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Numerical Simulation of Cryogenic Flow with Phase Change Using Sharp Interface Cut-Cell method

Permanent Link: http://ufdc.ufl.edu/UFE0041249/00001

Material Information

Title: Numerical Simulation of Cryogenic Flow with Phase Change Using Sharp Interface Cut-Cell method
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Agarwal, Alpana
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: annular, boiling, chilldown, cryogenic, film, internal, inverted, microgravity, multigrid, multiphase, phase, sharp, vaporization
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Cryogenic fluids find wide use in many different types of industries as well as space applications, where they may be used as the liquid fuel or the cryogen for other vital support systems. Therefore the transportation, handling and storage of cryogenic flow under microgravity in space missions is an important design concern. During their transportation through the pipes in a spacecraft, because of strong heat flux from wall, a rapid quenching process with voracious boiling of the cryogen takes place. This can subject the piping system to extreme thermal stresses due to sudden contraction. Due to strong vaporization and resulting two-phase flow, the mass flow rate of cryogenic flow will decrease. The insufficient flow rate can cause many problems in the spacecraft. Therefore a thorough physical understanding of the phase change phenomenon in cryogenic flow under microgravity is very important. Experimental investigation of cryogenic chill down under microgravity is not easy because of the difficult in creating reduced gravity conditions on earth. Most of the experiments on quenching under microgravity have been done on board special aircrafts like NASA's KC-135. The cost of performing these experiments is very high. Moreover, it is not easy to collect experimental data in flight. In this research work, numerical simulation techniques are used instead of experiments. In numerical simulation, the microgravity is easily modeled and the cost is much less than performing experiments. The sharp interface method (SIM) with cut-cell technique (SIMCC) is adopted to handle the two-phase flow computations. In SIM, the background grid is the Cartesian grid and the explicit interfaces are embedded in the computation domain dividing the entire domain into different sub-domains corresponding to various phases. In SIM, each phase has its own set of governing equations. The interfacial conditions act as the link between different phases. The cut-cell technique is utilized to handle the non-rectangular cells produced by the intersection of interfaces with the Cartesian grid. The conservative properties of the finite volume method can be satisfied better near the interface using cut-cells. The interface is treated as an entity with zero thickness with no volume association. With the explicit geometrical information about the interface and high resolution numerical schemes, the heat flux near the interface can be evaluated more accurately than by any other multiphase techniques. This research aims to expand the scope of the SIMCC method by several enhancements like multigrid methods and third-order upwind differencing scheme for the convective term. These enhance the performance and stability of the SIMCC and improve its capability to handle the very challenging task of simulating internal multiphase flows. The specific focus of this research is inverted annular film boiling regime. Various physical mechanisms that influence the flow patterns and heat transfer characteristics during the transportation under reduced gravity are investigated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alpana Agarwal.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Chung, Jacob N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041249:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041249/00001

Material Information

Title: Numerical Simulation of Cryogenic Flow with Phase Change Using Sharp Interface Cut-Cell method
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Agarwal, Alpana
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: annular, boiling, chilldown, cryogenic, film, internal, inverted, microgravity, multigrid, multiphase, phase, sharp, vaporization
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Cryogenic fluids find wide use in many different types of industries as well as space applications, where they may be used as the liquid fuel or the cryogen for other vital support systems. Therefore the transportation, handling and storage of cryogenic flow under microgravity in space missions is an important design concern. During their transportation through the pipes in a spacecraft, because of strong heat flux from wall, a rapid quenching process with voracious boiling of the cryogen takes place. This can subject the piping system to extreme thermal stresses due to sudden contraction. Due to strong vaporization and resulting two-phase flow, the mass flow rate of cryogenic flow will decrease. The insufficient flow rate can cause many problems in the spacecraft. Therefore a thorough physical understanding of the phase change phenomenon in cryogenic flow under microgravity is very important. Experimental investigation of cryogenic chill down under microgravity is not easy because of the difficult in creating reduced gravity conditions on earth. Most of the experiments on quenching under microgravity have been done on board special aircrafts like NASA's KC-135. The cost of performing these experiments is very high. Moreover, it is not easy to collect experimental data in flight. In this research work, numerical simulation techniques are used instead of experiments. In numerical simulation, the microgravity is easily modeled and the cost is much less than performing experiments. The sharp interface method (SIM) with cut-cell technique (SIMCC) is adopted to handle the two-phase flow computations. In SIM, the background grid is the Cartesian grid and the explicit interfaces are embedded in the computation domain dividing the entire domain into different sub-domains corresponding to various phases. In SIM, each phase has its own set of governing equations. The interfacial conditions act as the link between different phases. The cut-cell technique is utilized to handle the non-rectangular cells produced by the intersection of interfaces with the Cartesian grid. The conservative properties of the finite volume method can be satisfied better near the interface using cut-cells. The interface is treated as an entity with zero thickness with no volume association. With the explicit geometrical information about the interface and high resolution numerical schemes, the heat flux near the interface can be evaluated more accurately than by any other multiphase techniques. This research aims to expand the scope of the SIMCC method by several enhancements like multigrid methods and third-order upwind differencing scheme for the convective term. These enhance the performance and stability of the SIMCC and improve its capability to handle the very challenging task of simulating internal multiphase flows. The specific focus of this research is inverted annular film boiling regime. Various physical mechanisms that influence the flow patterns and heat transfer characteristics during the transportation under reduced gravity are investigated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Alpana Agarwal.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Chung, Jacob N.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-10-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041249:00001


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IwouldliketoexpressmysinceregratitudetomycommitteechairandadvisorDr.ChungforhisconstanthelpandsupportthroughoutmystudiesatUF.HeisoneofthebestresearchersIknow,verythoroughandfocused.Ilearnedalotunderhisabletutelage.IwouldalsoliketothankmyfriendandcolleaguePeterfromwhomIlearntalotaboutnumericalmethods.IamgratefultomyPh.D.committeemembersfortheirvaluablecommentsandsuggestionsandforagreeingtoserveonthecommittee.IwouldalsoliketothankmyfriendsJaya,Mohua,Midhun,Tanu,Rahul,RohitandZeewhohavealwaysbeenthereforme.Iwouldliketoacknowledgemyfamily,especiallymymotherMrs.MadhuKumarandmysisterArtifortheirconstantlove,supportandbeliefinme.NowordsareenoughtoexpresshowIfeel.LastbutnottheleastIamdeeplygratefultoSwamiji,whowasaconstantsourceofinspirationformeandsometimestheonlylightguidingmehome. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 1.1Overview .................................... 15 1.2RoleofCryogenicsinSpaceMissions ..................... 15 1.3CryogenicChillDownProcess ......................... 16 1.4StateoftheArt ................................. 17 1.5ObjectivesofthisResearch ........................... 17 1.6ScopeandStructure .............................. 18 2LITERATUREREVIEW .............................. 20 2.1TheBoilingCurveandPhaseChangeProcesses ............... 20 2.2ConvectiveBoilinginTubesandChannels .................. 21 2.3Two-PhaseFlowinMicrogravity:Experiments ................ 24 2.3.1Two-PhaseFlowPatternsinMicrogravity ............... 25 2.3.2HeatTransferinMicrogravity ..................... 27 2.4Two-PhaseFlowinMicrogravity:NumericalModels ............ 28 3PROBLEMSTATEMENTANDGOVERNINGEQUATIONS .......... 31 3.1AssumptionsintheNumericalModel ..................... 33 3.2GoverningEquations .............................. 34 3.2.1GoverningEquationsforBulkPhases ................. 34 3.2.2InterfacialConditions .......................... 35 3.3Non-Dimensionalization ............................ 36 3.4ComputationalDomain,InitialandBoundaryConditions .......... 37 4NUMERICALMETHODFORMODELLINGMULTIPHASEFLOWS ..... 40 4.1FractionalStepMethodforNavier-StokesEquation ............. 40 4.2CartesianGridwithCut-CellsforTreatmentofComplexGeometries ... 43 4.3SharpInterfaceMethodwithCut-cellTechnique(SIMCC) ......... 45 4.3.1InterfaceRepresentationandTracking ................. 47 4.3.2Cut-CellFormationandMergingProcedure .............. 48 4.3.3CalculationofFluxesforCut-Cells .................. 49 4.4MovingInterfaceAlgorithm .......................... 51 5

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............... 51 4.4.2UpdatingCellswhichhaveChangedPhase .............. 53 4.4.3InterfaceMotionDuetoPhaseChange ................ 54 4.5ChallengesinDirectNumericalSimulationofInternalTwo-PhaseFlowswithPhaseChange ............................... 56 4.6MultigridMethodforSIMCC ......................... 59 4.6.1TheMultigridTechnique ........................ 60 4.6.2ImplementationofMultigridMethodforSIMCC ........... 61 4.6.3TreatmentofCut-CellsonCoarseGrid ................ 63 4.6.4ResultswithMultigrid ......................... 65 4.7QUICKDierencingSchemefortheConvectiveTerm ............ 68 5VERIFICATIONANDVALIDATION ....................... 73 5.1TestCase1:LidDrivenCavityFlow ..................... 73 5.2TestCase2:FlowOveraSolidSphere .................... 75 5.3TestCase3:DeformableBubble ........................ 77 5.3.1BubbleShape .............................. 80 5.3.2ShapeTransition ............................ 80 5.3.3BubbleShapeAspectRatio ...................... 83 5.3.4Circulation ................................ 83 5.3.5TimeDependantDrag ......................... 83 5.4TestCase4:BrethertonProblem ....................... 87 5.5Conclusion .................................... 89 6NUMERICALRESULTSFORTWO-PHASECRYOGENICCHILLDOWNPROCESS ....................................... 91 6.1ScopeofCryogenicChillDown ........................ 91 6.2CryogenicChillDownofNitrogen ....................... 92 6.3CryogenicChillDownofOxygen ....................... 98 6.3.1EectofReynoldsNumber ....................... 107 6.3.2EectofWeberNumber ........................ 109 6.3.3EectofJakobNumber ......................... 113 6.4CryogenicChillDownofArgon ........................ 117 6.4.1EectofReynoldsNumber ....................... 119 6.4.2EectofWeberNumber ........................ 121 6.4.3EectofJakobNumber ......................... 122 6.5Conclusion .................................... 123 7ANALYSISOFRESULTS .............................. 127 7.1PredictionMethodsforInvertedAnnularFlow ................ 128 7.2EvaluationofHeatTransferEngineeringCorrelationsfortheSimulatedResults ...................................... 130 7.3Conclusion .................................... 137 6

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........................ 139 8.1Summary .................................... 139 8.2FutureDirections ................................ 141 REFERENCES ....................................... 143 BIOGRAPHICALSKETCH ................................ 150 7

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Table page 2-1Microgravityquenchingexperiments ........................ 25 5-1Dragondeformablebubble ............................. 86 6-1ParametersforN2chilldownsimulation ...................... 93 6-2ParametersforO2Reynoldsnumberstudy ..................... 108 6-3ParametersforO2Webernumberstudy ...................... 110 6-4ParametersforO2Jakobnumberstudy ....................... 114 6-5ParametersforargonReynoldsnumberstudy ................... 119 6-6ParametersforargonWebernumberstudy ..................... 122 6-7ParametersforargonJakobnumberstudy ..................... 123 8-1VericationandvalidationstudyforSIMCC .................... 139 8

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Figure page 2-1Atypicalboilingcurve ................................ 21 2-2Flowregimesforconvectiveboilinginahorizontaltube .............. 22 2-3Flowregimesforconvectiveboilinginaverticaltube ............... 22 2-4Flowpatternsduringquenchingofasupportedhottube ............. 24 2-5Flowpatternsatdierentmassowrates,Tw=235,inmicrogravity ..... 27 2-6Comparisonbetweenmeasuredandpredictedwalltemperaturesundermicrogravitywithowrateof40cc/s ............................... 29 3-1Simpliedschematicofcryogenicsystem ...................... 31 3-2Modelofinvertedannularlmboiling ....................... 32 3-3Controlvolumeforheattransfer ........................... 33 3-4Computationaldomain ................................ 38 4-1Non-staggeredgridsystem .............................. 41 4-2Anexampleofcutcellsformedneartheinterface ................. 43 4-3Exampleofmixedstructuredandunstructuredgrid ................ 44 4-4Fluxcomputationsforcut-cells ........................... 50 4-5Illustrationofinterfacialadvancingprocess ..................... 52 4-6Cellupdationprocedure ............................... 53 4-7Challengesininternalphasechangeows ...................... 58 4-8Restrictionandprolongationoperationsinmultigridtechnique .......... 60 4-9IllustrationofthemultigridtechniqueforSIMCC ................. 63 4-10Determinationofthephaseofcoarsegridcut-cells ................. 64 4-11Eciencyofmultigridbasedonnumberofiterations ............... 65 4-12EciencyofmultigridbasedonCPUtime ..................... 66 4-13Streamlineswithandwithoutarticialinterface .................. 67 4-14Centerlinevelocitieswithandwithoutarticialinterface ............. 67 4-15Performanceofmultigridforarticialinterfacecase ................ 68 9

