceslessthen5LM,I,z=15andz=20,thesolutionhasnotnotcollapsedtoasinglecurve.However,thedataaboveorapproximatelyequalto5LM,IthedatamatchesfairlywelltothepreviousexperimentalstudiesbyShabbirandGeorge[ 48 ]andZhouetal.[ 66 ]. 4.3.3SecondOrderStatisticsFigure 4-7 comparesthecenterlineturbulentintensityintheaxialdirection,p hu02zi=hUz,ci,alongtheowaxisofthepurejet,plume,andforcedplume.TheLESresultsofZhouetal.[ 66 ]foraforcedplumeandtheDNSresultsofPhametal.[ 40 ]forapureplumearealsoincludedinthegure.Theturbulentintensityalongthecenterlineofthepurejetsimulationofthecurrentstudywascomparedtopreviousstudiesinchapter 3 92

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Figure4-7. Comparisonofturbulentintensityalongtheaxisforthepurejet(solidline),pureplume(dashedline)andforcedplume(dash-dotline)ofthecurrentDNSstudywithpreviousstudiesbyPhametal.[ 40 ](thin-dashedlines)andZhouetal.[ 66 ](thin-dash-dotlines.) Thecenterlineturbulentintensityofthetwoowswithinitialmomentumux,thepurejetandtheforcedplume,evolveinasimilarmanner.Asdiscussedinchapter 3 theinitialperturbationsaddedtothetophataxialvelocityprolediesoutveryclosetothesource.Ataxialdistancesgreaterthanz>11bothnonzerocomponentsoftheturbulentintensityobtainconstantvaluesofhu0zu0zi=hUz,ci0.25andhu0ru0ri=hUz,ci0.21.Inthecaseofthepureplumetheaxialcomponentoftheturbulentintensityinthelimitasz!0isnonzero,suggestingthatasz!0theaxialcomponentoftheReynoldsstressapproacheszeroatthesamerateasthemeanaxialvelocity.Theturbulentintensitycontinuestogrowwithrespecttozuntilapproximatelyz9whenitreachesamaximumvalueofhu0zu0zi=hUz,ci0.49andthenbeginstodecrease.Aboveanaxialdistanceofapproximatelyz19theturbulentintensitybecomesrelativelyconstant,obtainingthesameconstantvalueasthepurejetandpureplume.ThisbehaviorwasnotseenintheearlierDNSstudyofpureplumesconductedbyPhametal.[ 40 ]whoreportedlargevaluesofp hu02zi=hUz,cinearthesourcebutnomaximaofsecondorderaxialvelocityawayfromthesource. 93

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Figure4-8. ComparisonoftheradialprolesofReynoldsstresstermsforthepurejet,pureplumeandforcedplumeofthecurrentDNSstudyattwoaxialdistancesfromthesource,z=15andz=30. Figure 4-8 comparestheradialprolesoftheReynoldsstresstermsofthethreedifferentowsattwoaxialdistances,z=15andz=30.Theaxialdistanceofz=30iscomparableto5LM,Iwherepureplumelikebehaviorisexpected.TheReynoldsstressprolesofthepurejetatthetwodifferentaxialdistanceswerefoundtoberoughlyidentical.Forthesakeofclarity,thejetprolesatz=30arenotincludedinthegure.TheReynoldsstressprolesofbothplumesarenarrowerthanthatofthepurejet,withthedifferencebeingmostpronouncedforthepureplumeatz=15.Theforcedplumeat 94

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Figure4-9. ComparisonofthesecondorderbuoyancytermsforthepureplumeandforcedplumeofthecurrentDNSstudyattwoaxialdistancesfromthesource,z=15andz=30withapreviouslaboratoryandapreviousnumericalstudy. z=30hasclearoff-axismaximaforallthreenormalstressterms,thiscanalsobeseeninthepurejetprolesbutislesspronounced.Ingure 4-7 itwasseenthatthecenterlinehu02zitermofthepureplumeismuchgreaterthanthatofthepurejetandforcedplumeforz<19.Thehu02zitermofthepureplumeatz=15hasclearnon-Gaussianbehaviorneartheplumecenterline,butbehavessimilartothatoftheforcedplumeawayfromthecenterline.Byz=30thisbehaviorhasdissipatedandthehu02ziproleofthepureplumebehavessimilarlytothat 95

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oftheforcedplume.Thissuggeststhatthemagnitudeofhu02zitermofapureplumebeinggreaterthanthatoftheothertwoowsisisolatednearthecenteroftheow.Inordertoprovideacomparisonofourcurrentforcedplumeresultswithpreviousstudies,curvetsofdatafromShabbirandGeorge[ 48 ]andLESresultsfromZhouetal.[ 66 ]areincludedingure 4-8 .Thejetsimulationwascomparedtopreviousstudiesinchapter 3 .Theradialprolesofsecondorderstatisticsinvolvingbuoyancyforallthreeowsaredisplayedingure 4-9 .TheplumedatafromthecurrentstudyiscomparedtoexperimentalcurvetsofShabbirandGeorge[ 48 ]andtheLESstudyofZhouetal.[ 66 ].Thesecondorderbuoyancyterm,h02i1=2=hci,oftheforcedplumehasaclearoffaxismaximumnotpresentineitherthepureplumeorreportedbytheearliertwostudies.Bothearlierstudiescanbeconsideredweaklyforcedplumes.Thisoffaxismaximumispresentatbothaxiallocationsz=15andz=30.Theforcedplumealsohasanoffaxismaximumfortheh0u0zi1=2=(hcihUz,zi)term,whichwasalsoreportedbyShabbirandGeorge.TheuctuatingbuoyancyandaxialvelocitytermforboththecurrentpureplumeandtheLESstudyofZhouetal.[ 66 ]doesnothaveanoffaxismaximumbutratherthecurvesareratheratnearthecenterline. 4.3.4ThirdOrderStatisticsDerivativesofthirdordermomentsofvelocityplayadominantroleintheturbulentdiffusionofturbulentkineticenergyandReynoldsstress.Figures 4-10 and 4-11 comparetheradialprolesofthethirdordermomentsofvelocityofthethreeaxisymmetricowssimulatedinthecurrentstudyatanaxialdistanceofz=15andz=30respectivelyawayfromtheowsource.InordertocomparethecurrentplumeresultstopreviousstudiesthecurvetsofShabbirandGeorge's[ 48 ]dataisincludedinbothgures.HerethestatisticforallthreeowsarescaledbyhUz,ci3.Atanaxialdistanceofz=15thepurejetandforcedplumehavesimilarthirdordermomentproleswhereastheprolesofthepureplumebehavequitedifferently.The 96

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Figure4-10. Comparisonoftheradialprolesofthethirdordervelocitystatisticsforthepurejet,pureplumeandforcedplumeofthecurrentDNSstudyatanaxialdistanceofz=15fromthesource.ThethirdorderstatisticsarescaledbyhUz,ci3. 97

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Figure4-11. Comparisonoftheradialprolesofthethirdordervelocitystatisticsforthepurejet,pureplumeandforcedplumeofthecurrentDNSstudyatanaxialdistanceofz=30fromthesource.ThethirdorderstatisticsarescaledbyhUz,ci3. 98

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Figure4-12. ComparisonoftheradialprolesofthethirdorderbuoyancystatisticsforthepureandforcedplumeofthecurrentDNSstudyataxialdistancesofz=15andz=30.ThedataofthecurrentstudyiscomparedtocurvetsofthedataobtainedbyShabbirandGeorge[ 48 ]whenavailable. 99

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centerlinehu03zitermofthepureplumehasavalueofhu03z,ci=0.38avaluethatistwoordersofmagnitudelargerthanthatforthepurejetandforcedplume.Ingeneraltheprolesofthepureplumetendtobenarrowerthanthoseofthetwoowswithinitialmomentumatthesource.Theintegrallengthandtimescalesgrowslinearlywithdistancefromtheowsourceandlargescaleeddiestakelongertoturnover.Assuchthefurtherawayfromtheowsourcethegreaternumbertimestepsareneededforhigherorderstatisticstoconverge.Thereforethethirdorderprolesatz=30comparedingure 4-11 appearlesssmooththenthecorrespondingprolesatz=15.Nevertheless,atz=30thethirdordermomentprolesofthepureplumeappearsimilartothatofthepurejetandforcedplume.Itisinterestingtonotethatwhileatz=15thecenterlinehu03zitermofthepurejetandforcedplumehadapositivevalue,butatz=30thecorrespondingvalueofbothowsisnegative.Thereisnosimilarchangeinthevaluesorshapesoftheother5non-zerothirdordervelocityprolesforthetwoows.Nevertheless,thatbothowshavedifferentcenterlinevaluesatz=15thantheydoatz=30suggeststhepossibilitythattheowshavenotobtainedselfsimilarsolutionsoftheirthirdordermomentsbyz=15eveniftheydohaveselfsimilarrstandsecondorderstatisticsatthataxialdistance.Thethirdordertermsinvolvingbuoyancyaredisplayedatbothaxialdistancesofz=15andz=30ingure 4-12 .CurvetsfromShabbirandGeorge's[ 48 ]datafortheh0u02ziandh02u0ritermsarealsoincludedinthegure.Thethirdorderbuoyancytermsofbothowsdonotappeartochangesignicantlybetweenthetwoaxialdistances.Thethirdorderbuoyancyterm,h03i,ofthepureplumehasalmostGaussianbehaviorwhereastheforcedplumehasit'smaximaintheshearlayerawayfromthecentralaxis. 4.3.5TurbulentKineticEnergyBalanceByperformingReynoldsdecompositionofthemomentumequations,subtractingofftheRANSequations,takingthedotproductwiththeuctuatingvelocityvectorand 100

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takingtheaverage,thetransportequationforturbulentkineticenergy,k,canbederivedas:0=)]TJ /F8 11.955 Tf 11.29 16.85 Td[(hUzi@k @z+hUri@k @r)]TJ /F8 11.955 Tf 11.96 16.85 Td[(1 r@ @r(rhp0u0ri)+@ @zhp0u0zi)]TJ /F8 11.955 Tf 11.95 16.86 Td[(hu0zu0ri@hUzi @r+@hUri @z+u02z@hUzi @z+u02r@hUri @r+hUri rhu02i)]TJ /F8 11.955 Tf 11.95 16.85 Td[(1 r@ @rr 2q2u0r+1 2@ @z\012q2u0z+[Rihu0z0i])]TJ /F8 11.955 Tf 11.95 20.44 Td[("1 ReJh@u0i @xj2i#. (4)Equation 4 isalsoreferredtoastheturbulentkineticenergybalance.Here k=1 2u02z+u02r+u02,(4)istheturbulentkineticenergyand, hq2uii=u02zu0i+u02ru0i+u02u0i.(4)Thesixtermsinbracketsrepresentthecontributionofadvection,pressurediffusion,production,turbulentdiffusion,buoyancy,anddissipationrespectively.Equation 4 isstatedintermsofthejetnon-dimensionalization.Theequationisthesameasthatforapureplumebutwithp GrreplacingtheReynoldsnumberandbysettingtheRichardsonnumbertoRi=1.ThecontributiontoturbulentdiffusionfromtheviscoustermisnegligibleduetotherelativelylargeReynoldsnumberusedinthesimulationsandisnotincludedinequation( 4 ).Figure 4-13 displaystheturbulentkineticenergybalanceofthethreeowsexaminedinthecurrentstudyatz=15andz=30.Theprolesofturbulentdiffusion,whicharedependentonthirdorderstatistics,appearlesssmooththanthecorrespondingprolesatz=15.Thisisduetoeddysturningoverataslowerrateatthisaxialdistance,leadingtotherequirementofaveragingoveralongertimescalefor 101

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Figure4-13. Turbulentkineticenergybalanceofapurejet,pureplume,andforcedplumeatz=15andz=30. 102

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thesestatisticstoconverge.Neverthelesstheshapeoftheturbulentdiffusionprolescanstillbediscernedatthislocation.Theturbulentkineticenergybalanceofthepurejetappearsfairlysimilaratbothaxialdistanceswiththeexceptionofthepressureandturbulentdiffusiontermsverynearthejetcenterline.Atanaxialdistanceofz=15bothdiffusiontermstendtowardzeroalongthejetaxis,butatz=30thepressuretermisclearlypositiveandisbalancedoutbytheturbulentdiffusiontermwhichisequalitynegative.Theturbulentkineticenergybalanceofthepureplumeatz=15isquitedifferentthanthatofthepurejet.Thebuoyancyterm,whichisnotpresentinapurejet,hasapositivecontributiontoturbulentkineticenergythroughouttheow.Whiletheadvectionproleofthejetisstrictlypositiveforall,thetermbecomesnegativeintheouterportionoftheplume,>0.07.Themagnitudeofthedissipationtermattheplume'saxisismuchgreaterthanthatofajet.Theoffaxismaximumofthedissipationproleofapurejet,discussedatsomelengthinchapter 3 ,isnotpresentinthepureplumeatthisaxialdistance.Atz=30thedifferencesbetweentheturbulentkineticenergybalanceofajetandplumearelesspronounced.Themagnitudeofthedissipationtermoftheplumeisdecreasedfromit'svalueatz=15andtheterm'sproleisatterneartheowcenterline.Thechangeisbalancedbytheadvectionterm,whichhasasmallermagnitudeatz=30thanatz=15.Atz=15theturbulentkineticenergybalanceoftheforcedplumeismoresimilartothatofthejetthanthepureplume.Whiletheoff-centermaximumofthedissipationproleisnotpresent,themagnitudeoftheproleissimilartothatofthejet.Theothertermsinthebalanceallhaveroughlythesameshapesandmagnitudesasthatofthejet.Thebuoyancyterm,whilepresentisnegligiblecomparedtotheotherterms.Atthefurtheraxialdistanceofz=30,theforcedplumesturbulentkineticenergybalanceissimilarthatofthepureplume.Atthisdistancebothplumeshaveamuchsmaller 103

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advectiontermthanthepurejet.Thebuoyancytermoftheforcedplume,whilestillsmallisnowcomparableinmagnitudetothatofthepressureandandadvectionterms. 4.3.6ReynoldsStressTransportEquationsTransportequationsfortheReynoldsstresstermscanbederivedinasimilarfashionastheturbulentkineticenergybalance.Fromsymmetryalltermsthatinvolveoddpowersofu0arezeroandthereforeonlythenormalstressesandthehu0zu0rishearstresstermneedtobeconsidered.Thesefourtransportequationsare: u2ztransportequation:0=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(hUzi@ @zu02z+hUri@ @ru02z)]TJ /F8 11.955 Tf 11.95 16.86 Td[(2hu0z@p0 @zi)]TJ /F8 11.955 Tf 11.95 16.86 Td[(2hu02zi@hUzi @z+2hu0zu0ri@hUzi @r)]TJ /F8 11.955 Tf 11.96 16.86 Td[(@hu03zi @z+1 r@ @r)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(rhu02zu0ri)]TJ /F7 11.955 Tf 17.2 8.09 Td[(2 Re"h@u0z @z2i+h@u0z @r2i+1 r2h@u0z @2i#+2[Rihu0z0i] (4) u2rtransportequation:0=)]TJ /F8 11.955 Tf 11.29 16.85 Td[(hUzi@ @zu02r+hUri@ @ru02r)]TJ /F8 11.955 Tf 11.95 16.85 Td[(2hu0r@p0 @ri)]TJ /F8 11.955 Tf 11.96 16.85 Td[(2hu02ri@hUri @r+2hu0zu0ri@hUri @z)]TJ /F8 11.955 Tf 11.95 16.86 Td[(@hu0zu02ri @z+1 r@ @r)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(rhu03ri)]TJ /F7 11.955 Tf 11.95 0 Td[(2hu0ru0i r)]TJ /F7 11.955 Tf 17.2 8.09 Td[(2 Re"h@u0r @z2i+h@u0r @r2i+1 r2h@u0r @2i# (4) u2transportequation:0=)]TJ /F8 11.955 Tf 11.29 16.85 Td[(hUzi@ @zu02+hUri@ @ru02+2hUri ru02)]TJ /F8 11.955 Tf 11.95 16.85 Td[(2hu0@p0 @i)]TJ /F8 11.955 Tf 11.96 16.85 Td[(2hu0zu0i@hUi @z+2hu0ru0i@hUi @z)]TJ /F8 11.955 Tf 11.96 16.85 Td[(@ @zhu0zu02i+1 r@ @rrhu0ru02i+2hu0ru02i r)]TJ /F7 11.955 Tf 17.2 8.09 Td[(2 Re"h@u0 @z2i+h@u0 @r2i+1 r2h@u0 @2i# (4) 104

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uzurtransportequation:0=)]TJ /F8 11.955 Tf 11.29 16.86 Td[(hUzi@ @zhu0zu0ri+hUri@ @rhu0zu0ri)]TJ /F8 11.955 Tf 11.96 16.86 Td[(hu0r@p0 @zi+hu0z@p0 @ri)]TJ /F8 11.955 Tf 11.96 16.85 Td[(hu0zu0ri@hUzi @z+hu02ri@hUzi @r+hu02zi@hUri @z+hu0zu0ri@hUri @r)]TJ /F8 11.955 Tf 11.96 16.85 Td[(@ @zhu02zu0ri+1 r@ @rrhu0zu02ri)]TJ 20.45 8.08 Td[(hu0zu02i r)]TJ /F7 11.955 Tf 17.2 8.09 Td[(2 Reh@u0z @z@u0r @zi+h@u0z @r@u0r @ri+1 r2h@u0z @@u0r @i+2[Rihu0r0i] (4)Similartotheturbulentkineticenergybalancetherstvetermsinbrackets,foreachequation 4 through 4 ,representthecontributionofadvection,pressurediffusion,production,turbulentdiffusionanddissipationrespectively.ThesixthterminbracketsrepresentsthegenerationofReynoldsstressduetobuoyancyandisonlypresentinthehu02ziandhu0zu0ritransportequations.ThecontributiontoturbulentdiffusionfromtheviscoustermisnegligibleduetotherelativelylargeReynoldsnumberusedinthesimulationandisnotincludedinequations 4 through 4 .Figures 4-14 through 4-17 comparethehu02zi,hu02ri,hu02iandhu0zu0riReynoldstresstransportbalancesofthepurejet,pureplumeandforcedplumeattwodifferentaxialdistancesfromtheowsourcez=15andz=30.Themagnitudeoftheprolesofthepressuretermintheturbulentkineticenergybalance,whichisderivedfromsummingthepressuretermofthethreenormalReynoldsstresspressuretermsmakesonlyasmallcontributiontothetransportofturbulentkineticenergy.Lookingonlyattheturbulentkineticenergybalanceingure 4-13 oneisleftwiththeimpressionthatthederivativesofthepressuretermsarequitesmall.Howeverfurtherexaminationofthehu02ziandhu02ritermsingures 4-14 and 4-15 ,showsthatthesetermsareactuallyquitesignicantinbothjetsandplumes.Thepressureterminthehu02zitransportequationhasasimilarmagnitudeasdissipation,andinthehu02riequationitissimilarinmagnitudetotheadvectionterm.Thepressureterminthehu02zi 105

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Figure4-14. Prolesofthetransportofhu02ziReynoldsstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. 106

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Figure4-15. Prolesofthetransportofhu02riReynoldsstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. 107

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Figure4-16. Prolesofthetransportofhu02iReynoldsstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. 108

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Figure4-17. Prolesofthetransportofhu0zu0riReynoldsshearstressforthepurejet,pureplume,andforcedplumesimulationsofthecurrentstudyatz=15andz=30. 109

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transportequationhasanegativecontributiontototalturbulentkineticenergyandthecorrespondingterminthehu02riequationhasapositivecontribution.ThetransportofReynoldsshearstress,hu0zu0ri,examinedingure 4-17 ,ismostlyabalancebetweenproductionanddissipation,withtheadvection,turbulentdiffusionandpressuretermsonlyplayingaminorrole.Atthesmalleraxialdistanceofz=15boththeproductionanddissipationofReynoldsshearstressaresmallneartheowcenterlineofboththejetandtheforcedplume.Theproductionanddissipationofofturbulentshearstressbeingmostlyisolatedintheannularshearlayer.Thisislesstrueforthepureplumeatthisaxialdistanceawayformthesource,wheretheamountofturbulentshearstressproducedanddissipatedisnearthatproducedintheshearlayer.Byz=30thisisnolongerthecaseandthepureplumes'shearstressbalanceissimilartothatofthepurejetandforcedplume. 4.3.7VorticalStructuresandEntrainmentPlourdeetal.[ 42 ]examinedtheeffectofindividualvorticalstructuresofanaxisymmetricshearowontheentrainmentofambientuidintotheow.Inordertovisualizethevorticalstructureofthejet,plumeandforcedplume,wecalculateci,theimaginarypartofthecomplexeigenvaluesofthelocalvelocitygradienttensor,assuggestedbyZhouetal.[ 64 ]andChakrabortyetal.[ 18 ].Asdiscussedintheirpapers,ifatanypointthevelocitygradienttensorhasallrealeigenvalues,thentheowhaszeroswirlatthatpoint.Ifontheotherhandthelocalowisswirling,thenthevelocitygradienttensorwillhavetwocomplexconjugateeigenvalues,andthevalueoftheimaginarypart,ci,isameasureofthestrengthofthelocalswirlingmotion.Figure 4-18 presentstheiso-surfacesofci=0.5forthethreedifferentowsexaminedinthecurrentstudy.Thevalueci=0.5waschosensothattheimportantvorticalstructuresofallthreeowsareclearlyseen.Nearthesourcethepurejetandforcedplumehavesimilarvorticalstructures.Instabilitiesareseenformingveryclosetotheoriginintheformofmostlyvaricose 110

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Figure4-18. Comparisonofiso-surfaceofci=0.5ofapurejet,pureplumeandforcedplume. vortexrings,aroundalaminarcore.Acloseexaminationofthesevortexringsrevealsthattheinstabilitiesareslightlyhelicalinnature.Byz=4,secondaryinstabilitiesbegintoforminthepurejet.Inthecaseoftheforcedplume,similarsecondaryinstabilitiesbegintoappearslightlyfurtherawayfromthesource,z=6.Awayfromthesourceoftheow,astheaxialvelocitydecreases,thestrengthofthelocalswirlingmotionbeginstovisibilityweaken,andthestructureswhereci=0.5appearlesscloselypacked.Forthepurejetwhoseaxialvelocitydecaysasz1themaximumstrengthoftheswirlingmotionbeginstodropbelowci=0.5aboveanaxialdistanceofapproximatelyz>17,.Intheforcedplume,wheretheaxialvelocityhasadecayrateofz1=3,thevorticalstructureswithci=0.5begintobecomelesscloselypackedafterz>24.Thevorticalstructuresofthepureplumenearthesourceappearquitedifferentthanthoseofthetwoowswithinitialmomentumux.Inthepureplumeinstabilitiesrstappearalongtheplume'scenterlineandnolaminarcoreispresent.Thevortical 111

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structuresremainisolatedalongtheplumeaxisuntilapproximatelyz=7andthenbegintoquicklywiden.Thevorticalstructuresoftheforcedplumeinthisstudyappeartobehavemoresimilarlytothatofapurejet,thanofapureplume.Itshouldbenotedthattheparametersoftheforcedplumeinthisstudywaschosensothattheinitialmomentumuxwouldbefairlylargecomparedtotheinitialbuoyancyux,andsothatthelengthscale5LM,Iwouldbeapproximatelyonehalftheaxialdomain.Itispossiblethatanotherforcedplume,withaweakerinitialmomentumux,wouldhavemoreplumelikevorticalstructuresnearthesource.Intheirseminalpaperonplumes,Morton,TaylorandTurner[ 35 ]suggestedthatthemeanradialvelocityattheedgeofanaxisymmetricboundarylayersuchasajetoraplumeisproportionaltotheaxialvelocity.Thatis,hUri=hUzi.Theysuggestedthatthecoefcientofentrainment,M,couldbeevaluatedas, M=d dzR10hUzirdr 2R10hUzi2rdr1=2.(4)Husseinetal.[ 26 ]showedthatforaselfsimilaraxisymmetricowequation 4 isequivalentto M=I1=2I1=22,(4)where, I1=2Z10hUzi hUz,cid,(4)and, I2=2Z10hUzi hUz,ci2d.(4)TheGaussiant, hUzi hUz,ci=exp )]TJ /F4 11.955 Tf 9.3 0 Td[(ln(1=2) 21=22!,(4)canbeusedfortheselfsimilaraxialvelocityprole.Thehalfwidthofthepurejet,forcedplume,andpureplumewereevaluatedas,1=2=0.94,0.101,and0.102respectively. 112