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.............. 69 4-17Comparisonofconvectiveschemes .......................... 72 5-1Uvelocitythroughverticalcenterline ........................ 74 5-2Vvelocitythroughhorizontalcenterline ...................... 74 5-3Streamlineplotsforliddrivencavityow ...................... 75 5-4Featuresofwakebehindsolidsphere ........................ 76 5-5Dragcoecientofsolidsphere ............................ 77 5-6Computedbubbleshapes ............................... 79 5-7BubbleshapesbyRyskinandLeal ......................... 79 5-8Comparisonofbubbleshape ............................. 80 5-9Transitioninshape .................................. 81 5-10InuenceofReonthecurvatureofbubble ..................... 82 5-11InuenceofWeonaspectratio ........................... 82 5-12Developmentofcirculationregion .......................... 84 5-13Draghistory,Re=10;We=3 ............................ 85 5-14Draghistory,Re=100;We=3 ........................... 85 5-15SetupforBrethertonproblem ............................ 88 5-16FloweldandpressurecontoursforRe=10;Ca=0:05 .............. 89 5-17PressureatthechannelwallforRe=10;Ca=0:05 ................ 90 5-18FilmthicknessforRe=10anddierentCa 90 6-1U-velocitycontoursofLiq.N2att=0.4 ....................... 94 6-2V-velocitycontoursofLiq.N2att=0.4 ....................... 94 6-3TemperaturecontoursofLiq.N2att=0.4 ..................... 95 6-4U-velocitycontoursofLiq.N2att=1:5fordierentRe ............. 96 6-5V-velocitycontoursofLiq.N2att=1:5fordierentRe ............. 97 6-6TemperaturecontoursofLiq.N2att=1:5fordierentRe ............ 98 6-7Temperatureatr=RiforLiq.N2 99 10

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99 6-9Non-dimensionalmassowrateatt=1:5forN2 100 6-10FloweldofO2att=8:0forRe=3000 ....................... 101 6-11DevelopmentoftheoweldforatypicalcaseofO2 102 6-12Temperaturecontoursneartheliquidfront,Re=3000 .............. 103 6-13PressureatcenterlineandwallforO2,t=8:0,Re=3000 ............ 103 6-14TimedependenceofwalltemperatureforO2,Re=3000 ............. 104 6-15TemperaturegradientatwallforO2,Re=3000 .................. 105 6-16Timedependenceofmassowatpipeexit,O2,Re=3000 ............ 106 6-17DependenceofwalltemperatureonReforO2att=8:0 ............. 108 6-18Non-dimensionalmassowrateatt=8:0forO2 109 6-19TemperaturegradientandNu1att=8:0forO2 110 6-20InterfaceshapefordierentWeatt=2:0 ..................... 111 6-21U-velocityproleofO2fordierentWeatt=2:0 ................. 111 6-22HeattransferfordierentWeatt=2:0 ...................... 112 6-23HeattransfercharacteristicsfordierentWeatt=8:0 .............. 112 6-24InterfaceshapefordierentJaatt=8:0 ...................... 114 6-25MassowhistoryfordierentJaforO2 115 6-26EectofJaonwalltemperatureofO2att=8:0 ................. 115 6-27HeattransfercharacteristicsfordierentJaatt=8:0 .............. 116 6-28FloweldofAratt=8:0forRe=2500 ....................... 117 6-29Developmentoftheoweldforatypicalcase ................... 118 6-30Temperaturecontoursneartheliquidfront ..................... 119 6-31ComparisonofmassowrateforArandO2 .................... 120 6-32DependenceofwalltemperatureonReforAratt=8:0 ............. 120 6-33HeattransferatwallforAr,t=8:0 ......................... 121 6-34InterfaceshapefordierentWeatt=1:0 ..................... 122 11

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...................... 124 6-36MassowhistoryfordierentJaforAr 124 6-37EectofJaonwalltemperatureofAratt=8:0 ................. 125 6-38EectofJaonNu1ofAratt=8:0 ........................ 125 7-1TemperaturecontoursforRe=3000,O2 130 7-2ComparisonofdierentdenitionsofNusseltnumber ............... 132 7-3Nu2foralltheoxygencases ............................. 134 7-4Nu2foralloxygenandargoncases ......................... 135 7-5ComparisonofNusseltnumberwithotherresearchers ............... 136 12

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1 ].Inspaceexplorations,thereisahugeincentiveforimprovingthetechnologyforthestorageandtransportofcryogenicuids.Forexample,theabilitytousehydrogenasfuelmeansthatagivenmissioncanbeaccomplishedwithasmallerquantityofpropellants(andasmallervehicle),oralternately,thatthemissioncanbeaccomplishedwithalargerpayloadthanispossiblewiththesamemassofconventionalpropellants.Theecientandsafeutilizationofcryogenicuidsinthermalmanagement,powerandpropulsion,andlifesupportsystemsofaspacecraftduringspacemissionsinvolvesthetransport,handling,andstorageoftheseuidsinreducedgravityconditions.Theuncertaintiesabouttheowpatternandheattransfercharacteristicsposedicultiesfordesignofequipment. 15

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2 { 4 ]itmaybecompletein20-30seconds.Thereforeitrequireshighlyskilledtechnicalknowledgetochilldownacryogenicsysteminasafeandecientmanner.Thehighlyunsteadychilldownprocessisextremelycomplexbecausewhenacryogenicliquidisintroducedintoasystemwhichisatambienttemperature,voraciousevaporationoccursandaveryhighvelocityvapormisttraversesthroughthesystem.Asthesystemcools,slugsofliquid,entrainedinthevapor,owthroughthesysteminatwo-phaselmboilingmode.Asthesystemcoolsfurther,aliquidquenchingfrontowsthroughthesystemandisaccompaniedbynucleateboilingandtwo-phaseow.Therateofheattransferinthenucleateboilingregimeisveryhighandthesystembeginstocooldownveryrapidly.Asthesystemrapidlycoolsdown,thetwo-phaseowpassesthroughseveralowregimetransitionstosingle-phaseliquidow.Theinherentdangerduringchilldownisthattwo-phaseowsareinherentlyunstableandcanexperienceextremeowandpressureuctuations.Thehardwaremaybesubjecttoextremethermalstressesduetothermalcontractionandmaynotbeabletosustainextremepressureuctuationsfromthecryogen.Eciencyofthechilldownprocessisalsoanimportantissuesincethecryogenusedtochilldownthesystemcannolongerbeusedforpropulsionorpowergeneration.Toeconomize,thechilldownmustbeaccomplishedwithaminimumconsumptionofcryogeninorderfortheoverallenergyeciencytobewithintolerablelimits.Inorderforliquidhydrogentobeadoptedasthefuelofchoice,itisimportanttofullyunderstandthe 16

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5 6 ].[ 7 ]attemptedadirectnumericalsimulationofthecryogenicchilldownprocesswithalimitedscope.Thepresentresearchfocusesonaddressingspecicfundamentalandengineeringissuesrelatedtothemicrogravitytwo-phaseowandboilingheattransferofcryogenicuids.Itprovidesphysicalunderstandingofthetransportphysicsofcryogenicboilingandtwo-phaseowsinreducedgravitybyusingnumericaltechniques. 17

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1. Toinvestigatetherateofvaporization(masslossofcryogenicliquid)basedondierentdrivingmechanisms(Ja,Reetc.)andtheeectofwallboundary. 2. Tocalculatethetransientlmboilingheattransfercoecientandtemperaturedistributionsofwallduringthechilldownprocess. 3. Tocalculatetheabovefordierenttypesofcryogenicuids,forexampleliquidnitrogen,oxygenandargon. 4. Todevelopthenecessarynumericaltechniquesforphasechangecomputationinthecontextofinternalows.Theoverallthrustistodevelopanaccurateandecientnumericalpackageforsimulatingthecryogenicowinmicrogravitytohelpinthedesignofthemostreliablecryogenictransportationsysteminspacemissionsorrelatedindustrialapplications.Inthisdissertationthereareeightchapters.Inthersttwochapters,theliteratureaboutheattransferandphase-changecharacteristicsofcryogenicow,theimpactofmicrogravityandtheowpatternsinthechilldownprocessesisreviewed.Inchapterthree,thephysicalmodelandrelatedgoverningequationsandinterfacialconditionsareexplained.Inchapterfour,thenumericaltechniquesaboutthesolverofgoverningequation,movinginterfacetechnique,thesharpinterfacemethodwithcut-celltreatment(SIMCC),phasechangecomputationandthemultigridtechniqueforSIMCCwillbeintroduced.Inchapterve,aseriesoftestcasesaredonetoensurethatthecurrent 18

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2-1 ,whichisaplotoftheheatuxq00versusthewallsuperheatTwTsatonthelogarithmicscale.TheliquidisassumedtobeatitssaturationtemperatureTsat.Asthewallsuperheatincreases,theuidgoesthroughnaturalconvection,formationofisolatedbubblesorpartialnucleateboiling(A-B),formationofslugsandcolumnsoffullydevelopednucleateboiling(B-C),transitionboiling(C-D)andnallylmboilingbeyondthepointDuptoE.ThepointConthecurvecorrespondstothecriticalheatux(CHF).Thisisthecasefortemperaturecontrolledboilingprocess.Inaboilingprocesswhereheatuxiscontrolled,iftheheatuxisincreasedbeyondtheCHF,theuidjumpshorizontallyfrompointCtopointEanddirectlyentersthelmboilingregimewithoutthetransitionboilingphase.AftertheCHFisreached,mostofthesurfaceiscoveredwithvaporandbecomesnearlyinsulated[ 8 ].Thismakesthesurfacetemperatureriseveryrapidly.ThereforetheCHFmarksthesafelimitofoperationformanyboilingsystems.Achilldownorquenchingprocessproceedsinthereversedirectionontheboilingcurve.ItusuallystartsabovepointEinthepost-CHFregionandthengoestowardspointDinthelmboilingregimeasthewalltemperaturedecreases.PointDiscalledtheLeidenfrostpointwhichsigniestheminimumheatertemperaturerequiredforthelmboiling.Forthelmboilingprocess,thewallissohotthatliquidwillvaporizebeforereachingtheheatersurfacethatcausestheheatertobealwaysincontactwithvapor.WhencoolingbeyondtheLeidenfrostpoint,ifaconstantheatuxheaterwereused,thentheboilingwouldshiftfromlmtonucleateboiling(somewherebetweenpointsAandB) 20

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Atypicalboilingcurve directlywithasubstantialdecreaseinthewalltemperaturebecausethetransitionboilingisanunstableprocess. 9 ].Figure 2-2 isanexamplethatshowsthedierenttwo-phaseowtypesinahorizontalpipe.Theowtypesmaybebubblyow,plugow,stratiedow,wavyow,slugoworannularow[ 10 ].Figure 2-2 [ 11 ]isanotherexampleandshowstheowregimesofatwophaseowinaverticaltubewithupwardowdirection.Inthiscase,thepossibleowtypesarebubblyow,slugow,churnow,annularowandmistow.Themaindierencebetweenthehorizontalandverticalowsistheeectofgravitythatcausesthehorizontalowtobecomenon-symmetricaltothetubecenterline. 21

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Flowregimesforconvectiveboilinginahorizontaltube Figure2-3. Flowregimesforconvectiveboilinginaverticaltube[ReprintedwithpermissionfromDziubinskiM.et.al.,2004.Flow-patternMapofaTwo-PhaseNon-NewtonianLiquidGasFlowinaVerticalPipe.(Page552,Figure1).InternationalJournalofMultiphaseFlow,30.] TheverticalquenchingofapipeinterrestrialgravityhasreceivedalotofattentionduetoitsimportanceintheLossOfCoolantAccident(LOCA)safetyconsiderationsinthenuclearpowerindustry.[ 12 ]studiedingreatdetailthetheowregimesandheattransfercharacteristicsinaverticalpipequenching.Theyfoundthatforsteadyinjectionrateofcoolant,theobservedowpatterndependedonwhethertheliquidwassubcooled 22

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2-4 .Forthecaseofsubcooledliquid,aninvertedannularlmboiling(IAFB)regimeisobservedabovetheliquidcolumn.Thisisnotthecaseforasaturatedliquid,wheretheowregimesshowanannularlmboilingtypepatternabovetheregionofnucleateboiling.Inboththecases,abovetheinvertedannularorannularowregion,dispersedowlmboiling(DFFB)regimewasobserved.DFFBischaracterizedbyvaryingsizedropsandglobsofuiddispersedinthevaporphase.Theheattransfermechanismislmboiling.Asthewalltemperaturereducestoacertaindegree,theliquidphaseisabletocontactthetubewallsomewhereupstreamofthelmboilingregion.Theleadingliquid-wallcontactpoint,whichisoftenreferredtoasthequenchingfront(QF)orsputteringregion,ischaracterizedbyvoraciousboilingwithlargedecreaseinthewalltemperature.Thequenchingfrontpropagatesdownstreamwiththeow.TheheattransfermechanismattheQFistransitionboilingandtheowisveryagitatedinthisregion.Thisre-establishmentofliquid-wallcontactiscalledrewettingphenomenonandhasbeenaresearchtopicforseveraldecades.Thus,studiesofquenchinginterrestrialgravity(1-g)showthattheowpatternsexistingarequitedierentfromthoseobservedunderadiabaticorevenboilingbutnon-quenchingconditions.TheowpatternsalsodependontheoodingrateatQFandtheinletwatervelocity[ 13 { 15 ].Iftheinletmassowrateislow,thenthevaporqualityattheQFishighwhichresultsinannularlmboilingregime.Conversely,iftheinletmassowrateishigh,thenthevaporqualityattheQFislowandIAFBexists.Therehavebeenrelativelyfewstudiesaboutthesequenchingowpatternswhichoccuratveryhighheatuxes(greaterthanCHF)andtheheattransfermechanismsarenotaswellunderstood.Agoodpaperdiscussingthehydrodynamicaspectsofpost-CHFowsis[ 16 ].[ 13 ]havereviewedsomeoftheheattransferaspectsandcorrelationworkforDFFB.IAFBinterrestrialgravityhasbeenstudiedusingtwo-uidmodels[ 17 18 ],homogeneousowmodels[ 19 20 ]andseparatedowmodels[ 10 ]. 23