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Thecorrespondingentrainmentcoefcientsforthepurejet,forcedplumeandpureplumeare,M=0.084,0.90,and0.091respectively.Thecoefcientofentrainmentforthepureandforcedplumesareapproximately16%largerthanthatofapurejet,suggestingbuoyantowsentrainmoreambientuidthannon-buoyantowswiththesameaxialvelocity.Sinceequation 4 isonlyvalidfarfromtheowsource,thiscanonlybesaidofplumesandjetsinthesimilarityregionoftheow.Phametal.[ 41 ]suggestedanalternateapproximationfortheentrainmentcoefcient, P(z)=d dzR10hUz(z)irdr (z)hUz,ci,(4)whereisthemomentumbalancelengthscaledenedas, 2(z)=2)]TJ 5.48 -.05 Td[(R10hUz(z)irdr2 R10hUz(z)i2rdr.(4)ForaselfsimilarjetPisequivalentto P=1 2p I2,(4)which,usingthedatafromthecurrentstudy,canbeevaluatedasP=0.042,0.45,and0.45,forapurejet,forcedplumeandpureplumerespectively.P(z)canalsobeevaluatednumerically,whichenablesustoexaminethechangesinentrainmentnearthesourceoftheowwherePisnotexpectedtobeconstant.Figure 4-19 comparesP(z)ofthethreeowsexaminedinthecurrentstudy.Itisinterestingtocomparetheevolutionofthebehaviorofentrainmentshowningure 4-19 withtheevolutionofthevorticalstructuresdisplayedingure 4-18 .ForboththepurejetandtheforcedplumeP(z)increasesrapidlyfrom0
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Figure4-19. ComparisonofthecoefcientofentrainmentasdenedbyPham[ 41 ],p(z).Forapurejet,pureplumeandaforcedplume. z=7.Atthesetwodistancesfromthesource,secondaryinstabilitiesarebeginningtoforminthetworespectiveows.From7.z.16,wheretheowstransitiontofullturbulence,therateofentrainmentincreases.Inthesimilarityregionabovez.16Pisroughlyconstant.Thebehaviorofentrainmentnearthesourceofthepureplumeisquitedifferentthanthatofthetwoowswithinitialmomentumux.Inthepureplumewheretheinitialaxialvelocityatz=0isuz(z=0)=0,theradialvelocityisofthesameorderastheaxialvelocity,thatistheplume'sentrainmentcoefcientisP(z1)1.Aszincreasestherateofentrainmentbeginstorapidlydecreaseuntilapproximatelyz4,wherePbeginstoleveloffinapproximatelythesameregionoftheowwherethininstabilitiesareformingalongtheplume'scenterline.Atz=6alocalminimumisreachedassecondaryinstabilitiesformneartheowsaxis.ThevalueofPcontinuestoincreaseuntilz25wheretherateofentrainmentbecomesroughlyconstant.This 114

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suggeststhatthepureplume,intermsofrateofentrainment,becomesselfsimilarfurtherawayfromthesourcethanapurejetoraforcedplume. 4.3.8SpectraVariousresearchers[ 13 21 41 42 56 ]haveobtainedinsightsintothebehavioroftheuctuatingvelocitycomponentsbyexaminingthetemporalspectraandusingtheTaylorhypothesistorelatetheresultstoonedimensionalspectraintheaxialdirection.Inthecurrentstudyonedimensionalspectraintheazimuthaldirectionwasobtaineddirectlybycalculatingthefouriertransform, ^ui(r,f,z,t)=1 p 2Z20ui(r,,z,t)e)]TJ /F5 7.97 Tf 6.59 0 Td[(ifd.(4)Numericalevaluationofthefouriertransformwaswith256evenlyspacedpointsinthetangentialdirectionatagivenlocationinther)]TJ /F4 11.955 Tf 12.54 0 Td[(zplane.Instantaneousspectrasarethenaveragedovertime.Inall,theenergyspectrawascalculatedateightdifferentpointslocatedatr=0.6,1.6andz=20,25,30,and35.Theshapeandmagnitudeofthethreevelocitycomponentsweresimilarforallthreeows.Inordertobettercomparethespectrabetweenowsandtobetterrelatetheenergyspectratoturbulentkineticenergywedene, h^ki=1 2j^uzj2+j^urj2+j^uj2.(4)Figure 4-20 plotstheenergyspectraofthethreeaxisymmetricowsattwodifferentradialpositionsr=0.6,1.6atz=30.Inthegurethefrequency,f,hasbeendividedbytheradialdistancefromthecenterlinesothatthefrequenciesatbothradialpositionsrepresentthesamewavelengthsintheazimuthaldirection.Thestraightlinesinthegurerepresentf)]TJ /F9 7.97 Tf 6.58 0 Td[(5=3,f)]TJ /F9 7.97 Tf 6.59 0 Td[(3andf)]TJ /F9 7.97 Tf 6.59 0 Td[(7.The)]TJ /F7 11.955 Tf 9.3 0 Td[(5=3slopeisrelevantintheinertialrangeandhereitcanbeseenthatthisbehaviordominatesonlyforverylowfrequencies.ThisissimilartotheresultsofBogeyandBailly[ 13 ]whocomparedtheonedimensionalspectrumoftheaxialvelocityalongthecenterlineofseveraljetswithvaryingReynoldsnumbers.Aslopesteeperthan)]TJ /F7 11.955 Tf 9.3 0 Td[(5=3isanindicationthatdissipationisplayingalarger 115

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Figure4-20. Onedimensionalvelocityspectraintheazimuthaldirectionforeachvelocitycomponent,attwodifferentlocationsonther)]TJ /F4 11.955 Tf 11.96 0 Td[(zplane roleintheenergycascade.Asdissipationbecomesdominantathigherfrequenciesaowdependentsteeperslopeiscanbeexpected.Forexample,inbuoyantjetsandplumesa)]TJ /F7 11.955 Tf 9.29 0 Td[(3slopeisobservedinthedissipativeregime[ 41 42 ].Thevalueof)]TJ /F7 11.955 Tf 9.3 0 Td[(7wasobservedbyFellouahetal.[ 21 ]inpurejets.Heretheforcedplume,whichcanalsobetermedasabuoyantjet,followsthebehaviorofthepureplumemorecloselythanthatofthepurejet. 4.4SummaryDNSstudiesofapurejet,apureplumeandaforcedplumewerecompletedandcompared.Aftereachsimulationbecamestatisticallysteady,turbulentstatisticswerecollectedoverenoughtimestepsforthirdorderstatisticstoconverge.Theconvergenceofthethirdorderstatisticsrequireslesstimestepsclosertothesourceoftheow.Themeanowbehaviorofthepurejet,pureplume,andforcedplumefarfromthesourcebehavedsimilarlytothatofpaststudiesandknownsimilaritysolutions.TheaxialdistancefromthesourceatwhichthesimulatedforcedplumeconformedtotheselfsimilarsolutionofapureplumeroughlymatchedthataspredictedbyMorton[ 34 ].Firstandsecondorderstatisticsofthethreeowswerefoundtoagreewellwithpreviousstudiesinthesimilarityregion. 116

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Theresultsregardingthetotalmomentumintegral,M(z),discussedinchapter 3 werereviewed.Inthecaseofbuoyantowsthemomentumintegralwillnotbeconserved,howeverbyintegratingthebuoyancyequation,thebuoyancyux,JB(z),isfoundtobeconstantforallaxialdistancesz.ThebuoyancyuxforthetwosimulatedbuoyantowswascalculatednumericallyandJBwasfoundtobeaconstanttowithin1%forallaxialdistancesgreaterthanz>10.Farfromthesourceofthesimulatedpureplumetheuctuatingtermhu0z0icontributedto17%ofthetotalbuoyancyux.Closertothesource,however,theuctuatingtermcontributedasmuchas32%.Inthecaseofthesimulatedforcedplumethecontributionoftheuctuatingtermtothetotalmomentumuxfarfromthesourceis13%.Theaxialturbulentintensityalongthejetandforcedplumesaxisevolvesinasimilarmanner,startingaszeroatzequalszeroandincreasinguntilreachingaconstantvalueofapproximatelyhu20zi1=2=hUz,ci=0.25byz=12.Theevolutionoftheaxialturbulentintensityofthepureplumehoweverbeginsasanonzerovalue,rapidlyincreasesandthendecreasesbeforenallybecomingaconstantofapproximatelyhu02zi1=2=hUz,ci=0.25atafurtheraxialdistancefromthesource,z=18,thantheothertwoowsexaminedinthisstudy.Anexaminationoftheradialproleofaxialturbulentintensityrevealsthattheadventofapureplumehavingagreateraxialvelocityturbulentintensitythanaowwithinitialmomentumux,appearsisolatedneartheowaxis.Awayfromtheowcenterlinetheturbulentintensityofapureplumefollowsasimilar,althoughslightlynarrower,Gaussianproleasthatofapurejetorforcedplume.Thethirdordervelocityproles,turbulentkineticenergyequationandReynoldsstressequationswerecalculatedandexaminedattwodifferentaxialdistancesfromthesourceoftheow,z=15andz=30.z=15beingnearthelowerboundofthesimilarityregionofthepurejetandpureplumeandz=30beingroughlythedistancepredictedbyMorton[ 34 ]atwhichaforcedplumecanbeconsideredtohavepureplumebehavior.Thethirdorderstatisticsofapureplumeatz=15whichincludeu0ztoany 117

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powerbehavequitedifferentlythanthecorrespondingstatisticsofapurejetorforcedplume.Atanaxialdistanceofz=30thepureplume'sthirdorderstatisticsaresimilartothatoftheothertwoowsdiscussedinthisstudy.Verylittlechangesbetweenthethirdordervelocityprolesofathepurejetandforcedplumeatthetworadialdistancesz=15andz=30.Theexceptiontothisisthehu03zitermwhichispositivealongthecenterlineatz=15buthasanegativevalueatthecenterlineatz=30,forbothows.TheturbulentkineticenergybalanceandReynoldsstresstransportequationsaredependentonthesecondandthirdorderstatisticsofthevariousaxisymmetricows.Assuchitisnotsurprisingthatliketherstandsecondorderstatisticsthetransportequationsofthepurejetandforcedplumeareroughlysimilaratbothaxialdistancesexamined.Thepureplumeatz=15hasdifferentbehaviorthantheothertwoows.Thisismostapparentinthedissipationtermwhichhasamuchlargermagnitudeinapureplumethaninaowwithinitialmomentumux.ThetransportofReynoldsshearstressismostlyabalancebetweenproductionanddissipationwithpressurediffusion,turbulentdiffusionandbuoyancyplayingalesserrole.Injetsandforcedplumesatz=15bothproductionanddissipationaresmallneartheowaxis,whereasthesetermsareneartheirmaximumvalueinapureplume.Thevorticalstructuresofthethreeowswereexaminedbycalculatingci,theimaginarypartofthecomplexeigenvaluesofthestresstensor,ateachgridpoint.Theevolutionofvorticalstructuresofthepurejetandpureplumeareverysimilar;mostlyhelicalvortexringsbegintoformveryclosetotheowsourcearoundalaminarcore.Thelaminarcorenarrowsasthevortexringsgrowandsecondaryinstabilitiesbegintoformaroundz=6,leadingtowardfullyturbulentow.Inthecaseofapureplume,however,instabilitiesrstappearalongthecenterlineoftheowandspreadoutward.Theentrainmentofambientuidwasdiscussed.Itwasdeterminedthatawayfromthesource,buoyantowsentrainslightlymoreuidforagivenaxialvelocitythannonbuoyantows.Veryclosetotheowsource,theentrainmentcoefcientsof 118

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axisymmetricowswithinitialmomentumareverysmall,whereastheradialvelocityofapureplumenearthesourceisonthesameorderastheaxialvelocity.Theonedimensionalvelocityspectraintheazimuthaldirectionwascalculateddirectly.The)]TJ /F7 11.955 Tf 9.3 0 Td[(5=3slopeindicativeoftheinertialregimeofturbulentowswasobservedforallthreeows.Inthedissipativeregimethecharacteristicslopeof)]TJ /F7 11.955 Tf 9.3 0 Td[(7whichwasobservedbyFellouahetal.[ 21 ]wasfoundtobepresentforthepurejet.Forboththepureandforcedplumetheslopeof)]TJ /F7 11.955 Tf 9.29 0 Td[(3associatedwithbuoyantowswasobserved.Inthenextphaseofthisresearchthecodeusedtoperformdirectnumericalsimulationsofthesinglephaseowsdiscussedinthischapterwillbemodiedinordertoexaminemultiphaseforcedplumes.Progresstowardthisgoalwillbediscussedinchapter 6 119

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CHAPTER5PIVEXPERIMENTSOFAMULTIPHASEPLUME 5.1BackgroundInthepreviouschapterswefocusedsinglephaseplumes,whereitwasassumedthattheplumeandtheambientenvironmentconsistedofthesameuid.Thebuoyancydrivingtheowwasassumedtobeduetoeitheratemperaturedifferenceorsomeotherscalarphenomenawhichcausedthedensityoftheuidinsidetheplumetobedifferentthanthatoftheambientuid.Inmultiphaseplumesbuoyancyisduetotheinjectionofamaterial,differentthantheambientuid,attheplume'ssource.Thematerialcanbeasolid,liquidorgaswhichformsparticles,dropsorbubblesrespectivelyastheplumerisesorfalls.Itiscommontorefertotheambientuidasthecontinuousphaseandtheinjectedparticles,bubblesordropsasthedispersedphase.Forthesakeofbrevity,throughouttherestofthischapteranysmallpacketofthedispersedphaseshalebereferredtoasaparticle.Itistobeunderstoodthatinthiscontext,aparticlecanbeeitherasolid,liquid,oragas,andcanbeeithermoreorlessdensethantheambientuid.Atthesourcewherethedispersedphaseisinjectedintotheambientuid,thedensitydifferencebetweentheindividualparticlesandtheambientuidwillcausetheparticlestoriseorfall.Astheydosotheparticleswillentrainingtheambientuidaroundthem.Thiscausestheuidparticlemixturetohaveadifferentbulkdensitythantheambientuidwhereparticlesarenotpresent.Theentraineduidbeginstoriseorfallenmassewiththeparticles,formingaplume.Itisinterestingtonotethattheresultingplumewillbehaveasifitistheresultofthedensitydifferencebetweentheparticle-uidmixtureandtheambientuid,asopposedtothedensitydifferencebetweentheindividualparticlesandtheambientuid.Assuch,thebulkvelocityoftheplumewillbeverydifferentthantheterminalvelocityof 120

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Figure5-1. Diagramofamultiphaseplume theindividualparticles.Themixtureofparticlesanduidcanoftenbetreatedasacontinuum.Aschematicofamultiphaseplumeispresentedingure 5-1 .Hereparticlesofdiameterdwithadensity,,lessthantheambientdensity,aareinjectedintotheambientenvironmentthroughacircularopeningofdiameterD.Multiphaseplumesarecommoninenvironmentalandindustrialapplications.AmajormotivationforthiscurrentworkwastheDeepWaterHorizonOilspillinthegulfofMexico.Indeepwateroilspillsamixtureofoilandnaturalgasunderintensepressurecanescapeintotheocean.Thereleasedoilandgasquicklybreakupintosmalldropletsformingabubblyplumewiththeentrainedseawater.Otherexamplescanbefoundinaerationsystems,carbonsequestrationschemes,ashrisingfromavolcano,tonameafew.Agreatdealoftheunderliningphysicsofmultiphaseplumesisnotyetwellunderstoodandresearchintothetopicisongoing.Kobus[ 27 ]performedsomeoftheearliestworkonmultiphaseplumes.Usingairbubblesasabuoyancysource,heperformedlaboratoryexperimentsina4.7mtank.HepostulatedGaussianprolesand 121

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alongwithanentrainmentassumptionsimilartothatofMorton,etal.[ 35 ],proposedanintegralmodelforsuchows.VariousothersincludingMcDougall[ 33 ],Wuest[ 60 ],andAsaedaandImberger[ 6 ]improvedandextendedonKobus'earlymodeltoplumesinastratiedenvironment.SocolofskyandAdams[ 52 ]exploredtheroleofsettlingorslipvelocityonthedynamicsofamultiphaseplume.Seoletal.[ 47 ]conductedParticleImageVelocimetry(PIV)experimentsandreportedonthemeanowcharacteristicsofabubbleplume.InthiscurrentstudyaPIVexperimentiscompletedofaplumecausedbytheinjectionofparticlesheavierthantheambientwaterintoa1008080cmtank.FluorescenttracerparticleandanopticallterwereusedinordertodistinguishbetweenthecontinuousanddispersedphasevelocityeldsusinganimagecaptureseparationtechniquesimilartothatdescribedbySridharetal.[ 53 ],Hilgersetal[ 25 ]andDeanetal.[ 20 ].2000imageswerecapturedineachviewingwindow.Inthismannerboththerstandsecondorderstatisticsofboththecontinuousanddispersedvelocityeldscouldbeanalysed.Inordertoavoidconfusion,throughouttheremainderofthischapter,theuorescenttracerparticlesaddedinordertocapturethevelocityeldofthecontinuesphaseshalebereferredtoas'tracerparticles.'Thedenserthanwaterparticlesinjectedintotheexperimentaltank,whichmakeupthedispersedphaseofthemultiphaseplumeshalebereferredtoas'buoyancyparticles.'Webegininsection 5.2 wheretheexperimentalsetupanddetailsoftheproceduresusedtoacquireandanalyzethedatewillbediscussed.Resultswillbepresentedinsection 5.3 .Conclusionsandfutureworkwillbediscussedinsection 5.4 5.2ExperimentalSetupExperimentswereconductedatEcoleNationaleSuprieuredeMecaniqueetd'Aerotechnique(ENSMA)nearPoitiersFranceinafacilitydesignedforthispurpose.Figure 5-2 isaphotographofthefacilitywhichconsistsofatypicalPIVsetupincluding 122

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Figure5-2. Photographoftheexperimentalsetup. aNewWaveResearchSoloIII-15hzlaserandaDantecHisenesPIVcamerabothmountedonCharlyrobotC142roboticracks.TheroboticrackswerecontrolledbyLabVIEWandallowedthepositionofthecameraandlasertobealteredwithaprecisionof0.1mm.Thecameraandlaserwerefocused,ata90degreeangletooneanother,ata1000800800mmwatertankwith8viewingwindows.ThetimingofthelaserandcameraweresyncedbyaDANTECDK-2740PIVcontroller.DatawascapturedonaPCcomputerwithDantecFlowManagerVersion4.71softwareandlateranalyzedusingDantecDynamicStudioversion3.30.49Figure 5-3 detailsoftheconstructionanddimensionsoftheexperimentalwatertankwhichwasconstructedwithtwospillwaysandgutters,whichfedintoacommoncollector.Inthiswaythedepthofthewaterremainedconstantthroughouttheexperiment.Thevolumeowrateofthewater-particlemixtureinjectedintothewatertankcouldbe 123

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Figure5-3. ExperimentalWaterTank veriedbydeterminingthevolumeofwaterwhichspilledoverthespillwaysoveragivenperiodoftime.Inordertoexaminetheeffectofvariousinitialmomentumandbuoyancyuxesinrelationtooneanother,amechanismwasdesignedinordertocontroltherateofinjectionofthebuoyancyparticlesandlabwater,whichwasusedasthecontinuousphase,separatelythroughthesame12mmopening.Figure 5-4 isaphotographoftheinjectionsystem.Thebuoyancyparticles,whichwereglassbeads200to300mindiameterwithadensityof1.51kilogramsperliter,weredroppedthroughanhourglasstypedevicewhichconsistedofasyringewithanopenofapproximately3mm.Thehourglassdevicereleased29.75mlperminofbuoyancyparticles,asmeasuredduringtheexperiment.Infutureexperimentsthesyringeistobereplacedwithanaperturedevicewhichwillallowfordifferentsizedopeningsandthereforedifferentowrates.Thereleasedparticlesfellintoafunnelwhichwashalfsubmergedinthesurface 124

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Figure5-4. Waterandbuoyancyparticleinjectionsystem. oftheexperimentaltank.A12cmlongcopperpipewitha12mminternaldiameterwasalignednormaltothethewater'ssurfaceandattachedtothebottomofthefunnel.Asecondfunnelwasplacedabovethesyringeinordertoserveasahopper.Asecond'reservoir'watertankmeasuring600600400mmwasplacedapproximately5metersabovetheexperimentaltank.Waterwasfedfromthereservoirtanktothehalfsubmergedfunnelthroughaplastichose.Twovalvesalongtheplastichosewereusedtocontroltheowrateofwaterbeingfedtothefunnelandthereforethroughthecoppertube,theendofwhichservesatthesourceoftheplume.Carewastakentomakesuretheheightofthewaterinthereservoirtankremainedroughlyconstantthroughouttheexperiment.Theowrateoftheinjectedwater,buoyancyparticlesandwaterparticlemixturewasmeasuredperiodicallythroughout 125

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theexperimenttocheckforconsistency.Thiswasdonebothbymeasuringtheowrateofparticlesreleasedfromthesyringeandtheowrateofthewaterwhichoverowedoverthesidesoftheexperimentaltankdirectly,andbymeasuringthevelocityoftheuidparticlemixtureattheplumessourceusingthePIVsystem.Theowrateofthewaterwasfoundtobeconsistentuptoa7%errorandtheparticleowratewasconstantuptoa18%error,whichwasfoundtobesufcientforthisexperiment.WhenperformingPIVonmultiphaseaow,caremustbetakentodistinguishbetweentheparticleswhichmakeupthedispersedphaseandthetracerparticlesaddedtotothecontinuousphase.Techniquestodothisgenerallyfallintotwocategories,imagecaptureseparationtechniqueswheretwodifferentimagesarecapturedatthetimeoftheexperiment,andpostprocessingtechniqueswhichattempttodistinguishbetweenthetwoparticlesbasedonsizeandorlightintensityduringtheimageprocessingphase[ 47 ].InthiscurrentstudythecontinuousanddispersedphasevelocityeldswereweredistinguishedbyanimagecaptureseparationtechniquesimilartothatdescribedbySridharetal.[ 53 ],Hilgersetal[ 25 ]andDeanetal.[ 20 ].FluorescenttracerparticleswereaddedtotheuidandanopticallterwhichissensitivetothewavelengthoflightatwhichthetracerparticlesuorescewasplacedinfrontofthePIVcamera.Withthelterpresentonlythetracerparticleswerepickedupbythecameraandthecontinuousphasevelocityeldcouldbecalculated.Sincetheintensityofnonlteredlaserlightreectingoffthebuoyancyparticlesismuchgreaterthanthatofthetracerparticles,whenthelterisremovedonlythebuoyancyparticlesarepickedupbythePIVcamera'sCCD,andthebuoyancyparticlevelocityeldcanbecalculated.PolymethylmethacrylateparticlescoatedinRhodamineBwithadiameterbetween1and20mandadensityof1.19kilogramsperliterwereusedasthetracerparticlesandaddedtoboththeexperimentalandreservoirtank.AsderivedbyStokesanddescribedinWhite[ 57 ],thesettlingvelocity,usofasphericalparticlecan 126

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beapproximatedas, us=gd2 18,(5)wheregistheaccelerationduetogravity,disthediameteroftheparticle,isthekinematicviscosityoftheuidandisdenedas, =d)]TJ /F3 11.955 Tf 11.95 0 Td[(c c.(5)Here,dreferstothedensityofthedispersedphaseparticlesandcrepresentsthedensityofthecontinuousphase.Thereforethesettlingvelocitiesoftheaveragebuoyancyandlargesttracerparticlesareus,b=1.1810)]TJ /F9 7.97 Tf 6.58 0 Td[(2m=sandus,t=2.7610)]TJ /F9 7.97 Tf 6.58 0 Td[(5m=srespectively.ThesettlingvelocityofthetracerparticlesisthreeordersofmagnitudelessthanthatofthebuoyancyparticlesandismuchlessthanthesensitivityofthePIVsystem.Assuch,thedensitydifferencebetweenthetracerparticlesandthecontinuousphaseshouldnotadverselyeffecttheexperimentalresults.ThePIVsystemcaptured4imagepairspersecondandcouldstore500imagesinit'sbufferbeforerecordingtotheharddrive,consideredoneimageburst.Inordertoconservedrycleanbuoyancyparticles,betweeneachimageburst,theowofparticleswouldbestopped.Afterrestartingtheowofparticles,60secondswereallowedtopassbeforeimagerecordingwasresumed.Asthesettlingvelocityofofthebuoyancyparticleswasontheorderof1cm=sec,andviewabledomainmeasuredwas25cm,thiswasdeemedlongenoughfortheplumetobecomestatisticallysteady.After2000imagesofeachviewingareawerecaptured,thevolumeowrateoftheparticlesandwater-particlemixturewouldbemeasuredandthecameraandlaserwouldbemovedtothenextviewingarea.ThePIVcamera,hadaresolution1,280by1,024pixels.Eachviewingareawas128.6mmby102.9mm,withanoverlapof10mmbetweeneachviewingarea.Theinterrogationzoneswere88pixelswitha50%overlap.Threeviewingareaseachcenteredonandshiftedalongtheplumeaxiswereusedtorecordthetracerparticles.Theopticallterwasthenremovedandthesimilarviewing 127

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areaswereusedtorecordthebuoyancyparticles.Thisallowedeachvelocityeldtobecalculatedinadomainstretchingfromtheplumessourceto250mmalongtheplumesaxis.Duetothesupportbeamlocatedbetweentheupperandlowerviewingwindow,aregionbetween135mmand166mmalongtheplumeaxiswasnotrecorded.Thetimebetweenlaserpulsepairswasresetforeachviewingwindowinordertobestcapturetheaxialvelocityoftheplumeatthatlocation.Asistypicalofaxisymmetricshearows,themagnitudeoftheradialvelocitycomponentismuchsmallerthanthatofthetheaxialcomponent.AlimitationofthePIVsetupistheradialcomponentofvelocityinashearow,cannotbecapturedwithanyaccuracyatthesametimeastheaxialvelocity.Assuch,whilebothplanercomponentsofvelocitywererecorded,onlytheaxialvelocitycomponentswillbediscussedbelow.Asanotherlimitationofthecurrentexperimentalsetup,noattemptwasmadetomeasurethevolumefractionoftheparticlesatanylocationotherthantheplume'ssource. 5.3ResultsTwoexperimentswerecompleted.Apurejetexperimentwherenobuoyancyparticleswereinjected,andamultiphaseplumeexperiment. 5.3.1JetExperimentInordertogaincondenceinthePIVandexperimentalsetup,apurejetexperimentwascompetedbeforeperforminganexperimentofamultiphaseplume.Theaxialvelocityofthewaterinjectedthroughthe12mmcopperpipewasUI=0.380m=s.ThisyieldsaReynoldsnumberofRe=3,040whichiscomparabletothatofthepurejetsimulatedinchapter 3 .Sincebuoyancyparticlesdidnotneedtobeinjectedinthejetexperiment,theparticleinjectionmechanismdescribedintheprevioussectionwasnotneeded.Therefore,thecopperpipeusedasthejetsourcecouldbeplacedclosertotheexperimentaltankssurface.Thisallowedtheexperimentaljettobeexaminedinan 128