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Flowpatternsduringquenchingofasupportedhottube 21 { 23 ].Two-phaseowunderadiabaticconditionsinmicrogravityhasalsobeenstudiedforsometime.[ 24 { 26 ]havefocusedonthetwo-phaseowregimes,voidfractionandpressuredropundersteadystateconditions.Thegeneralunderstandingfromtheseexperimentsisthatin-gthedistributedowregimessuchasbubblyanddispersedowoccuroverawiderrangeofqualitiesthanundernormalgravity(1-g).However,theseexperimentsonsteady-stateoworpoolboilingarenotveryusefulforunderstandingquenchingprocessesinmicrogravity,aseveninterrestrialgravityquenchingowpatternsarequitedierentfromadiabaticorevenboilingows[ 27 ].Quenchingexperimentsconductedunder-gshowthattherearecertainsimilaritiesanddierencesintheowcharacteristicswhichwouldbeimportantinthedesignofthermalcomponentsforspaceapplications.MostoftheseexperimentshavebeenperformedonboardNASA'sKC-135aircraft[ 2 { 4 28 ].Twotypesofdatawereobtainedfromtheseexperiments-aset 24

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2-1 summarizessomeoftheexperimentsonquenchinginmicrogravityperformedbyvariousresearchers. Table2-1. Microgravityquenchingexperiments ResearchersEquipmentandCoolantObservedFlowPattern [ 28 ]14mmID,1.2mlongquartztubewithFreon(R-113)atatmosphericpressureIAFB,DFFB[ 3 29 ]1.05cmID,1.274cmOD,60cmlongquartztubeand0.432cmID,0.635cmOD,70cmlongSStubewithLiquidN2Filamentaryow[ 30 ]SSatsurface,usingamicroheatuxandsurfacetemperaturesensorwithFreon(R-113)IAFB[ 31 ]6mmIDpyrextubewithtransparent100nmIndium-TinOxidecoatingforheating,FC-72liquidIAFBandbubblyow 25

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28 ]andR-113werepredominantlyinvertedannularowanddisperseddropletowregimes.Theinvertedannular-likeowregimein-gshowedamuchthickervaporlmascomparedto1-gconditions.Theyobservedathickliquidlamentowingfreelythroughthethickvaporlayer.Thelamentwasnotaxisymmetric,duetosmalluctuationsintheg-level.Therewasmuchlessbubbleentrainmentduetohighsubcoolingoftheliquid.Theliquidcorewasoftenmuchthinner,resemblingaliquidlamentratherthanthetypicalIAFBunderterrestrialgravity.`Itwassmoothandcontinuous,andowedmostlyinthemiddleofthetube,sometimesalmostllingupthewholevolume'[ 28 ].Intheentiredurationoftheirtests,rewettingofthewallbytheliquiddidnottakeplace.Anotherimportantexperimentwasperformedby[ 29 ].TheyusedliquidN2intheirexperiments,whichwasinjectedatvaryingpressurestocontrolthecoolantowrateandatdierentwalltemperaturesintoaquartztube.Theyinitiallyobservedonlyvapor,followedforashortperiodbydispersedow,whichwasthenreplacedbywhattheycallas\lamentary"ow.Theliquidtooktheformoflongliquidlamentssurroundedbyavaporblanketseparatingthemfromthetubewall.Theywereabletoobservequenchingintheirexperiments,duringwhichtheowpatternwasnotveryclear.ThepassageoftheQFwasfollowedbynucleateboilingregionandfullyliquidregime.Inanotherexperiment[ 3 ]reportthatthesurfaceofthelamentappearedtobehighlyturbulent.Theliquidseemedtobeconnedtotheselaments,withnoappreciabledropletsorliquidmasses.Thelamentaryowwasobservedinnearlyallofthelowgravitycryogenicchilldowntests,andoccurredforallthecoolantowratestheytested.Inarecentstudyby[ 31 ]withFC-72liquid,itwasfoundthatthemassowrateoftheliquidhadasignicanteectonthestabilityoftheliquidcoreinIAFB,bothunderterrestrialgravityandmicrogravity.Figure 2-5 showstheobservedowpatterns 26

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BHighmassowrateFigure2-5. Flowpatternsatdierentmassowrates,Tw=235,inmicrogravity.[ReprintedwithpermissionfromCelataG.et.al.2009.Quenchingexperimentsinside6.0mmtubeatreducedgravity.(Page2813Figure16and17).InternationalJournalofheatandmasstransfer,52.] intheirstudyformicrogravity.Forlowmassowratesand1-g,theinvertedannularowwasobservedbuttheinterfacewascharacterizedbyhighlevelanddisorderandwasveryunstable.Athighermassowrates,theIAFBwasmuchmorestable.In-g,theliquidcorewasmuchmorestablethanin1-g,withthehighermassowratecasebeingmorestable.Thestabilityoftheliquidcoreisattributedtothesmalldierencerelative-velocitiesoftheliquidandvaporphaseforthehighermassowrates.Asdiscussedin[ 10 ],theprimarymechanismsofinterfaceinstabilityaretheRayleigh-TaylorandKelvin-Helmholtzinstabilities.Thereforeitistobeexpectedthatintheabsenceofgravityandforlowerrelativevelocityofthetwophases,theliquidcorewillbemorestable. 30 ]haveinvestigatedtheowlmboilingduringquenchingofR-113onahot 27

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3 ]havereportedthatthequenchprocessisdelayedinlowgravityandthetubewallcoolingratewasdiminishedundermicrogravityconditions.TheabsenceofRayleigh-Taylorinstabilityinmicrogravityenhancesthestabilityoftheinterface,contributingtotheloweringoftherewettingtemperature.[ 4 ]havereportedadrasticdecreaseinthelmboilingheattransferratesduringthequenchingofatubeundermicrogravity. 32 { 35 ].Therehavebeenseveralattemptstomodelthecryogenicchilldownprocessundernormalgravity.[ 5 ]lookatstratiedowlmboilinginhorizontalpipes.Theylookedat 28

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17 18 36 ]. Figure2-6. Comparisonbetweenmeasuredandpredictedwalltemperaturesundermicrogravitywithowrateof40cc/s[ReprintedwithpermissionfromYuanK.etal.2009,Numericalmodelingofcryogenicchilldownprocessinterrestrialgravityandmicrogravity(Page51Figure5).InternationalJournalofHeatandFluidFlow,30] Arecentpaperby[ 6 ]isoneoftherstattemptstomodelcryogenictwo-phaseowinreducedgravity.Intheirmodel,theyhaveincludedIAFBandDFFBregionsaswellasthesingle-phaseliquidwithnucleateboilingandpurevaporzonessincethesearetheowpatternsobservedin-gconditionbyresearchers.Theyuseatwo-uidmodelforIAFBandDFFB.Themodelresultsindicatethatlmboilingheattransferdecreaseswithdecreasinggravitylevel.Theyachievedgoodagreementwithexperimentalresultsintheirmodel.Figure 2-6 showstheirresultsforthewalltemperatureasafunctionoftime. 29

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37 ]wereattempted.However,duetoanexponentialincreaseincomputingpowerintherecentyears,theycannowbesolvedbydirectnumericalsimulation.Insomeoftheearlyresearchdoneonmultiphaseows,manyresearcherspreferredtousecurvilineargrids[ 38 { 42 ].Thisapproachcanbeusedforverysimplemultiphaseowswithonlyasingleembeddedobject.Inordertodescribethedeformationoftheinterfacepowerfulgridgenerationisrequired.Thegridhastobeupdatedfrequentlytoobtainaconvergentsolution,makingitverycomputationallyintensive.Inrecentyears,severalCartesiangridmethodshavebeenintroduced:theSharpInterfaceMethod(SIM)[ 43 44 ],theImmersedBoundaryMethod(IBM)[ 45 46 ],theVolume-of-Fluid(VOF)method[ 47 48 ],theLevel-Setmethod[ 49 50 ],thehybridLevel-SetandVolume-of-Fluidmethod[ 51 52 ],thePhase-Fieldmethod[ 53 ]etc.Basedonthecomputationalframework,SIMandIBMareclassiedasmixedEulerian-LagrangianwhiletheLevel-set,VOFandPhase-FieldmethodsareintheEuleriancategory[ 54 ].[ 55 ]providesagoodreviewabouttherespectivestrengthsandadvantagesoftheSharpInterfaceMethods(SIM)andtheContinuousInterfaceMethods(CIM)formicrogravityapplications.Tothebestoftheauthor'sknowledge,theonlyattemptatdirectlysimulatingcryogenicowsinmicrogravityusingthefullsetofequationsforuidowandheattransferisby[ 7 ]withalimitedscope.Thisworkwillbereferredtotimeandagaininthisresearch,asthebasicnumericaltechniquesarethesame,withdeviationsandenhancementswhicharespecictothisresearch. 30

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3-1 isaschematicofacryogenicsystem,whereliquidcryogenisstoredinthetankandowsthroughapipetobeusedinotherdevices.Whenthecoolantentersthehottube,thewalltemperatureismuchhigherthantheLeidenfrostpoint,andtheliquidevaporatesveryquicklyformingavaporlm,whichdoesnotallowtheliquidtocontactthewallofthetube,thusleadingtoinvertedannularlmboiling(IAFB).Therefore,itisreasonabletosimplifytheproblemandstudyonlytheinvertedannularow.Neartheinletofthetube,aquenchingfrontformsthatisfollowedbyaninverted Figure3-1. Simpliedschematicofcryogenicsystem annularowpatternwithvaporphasenexttothepipewallandaliquidcoreinthecenter[ 3 ].Astheliquidvaporizes,theradiusoftheliquidcorewoulddecreaseasittravelsdownstream.Figure 3-2 illustratestheproposedphysicalmodelofIAFBinmicrogavity.InFigure 3-2 thetoppictureshowsrealinvertedannularowwhilethepictureinthebottomisthatofidealizedinvertedannularowwithasmoothinterface.Intherealcase,theinterfaceisnotsmoothduetomanyreasonsliketurbulenceandinstabilitiessuchastheKevin-Helmholtzinstability. 31

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Modelofinvertedannularlmboiling Thereareseveralpublicationsaboutpost-CHFowregimes,i.e.,IAFBandDFFB[ 14 17 18 56 { 58 ].Allofthemrelyoncorrelationstocalculatetheheattransferatthewallandinterface.Inthepresentwork,noassumptionhasbeenmadeaboutthetemperatureorheatuxapriori.Theonlyassumptionisthatthepipeisinsulatedfromthesurroundings.Thisisreasonabletoassumebecauseinmanyapplicationsitisofinteresttominimizethelossofcryogenbyevaporation.Thereforeinthisresearchthetemperatureofthewall'sinnersurfacewillvaryandsowilltheheatux.Theheattransfercoecientwillbecalculatedfromthetemperatureeldobtainedfromthesimulations.Aconjugateheattransfermodelhasbeenusedtocouplethetemperatureeldsinthewallandtheuid.Theprocessedoccuringinsidethetankandotherdevicesdownstreamofthepipeexitwillnotbeincludedinthecurrentsimulation.Onlythestraightsectionofpipeconnectingthesedeviceswillbeconsideredinthisresearch.Intheabsenceofgravity,itisreasonabletoassumetheinvertedannularowisaxisymmetric.Figure 3-2 showsthemodelfortheidealizedcontinuousinvertedannularowwhichwillbeusedinthecurrentnumericalsimulation. 32

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Controlvolumeforheattransfer Tocompletetheentireheattransferpath,thetwo-phaseowinsidethepipemustbeconnectedtotheheatsource,thepipewall.Thewallprovidesheatbythreemodes.Therearetheregularheatconductionandconvectionmechanisms.Duetothehightemperaturedierence,therewillalsoberadiationfromtheinternalsurfaceofthewalltotheliquidcore.Figure 3-3 showsthemodelforconjugateheattransferofwallanduid.Theexternalsurfaceofthewallisassumedtobetotallyinsulated.ThereforeontheoutsideofwalltheNeumannboundaryconditionwillbeassigned. 33

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34

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_q00rad=S(T4wT4sat) _q00rad;w=S(T4wT4sat) 35

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36

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We+v (3{14) Pekv klT(3{15) Reynoldsnumber:Re=lUD lPecletnumber:Pe=lCplUD klWebernumber:We=lU2d Jakobnumber:Ja=lCpl(TwTsat) 37