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Figure5-5. Contourplotoftheaxialvelocity,hUzioftheexperimentaljet. uninterrupteddomain,entirelythroughtheupperviewingwindow,stretchingfromthejetssourceto23inletdiametersdownstreamintheaxialdirection.Figure 5-5 isacontourplotoftheaxialvelocityscaledbytheinletvelocity,hUzi=UI.Heretheaxialcoordinate,z,hasbeeninvertedsothatthesourceofthejetappearsonthebottomandsothejethasthesameorientationasthesimulatedjetsandplumesdiscussedinthepreviouschapters.Figure 5-6 depictsthedecayofcenterlineaxialvelocityofthecurrentexperimentalstudycomparedtotheexperimentalstudiesbyHusseinetal.[ 26 ]andPanchapakesanetal.[ 39 ]aswellasthenumericalstudyperformedbyKung[ 28 ]andthesimulatedjet 129

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Figure5-6. Initialinletvelocity,UI,overmeancenterlineaxialvelocity,hUz,ci,ofcurrentexperimentalstudy(lledindiamonds)comparedtocurrentDNS(hallowcircles),Husseinetal.[ 26 ](solidline),Panchapakesanetal.[ 39 ](dashedline)andKung[ 28 ](opensquares). whichwasdiscussedinchapter 3 .Aswasdiscussedinchapter 3 thereciprocalofthecenterlinevelocityscaleslinearlywithrespecttoz.Fromthatchapterrecall, Uc(z)Cd (z+zv).(5)Forthecurrentexperimentaljet,thecoefcientofdecayisfoundtobeCd=5.88andthevirtualoriginisfoundtobelocatedatzv=2.9.Ascanbeseeninthegure,thesearesimilartothevaluesdiscussedinsection 3.3.1 .Figure 5-7 comparestheexperimentaljet'saxialvelocityproleatz=D=15withthatofthesimulatedjetdiscussedinchapter 3 andotherpreviousexperimentalstudies.Theaxialvelocityhasbeenscaledbyit'speakvalueatthisaxialdistance.Thejetisslightlynarrowerthanthejetsfrompreviousstudieswithajethalfwidthof 130

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Figure5-7. Comparisonofexperimentalstudy'stimeaveragedaxialvelocityprole(lleddiamonds)atanaxialdistanceof,z=D=15,withDNS(hallowcirclesandsquares)andcurvetsofexperimentalresultsbyHusseinetal.[ 26 ](solidlines),Panchapakesanetal.[ 39 ](dashedlines)andWygnanskiandFiedler[ 61 ](dash-dotlines). 1=2=0.081ascomparedtothevalueof1=2=0.094foundforthesimulatedjetandtheexperimentaljetexaminedbyHusseinetal.[ 26 ].Figure 5-8 comparestheuctuationsoftheaxialvelocity,p hu02zinormalizedbythecenterlinemeanaxialvelocity,withpreviousstudiesandtheDNSjetdiscussedinchapters 3 and 4 .TheevolutionofaxialturbulentintensityalongthejetaxisbehavedsimilarlytothatofthejetexaminedbyAmielhetal.[ 5 ]whichobtainsaconstantvalueofapproximatelyp hu02zi=hUz,ci=0.25forlargevaluesofz.ThisisthesamefareldvalueobtainedbythenumericalsimulationsperformedbyKung[ 28 ]anddiscussedinchapter 3 .Thenumericalsimulationsapproachthisconstantvaluemuchclosertothejetsourcethanthetwoexperimentalstudiesdisplayedinthegure.Therightsideofgure 5-8 comparesthatproleofaxialturbulentintensityatanaxialdistanceofz=D=15.Similartowhatwasfoundinpreviousstudiesincludedinthegure,themaximumaxialturbulentintensitywasnotfoundalongthejet'scenterline. 131

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Figure5-8. Comparisonofhu02zi=hUz,ciofthecurrentexperimentaljetwithpreviousnumericalandexperimentalstudies.Alongthejetscenterline(left)andtheradialproleatanaxialdistanceofz=D=15(right). Onawholethejetexaminedinthiscurrentexperimentalstudybehavedsimilarlytothatofpreviousstudies,givinguscondenceinourexperimentalsetupanddataacquisitiontechniques. 5.3.2MultiphasePlumeForthemultiphaseplumeexperimentglassbeadswithanaveragediameterofd=250mwereinjectedintothewatertankataowrateof29.75.0ml=min.Theparticleshadadensityof1.51kg=lwhichyields=0.52withlabwaterastheambientuid.Themixtureofwaterandparticleswasinjectedintothewatertankwithaninjectionvelocityof0.082m=s.Thevolumefractionofparticlesintheinjectedmixtureatthesourcewas5.1%.Inmanynumericalstudiesofmultiphaseowssuchas[ 8 17 50 ]wherethevolumefractionislowandwherethestokesnumberissmallbutnon-negligible,thevelocityeldofthedispersedphaseisassumedtobeequaltothevelocityeldofthecontinuousphaseplusthesettlingvelocity, hVihUi+us,(5) 132

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Figure5-9. ImagefromPIVcameraTracerParticles(left)andBuoyancyParticles(right). WherehViisthemeanowvelocityeldofthedispersedphase.Heretheparticletimescaleis, p=d2 18=1.210)]TJ /F9 7.97 Tf 6.58 0 Td[(3sec,(5)andthetimescaleofthesmallesteddyisapproximatedtobenomorethan=0.017secyieldingaKolomorgovStokesnumberofSt=0.069.Assuch,thedifferencebetweentheaxialvelocitycomponentofthecontinuousanddispersedphaseisapproximatedtobeequalto, (hVzi)-222(hUzi)us1.18x10)]TJ /F9 7.97 Tf 6.59 0 Td[(2m=s.(5)Figure 5-9 displaysunprocessedimagesofthetracerparticles(left)andthebuoyancyparticles(right)neartheplumesource.Intheleftimage,wheretheopticallterblockedalllightnotofthewavelengthatwhichthetracerparticlesuoresced,theimageappearsfarbrighter,andmoreindividualtracerparticlescanbeseenontherightsideoftheimage.Thelaserwhichilluminatesthetracerparticlesislocatedontherightsideofthisimage.Evenwithalocalbuoyancyparticlevolumefractionoflessthan5%asignicantamountoflaserlightisblockedbythebuoyancyparticlesintheow.Assuchtheleftsideifthisimageappearsdarkerthantheright. 133

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Figure5-10. Contourplotofaxialvelocity,hUzi,ofthecontinuousphase(left)anddispersedphase(right). Inordertogetaqualitativesenseofthebehavioroftheplume,contourplotsoftheaxialvelocityforboththecontinuousanddispersedvelocityeldcanbeseeningure 5-10 .Aswasdoneforthecontourplotofthejetaxialvelocity,theimageshavebeeninvertedsothattheplumesourceappearsonthebottom,inlinewiththesimulatedjetsandplumesdiscussedinearlierchapters.Anareabetween13
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areaforthebuoyancyparticles.Assuchonlytheviewingareaclosestandfurthestawayfromthesourceisincludedinthecontourplotofthedispersedphaseaxialvelocity.Ascanbeseeninthegureboththecontinuousanddispersedphasesinitiallyaccelerateafterbeinginjectedintotheexperimentaltank.Afterreachingamaximumvelocityafewinletdiametersdownstreamfromthesourcebothphasesbegintodecelerate,similartothebehaviorseeninthesinglephaseforcedplumediscussedinchapter 4 .Fromtheappearanceofthecontourplots,itisassumedthatfarfromthesourcetheforcedplumebecomesselfsimilar,andthatthemeanaxialvelocityofbothphasescanbedescribedbyanequationoftheform, hUzi=AhUz,Ii (z+zv)exp ln1 2 21=22!.(5)Here,isthesimilaritycoordinatedenedas, =r z+zv,(5)and1=2istheplumehalfwidth.Theexponentsigniestherateofdecayofaxialvelocitywithrespecttoz,andthecoefcientAistheslopeofthelinearizedcurvet, hUz,ci=A(z+zv))]TJ /F6 7.97 Tf 6.59 0 Td[(.(5)Theleftsideofgure 5-11 displaystheevolutionoftheplume'scenterlineaxialvelocity,normalizedbytheinjectionvelocity,onalog-logplot.Inasinglephaseplumeisknowntobeequalto1=3[ 10 34 35 46 ],butinthemultiphasephaseplumeexaminedherethecenterlineaxialvelocityofbothphasesdecayedasz1=6.ThecoefcientsAc=2.27andAd=2.65canbedeterminedfromthemaximumvalueoftheGaussianprolesdepictedontherightsideofgure 5-11 ,whichdepictsthenormalizedaxialvelocityprolesataxialdistancesofz=D=20andz=D=23ofboththecontinuousanddispersedphases.Thehalfwidthsaredeterminedtobe1=2,c=0.075 135

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Figure5-11. MultiphasePlumeExperiment,meanaxialvelocity,hUzi. Figure5-12. Differencebetweendispersedandcontinuousmeanaxialvelocitycomparedtotheoreticalsettingvelocity,alongthecenterline(left)andalongaradialproleatanaxialdistanceofz=D=20(right). and1=2,d=0.090.Novirtualoriginwasused,zv=0.Thecurve2.27(z)]TJ /F4 11.955 Tf 12.34 0 Td[(zv))]TJ /F9 7.97 Tf 6.58 0 Td[(1=6hasbeenincludedontheleftsideofthegureasareference.Inordertoexaminetheassumptionthatthevelocityeldofthedispersedphaseisapproximatelyequaltothecontinuousphasevelocityeldplusthesettingvelocity,gure 5-12 displaysthedifferencebetweenthetwophasesaxialvelocitiesalongwith 136

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Figure5-13. Scaledhu02zialongthecenterline(left)andradialproleatz=D=15(right)ofamultiphaseplume. thetheoreticalsettingvelocityasderivedbyStokesandequation 5 alongtheplumescenterline(left)andalongaradialproleatanaxialdistanceofz=D=20(right).Closetotheplumessourcewherez=D<10andalongthecenterlinetheassumptionhVzi=hUzi+usappearsquiteaccurate.Furtherfromthesourceagreatervariationexistsbetweenthetruedifferencebetweenthevelocityeldofthetwophasesandthesettlingvelocity.Itisimportanttonotehoweverthatagreementappearsverygoodintherstviewingareaandthedivergencefromtheoryismoreclearintheviewingareafurthestfromthesource.Thissuggeststhepossibilitythatthediscrepancynotedisduetoerrorsincalibrationofdataacquisitionsystemandorinthedataanalysis.Thisndingshouldbefurtherexaminedandveriedbyfutureexperiments.Theleftsideofgure 5-13 displaysthecenterlineaxialReynoldsstress,p hu02z,ci,andtherightsidedisplaystheradialproleofaxialReynoldsstressatanaxialdistanceofz=D=20.Bothguresarescaledbythemeancontinuousphasecenterlineaxialvelocity.Farformthesourcethecenterlineturbulentintensityalongthecenterlineofbothphasesbehavessimilarlytothatofasinglephaseplumeasseeningure 4-7 inchapter 4 ,obtaininganearlyconstantvalueofapproximately2.7,asimilarvalueas 137

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foundforasinglephaseplume.Thisvalueremainsfairlyconstantevenveryclosetheplume'ssource.Thiscontrastswiththesimulatedsinglephaseforcedplumewhichwasnearlylaminaralongthecenterlinenearthesource.Likesinglephaseshearowsthemaximumvalueofp hu02z,ciisnotfoundalongthemultiphaseplumesaxis.Atanaxialdistanceofz=D=20boththecontinuousanddispersedphasesaxialReynoldsstresstermsreachtheirmaximumoffoftheplume'scenterline.ThemaximumvalueofReynoldsaxialstressofthedispersedphaseisfoundfurtherawayfromthecenterlineandit'smagnitudeiseverywherelargerthanthatofthecontinuousphase. 5.4SummaryAnexperimentalsetupwasconstructedinordertoexaminemultiphaseplumesusingPIVvisualizationtechniques.Florescenttracerparticleswereaddedtothecontinuousphaseandthebuoyancyparticlesthemselveswereusedinordertotrackthevelocityeldofthedispersedphase.Amechanismwasconstructedinordertocontroltheinjectionrateofparticlesandwaterseparatelythoughtheplumesourcesothatplumeswithvariousinitialmomentumandbuoyancyuxescouldbeexamined.Twoexperimentsoneonapurejetandoneonamultiphaseforcedplumewerecompleted.2000imagesweretakenofeachviewingwindowandthemeanowanductuatingaxialvelocityofboththecontinuousanddispersedphasewerecalculated.ThejetexperimentwascompetedinordertoverifytheexperimentalsetupandPIVsystemandwasdeterminedtobesimilartopreviousnumericalandexperimentalresults.Theaxialvelocityofboththecontinuousandthedispersedphasewasfoundfoundtodecayasz1=6asopposedtoz1=3whichistheknownvalueforsinglephaseplumes,[ 10 34 35 46 ]butwasneverthelessfoundtobeselfsimilarfarfromthesource.TheassumptionthatformultiphaseowswherethevolumefractionofparticlesandtheStokesnumberissmall,thatthevelocityeldofthedispersedphasecanbe 138

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approximatedbyaddingthesettingvelocityoftheparticlestothecontinuousphasewasexamined.Goodagreementwasfoundforz=D<10alongtheplume'scenterline.Furtherfromtheplume'ssourcetheaxialvelocityofthedispersedphasewasfoundtobegreaterthanthecontinuousphasebyasmuchastwicethesettlingvelocity.Itispossiblehoweverthatthisisduetoerrorsincalibratingthedataacquisitionsystemandmoreexperimentsarewarranted.Theseweretherstexperimentscompletedusingthisexperimentalsetup.Inadditiontoexaminingmultiphaseplumes,theintentoftheseexperimentswastogainexperiencewiththesystemandndwhereimprovementsneedtobemade.Intheimmediatefuturethesyringeusedasahourglassdevice,forcontrollingtheowrateofparticle,willbereplacedwithamorerobustsystem,whichwillallowfordifferentparticleinjectionrates.Futureexperimentswillaccuratelyacquireboththeaxialandradialvelocity.Inthenextroundofexperimentsweintendtocomparevariousplumeswiththesamebuoyancyuxbutwithdifferentinitialmomentumux.MultiphaseplumeswithdifferentbuoyancyparticlesizesandthereforesettlingvelocitiesandStokesnumbersarealsotobeexamined.ThesefuturemultiphaseplumePIVexperimentsaretotakeplaceinconjunctionwithmultiphaseplumenumericsimulationsaswillbediscussedinthenextchapter. 139

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CHAPTER6PRELIMINARYRESULTSTOWARDMULTIPHASEPLUMESIMULATIONS 6.1MotivationAgreatdealofinformationregardingmultiphaseplumescanbeobtainedbyperforminglaboratoryexperimentssuchasthosediscussedinchapter 5 .Caremustbetakenwhendesigning,settingup,andperformingsuchexperimentstoensurethedataobtainedisofhighqualityandthattheparametersdeningthemultiphaseplumeareinfactthoseonewishestostudy.Ifthegoaloftheresearchistoexaminethebehaviorofvariousmultiphaseplumesundervaryingconditions,thenmultipleexperimentalsetupsmaybenecessary.Insomecasesdesigninganexperimentalsetuptosimulatespecicconditionsmaynotbepracticalduetocostandotherconstraints.Evenwhenitispossibletodesignanexperimentforamultiphaseplumewithspecicconditions,theremaybelimitationstowhatrelevantdatacanbecollected.Forexample,intheexperimentsdescribedinchapter 5 ,whiletheradialvelocityandvolumefractionofthedispersedphaseareofinterest,onlyaxialvelocityina2Dplanewasobtainedwithanyaccuracy.Inordertoefcientlyexamineavarietyofmultiphaseplumeswithvaryingparametersandinordertoobtainalldataofinterest,weturntonumericalsimulations.Numericalsimulationsallowforagreatvarietyofmultiphaseplumestobeexaminedwithcomparativelylittleexpenseintermsoftimeandmoney.Inmostcasestheabilitytoobtainrelevantdataisonlylimitedbytheabilitytoefcientlystoretheverylargeamountofdatawhichcanbegeneratedbythesimulation.Further,alldataofinterestcaneasilybecollectednoninvasively,withoutaffectingthebehaviorofthesimulatedow.Sincethegoverningequationsforincompressibleowaroundaspherearewelldocumented,inanidealworld,itwouldbepossibletosimulateamultiphaseplumebyfullyresolvingtheowaroundeachsphericalparticleintheow.Unfortunatelysimulatingmultiphaseplumesinthismannerisprohibitivelyexpensiveforallexceptverysimplecaseswherethenumberofparticlesislimited.Inordertoefcientlysimulate 140

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multiphaseplumesinacosteffectivemanner,modelsmustbeusedwhichaccountfortherelevantphysicsoftheowwithoutthecomputationalcostassociatedwithfullyresolvedsimulations.Whichmodeltouseisoftenatradeoffbetweenwhichphysicstoincludeandcomputationalcost.Ifthevolumefractionofparticlesissmallenoughsothattheforcesduetoparticle-particleinteractioncaneitherbeignoredormodeledinasimplemanner,thanonepossiblemodelingmethodistheLagrangianpointparticleapproach.Inthismethodeachparticleistrackedindependentlybutassumedtotakeupnospace.Theforceseachparticleassertsontheuidandtheuidontheindividualparticlescanthenbemodeledbasedontheaccelerationandvelocityofeachparticlerelativetotheuidsurroundingitateachtimestep.WhiletheLagrangianpointparticleapproachislesscomputationallyexpensivethanfullyresolvedow,itstillrequiresagreatdealofcomputationalpowerandmemory,asateachtimestepboththepositionandvelocityofeachparticlemustbecalculatedindividually.Thecostofsimulatingmultiphaseowcanbefurtherreducedbytreatingthedispersedphaseasacontinuumratherthanacollectionofparticles.ThisisreferredtoastheEulerian-Eulerianapproach.Insuchanapproachthevelocityeldandvolumefractionofthedispersedphaseateachpointinthedomainiscalculatedratherthantrackingthemotionofindividualparticles.Inotherwords,thevelocityaparticlewouldhaveifitwerelocatedatapointinspaceiscalculated,ratherthanthelocationandvelocityofeachparticle.Thismethodrequiressolvingacoupledsystemoffourpartialdifferentialequationsforthecontinuousphaseandfouradditionalpartialdifferentialequationsforeachtypeofdispersedparticlesintheow.IfthevolumefractionofthedispersedphaseandtheStokesnumberarebothsmallenoughsothatonlythequasi-steadyforceplaysasignicantroleintheinteractionbetweenthedispersedandcontinuousphases,thantheEquilibrium-EulerianmodelrstdevelopedbyFerryandBalachandar[ 22 ]canbeused.Inthismethodthethreepartial 141

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differentialequationsusedintheEulerian-Eulerianmethodtosolveforthevelocityeldofthedispersedphasearereplacedwithasimplealgebraicexpressionwhichdescribestherelationshipbetweenthevelocityeldofthedispersedphasewiththatofthecontinuousphase.Assuchthesystemcanbecompletelydescribedbysolvingacoupledsystemfourpartialdifferentialequationsforthecontinuousphaseandonedifferentialandthreealgebraicequationsforeachtypeofparticlesincludedinthedispersedphase.InthischaptertheworkinprogresstowardsimulatingmultiphaseplumesusingtheEquilibrium-Euleriantechniqueisdescribed.Forthesakeofbrevity,asinchapter 5 ,thegenericterm'particles'willbeusedwhenreferringtoadiscretepacketofthedispersedphase.Thereadershouldunderstandhoweverthatthemethodologydiscussedinthischapterisnotlimitedtoowswherethedispersedphaseisinsolidformandthattheterm'particle'mayalsorefertoabubbleoradropofgasorliquid.Thesinglephasesoftwarecodeusedtosimulatesinglephaseforcedplumesdiscussedindetailinchapter 4 hasbeenmodiedtosolvetheEquilibrium-Eulerianequations.Onemultiphasesimulation,whichwasdesignedtohavesimilarparametersasthemultiphaseplumeanalyzedintheexperimentsdescribedinchapter 5 ,hasbeencompleted.Thereissomediscrepancyinthetherecordedmeanowbehaviorofthesimulatedplumeandthedataobtainedfromtheexperimentdescribedinthepreviouschapter.Assuch,moreverication,testing,debuggingisnecessarybeforeweproceedwithfurthersimulationsandtheworkpresentedinthischaptershouldbeconsideredpreliminary.Theremainderofthischapterwillproceedasfollows.Insection 6.2 theEquilibriumEulerianequationsastheypertaintomultiphaseplumeswillbederivedfromtheEulerian-Eulerianequations.AshortdescriptionwillbegivenonhowthesinglephaseplumesimulationsoftwarewasmodiedtosolvefortheEquilibrium-Eulerianequations.Insection 6.3 theparametersofthecompletedmultiphaseplumesimulationwillbe 142

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Figure6-1. Schematicofthecomputationaldomain. discussed.Preliminaryresultswillbepresentedandcomparedtotheresultsobtainedfromthemultiphaseplumeexperimentpresentedinchapter 5 andsinglephaseplumesimulationswiththesameReynoldsandRichardsonnumbers. 6.2MethodologyWeassumeamultiphaseplume,withboththecontinuousanddispersedphaseemergingfromacircularpipeofdiameter~D.Theuidparticlemixture,consistingofboththecontinuousphaseanddispersedparticlesofdiameter,~d,exitthecircularpipeandentrainambientuidabove.Figure 6-1 showsthecomputationaldomainwhichisarectangularregionwithdimensionlesslength,widthandheight,ofLx,LyandLz.Thequantities~r,ez,andedenotetheradial,axialandazimuthaldirectionalvectorsinthecylindricalcoordinatesystem.Thequantitiesex,ey,andezarethedirectionalvectorsintheCartesiancoordinatesystem.Theoriginofthecoordinatesispositionedatthecenteroftheforcedplumeinowboundary.Thegravitationalacceleration,~g,actingontheuidisdirectedinthenegativez-direction.Thetime-dependentincompressibleowcanbemodeledwiththegoverningequationsinEulerian-Eulerianformas, @~c @~t+~r~c~u=0(6) 143

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@~d @~t+~r~d~v=0(6) ~c~c@~u @~t+~u~r~u=)]TJ /F7 11.955 Tf 10.6 2.66 Td[(~c~r~p+~c~r2~u+~c~c~g^k)]TJ /F7 11.955 Tf 14.45 10.75 Td[(~d ~8~Fdc(6) ~d~d@~v @~t+~v~r~v=)]TJ /F7 11.955 Tf 10.6 2.66 Td[(~d~r~p+~d~r2~u+~d~d~g^k+~d ~8~Fdc,(6)where~udenotesthevelocityeldofthecontinuousphaseand~vdenotesthevelocityeldofthedispersedphase.~representsvolumefractionsandpressureisdenotedas~p.Thesubscripts,canddsignifythecontinuousanddispersedphasesrespectively.Thedensityofthetwophases,~cand~dareconsideredtobeconstantsinthisanalysisasisthedynamicviscosity,.Equations 6 and 6 arecoupledbytheforcesduetothedispersedphaseactingonthecontinuousphasedenotedas~Fdc.Alternatively)]TJ /F7 11.955 Tf 10.52 2.65 Td[(~Fdccanbedescribedasforcesduetothecontinuousphaseactingonthedispersedphase.Thesymbol~8denotesthevolumeofanindividualparticle.Herethetildesigniesadimensionalquantity.WhiletheEulerian-Eulerianmodelrepresentedaboveisfarmoretractablethantreatingeachindividualparticleseparately,itstillconsistsofeightcoupledpartialdifferentialequations,whereasonlyveequationswereneededtosolveforasingleplume.FerryandBalachandar[ 22 ]notedthatiftheStokesnumberissmall,thantheforcesduetointeractionsbetweenthecontinuousanddispersedphasescanbewrittenas, ~Fdc=3(Red)[~u)]TJ /F7 11.955 Tf 11.9 0 Td[(~v]+.(6)Herethersttermdenotesthequasi-steadyforce.isthesumoftheaddedmass,BassetHistoryandotherforcesandisassumedtobenegligibleformultiphaseowswheretheStokesnumber,St1.Thefunction(Red)correctsthequasi-steadyforcefornon-Stokesowandhasthevalueof1.0iftheReynoldsnumberbasedonparticlediameterandsettlingvelocity,Redislessthan1. 144