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3-4 .Thegureisnottoscale. Figure3-4. Computationaldomainfornumericalsimulationofcryogenicowthroughpipe Initially,thetemperatureofthewallandvaporinsidethepipeissetasTw=1:0andTv=1:0.Thecryogenenteringthetubeisassumedtobeatitssaturationtemperature.Therefore,fortheliquidphase,Tl=0:0issetastheinitialcondition.Attheinletboththevaporandtheliquidareprescribedaunitvelocity,whereasthevaporintheowdomainisassumedtobequiescentinitially.Attheoutlet,massconservationofthevaporphaseisenforced.Forthispurpose,theinletvapormassowandthemassgeneratedattheliquidsurfacearetakenintoaccount.Bytakingaverylongowdomain(forexampleL=44forRe=3000),thefullydevelopedboundaryconditionisensuredatthepipeexit.Thetopboundaryisawall,thereforeweusetheno-slipboundaryconditionforvelocity.Thebottomboundaryissymmetric.Thetemperatureconditionneedsmoreattention,sincethewallisalsoincludedinthecomputationaldomain.Forthevapor-wallsurface,wehave @r=kv@T @r+_q00rad;wD klTatr=Rwi=D(3{18) 38

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@r=0atr=Rwo=D(3{19) 39

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7 59 ]. 4-1 .Theintegralformsofgoverningequationsaregiven:Continuityequation: @tdV+Zcs~u(~u~n)dS=Zcsp~ndS+1 60 ]ontheCartesiangridsystemisadopted,theprimitivevariables(velocity,pressureandtemperature)aredenedatthecellcentersandtheprimaryvariablesneededatthecellfacesareevaluatedbyinterpolationfromrespectivevariablesatcellcentersasshowninFigure 4-1 .The 40

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Non-staggeredgridsystem fractionalstepmethodisabranchofpressurebasedpredictor-correctormethods.Thepredict-correctprocedureisnotdetermineduniquelyandcanbeconstructedbydierentcombinationsofpredictionandcorrectionprocedures.Thekeypointisthatthegoverningequationscannotbemodiedduringthepredictionprocedureandthecontinuityequationmustbeincludedinlastcorrectionproceduresincethecontinuityequationisnotsolvedseparately.Here,asecondorderaccuratetwo-stepfractionalstepmethod[ 61 { 63 ]isusedforadvancingthesolutionsoftheintegralunsteadygoverningequationsintime.Inthisapproach,thesolutionisadvancedfromtimestepnton+1throughanintermediatediusion-convectionstep.Intheintermediatestep,themomentumequationswithoutthepressuregradienttermsarerstsolvedandadvanced.Theintermediatediusion-convectionmomentumequationcanbediscretizedas:Zcv~u~un 2Zcsh3~un~Un~n~un1~Un1~nidS+1 2ReZcs~r~u+~r~un~ndS 41

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2Zcsh3Tn~Un~nTn1~Un1~nidS+1 2PeZcs~rTn+1+~rTn~ndS 42

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Figure4-2. Anexampleofcutcellsformedneartheinterface 38 { 42 ].Thisapproachcanbeusedforverysimplemultiphaseowswithonlyasingleembeddedobject.Inordertodescribethedeformationoftheinterfacepowerfulgridgenerationisrequired.Thegridhastobeupdatedfrequentlytoobtainaconvergentsolution,makingitverycomputationallyintensive.Inrecentmultiphasecomputations,severalCartesiangridmethodshavebeenintroduced:theSharpInterfaceMethod(SIM)[ 43 44 ],theImmersedBoundaryMethod(IBM)[ 45 46 ],theVolume-of-Fluid(VOF)method[ 47 48 ],theLevel-Setmethod[ 49 50 ],thehybridLevel-SetandVolume-of-Fluidmethod[ 51 52 ],thePhase-Fieldmethod[ 53 ]etc.Basedonthecomputationalframework,SIMandIBMareclassiedasmixedEulerian-LagrangianwhiletheLevel-set,VOFandPhase-FieldmethodsareintheEuleriancategory[ 54 ].Inthisresearch,theSharpInterfaceMethod(SIM)whichistypeofmixedEulerian-LagrangianCartesiangridmethodisemployedtosimulatecomplexgeometries.InSIM,aCartesiangridformsthebackgroundmeshandexplicitinterfacesareusedtodescribetheshapesoftheobjectsembeddedinthisbackgroundgrid.Theinterfaceisexplicitandisconstructed 43

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Figure4-3. Exampleofmixedstructuredandunstructuredgrid InSIM,oneneedstodenetherelationshipbetweenthebackgroundgridandtheexplicitinterface.Becausetheinterfacedoesnotconformtothegrid,thecellscontainingtheinterfacewillbecutandformnon-rectangularcut-cells.Thesecut-cellsneedtobetreateddierentlyfromordinaryrectangularcells.Inthisresearch,acut-cellprocedurebasedon[ 44 64 { 66 ]isemployedtotreatthethecellsneartheinterface.Inthecut-cellapproach,eachsegmentofthecut-cellismergedintoaneighboringcellorassignedtheidentityoftheoriginalCartesiancell.Hence,eventhoughtheunderlyinggridisCartesian,thecutcellsarereconstructedtobecomethenon-rectangularcellsandthecutsideswillformtheinterface.Afterthereconstruction,theentiregridislledwiththerectangular 44

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44 65 ].AmongtheEulerian-Lagrangianapproaches,SIMCCgivesthebestaccuracy,especiallynearasolidboundary.Figure 4-3 isanexampleofthetypeofgridsystemusedinSIMCC.Anexplicitinterfaceconstructedbymarkerpointswilldividetheentiredomainintodierentphases.Thegridhastwotypesofcells,rectangularcellsawayfromtheinterfaceandcut-cellsinitsvicinity.Thecut-cellapproachalsoensuresthatthetotalnumberofcellsdoesnotchangeduringthecomputation. 45

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64 ]isusedtohandleirregularintersectionsbetweenaninterfaceandtheCartesiangrid.Becauseoftheinterface,somecellsarecutandcannotmaintaintheirrectangularshapeanymoreandrequirespecialtreatment.Thesecellsarecalled`cut-cells'.Figure 4-2 illustratestheformationofcut-cellswheretheregulargridiscutbytheinterface.Forinterfacialtracking,informationabouttheinterfaceneedstobestoredinawaywhichallowsforeasycomputations.Inthepresentresearch,positionsofmarkerpointsarettedbypiecewise-quadraticcurves.Fromthesecurves,variousgeometricalquantitieslikethenormal,curvatureandtheintersectionsbetweentheinterfaceandbackgroundgridcanbeeasilycalculated.Informationabouttheintersectionsisneededtoknowwhichcellshavebeencutbytheinterfaceandneedtoberecongured.Thisinformationisalsousedforthenextstep,themergingprocedure.Usingthemergingprocedure,thefragmentsofcellsintersectedbytheinterfacecanbemergedwithneighboringcellsorwithlargerfragmentstoformthecut-cells.Forthepurposeofformingcut-cells,theinterfaceisassumedtobeformedbyasequenceofstraightlinesections.Thepossibleshapesofcut-cellsthatcanbeformedafterthemergingprocedurearetrapezoidal,triangularandpentagonal.Thenewgridisformedbytheregularcellsandthespecialcut-cells.Mostcellsstillkeeptheoriginalshapes.Also,thenumberofcellsdoesnotchange.Itisimportanttoknowtheshapeofthenewlyformedcut-cellsforthethirdtechnique,whichisuxcomputations.Itisalsoimportanttoknowonwhichsideofthecut-celltheinterfacelies.Aspecialinterpolationschemewithahigherorderaccuracyisrequiredtohandlethecomplicatedcut-cellstogettheaccurateprimaryvariablesor 46

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4-3 isanexampleofthemixedstructuredandunstructuredtypegridobtainedafterthemergingprocedure.Thesealgorithmsandtheirimplementationinthenumericalcodeisexplainedinthefollowingsections.Formoredetails,pleasereferto[ 7 59 ]. 4-3 showsthemarkerpointsdistributedontheinterface.Thelocationofmarkerpointsisstoredparametricallyasafunctionofthearclengthmeasuredfromsomereferencepointontheinterface.Sincequadraticcurveshavebeenused,thereforethefunctionalformoftheparametricequationsisalsoquadratic.Infact,alltheprimaryvariablesarestoredinthesameformat, 47

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66 ],thecurvatureiscalculatedfrom 4-2 illustratestheformationofinterfacialcellswherecells1to4arecutbyaninterface.Accordingtothepresentcut-celltechnique,thesegmentsofaninterfacialcellnotcontainingtheoriginalcellcenterareabsorbedbytheirneighboringcells;thesegmentscontainingtheoriginalcellcenteraregiventhesameindexastheoriginalcell.Forexample,inFigure 4-2 ,theuppersegmentofcell3isabsorbedintocell5toformanewtrapezoidcell.Thefractionofcell3withcellcenterbecomesanewindependenttrapezoidcell.Themainsegmentofcell1thatcontainstheoriginalcellcenterwillabsorbthesmallsegmentsofcells4and2toformanewtriangularcell.Theremainingsegmentofcell4containingitsoriginalcellcenternowbecomesanindependentpentagonalcell.Withthesecut-and-absorptionprocedures,theinterfacialcellsarereorganizedalongwiththeirneighboringcellstoformnewcellswithtriangular,trapezoidal,andpentagonal 48

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4-2 showsanexampleofthecut-cellsafterre-construction.Whentheareaofasegmentislessthathalfareaofanormalcell,itwillbeabsorbed.Thismaintainsagoodratiobetweenthevolumesofthecells,andalsoensuresthatthetotalnumberofcellsremainsunaltered.Afterthisprocedure,eachnewlydenedcellmaintainsauniqueindexandcellcentertosupporttheneededdatastructure.TheoriginalCartesiangridwillnowbecomeamixedgridwhichincludessomeoriginalstructuredgrid,andtheunstructuredgridduetothenewlygeneratedirregularlyshapedcut-cells.Figure 4-3 isanexampleofthemixedstructuredandunstructuredtypegridobtainedafterthemergingprocedure.Anyobjectwithoutsharpedgescanbemodeledusingthecurrentcut-celltechniques. 4-4 showsaCartesiangridwithcellswhicharecutbyaninterface.Thesolidsquaresindicatethecentersofcells.Becauseoftheinterface,theoriginalABCDcellwiththecellcenter1willabsorbthefragmentfromanothercelltoformatrapezoidalcellBCEF.Theoriginalgridline 4-4 .Onecanconstructasecond-orderaccurateintegrationprocedureasfollows: 49

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Fluxcomputationsforcut-cells secondorderaccuracy:CD=11+3(11)1=x1xAC @xCD=13 50

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4-4 .Thusthevariableatthecenteroftheface @xDE=c1y2DE+c3yDE+c5(4{15)Asimilarapproachisusedtocomputetheowvariablesortheirnormalgradientsontheremainingsegments.Oncetheprimitivevariablesandthederivativesatallthemodiedfaceshavebeendetermined,thecoecientsofmatrixforthecut-cellscanbemodiedbasedonthisinformationandthesolverwillbeinvokedtoobtainthesolution.Itmustbeemphasizedthattheequationobtainedaboveisonlyforthisspeciccase.Theinterpolationpolynomialisquadraticinydirectionandlinearinxdirection.Foradierentinterfaceconguration,thepolynomialmaybequadraticinx. 38 ]andthesecondoneistoupdatethecellsbecauseofchangeofphase[ 66 ]. 38 ]isusedtodeterminethenewlocationoftheinterfaceinordertosatisfytheforcebalanceattheinterface.Thestressesactingattheinterfacecanbedividedintothenormalandtangentialcomponents.Intheseowcomputations,theReynoldsnumber 51

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Illustrationofinterfacialadvancingprocess beingveryhigh,thetangentialshearstressesarequitesmall.Thereforethedisplacementofinterfaceisgovernedonlybythenormalforcesattheinterface.Thenewlocationofinterfaceisdeterminediteratively.Ineachiteration,theresidualoftheforcebalanceinthenormaldirectionwillbecomputedandthedisplacementofmarkerpointsisassumedproportionaltothisresidual: 4-5 illustratesforthisinterfacialadvancingprocess.InFigure 4-5 ,amarkerpointAoislocatedattheinitialinterfaceandtheresidualofforcebalanceinthenormaldirection1willbecomputed.BasedonthelocationofAoandthevalueof1,themarkerpointwillbepushedtoA1.Inthismoment,theresidualofforcebalanceinthenormaldirectionwillbechecked.Ifthenewresidual2isstilllarge,themarkerpointA1willbepushedtoA2.Oncetheresidualissmallenoughwhichmeansthattheforcebalanceinthenormaldirectionhasachievedconvergence,theiterationwillbestopped 52

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Figure4-6. Cellupdationprocedure 4-6 ,thephaseofcellcenter`5'isdierentfromcellcalled`1',`2',`3'and`4'initiallyattimet.Aftertheinterfacemovestothenewlocationatt+t,thephaseofcell`5'willchangeasthewillchangeasthecellcenternowliesin`Phase2'.Inthiscase,theuidpropertiesofcellcenterwillchangetothecorrespondingphase.Allnewprimaryvariablesofcellcenter`5'shouldnowbeobtainedbyfollowingsteps: 53