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Byaddingequations 6 and 6 andsolvingforthediffusionterm,FerryandBala[ 22 ]derived, ~r2~u=~c~cD~u D~t+~p~cD~v D~t)]TJ /F7 11.955 Tf 12.81 0 Td[(~m~g+~r~p,(6)wheremisthedensityofthemixturedenedas, ~m=~c~c+~d~d.(6)Bysubstitutingequation 6 forthediffusiontermandequation 6 for~Fdcintothedispersedphasemomentumequation 6 FerryandBalachandar[ 22 ]derivedanalgebraicequationforthedispersedphasevelocityeld, ~v=~u+~d~g^k)]TJ /F4 11.955 Tf 13.15 8.09 Td[(D~u D~t.(6)Here~distherelaxationtimeoftheparticlesdenedas, ~d=~d2~d ~c+1 2 18(Red),(6)where~disthediameterofanindividualparticleofthedispersedphaseandisthekinematicviscosity.Ifthevolumefractionofthedispersedphaseissmallthanthevolumefractionofthecontinuousphaseisalmostunityandequation 6 canbesimpliedtostatethatthecontinuousphaseissolenoidal.Withthesesimplicationsinmind,theEquilibrium-Eulerianmodelcanbewrittenas, ~r~u=0(6) @~d @~t+~r~d~v=0(6) @~u @~t+~u~r~u=)]TJ /F7 11.955 Tf 12.95 8.09 Td[(1 ~c~r~p+~r2~u+~g~^k.(6) ~v=~u+~d~g^k)]TJ /F4 11.955 Tf 13.15 8.09 Td[(D~u D~t,(6) 145

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whichconsistsofonlyvepartialdifferentialequationsandthreealgebraicequations.Hereistherelationshipbetweenthedensityofthetwophasesdenedas, =~c)]TJ /F7 11.955 Tf 12.81 0 Td[(~d ~c.(6)Thevariablesarescaledbytheplume'sinletdiameter,~Dandthevelocityoftheinjecteduidattheinlet,UI.ThevolumefractionofthedispersedphaseisrenormalizedbydividingbyI,theparticlevolumefractionattheinlet.Theresultingdimensionlessvariablesarexi=exi=D,ui=eui=UI,p=ep=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(cU2I,=~p=Iandt=UIet=D.Atophatinitialvelocityprole,UI,andvolumefractionprole,I,isassumed.Withappropriatenon-dimensionalization,theEquilibrium-EulerianModeliswrittenas, ru=0(6) @ @t+r(v)=0(6) @u @t+uru=rp+1 ReIr2u+Ri^k(6) v=u+W^k)]TJ /F4 11.955 Tf 11.95 0 Td[(StDu Dt.(6)Thedimensionlessparametersappearingintheaboveequation,theReynoldsnumber,ReI,theRichardsonnumber,Ri,thenormalizedsettlingvelocity,W,andtheStokesnumber,St,aredenedas: ReI=UID ,Ri=~gID U2I,W=~p~g UI,St=~dUI D.(6)Thenon-dimensionalEquilibrium-Eulerianequations 6 through 6 arevarysimilartothesinglephaseplumeequations, 4 through 4 ,discussedinchapter 4 .Theonlydifferenceisthatthebuoyancyequationhasbeenreplacedwiththeparticlevolumefractionequationandanalgebraicexpressionforthevelocityeldofthedispersedphasehasbeenadded.. 146

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ThegoverningequationsweresolvednumericallyonaCartesiangridwithgridpointsclusteredclosetothesourceofmomentumandbuoyancy.Asecond-orderaccuratecentraldifferenceschemeisemployedforthespatialdiscretizationofthegoverningequationsonanon-staggeredgrid.Afractionalstepmethodisusedfortimeadvancement.Intheadvection-diffusionstep,thenonlineartermsaretreatedexplicitlyusingasecond-orderAdams-BashforthschemeandthediffusiontermsaretreatedimplicitlywiththeCrank-Nicholsonscheme.ThepressurePoissonequationissolvedasapressurecorrectionstep.Analdivergence-freevelocityisobtainedateachtime-stepwithapressurecorrectionstep.Theboundaryconditionsweresimilartothoseusedinthesinglephasesimulations.Attheinowboundary,atop-hatproleisusedforbothaxialvelocity,uz,anddispersedphasevolumefraction,,forr=p x2+y2<1=2.Randomuctuationsrangingbetween0.05UIwereaddedtothetophatinletvelocityproleinordertospeedupthetransitiontofullyturbulentbehavior.Thelowerboundaryoutsidetheplumeinlet,r>1=2,isconsideredasano-slipwall.Weallowtheuidtofreelyenterorleavethelateralboundaries,x=1 2Lx,y=1 2Ly,byapplyingNeumannboundaryconditions.Neumannboundaryconditionswerealsosetatthetopofthedomain,z=Lz. 6.3PreliminaryResultsInordertoverifythemultiphaseforcedplumecode,asimulationwasrunwithparameterssimilartothatofthelaboratorymultiphaseexperimentdescribedinchapter 5 .Fromthepropertiesoflaboratorywateratstandardatmosphericconditionsandthephysicalparameters,~D,~d,~UI,~d,~dand~Idescribedinchapter 5 ,thecorrespondingvaluesofthenon-dimensionalparametersdenedinequation 6 are, ReI=656,Ri=0.455,W=0.144,St=0.0082.(6)Thesimulationwasrunoncartesiangridwithadomainsizeandresolutionof252550(LxLyLz)and251251750(NxNyNz),respectively.Thesimulationran 147

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Figure6-2. Continuousphaseaxialvelocityofasimulatedmultiphaseplumecomparedtoexperimentalresults. forover500timeunits,(=~D=UI),beyondtimetimeittookthesimulationtobecomestatisticallysteadystate.Asinthesimulationsofsinglephaseplumesdiscussedinpreviouschapters,averagesofuctuatingvariableswerecalculatedazimuthallyaswellastemporally.Figure 6-2 comparesthemeanowaxialvelocityofthecontinuousphaseofthecurrentnumericalsimulationwiththeresultsfromthemultiphaseplumeexperimentfromchapter 5 .ThecenterlineaxialvelocityofasinglephaseplumesimulatedwiththemultiphasesoftwarecodebutwithWandStsettozeroandwiththesameReynoldsandRichardsonnumberasthesimulatedmultiphaseplumeisalsoincludedinthegure.Asdiscussedinpreviouschaptersthemeanowaxialvelocityfarfromthesourceisexpectedtobeselfsimilarandcanbeapproximatedbetheequation, hUzi=AhUz,Ii (z+zv)exp ln1 2 21=22!,(6)wherethesimilaritycoordinateisdenedas, =r z+zv,(6) 148

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and1=2istheplumehalfwidth.Asdiscussedinchapter 5 theexponentsigniestherateofdecayofaxialvelocitywithrespecttoz,andthecoefcientAistheslopeofthelinearizedcurvet, hUz,ci=A(z+zv))]TJ /F6 7.97 Tf 6.59 0 Td[(.(6)Therightsideofgure 6-2 displaystheevolutionofthecenterlineaxialcontinuousphasevelocityofthesimulatedmultiphaseplume(thicksolidline)whichinitiallyacceleratesuntilapproximatelyz=D=7.Thisissimilartotheinitialbehavioroftheaxialvelocityreportedforthemultiphaseplumeexaminedinthelaboratory(symbols)whichaccelerateduntilaboutz=D=3.4.Aftertheinitialaccelerationthebehaviorofthesimulatedandexperimentalmultiphaseplumesdiverge.BothowsbegintodeceleratebutwhereasbothphasesofthelaboratorymultiphaseplumeweredeterminedtohaveadecayrateofExp.=1=6,thedecayrateofthesimulatedmultiphaseplumewasdeterminedtobeDNS=3=4.SinglephaseplumesareknowntodecayatarateofS.P.=1=3.Thismeansthatthesimulatedmultiphaseplume'saxialvelocitydecaysmuchfasterthanthatofasinglephaseplume,andthatthemultiphaseplumeexaminedinthePIVexperimentdecayedmuchslowerthanasinglephaseplume.ThecoefcientArepresentingslopeofthelinearizedcurvethasavalueofADNS7.1.Thecurveswhichrepresentthebesttoftheaxialvelocityfarfromthesourcehavebeenincludedinthegure(thinlines).Novirtualoriginwasusedinthesecurvets,zv=0.Theleftsideofgure 6-2 depictsthenormalizedaxialvelocityprolesatvariousaxialdistancesfromthesimulatedmultiphaseplume'ssourceandcomparestheprolestothatofthemultiphaseplumeexaminedinthelaboratory.LiketheexperimentalplumetheaxialvelocitycollapsesfairlywellontotheGaussiantgivenbyequation 6 .Thesimulatedmultiphaseplumewasfoundtohaveanapproximatehalfwidthof1=20.086.Atthistimeitisnotclearwhetherthedifferenceinthemeanowbehaviorofthesimulatedmultiphaseplumefromthatofthemultiphaseplumeexaminedinthe 149

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laboratoryisduetosomerelevantphysicswhichisnotcapturedbythesimulation.Thatneithertheexperimentalnorsimulatedmultiphaseplumedeceleratedatasimilarrateasthesinglephaseplumesdiscussedinchapter 4 makesitmoredifculttodecernwhythetwomultiphaseplumesdonotbehaveinasimilarfashion.Severaltheorieshavebeensuggested.Itisknownthattheaxialvelocityoflaminarplumesdonotdecay[ 15 63 ].Thepresentsofparticlesinmultiphaseowhasbeenknowntosuppressturbulence[ 50 ].Itmaybethatforthisreason,themultiphaseplumeexaminedintheexperimentdiscussedinchapter 5 isnotfullyturbulentandthereforedecaysslowerthanasimilarsinglephaseplume.Itisfurtherpossiblethatthisphysicsisnotcapturedbythemultiphasesimulation.Itisalsopossiblethatthedomainofboththeexperimentandthesimulationistosmallinordertoascertainthefareldbehavioroftheplumes.Issueswiththeoutowboundaryconditionsmayalsomakethesimulatedplumebehaveinawaynotaccuratelydescribedbythephysics. 6.4FutureWorkTheresultsofthemultiphasesimulationdiscussedinthischapterarepreliminary.Moretestingneedstobedoneinordertoverifythesoftwarecode.Theresultssimilarsimulations,butwithdifferentoutowboundaryconditionswillbecompared,bothtoeachotherandtopreviousworks.Inthecurrentsimulationaverysmallamountofarticialdiffusionwasincludedinthedispersedphasevolumefractionequation.Inthefuturewewillperformsimulationswhichincludeaneffectiveparticlediffusiontermthataccountsfortheeffectofnumberdensityinduceddispersionofparticles.Aftertheoutowboundaryconditionshavebeenexaminedandcodehasbeenfurtherveried,thecodewillbeusedtosimulatevariousplumeswiththesamebuoyancyuxbutwithdifferentinitialmomentumux.MultiphaseplumeswithdifferentbuoyancyparticlesizesandthereforesettlingvelocitiesandStokesnumbersarealso 150

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tobeexamined.ThesefuturesimulationsaretobedoneinconjunctionwithfurtherPIVexperimentsaswasdiscussedattheendofchapter 5 151

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CHAPTER7CONCLUSIONSAxisymmetricshearowsdrivenbyinitialmomentumandorbuoyancyarecommoninnatureandhavemanyengineeringapplications.Whenbuoyancyispresentinsuchowsitiscausedbyadensitydifferencebetweentheambientuidandtheuidandothermaterialcontainedwithintheplume.Iftheambientenvironmentandtheplumearemadeofthesameuid,werefertotheplumeassinglephase.Ifontheotherhandtheplumecontainsadifferentuidthantheambientenvironmentorifitcontainssolidparticles,wesaytheplumeisamultiphaseow.Inbothsingleandmultiphaseplumesambientuidisentrainedintotheplumeasitrisesorfallsthroughitsenvironment.Thisimpliesthatallforcedmultiphaseplumescontainboththematerialwhichemanatesfromtheowsourceandtheambientuid.AstherecentDeepWaterHorizonoilspillintheGulfofMexicodemonstratedtheseowscanbequitecomplicatedandagreatdealoftheirunderliningphysicsisnotwellunderstood.Manydifferentspeciesofmaterialcanemanatefromtheowsource.Thesespeciescanbeintheliquid,gasorsolidphaseandcanbeeithermonoorpolydispersed.Bubblesanddropscandissolveintotheambientuidastheforcedplumerisesorfallsthroughitsenvironment.Individualbubblesanddropsmaybreakupormergetogether.Iftheambientuiddoesnothaveauniformdensity,thesizeofindividualbubbleswillnotremainconstant.Ifthemeandensityoftheplumeisequaltothedensityoftheambientuidatsomeaxialdistanceawayfromthesource,intrusionscanform.Crossowsintheambientuidarecommonandcandramaticallyaffectthebehaviorofsuchshearows.Thisstudywaslimitedtoexaminingsinglephasejetsandplumesandtomonodispersedmultiphaseplumeswithasmalldispersedphasevolumefraction.Theambientuidwasassumedtohaveauniformdensityandnocrossowwaspresent. 152