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Anormalprobepassingthroughthecellcenter`5'istaken,andtheintersectionofthisprobewiththenewinterfaceisdenotedbypoint`B'intheFigure 4-6 2. Thesameprobeisextendedintheoppositedirectiontolocateanotherpoint`A'suchthatthedistancebetween`A'nd`B'is1:5x. 3. Findthefourcellcenters`1',`2',`3'and`4'whichsurroundthepoint`A'.Usethevaluesoftheprimaryvariablesatthesefourcellcentersandbilinearinterpolatiotondthevalueoftheprimaryvariablesatpoint`A'. 4. Uselinearinterpolationtoobtaintheprimaryvariablesofcell`5'usingtheirvaluesatpoints`A'and`B'. 5. Updatetheuidpropertiesofcellcenter`5'.Thereforeduetothesharpinterfacemethod,thevaluesofprimaryvariables`jump'whenacellchangesphase.ThisalgorithmisnecessarytomaintaintheconsistencyofSIMformulation. 4{19 ,thenormalcomponentofthevelocityneartheinterfaceofeachphasecanbeobtained.Thevelocitiesofeachphase 54

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_q00=kv@Tv m=e=(4{22)Fromthemasschangeonecancalculatethecorrespondingvolumechangeoftheliquid.Thiswouldbethetotalchangeinthevolumeoftheliquidphase,andthenewpositionoftheinterfaceshouldsatisfythiscondition.Themarkerpointsaregivenalocaldisplacementinproportiontothelocalheatuxversusthetotalheatux.Thenewpositionisdeterminediterativelybygivingsmallincrementsinthenormaldirection 55

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4{24 aninitialdisplacementduetothephasechangehasalreadybeenappliedbeforetheiterationstarts.Ineachtimestep,thetotaldisplacementisgivenasthedierencebetweentheoriginalinterfacepositionandthenalinterfaceposition.Therefore,thenewnormalcomponentofinterfacialvelocitycanbeobtainedby: 7 ].Heemphasizestheimportanceofobtainingaccuratemasstransfergeneratedattheinterface,forwhichaccuratecalculationtheheatuxesiscrucial.Inthisregard,SIMCCisabletoperformbetterthancontinuousinterfacemethods(CIM)becauseofitstreatmentoftheinterfaceasanentitywithzerothickness,whichismuchclosertothetheoryofcontinuum. 56

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67 ])sothattheowcanbecomefullydeveloped.However,noneofthesecharacteristicsofinternalowsposeanyseriousproblemsforsinglephasesimulations.Inmultiphaseinternalowswithphasechange,thesituationisextremelycomplicatedsincetheproblemishighlyunsteady.Inthecontextoftheinvertedannularowencounteredincryogenicchilldownprocess,theadvancementoftheliquidphaseinthepipeconstantlypushesoutthevapor,andtheareaoccupiedbythevaporphasedecreasescontinuously.Nomatterwhatlengthofthepipeistaken,itwillultimatelyviolatethefullydevelopedboundaryconditionattheoutletbecauseoftheconstantdecreaseinthelengthofthevaporcolumn.Sothesimulationsneedtobestoppedafterthefullydevelopedconditioncannotbesatisedattheoutlet.Itbecomesimportanttoestimatethelengthofthecomputationaldomain,whichisdiculttodoaprioribecauseeventheReynoldsnumberofthevaporphaseiscontinuouslyincreasingbecauseofphasechange.Ifthelengthisunderestimated,onewouldnotbeabletocarryoutthesimulationtoitsdesiredconclusion,whereasifitisoverestimated,itwillunnecessarilyslowdownthecomputations.Thegeometryofcryogenicinvertedannularowalsoposessomeuniquenumericalchallenges,eventhoughitisoneofthesimplerowregimesofthoseencounteredin 57

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Challengesininternalphasechangeows two-phaseows.Theissueisthatthegapbetweentheliquidcoreandthehottubewallisextremelynarrow.Thiscreatesproblemsintwoways:becauseoftheproximityofthewall,mostofthephasechangehappensinthisregion,secondlyduetothehighmasstransferrate,alocalizedregionofveryhighvelocityiscreated,sincealltheliquidmasswhichgetsconvertedintovaporhastobeconvectedoutofthisconstrictedspace.Thehighvelocitiesandsteepgradientsinthevaporphaseinthiszonecreatenumericalstabilityissuesasthesimulationprogresses.Thisisespeciallytrueforthecurrentschemewhichusescentraldierencingfortheconvectiveterminthemomentumtransportequation.TheseissuesareillustratedinFigure 4-7 .Thenarrowhighvaporvelocityregionshowninthegurewhichisduetotheslightbulgeintheadvancingliquidfrontwillhenceforthbereferredtoasthe`bulge'region.Toaddresstheseconcernsinthesimulationoftwo-phaseinternalows,itisproposedtotakethefollowingactions.Firstandforemost,thereisarealneedtoincreasetheperformanceofthecurrentSIMCCcodetoenablehighaccuracycalculationsoflargerdomainsthancancurrentlybehandled.Therefore,multigridaccelerationtechniquesbecomeanecessity.Multigridmethodsareveryecientandcansolveadiscretesystem 58

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68 ].Secondly,itisextremelyimportanttoaddressthenumericalissueofconvectiveinstabilityduetothehighvaporvelocityandgradientsinthe`bulge'region.Therefore,itisproposedtouseanupwinddierencingschemetomitigatetheconvectiveinstabilityeects[ 60 69 ].ThespecicschemesuggestedisthemodiedQUICKby[ 70 ],whichisastableandfastconvergingvariantoftheoriginalQUICKschemeby[ 71 ].Allofthesemethodswillbediscussedinmoredetailinthefollowingsections. 72 ]andaddressedthesolutionofpressure-Poissonequationonaunitsquare.Sincethen,theresearchinmultigridmethodshasproceededatarapidpace.Afewnotableworksinthisregardare[ 68 73 { 75 ].Multigridmethodsarenowusedtosolveawiderangeoflinearandnonlinearboundaryvalueproblems.MultigridmethodsareveryecientandcansolveadiscretesystemofnequationstothedesiredaccuracyinO(n)computeroperations[ 68 ].Intherecentyears,someresearchershavedevelopedmultigridtechniquesformultiphaseows[ 65 76 77 ].However,mostresearchersusetheAlgebraicMultigridmethod(AMG)whichignoresthepresenceoftheinterfaceonthecoarsergrids.SincetheSIMCCmakesuseofanorderedbutpossiblynon-uniform(butwithnonegrid`patches')andunstructuredgrid,itispossibletoimplementageometricmultigridschemeforSIMCC. 59

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Figure4-8. Restrictionandprolongationoperationsinmultigridtechnique largesparsesystemofequationsgivenby 4{28 onthecoarsegrid.Forthispurpose,theresidualis`restricted'ontothecoarser 60

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4-8 ,wherethreegridlevelshavebeentakenforillustration. 61

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78 ].Figure 4-9 showsatypicalcase.Inthegure,thenegridisshownbythelightercolorlines,whilethecoarsegridissuperimposedinadarkerhue.Thecentersofthecoarsegridcontrolvolumesshownbysquareshapedpointsfallattheintersectionofthenegridlines.Thecellcentersofthenegridareshownbylighterdots.Thegureshowsafewcut-cellsformedbythecoarsegridwhichareshadedandmarkedaandb.Forregularcellslikec,fournegridcellsformasinglecoarsegridcell.ThematrixAkwouldbeformedbydiscretizingthepressureequationonthiscoarsegrid: tZcs~Uk~ndS(4{33)Foraregularcellinthebulkphase,thiswouldleadtoanexpressionlike:pP2dyk 4-9 ,itcanbeseenthatforeveryne-gridcellinthebulk,denotedbygraydots,therecanbefoundfourcoarse-gridcell-centersinthesamephasesurroundingit.Thesameistrueforcoarse-gridcellcentersawayfromtheinterface-therecanbefoundfourne-gridcellcentersarounditwhichareinthesamephase.Thesefourpointsareusedtoconstructabilinearpolynomialforrestrictionor 62

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Figure4-9. IllustrationofthemultigridtechniqueforSIMCC 4-10 ,itcanbeseenthatacoarsecellissuperimposedonfournegridcells,labeled`a',`b',`c'and`d'intheclockwisedirectionfromthetop-leftcorner.Thesecellshavethephase`1'or`0'dependingonwhethertheyareintheimmersedphaseormotherphase,respectively.Todeterminethephaseofthecoarse-gridcell,weconstructavariablefromthevaluesofa,b,c,d.Essentially,thisvariablestorestheinformationaboutthephasesofthefournegridcells,andmaybecalled`abcd'.Thisinformationcanbeparsedtodeterminethephaseandtypeofthecoarse-gridcut-cell.Ifallfourcellshavethesamephase,i.e.,`abcd'iseither`1111'or`0000',thenit 63

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Figure4-10. Determinationofthephaseofcoarsegridcut-cells Aftertherestrictionoperation(Eq. 4{29 ),oneneedstoformulatethematrixAkonthecoarsegrid.Regularcellsaretreatedasdiscussedintheprevioussection.Forcut-cells,itisimportanttoreorganizethemintoaconsistentmosaicusingthemergingprocedure.SoexactlythesamesetofcalculationsneedtobeperformedasforthenegriddiscussedinSection 4.3.2 and 4.3.3 .Afterthisprocedure,theEq. 4{30 canbesolvedonthecoarsegridtogetek.Theprolongationoperator(Eq. 4{31 )alsousesinformationfromthenegridwhereverpossible,andisdesignedtohonorthepresenceoftheinterface.Prolongationistheinverseoperation,wherethesolutionfromthecoarsegridcellanditsneighborslyinginthesamephasewillbeusedforlinearinterpolationatthenegridcellcenters.However,nowfour 64

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4-9 .Therefore,thissituationismorecomplexthanrestriction,buttheapproachtowardspolynomialinterpolationissimilar.Onlycellsfromthesamephasewillareusedfortheinterpolation,andonesidedinterpolationisneededformanycut-cells. 4-11 ,Re=100,dt=0:01anda128128gridhasbeenused.ItisseenthatthereisasignicantadvantageobtainedbyusingmultigridbothintermsoftheCPU(centralprocessingunit)timeandthenumberofiterationsonthenegrid.Forgridlevelsgreaterthantwo,thenumberofiterationsonthenegridremainsalmostthesame,butthereisstillasignicantdecreaseintheCPUtimeasseeninFigure 4-12 Figure4-11. Eciencyofmultigridbasedonnumberofiterations Next,thegeometricmultigridschemeforSIMCCwastestedwithanarticialinterfaceembeddedintheliddrivencavityow.Thisteststhemultigridsubroutineswhichhandlethecut-cellsneartheinterface,suchasrestrictionandprolongationfor 65

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EciencyofmultigridbasedonCPUtime coarsegridcut-cellsdescribedinSection 4.6.3 .TheinterfacehasaradiusR=0:1andislocatedatthecenterofthecavitywhichisaunitsquare.Theviscosityanddensityofthephasesbothinsideandoutsidethearticialinterfacearesetasequal.Thesetupoftheproblemissimilartothatusedby[ 65 ]totesttheirmultigridmethod.Theboundaryconditionsattheinterfacearenormalstressbalance,shearstressbalanceandthepressureissameonbothsidesoftheinterface.ThesimulationwasdoneforRe=100,dt=0:01anda128128gridasbefore.Twogridlevelsareusedfortheresultsthatfollow,oneisthenegridwithspacingofx=y=0:007813andanothercoarsergridwithx=y=0:015625.Figure 4-13 showsthestreamlinesforthebaselinecasewhichissimplelid-drivencavityowwithoutinterfaceandwithoutmultigrid,andthetestcasewithembeddedarticialinterfaceandmultigrid.Thisteststheaccuracyofcut-cellsroutinesaswellasthemultigridroutines.Itcanbeseenthatthestreamlinescloselymatch.TheeectoftheinterfaceonthestreamlinesinFigure 4-13B isminimal.Thestreamlinespassthroughtheinterfacewithoutanydistortionorbending.ThisshowsthehighlevelofaccuracypossiblewiththecurrentgeometricmultigridmethodcoupledwithSIMCCprocedures. 66

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BWitharticialinterfaceandMultigridFigure4-13. Streamlineswithandwithoutarticialinterface BV-velocityathorizontalcenterlineFigure4-14. Centerlinevelocitieswithandwithoutarticialinterface Tofurthercomparethevelocities,theu-velocityonverticalcenterlineandthev-velocityonhorizontalcenterlineareshowninFigure 4-14 forthesametwocases.Thevelocitycalculatedwithmultigridmethodandarticialinterfaceisshownbythedashedline.It 67

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BCPUtimeFigure4-15. Performanceofmultigridforarticialinterfacecase Figure 4-15 showstheperformanceofmultigridforthetestcaseofliddrivencavityowwitharticialinterface.Thebaselinecaseforthisisarticialinterfaceproblemwithoutanymultigridacceleration,solvedusingonlytheSIMCCtechniques.Theresidualisplottedonthey-axis.ItcanbeseenthatbothintermsofthenegriditerationsandtheCPUtime,multigrid(2levelV-cycle)faroutperformsSIMCCwithoutmultigridenhancement.Withoutmultigrid,SIMCCtakesabout540secondstoreducetheresidualbyfourordersofmagnitude.Withmultigridacceleration,thisoperationonlytakesabout200seconds.Thisisaanimprovementinspeedbyafactorofmorethan2.5,maintainingthesamelevelofaccuracy. 68

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69 ]),italsohascertaindrawbackswhenappliedtointernalowsituations.ThereasonisthatCDSdoesnotrecognizethedirectionofoworthestrengthoftheconvectiontermrelativetothediusionterm. Figure4-16. SchematicforQUICKdiscretizationofconvectiveterm Ingeneral,theconvectivetermforanyintensiveuidproperty'canbediscretizedas: (4{36) 69

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79 ].Ifthiscannotbesatised,itleadstophysicallyunrealizablesolutionswithoscillations.Upwinddierencingschemesgivemoreweightagetotheupstreamnodewhencalculatingtheconvectiveuxatthecontrolvolumefaceandareabletoalleviatetheunboundednessproblemtovariousextents.Therearevarioustypesofupwinddierencingschemesavailable,includingtherstorderupwinddierencing(FOU),thehybridscheme,thepowerlawscheme,thethirdorderQUICK(QuadraticUpwindInterpolationforConvectiveKinematics)schememadepopularby[ 71 ]etc.FOUschemeisstableforallPecellbuthasloweraccuracythanCDSandsuersfromfalsediusion.Inthisresearch,itisproposedtouseavariantoftheQUICKschemeby[ 70 ]whichismoreconsistentlyformulated.ThisschemehastheadvantagethatitismorestablethantheoriginalQUICKschemeandsatisestherequirementsofconservativeness,boundednessandtransportiveness.Itachievesthisbyplacingtroublesomenegativecoecientsinthesourceterm,andretainingonlypositivecoecientsinthemainmatrix.Forauniformgrid,thex-directiontermsdiscretizedusingtheQUICKschemeby[ 70 ]canbewrittenas: 8[3'P2'W'WW]forFw>0'e='P+1 8[3'E2'P'W]forFe>0'w='P+1 8[3'W2'P'E]forFw<0'e='E+1 8[3'P2'E'EE]forFe<0(4{37)wherethesubscriptsP;E;W;EE;WWindicatethecell,anditseast,west,fareastandfarwestneighborsrespectively.Theconvectivetermisnowhandledimplicitly.Only 70

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60 ].Anotherreasontodothisistoavoidgettingaverylargecomputationalmolecule,whichwouldhappenifallthetermsweretreatedimplicitly.Figure 4-17 showssomesimulationresultswithCDSandQUICK.InFigure 4-17A ,Pecell>2andCDScreatesoscillationsshortlyafterwards.InFigure 4-17B ,Pecell>4:5butQUICKschemeisstillstable.Infact,itwaspossibletorunalmostallofthecasesforPecell>8:0,withoutchangingthegridspacing.ThisshowsthattheQUICKschemeisveryeectiveinimprovingthestabilityofthecodeduetotheconvectiveterm. 71

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BWithQUICKFigure4-17. Comparisonofconvectiveschemes 72

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80 ],sinceitincludestabularresultsforvariousReynoldsnumbers.ItisalsogivesdataforawiderangeofRe.Theyusethestreamfunctionandvorticityformulation.Therefore,wehavechosenthistestcasetobenchmarkourcode. 73

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Uvelocitythroughverticalcenterline Figure5-2. Vvelocitythroughhorizontalcenterline Thegeometryconsistsofasquareorrectangularcavity,withDirichletboundaryconditionsonallsides.Thetopmovingwallisgivenaunitvelocity,whichdrivestheow.ThepresentcodeimplementsthenitevolumediscretizationandthefractionalstepalgorithmtosolvetheNavierStokesequationinanon-staggeredgrid.ItwastestedforaccuracyatvariousReynoldsnumbersbetween100and5000.Iterationswereperformeduntilconvergencewasreachedwhenthevelocityandpressureresidualsbecameless 74

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BRe=400Figure5-3. Streamlineplotsforliddrivencavityow than1.0E-8.InFigure 5-1 ,theuvelocityattheverticalcenterlinehasbeenplotted.Figure 5-2 showsthevvelocityatthehorizontalcenterline.Boththecaseshavebeenplottedagainstresultsof[ 80 ].Thevelocityplotshavebeenstaggeredby0:2unitsalongtheyaxisforbetterclarity.Gridresolutionof6464hasbeenusedforReynoldsnumbersupto1000,andtheresultsagreewellwith[ 80 ].AthigherReynoldsnumbers,theresultswitha6464gridwerenotfoundtobeaccurateenough.Therefore,anergridof128128hasbeenused.Inallthecases,thepressureandvelocityresidualshadreducedtolessthan1:0E8.Theresultsshowgoodquantitativeagreementwiththebenchmarkdata.Figure 5-3 showsthestreamlineplotsforthecasesRe=100andRe=400. 75

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BCenterofvortexFigure5-4. Featuresofwakebehindsolidsphere vortexsheddingoccurs[ 81 ].Hence,wehaverestrictedourselvestothisrangeofReynoldsnumber.Thecomputationaldomainis15dinlengthand5dinwidth,wheredisthespherediameter.Ontheleftboundary,aunitinletvelocityisprescribed.Thelowerboundaryissymmetricduetotheassumptionofaxisymmetry.Thefar-eldboundaryconditionmust 76

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Dragcoecientofsolidsphere beusedforthetopandrightboundaryforaccurateresults.Auniformgridresolutionof0.05hasbeenused.Thisgridspacingwaschosenonthebasisofagridrenementstudy.SimulationwasdoneforReintherangeof1to100.ItwasfoundthatowseparationoccursatRe=25.Thisisinagreementwith[ 81 ]whofoundanexperimentalvalueofRe=24atwhichseparationoccurred.ForRe=30andhigher,wakelengthandthevortexcenterhavebeencomparedwiththeexperimentalresultsof[ 81 ].Figures 5-4A and 5-4B showgoodagreementbothintermsofthetrendandtheactualcalculatedvalues.InFigure 5-5 ,thecalculatedcoecientofdragiscomparedwiththenumericalresultsof[ 82 ].Theerrorisfoundtobewithin3percentforallthesixcases.ThisprovesthatthepresentcodecancalculatethedragoverasurfaceveryaccuratelyusingSIMCC,andatteststothehighaccuracypossiblewithSIMCC.Therefore,fromthistestcasewecanconcludethatthetechniquesforhandingthestationaryinterfaceincludingthegoverningequationssolverandSIMCCaresucientlyveried. 77

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83 84 ].Thiscaseconclusivelyprovestheabilityofthepresentcodetohandledeformableinterfacesintwo-phaseowswithhighaccuracy.Inthiscomputation,waterisadoptedastheambientliquidandtheisothermalbubblecontainswatervaporatthesametemperatureastheliquid,sothedensityratio0.0006andviscosityratiois0.045.TherangeofReynoldsnumbersinvestigatedis1
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Re 1 10 100 Figure5-6. Computedbubbleshapes Figure5-7. Bubbleshapesby[ 39 ].[ReprintedwithpermissionfromG.RyskinandL.G.Leal,1984.NumericalSolutionoffree-boundaryproblemsinuidmechanics,Part2:Buoyancydrivenmotionofagasbubblethroughaquiescentliquid.(Page25,Figure2).JournalofFluidMechanics(148)] themovinganddeformingbubble.Resultswerepresentedintheformofstreamlineplots,bubbleshapeaspectratioanddrag. 79

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5-6 .Theshapesareingoodagreementwiththosecalculatedby[ 39 ]whichareshowninFigure 5-7 .ResultsforRe=0.5,We=0.5,andRe=1.0,We=1.0aregiveninFigure 5-8 ,andcomparedtotheshapescalculatedforthesameparametersusingtheasymptoticformulaof[ 37 ].Thesolutionby[ 37 ]hasbeenderivedundertheassumptionsthatRe<<1,We<<1,andRe2<
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BRe=10;We=3 CRe=50;We=3 DRe=100;We=3Figure5-9. Transitioninshape ofthebubblesbetweenRe=10andRe=100,forthewholerangeofWestudied.AtRe=100,thebubblesbegintohavegreaterdeformationatthefrontend.ThisbecomesmoreandmoreobviousasWeincreases.AtWe=10thereisactuallyanindentationatthefrontendofthebubble.Tounderstandthistransition,acasewithRe=50;We=3isalsopresented.TheresultsareshowninFigure 5-9C .Itisseenthattheshapeofthebubbleisnearlysymmetrical.Thisndingisconsistentwiththeresultsreportedby[ 39 ] 81

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InuenceofReonthecurvatureofbubble Figure5-11. InuenceofWeonaspectratio whoalsofoundthatthiskindofshapetransitionoccursforRe=100.TheytooreportthatRe=50istheborderlinecaseandseemstohavefore-aftsymmetry.Tocharacterizethistransitionfurther,theresearcherlookedatthesurfacetensiondistributiononthebubblesforWe=3,atthefourdierentReinFigure 5-10 .The 82

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5-11 theaspectratioofthebubblesiscomparedto[ 85 ]forRe=10and100.Thecomparisonisveryfavorable.ItisseenthattheaspectratioisaverystrongfunctionoftheWebernumber.HighWeimpliesweakersurfacetensionwhichgiveshigherdeformation,whilethelimitWe=0correspondstoinnitesurfacetensionandperfectlysphericalbubble.TheaspectratioisalsoinuencedbyRe,higherRegivinghigherdeformationaswell. 5-12 showsthewakebehindthebubblefordierentRe.ThecirculationisjustevidentatRe=1,whilebyRe=100thereisafullyformeddetachedcirculationzonebehindthebubble. 83

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BRe=10;We=10 CRe=100;We=10Figure5-12. Developmentofcirculationregion normaldragandsheardragrespectively.Thecoecientofdragisdenedasfollows: 5-13 and 5-14 fortworepresentativecases.Theguresshowhowtheforcesonthebubbledevelopduringtheunsteadysimulation.Whenthedragonthebubblebecomesrelativelyconstantitmeansthatthebubblehasreacheditsterminalvelocityandshape.Inthesecomputations,theinitialdragforceonthebubbleisnotzero.Thisisbecausethebubbleisinitiallystationaryinagravitationalforceeld.Thereforeinnumericalsimulations,the`pushandpull'strategyattemptstobalancethisgravitationalforceonthebubblebythepressureforces.Inthegures,thetotaldragcoecientmatcheswellwithotherpublishedresults.ForRe=10,We=3weobtainCD=3:4,whichcompareswellwiththevalueof3.3calculatedby[ 39 ].ForRe=100,We=3,[ 39 ]predictavalueofCD=0:62.Inoursimulations,weobtainavalueof0.61.FromFigure 5-13 and 5-14 ,severalobservationscanbemade.Firstofall,itistobenotedthatthenormaldragandthepressuredragseemtobecomplementarytoeachother.Atthestartofthesimulation,sincethebubbleisstationary,thenormalandshear 84

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Draghistory,Re=10;We=3 Figure5-14. Draghistory,Re=100;We=3 stressonthebubblearenegligible.Duringthesimulation,asthebubbleacceleratesanddeformsthecontributionofthepressureforcestothetotaldragdecreases,whilethenormalandshearforcesincreaseinmagnitude.AnotherfeatureoftheowsisthatthecontributionofthesheardragtothetotaldragdecreaseswithincreasingReynolds 85

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Dragondeformablebubble 127.3428.3023.8021.2219.22120.57119.10{103.013.403.232.782.6413.2313.674.161000.450.611.282.150.4210.621.232.77 number.AtRe=10,itisfoundthatsheardragisresponsibleforalmosttenpercentofthetotaldrag.AtRe=100,thiscontributiondropstoaboutfourpercent.Thisndingisconsistentwiththephysicsoftheproblem,sincetheviscousstressesarelowerathigherRe.AlsonotethatthecontributionofthenormalviscousstressesissignicantlyreducedwithincreasingRe.AtRe=10,pressureandnormaldragsseemtohaveanalmostequalcontributiontothetotaldragonthebubble,butatRe=100,thenormaldragisonlyabouttwentypercentofthetotaldrag.Infact,atRe=100,pressuredragdominates,andthistrendissimilartowhatisobservedwithowoverrigidspheres.InTable 5-1 ,weshowtheresultsforthetotaldragonthebubble.TherangeforReynoldsnumbersinvestigatedis1
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32 { 35 86 87 ].Thisproblemcanbeusedforvalidationofournumericalmethodsincethegeometryisverysimilartothecryogenicchilldownowinpipes,theonlydierencebeingthatthisisanisothermalmultiphaseow.Theworkof[ 35 ]hasbeenselectedtocomparetheresults.TheschematicoftheproblemisshowninFigure 5-15 .Inthereferenceframexedtothewalls,thisproblemisinherentlyanunsteadyproblemduetothetranslationoftheairbubble.Ifhoweveramovingreferenceframeattachedtothetipoftheairbubbleisselected,ittransformsintoasteadystateproblem.Sincetheairchannelisinniteinbothdirectionsandtheair-ngeralsoextendstonegativeinnity,thiscanbedone.IntheoriginalreferenceframeiftheterminalvelocityofairbubblewasU,theninthenewreferenceframethetopwalltranslatesatavelocityu=U.Thegasphaseremainsstationaryandhasconstantpressure.Atboththeleftandtherightboundaries,thex-gradientsofvelocityarenegligiblesincetheowisfullydeveloped.Theinterfaceisassumedtobehavelifeafreesurfacewithzeronormalvelocity.Plugowcanbeassumedneartheleftboundary.Sincetheoweldissymmetriconlytheupperhalfismodeled.Thebottomwallisgiventhesymmetryboundaryconditionwherebygradientsand 87