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Thedensitydifferencewasassumedtobesmallenoughsothatthedensityterm,,canbeignoredexceptwhenitismultipliedbythegravityterm,g.Inthelaminarregimetheeffectofswirlonjetsandplumeswasexamined.ThroughanasymptoticapproachthesimilaritysolutionforthelaminarplumerstderivedbyYih[ 63 ]wasextendedtothecasewhereasmallamountofangularmomentumuxispresentintheow.Thesimilaritysolutionwascomparedtonumericalsimulationandfoundtobevalidevenforplumeswithsubstantialabove-criticalswirl.Inparticular,theaxialvelocityandtemperatureproleswereobservedtobelittleinuencedbyswirlandcomparedwellwiththeclassicalsimilaritysolution.Thiscontrastswiththesimilaritysolutionfortheswirlingjet,whereswirlhasastrongeffectonthejetow.Revueltaetal.[ 43 ]establishedthetheoreticalupperandlowerboundsforthevalueofcriticalswirl,Sc,forpurejets.ThevalueofScfoundinthecurrentnumericalstudyisinagreementwiththeirresults.UsingtheoreticalargumentsasimilarcriterionforanestimateofScforplumeswasderived.Acomparisonwasmadeinthebehavioroflaminarjetsandplumeswithabovecriticalswirl.Inpurejets,oncevortexbreakdownhasoccurred,theassumptionofanarrowowisnolongervalidandalmostalljetlikebehaviorislost.Twodifferentowstructuresarepossibleforjetswithabovecriticalswirl.Theowpatternthatresultsinasimulationdependedontheinitialcondition.Inplumesevenalargeamountofswirldoesnotgreatlyeffecttheoverallbehaviorinthefareld.Thevalueoftheconservedquantitybuoyancyux,JB,decreaseswithincreasedswirl,buttheoverallbehaviorremainsunchanged.Byaddingasmallamountofbuoyancytoajetorasmallinowtoaplumetheamountofswirlrequiredforvortexbreakdowntooccurincreased.ThiscaneasilybeexplainedintermsoftheBernoulliequationappliedalongthecenterline.Further,ajetwithevenaverysmallamountofbuoyancynolongerundergoeshysteresisandbecomestablewithonlyonepossibleowpattern. 153

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Intheturbulentregimethreedirectnumericalsimulationswerecompleted,apurejet,apureplumeandaforcedplume.Eachofthesimulationswasrunlongenoughforthethirdorderstatisticstoconverge.Byobtainingthethirdordermomentsofthevelocityeld,eachtermoftheturbulentkineticenergybalanceandReynoldsstresstransportequationscouldbecalculateddirectlywithoutrelyingonassumptionsormodels.Specialattentionwaspaidtothepurejet.ThemoderateReynoldsnumberjetsimulatedinthecurrentstudywasfoundtoachieveconstantcenterlineturbulentintensitiesclosertotheinlet,thanpreviousexperimentalstudieswherejetswithlargerReynoldsnumberswereexamined.ThisagreeswithanassertionbyBogeyandBailly[ 13 ]thatmoderateReynoldsnumberturbulentjetsachievesimilarityclosertothesourcethanlargerReynoldsnumberjets.Indeed,thejetinthecurrentstudy(ReJ=2000)achievedselfsimilaritywithin15inletdiametersfromthejetsource,whereasthejetstudiedbyPanchapakesanetal.[ 39 ](ReJ=11,000)didnotachieveselfsimilarityuntilz=70.ThetotalmomentumintegralwascalculatedandM(z)wasfoundtobewithin1%ofMJforallzvaluesabovez>10.Themomentumuxcontributedbythemeanowwasfoundtoaccountforapproximately93%ofthetotalmomentumintegral.Previousstudiesmadeuseofvariousassumptionsinordertoapproximatethepressurediffusionanddissipationtermsintheturbulentkineticenergybalanceofapurejet.Therehasbeensomedisagreementastotheshapeofthedissipationprolenearthecenterline.WithsomeresearchersreportingaGaussianprole,otherssuggestinganoffaxismaximumvalue,andstillotherssuggestingaratheratprolenearthecenterline.ThecurrentstudyagreedwithPanchapakesanetal.[ 39 ]inndingtheproleatnearthecenterline,butfoundthemagnitudenearthecenterlineclosertothatreportedbyHusseinetal.[ 26 ].FurtherawayfromthecenterlinetheshapeandmagnitudeofthedissipationprolewasclosesttothatreportedbyTaulbeeetal.[ 55 ]. 154

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Thebalanceofhu02zitransportinthecurrentstudyyieldsroughlysimilarresultstoboth,Husseinetal.[ 26 ]andPanchapakesanetal.[ 39 ],however,inthecurrentstudythepressureandturbulentdiffusiontermsarefoundtobeofapproximatelythesameshapeandmagnitude,bothpreviousstudiesreportedalargerpressuretermandasmallerturbulentdiffusionterm.Also,Husseinetal.[ 26 ]reportedaclearoffaxispeakfordissipation.InboththecurrentstudyandthatofPanchapakesanetal.[ 39 ]thisfeatureisnotclearandthedissipationprolenearthejetaxisappearsfairlyat.Thebalanceofthehu02riandhu02iReynoldsnormalstresstermsinthecurrentstudyaresimilartothatofPanchapakesanetal.[ 39 ],buttheReynoldsshearstresstermhu0zu0riagreesmorewithHusseinetal.[ 26 ].HopefullyabetterunderstandingofthetransportofallsixReynoldsstresstermscanbeusedtoimproveLESandRANSsimulationswhichrelyonmodelingtheseterms.Sinceaxialvelocityscalesasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1forajet,butasz)]TJ /F9 7.97 Tf 6.59 0 Td[(1=3foraplume,plumelikebehaviorwilldominateinthefareldofaforcedplume.BasedonasimpleboxmodelMorton[ 34 ]suggestedthatthebehaviorofthemeanowvelocityofaforcedplumewillbeidenticaltothatofapureplumebeyondadistanceofz=5J3=4M,I=J1=2B,Ior5LM.Byexaminingthemeanowandturbulentstatisticsatbothanaxialdistanceofz=15whichlessthenz=5LM,andz=30whichisapproximatelyequalto5LMthevalidityofMorton'ssuggestionwasexamined.TheaxialdistancefromthesourceatwhichthesimulatedforcedplumeconformedtotheselfsimilarsolutionofapureplumeroughlymatchedthataspredictedbyMorton[ 34 ].Thesecondandthirdordervelocityproles,turbulentkineticenergyequationandReynoldsstressequationswerecalculatedatbothaxialdistances.Thethirdorderstatisticsofapureplumeatz=15whichincludeu0ztoanypowerbehavequitedifferentlythanthecorrespondingstatisticsofapurejetorforcedplume.Atanaxialdistanceofz=30thepureplume'sthirdorderstatisticsaresimilartothatoftheothertwoowsdiscussedinthisstudy.Verylittlechangesbetweenthethirdordervelocity 155

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prolesofathepurejetandforcedplumeatthetworadialdistancesz=15andz=30.Theexceptiontothisisthehu03zitermwhichispositivealongthecenterlineatz=15buthasanegativevalueatthecenterlineatz=30,forbothows.TheturbulentkineticenergybalanceandReynoldsstresstransportequationsaredependentonthesecondandthirdorderstatisticsofthevariousaxisymmetricows.Assuchitisnotsurprisingthatliketherstandsecondorderstatisticsthetransportequationsofthepurejetandforcedplumeareroughlysimilaratbothaxialdistancesexamined.Thepureplumeatz=15hasdifferentbehaviorthantheothertwoows.Thisismostapparentinthedissipationtermwhichhasamuchlargermagnitudeinapureplumethaninaowwithinitialmomentumux.ThetransportofReynoldsshearstressismostlyabalancebetweenproductionanddissipationwithpressurediffusion,turbulentdiffusionandbuoyancyplayingalesserrole.Injetsandforcedplumesatz=15bothproductionanddissipationaresmallneartheowaxis,whereasthesetermsareneartheirmaximumvalueinapureplume.Inexaminingtheturbulentstatisticsalongthecenterlineofthethreeows,itisfoundthataxialturbulentintensity,p hu02zi=hUz,ci,ofthejetandforcedplumeevolveinasimilarmanner.Bothstartoutequalzeroatzequalszeroandincreaseuntilreachingaconstantvalueofapproximatelyhu20zi1=2=hUz,ci=0.25byz=12.Theevolutionoftheaxialturbulentintensityofthepureplumehoweverbeginsasanonzerovalue,rapidlyincreasesandthendecreasesbeforenallybecomingaconstantofapproximatelyhu02zi1=2=hUz,ci=0.25atafurtheraxialdistancefromthesource,z=18.Anexaminationoftheradialproleofaxialturbulentintensityrevealsthatthisbehaviorisisolatedneartheaxisofthepureplume.Awayfromthecenterline,theradialproleofaxialturbulentintensitybehavesinasimilarGaussianmannerwithasimilarmagnitudeasajetorforcedplume.Awayfromtheowcenterlinetheturbulentintensityofapureplumefollowsasimilar,althoughslightlynarrower,Gaussianproleasthatofapurejetorforcedplume. 156

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LaboratoryexperimentsonjetsandmultiphaseplumeswereconductedatEcoleNationaleSuperieureDeMecaniqueetD'Aerotechnique(ENSMA)locatedinPoitiers,FranceusingParticleImageVelocimetry(PIV)techniques.Apurejetexperimentwasconductedandresultswerecomparedtopast,theoretical,experimentalandnumericalstudiesinordertoverifytheaccuracyofthePIVsystemandexperimentalsetup.Inthemultiphaseplumeexperimentorescenttracerparticleswereaddedtothecontinuousphaseandthebuoyancyparticlesthemselveswereusedinordertotrackthevelocityofthedispersedphases.Theaxialvelocityofboththecontinuousandthedispersedphaseswasfoundfoundtodecayasz1=6asopposedtoz1=3whichistheknownvalueforsinglephaseplumes,[ 10 34 35 46 ]butwasneverthelessfoundtobeselfsimilarfarfromthesource.Thereporteddifferenceinbehaviorbetweensingleandforcedplumesmayberelatedtothesettlingvelocityofthebuoyancyparticles.Itisalsopossiblethattheadditionofneparticlesintoaplumesuppressesturbulence.Itisalsopossiblethatthevisualizationareaofthecurrentexperimentalsetupisnotlargeenoughtocapturethefareldbehaviorofthemultiphaseplumeexaminedintheexperiment.Thesinglephasecodeusedtosimulatemultiphasejetsandplumesinchapters 3 and 4 wasmodiedinordertoexaminemono-dispersedmultiphaseforcedplumes.Preliminaryresultswerepresented.Asacontinuationofthepresentwork,variousoutowboundaryconditionsandtreatmentsforimplementingtheeffectiveparticlediffusionofthedispersedphasethataccountsfortheeffectofnumberdensityinduceddispersionofparticleswillbetested.Inconjunctionwithoneanother,numericalsimulationsandlaboratoryexperimentswillbeconductedonvariousmultiphaseplumes.Inoneplannedtestingregime,plumeswiththesamebuoyancyuxbutwithdifferentinitialmomentumuxwillbeexamined.InordertoexaminetheeffectofStokesnumberandsettlingvelocity,plumeswiththesamebuoyancyandmomentumuxbutwithdifferentparticlesizeswillbecompared. 157

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Infuturestudies,boththesimulationsoftwareandlaboratoryfacilityaretobemodied,sothatmultiphaseforcedplumesinastratiedenvironmentandorwhereacrossowispresentcanbeexamined. 158

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BIOGRAPHICALSKETCH GordonTaubwasbornandraisedinSeattleWashington.Hereceivedhisrstbachelor'sdegreeindramafromtheUniversityofWashingtonin1994.AftergraduationhemovedtoLosAngeleswherehebeganworkingin,whatatthetime,wasaneweld;themanufacturing,sellingandmaintenanceofcomputervideoeditingsystemsfortelevisionandlm.Gordonworkedinthiseldinmanycapacitiesincluding,inventoryclerk,assistanteditor,hardwaremanagerfortheLosAngelesarea,sales,trainingandcustomersupport.OverthecourseofsevenyearsGordonwatchedastheindustrydevelopedfromitsnearinfancytovideoeditingbeingabasicfunctionofalllaptopcomputers.WhileGordonwaslookingforaneweldhecouldwatchandhelpmatureherealizedwhathereallywantedtodowastocreatecomputerandmathematicalmodelsofhowairandwaterowaroundrenewableenergysystemslikewindfarms,underwaterturbinesandsolarchimneys.Itwasatthispointthatherealizedwhathereallywantedtobewaseitheranappliedmathematicianoramechanicalengineer.HeenrolledhimselfincommunitycollegeandtookayearandahalfofcalculusandphysicswiththegoalofconvincingtheUniversityofWashingtontore-enrollhimasapostbaccalaureate.Hegraduatedwithasecondbachelor'sdegreeinAppliedandComputationalMathSciencesfromtheUniversityofWashingtonin2007beforebeingacceptedtothegraduateprogram.Hereceivedhismaster'sinappliedmathin2008andwasthenacceptedasaPhD.studentbytheMechanicalandAerospaceEngineeringDepartmentattheUniversityofFlorida. 164