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l;Ca=Ul (5{2)Thecomputationaldomainconsistsofarectangularareawithnon-dimensionalunitsofL=20andH=1.Theinitialshapeoftheinterfaceisconstructedfromastraightlineandahalfcircle,withthetipoftheinterfacebeingxedatx=3:0.Duringthecourseofsimulations,theshapeofthebubblechangesastheowelddevelopsaroundthebubble.Pressureinthegasphaseisalsocalculatedduringthenumericalprocedure. Figure5-15. SetupforBrethertonproblem Figure 5-16 showsthepressurecontoursandstreamlineswhensteadystateisreached.Thenalpositionoftheinterfaceisy=0:8838atx=0.Thiscorrespondstoalmthicknessofho=0:1162.Thiscompareswellwiththevaluecalculatedby[ 35 ],ho=0:123withlessthansixpercenterror.Thereisavortexformednearthetipofthebubble.Thepressureonthetopwall(y=1.0)isshowninFigure 5-17 .Thereferencelevelforpressureisshiftedtothedownstreamlocationx=13anditismultipliedbytheReynoldsnumbertocomparewith[ 35 ].Thepressuredistributionshowsbothqualitativeandquantitativeagreement.Intheresultsby[ 35 ],thepressureneartheinletisThepressurestaysrelativelyconstantinthelmregionneartheleftboundarybecausethevelocity 88

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35 ]reportsapressureofPwall=49forCa=0:05andRe=0andPwall=55:4forRe=70.Inourtestcase,weobtainPwall=52:5forRe=10andCa=0:05.Figure 5-18 showsthevaluesoflmthicknessobtainedforthreedierentCa-0:05,0:1and0:25.Theresultshavethesametrendas[ 35 ]andarewithinsevenpercentforallthecases. 38 40 85 ]forsimulatingmultiphaseows. Figure5-16. FloweldandpressurecontoursforRe=10;Ca=0:05 89

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PressureatthechannelwallforRe=10;Ca=0:05 Figure5-18. FilmthicknessforRe=10anddierentCa

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7 ].Bystudyingthemomentumandheattransfercharacteristicsofnitrogen,oxygenandargoninthiswork,liquidairwhichiscomposedofmainlythesethreeisalsocovered.Thecorrelationswhicharederivedfromthestudyofthesethreeconstituentswouldalsobeapplicabletoliqueedair.Fromthermodynamicconsiderations,thetemperatureoftheinterfacebetweenliquidanditsvaporisassumedtobethesaturationtemperatureoftheliquidatatmosphericpressure.Thetemperatureofthepipeissetapproximatelyattheroomtemperature.Theliquidphasedoesnothaveanysubcooling.Thetemperatureoftheinterfaceisthesameassaturationtemperatureoftheliquidphase.Thesesetsofassumptionscloselyresemblerealisticconditionsincryogenicchilldown.Duringthechilldownprocess,rapidheattransferhappensthroughlmboiling,sincethetemperaturedierencebetweentheliquidandthewallexceedstheLeidenfrostpoint.Asmoreliquidentersthepipe,largersurfaceareawillbeavailableforthephasechange.Duetothis,thevelocityofthevaporphaseandthemassowratewillincreasecontinuously.In[ 7 ],thecaseofconstantwalltemperaturewasalsosimulated.However,thisisthelimitingcaseforheattransferwhenthepipehasaninniteheatcapacityandultimatelyalloftheliquidshouldgetvaporized.Thisisnotwhatwillhappenintherealsituation.Thepipewillbequenchedandultimatelycometothesametemperatureasthe 91

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7 ].Thepropertiesofthismaterialare=4450kg=m3,Cp=4200J=kg:Kandk=4:8W=m:K.ThethicknessofthepipeischosentobeR=RwoRwi=0:02R.Thenon-dimensionalparametersinthisstudyaretheReynoldsnumber,theJakobnumber,thePecletnumberandtheWebernumber.Theratioofuidpropertiesalsoplaysarole.ItistobenotedthatthedenitionofJaisdierentfrom[ 7 ]inthisstudy,andcorrespondstothestandarddenitiongivenin[ 10 ].ForexampleJa=1630ischosenforallthecasesofN2chilldown.ThiscorrespondstothecasewithJa=0:42in[ 7 ].SeeEq. 3{17 forthedenitionoftheseparameters.Theeectsoftheseparametershavebeenstudiedinmoredetailthan[ 7 ].Also,inthepresentstudythesimulationsaredoneforamuchlongertimeandthereforegivemoreinsightintothephysicsofcryogenicchilldown. 7 ]indetail.Inourresearch,onlyafewcasesofN2willbestudiedandtheywillbecomparedwith[ 7 ].Thevaluesofthenon-dimensionalparametersforthesecasesarelistedintheTable 6-1 foreasyreference.Theparametersarechosentomatchthevaluesinthestudyby[ 7 ].ThepropertiesofN2aretakenat300Kand77Kunder1atm.ThereforetheratioofuidpropertiesisconstantforN2study.Theinletvelocityoftheliquidandvaporphaseisalsoconstantinthesecomputationsandsettobe10cm=s.Inthisstudy,theroleoftheReynoldsnumberonheattransfercharacteristicsandphasechangehasbeenstudied.Fourcaseshavebeenstudied:Re=1000,Re=1500,Re=2043andRe=3000.ThevalueofRe=2043waschosentocomparetheresultswith[ 7 ].ItistobenotedthateventhoughtheRe=3000,theowhasbeenassumedtobelaminar.This 92

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88 89 ].ThePecletnumbercannotbechangedindependantly,anditbecomesdeterminedoncetheReynoldsnumberandtheliquidischosen.ThepipediametercorrespondingtoRe=2043caseis4mm.ThethicknessofthepipeischosentobeR=RwoRwi=0:02R.Inthepresentcases,radiationeectshavenotbeenincluded. Table6-1. ParametersforN2chilldownsimulation ReJaPeWe Case11000163023201.84Case21500163034802.77Case32043163047393.69Case43000163069605.53 Figures 6-1 6-2 and 6-3 showthecontoursofUandVcomponentsofvelocityandthetemperatureattimet=0:4forCase3.Thisresultsfrom[ 7 ]areshowedalongsideforcomparisonwhichusethesameRe,Pe,WeandPe.Theowpatternslookalmostidentical.ItisseenthatthemaximuminUandVisveryclosetothatobtainedby[ 7 ],withthepresentsimulationsproducingslightlylowervelocities.Thisisbecauseinthepresentcase,theeectsofradiationfromthewallhavenotbeenincluded,whereastheyhavebeenmodeledin[ 7 ].Next,resultsofUandVvelocitycomponentsandtemperatureforthefourcaseswithdierentReynoldsnumberarepresentedinFigures 6-4 6-5 and 6-6 .ExceptforthecaseRe=2043,theothercaseshaveonlybeenstudiedby[ 7 ]fortheconstantwalltemperaturecase.Theseresultshighlightthefactthatmaximumphasechangetakesplaceinthe 93

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BResultsby[ 7 ]Figure6-1. U-velocitycontoursofLiq.N2att=0.4[ReprintedwithpermissionfromTai,C.F.2008.Cryogenictwo-phaseowandphase-changeheattransferinmicrogravity,PhDDissertation(Page162Figure7-29).UniversityofFlorida,Gainesville,Florida.] BResultsby[ 7 ]Figure6-2. V-velocitycontoursofLiq.N2att=0.4[ReprintedwithpermissionfromTai,C.F.2008.Cryogenictwo-phaseowandphase-changeheattransferinmicrogravity,PhDDissertation(Page163Figure7-30).UniversityofFlorida,Gainesville,Florida.] narrowgapbetweentheliquidcoreandthewallofthepipe.Thelocalizedhighvelocities 94

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BResultsby[ 7 ]Figure6-3. TemperaturecontoursofLiq.N2att=0.4[ReprintedwithpermissionfromTai,C.F.2008.Cryogenictwo-phaseowandphase-changeheattransferinmicrogravity,PhDDissertation(Page163Figure7-31).UniversityofFlorida,Gainesville,Florida.] occurintheconstrictionzonewheretheslightbulgeintheliquidfrontispresent.Atthislocation,themassowrateishighest,andtheareaavailableforitisthemostnarrow.Althoughthebasicowpatternsareverysimilarforallthecases,themaximumvaluesofUandVvelocitiesattainedarevastlydierent.ItisseenthataninverserelationshipexistsbetweenthevelocitiesandtheReynoldsnumberoftheow.Thisisanexpectedoutcomebecausethetemperaturegradientsforallthefourcasesareapproximatelythesame,ascanbeseeninFigure 6-6 .Therefore,fromEq. 3{15 ,themassuxgeneratedattheinterfaceisinverselyproportionaltothePecletnumber,i.e. _m00/Ja Pe(6{1)SincetheJaissameforallthecases,andPe=PrRe,sotherateofmassgenerationshouldvaryinverselyasRe.Figure 6-7 showsthetemperatureattheinnerwallofthepipeforRe=1000andRe=2043atthreedierenttimeinstants.Herethewallchilldowneectcanbeclearly 95

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BRe=1500(Case2) CRe=2043(Case3) DRe=3000(Case4)Figure6-4. U-velocitycontoursofLiq.N2att=1:5fordierentRe seen.Theyshowgoodmatchwiththeresultsof[ 7 ].Astheliquidentersthepipe,moreandmorecoolingtakesplace.Sincetheliquidisatsaturationtemperature,sotheheattransferfromthepipewalltotheliquidcausesevaporationattheinterface.TherateofheattransferdierentforthetwocasesseeninFigure 6-7A and 6-7B .However,adirectcomparisonisnotvaluableherebecausebothReandWearedierentforthetwocases.Theeectseenhereisthereforethecombinedeectofthetwoparameters.Forthis 96

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BRe=1500(Case2) CRe=2043(Case3) DRe=3000(Case4)Figure6-5. V-velocitycontoursofLiq.N2att=1:5fordierentRe reason,latertheireectwillbeinvestigatedseparatelybyvaryingonlyoneofthematatime.Figure 6-8 showsthenon-dimensionalmassowrateatthepipeexitforthefourcasesasafunctionoftime.Theowrateconstantlyincreasesduringthechilldown,asmoreandmoreliquidentersthepipe.ItcanbeclearlyseenthattherateofevaporationisinuencedbytheReoftheow.Toinvestigatethiseectfurther,themassowatt=1:5 97

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BRe=1500(Case2) CRe=2043(Case3) DRe=3000(Case4)Figure6-6. TemperaturecontoursofLiq.N2att=1:5fordierentRe istakenandplottedasafunctionofReinFigure 6-9 .Inthisgure,itisclearthattheheattransferisinverselyproportionaltotheReoftheow. 98

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BRe=2043Figure6-7. Temperatureatr=RiforLiq.N2 Non-dimensionalmassowratehistoryforN2 99

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Non-dimensionalmassowrateatt=1:5forN2 6-10 showsthetypicalowpatternsobservedforO2inthesimulationsperformedinourstudy.ThecaseselectedisCase1fromTable 6-2 withRe=3000.AftertheimplementationoftheQUICKscheme,thesimulationscanbeperformedevenwhenPe>>2.Thisgureshowsthevelocityandtemperaturecontours.Theinterfaceshowsabulgenearthetipduetotheeectofsurfacetension.Thedistancebetweentheinterfaceandthewallisrelativelyconstantforalongsection.Thevelocityoftheliquidslugisnearlyconstantthroughout.Thevelocityofthevaporphaseshowsalotofvariationintheow.Attheinlet,thevaporstartswithavelocityequaltothatoftheliquidphase.Duetothestrongtemperaturegradientsetupinthenarrowspacebetweentheinterfaceandthewall,lmboilingtakesplace.Sincetheliquidisnotsubcoooled,soalltheheattransferredtotheinterfacefromthewall(viaconvectionandradiation)isusedupforphasechange.Thiscausesasteadyincreaseinthevaporvelocityaswetravelalongthex-direction.Nearthetipoftheinterfacethevelocityofthevaporphase 100

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BV-velocity CTemperatureFigure6-10. FloweldofO2att=8:0forRe=3000 ismaximum.Thisispartlyduetothebulgeintheinterface,whichacceleratesthevaporfurther.Thetipoftheinterfaceislocatedatapproximatelyx=8:3att=8:0forthecaseshown.Theveryhighvaporvelocitynearthetip,andthesubsequentdiusionintoamuchwiderspace,givesaneectsimilartotheowovera(smooth)backwardfacingstep.Thereisavortexformedattheliquidfrontwhenthevaporvelocitybecomeshigher.ThenegativeU-velocitiesinthiszoneareduetotherecirculation.ThiscanalsobeseenintheV-velocityplot,Figure 6-10B .Thevelocitychangesfrompositivetonegativeinthespace 101

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6-10C .Itcanbeseenthatthewallissignicantlychilleddownneartheentranceofthepipe. Figure6-11. DevelopmentoftheoweldforatypicalcaseofO2 6-11 showssnapshotsoftheoweldatdierenttimeinstants.Itcanbeseenthatthereisnovortexatt=2:0,itisstartingtodevelopatt=5:0andgrowsbiggerinsizebyt=8:0.Thisisbecauseatt=2:0thevaporvelocityisnothighenough.Asitincreasesfurther,avortexstartsdeveloping.Duetothepresenceoftherecirculationzone,moremixingcantakeplaceattheliquidfrontwithenhancedheattransfer.Figure 6-12 showsthetemperaturecontoursforthesamecaseatt=8:0,inwhichitisclearthatthevaporphaseissignicantlycoolerneartheliquidfrontduetothepresenceofthevortex.Figure 6-13 showsthepressureatthecenterline(y=0:0)andthewall(y=0:5)att=8:0forthesamecase.Thepressureatthecenterlineisactuallythepressureoftheliquidphaseinthetwo-phaseregion(uptoaboutx=8:3).Thepressureatthewallcorrespondstothepressureofthevaporphase.Itcanbeseenthatthepressuregradientisnotconstantinthetwo-phaseregion.Thisisexpected,becausethemassowrateof 102

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Temperaturecontoursneartheliquidfront,Re=3000 Figure6-13. PressureatcenterlineandwallforO2,t=8:0,Re=3000 vaporiscontinuouslyincreasinguptoaboutx=8:3.Thex-gradientofvaporvelocityispositivewhichgivesanegativepressuregradientwhichbecomessteeperwithx.Thepressureintheliquidphaseisrelatedtothepressureinthevaporphasebytheinterfacialnormalstressbalance.Thedierencebetweenthetwoisduetotheeectofsurfacetensionandtherelativelysmallnormalstressesattheinterface.Thereisasharppeakinthepressureoftheliquidnearthetipoftheinterfacebecauseofthehighcurvatureand 103

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Figure6-14. TimedependenceofwalltemperatureforO2,Re=3000 Figure 6-14 showsthetemperatureofthesolid-gasinterfaceasafunctionoftime.Initially,thewallstartsoutatauniformnon-dimensionaltemperatureofT=1:0.Att=2:0,theinterfaceislocatedatapproximatelyx=2:4ascanbeseeninFigure 6-11 .Itisseenthatsignicantcoolingofthewallhasoccurreduptoaboutx=2:4,butbeyondthatregionthetemperatureofthewallisstillclosetotheinitialvalue.Astheinterfaceadvancesfurther,moreandmoreofthewallchillsdown.Signicantcoolingtakesplaceinthetwo-phaseregion,butastimeincreases,therestofthewallalsoseessomecoolingeect.Att=5:0,eventhoughtheinterfaceislocatedatx=5:3,thewalltemperatureuptox=10:0hasstartedtodecrease.Thisisduetothecoolervaporevaporatingfromtheliquidsurfaceandtravelingdownstream.Figure 6-15 showshowthetemperaturegradientatthewallchangeswithtime.Forthepurposeofanalysis,thegurecanbedividedintothreesections.Intheinitial 104

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TemperaturegradientatwallforO2,Re=3000 sectionwherethetemperaturegradientishighisthetwo-phaseregion.Afterthiscomesthetransitionzonewherethegradientrstpeaksandthenthereisasharpdropastheowtransitionsintosinglephase.Inthethirdsection,theowconsistsofonlythevaporphaseandthegradientsaremuchlower.Veryclosetotheinlet,thetemperaturegradientisextremelyhighduetoentranceeectandshouldbeignored.Thetemperaturegradientdropsveryquicklyandreachesavaluearound8:0fort=2:0case.Inthesecondpartofthecurve,thereisapeakinthegradientnearthetipoftheinterface.Thisisbecauseofthedeformationoftheinterfacebysurfacetension,whichbringsitclosertothewallcausingabulge.Thegradientdropsafterthisasthedistanceoftheinterfacefromthewallincreasessharplyandbecomeszeroatthetip.Inthesinglephaseregion,thegradientismuchlower.Itisinterestingtonotehowthegradientchangeswithtime.Fort=2:0case,thegradientishigherintheinitialtwo-phaseregionthantheothertwotimeinstants.Thisisbecausethewalltemperatureismuchhigherinitially.Asthewallchillsdownwithtime,thegradientbecomeslowerinthissection.Inthesingle-phaseregion,thetrendisopposite.Thegradientismuchloweratt=2:0thanatt=4:0ort=8:0.Thiscanbeexplainedbythefactthatthevaporphaseisgettingcooleddownin 105

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6-14 .Thiscombinedeectreducesthemaximumtemperaturegradientobservedatt=8:0. Figure6-16. Timedependenceofmassowatpipeexit,O2,Re=3000 Figure 6-16 showsthevapormassowrateatthepipeexit.Theowratepresentedisnon-dimensional.Itisseenthatthequantityofvaporexitingthepipeisalmostlinearlyincreasingwithtime.Thelmboilingfromtheliquidsurfacefeedsintothevapormass.Asmoreandmoresurfaceareaisavailable,therateofphasechangewouldincrease.Thereisaslightnon-lineareectdetectedattheendofthesimulationneart=8:0with 106

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6-2 ,onlytheRe(andPe)ischanged.Theinitialwalltemperatureforallthreecasesis300K.TounderstandtheroleoftheRe,weneedtoseewhatishappeningattheinterface.CombiningEq. 3{13 and 3{15 gives _m00=vJa Pekv klT(6{2)Inthenon-dimensionalproblemstudiedhere,thetemperaturegradientistheapproximatelysameforallthecases,providedtheinterfacedeformationisnotverydierent.SincetheWeissameforallthecases,itisexpectedthattheinterfaceshapeisgoingtobesimilar.Sinceliquidtemperatureisconstant,thefactorsaectingthephasechangeareJa,Peandvaporandliquidproperties.Radiationalsoplayssomerole.Figure 6-18 showsthemassowrateatt=8:0asafunctionofRe.Themassowrateisanindicatoroftheevaporationtakingplaceattheinterface.ThetrendissimilartothatobservedwithN2anditisseenthatthenon-dimensionalrateofevaporationisinverselyproportionaltoRe.Figure 6-17 showsthewalltemperatureatt=8:0forthethreecases.TheRedoesnothaveasignicantinuenceonthewallcooling.LowerReynoldsnumberproducesmorewallcooling,butthedierencebetweenthethreetemperatureprolesisnotbig.Thiscanbeexplainedbythefactthattheowisassumedtobelaminar.Inlaminarow,theNudoesnotdependonRe.ThiscanbeseenmoreclearlyinFigure 6-19 107

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ParametersforO2Reynoldsnumberstudy Diameter(m)Velocity(m/s)ReWeJaPe Case10.01005.15E-230002.2914686585Case20.00696.18E-225002.2914685487Case30.00447.72E-220002.2914684390 Figure6-17. DependenceofwalltemperatureonReforO2att=8:0 InFigure 6-19B ,itcanbeseenthatatt=8:0,theNusseltnumberforallthethreecasesisveryclose.HeretheNu1numberisdenedas kv(6{3)where_q00wvistheuxfromthewalltovaporphase,TwisthetemperatureofthewallandTvisthemixingcuporbulktemperatureofthevaporphasedenedby[ 90 ].AlthoughRehasaneectonthetemperaturegradient,itdoesnotaectNu1.ThisisbecausethereferencetemperatureisnowthebulktemperatureofthevaporphaseTv.Asthevapor 108

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Non-dimensionalmassowrateatt=8:0forO2 6-19A ,inthesinglephaseregion(afteraboutx=8:3)thegradientiscontinuouslydecreasing.However,inFigure 6-19B 6-17 ,thecoolingeectfarfromtheinterfaceissmallandthepipewallisstillclosetotheinitialtemperature,mimickingaconstanttemperaturewall.Thetwo-phaseowenhancestheheattransferwithNu116,whichisaverylargeincrease.Nu1reachesalmostthevalueof19neartheliquidfrontbecausethedistancebetweenthewallandtheinterfaceislesser. 7 ].Thisbringstheinterfaceclosertothehotpipewallandenhancestheheattransferandrateofevaporation 109

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BNusseltnumberFigure6-19. TemperaturegradientandNu1att=8:0forO2 6-3 .Thewalltemperatureis300Kforallthethreecases. Table6-3. ParametersforO2Webernumberstudy Diameter(m)Velocity(m/s)ReWeJaPe Case10.0058.58E-225003.1914685487Case20.014.29E-225001.5914685487Case30.058.58E-325000.31914685487 Figure 6-20 showstheinterfaceshapesattimet=2:0.ThelowestWe(highestsurfacetension)casehasthegreatestdeformationneartheliquidfront.Sincetheliquidhasnotspentenoughtimeinsidethepipeforsignicantevaporationtotakeplace,thechangeinshapecanbeattributedalmostentirelytothesurfacetensionforce.Itcanalsobeobservedthatduetoconservationofmassandconstantliquidinux,thelocationof 110

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6-21 showsthevelocityeldforthetwoextremecases. Figure6-20. InterfaceshapefordierentWeatt=2:0 BWe=0:319Figure6-21. U-velocityproleofO2fordierentWeatt=2:0 111

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BTemperatureatwallFigure6-22. HeattransferfordierentWeatt=2:0 BNusseltnumberatwallFigure6-23. HeattransfercharacteristicsfordierentWeatt=8:0 Figure 6-22A showsthetemperaturegradientattheinterfaceatthesametimeinstant.Thenumberofmarkerpointsisdierentforthetwocasesbecauseofthedierentinterfaciallength.Thepeakinthegradientoccursatthepositionwherethelocalcurvatureishighduetosurfacetension.Temperaturegradientismuchhigherfor 112

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6-22B itcanbeseenthatthecoolingeectismuchmoreduetolowerWeandhigherdeformationoftheinterface.InFigure 6-23 ,thetemperaturegradientandNusseltnumberfortwoWeareplotted.ThecaseofWe=0:319becomesunstablebeforet=8:0isreached.Therefore,itsresultsareexcludedfromthissetofgures.ItisseenthattheWeaectstheNusseltnumberonlylocallyneartheliquidfront.Inthiscasetoo,itisthedistanceoftheinterfacefromthewallwhichisthecontrollingmechanisminincreasingthetemperaturegradientandNu1.Therefore,itcanbeconcludedthatWeplaysanimportantrolebyenhancingthelocalheattransferneartheliquidfront. 6{2 showstheroleofJaontheevaporationrate.Incryogenicchilldownows,Jaisveryhigh.TostudytheinuenceofJaonheattransfer,thedegreeofwallsuperheathasbeenvaried.Thebaselinecasehas300Kastheinitialwalltemperature,andisthesameasCase2intheWenumberstudy.TheothertwocaseshavethesameRe,PeandWe.Thediameteris1cmandthevelocityis4.29cm=sforallthecases.TheyarelistedinTable 6-4 .Itistobenotedthatthepropertiesofthegasphasearedierentforthesethreecases.Figure 6-24 showstheinterfaceshapesforthethreecasesatt=8:0.SincetheWeisconstant,theshapechangehereisalmostcompletelyduetophasechangeattheinterface.ItcanbeseenthatthecasewiththehighestJahastheattestinterfaceshape.Figure 6-25 showsthemassowratesatthepipeexitasfunctionoftimeforallthethreecases.Thetrendagreeswiththepredictionfromtheory.ThemassowrateisdirectlyproportionaltotheJa.Thenon-dimensionalmassowrateincreaseslinearlywithtime.Figure 6-26 showsthewallcoolingasafunctionofJa.Itisinterestingtonote 113

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ParametersforO2Jakobnumberstudy WallTemperature(K)ReWeJaPe Case125025001.599315487Case230025001.5914685487Case325025001.5921225487 Figure6-24. InterfaceshapefordierentJaatt=8:0 thatJahasamuchmorepronouncedeectonthewallchilldown,ascomparedtoRe.BoththeJaandReinuencethemassowrate,buttheirinuenceonwallchilldownisdisparate.Tostudytheeectonheattransfercharacteristicsfurther,thetemperaturegradientandNu1areshowninFigure 6-27 .Forthepurposeofanalysis,thegraphscanbedividedintothreedistinctsections:theinitialsectionofthecurve,whichcorrespondstothesectionoftheinterfacealmostparalleltothewallandwithnosignicantdeformation;themiddlesectionwherethereisasharpchangeinthegradientwhichcorrespondstothe 114

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MassowhistoryfordierentJaforO2 EectofJaonwalltemperatureofO2att=8:0 curvedliquidfront;andthenalsectionwherethereisgradientsaremuchlowerbecauseitrepresentsthesinglephaseowregime.InFigure 6-27A ,fortherstsectionofthecurve,thegradientisaboutthesameforallthethreecases.Thisisbecausethedistanceoftheinterfacefromthewallissimilar.However,theNusseltnumberNu1isdierentinthesamesectionofthecurve.ThisisbecausethedrivingtemperaturedierenceTwTvisdierentforthethreecases.Forexample,walltemperatureforJa=2122ismuchlower 115

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BNusseltnumberatwallFigure6-27. HeattransfercharacteristicsfordierentJaatt=8:0 thanforJa=931.Ifthemeantemperatureofthevaporissimilar,thenthiswouldgiveahigherNu1forthehigherJa.Inthesecondsectionofthecurves,i.e.,nearthemaximaorpeak,Nu1trendisreversed.Thisisprimarilyduetothedierentshapeoftheinterfaceforthethreecases,ascanbenotedinFigure 6-24 .Theinterfaceshapeiscontrolledbytherateofevaporation,whichisdirectlyproportionaltoJa.TheatterproleforhigherJainthisregionatt=8:0givesarelativelylowerNusseltnumber.AnotherfeaturetobenotedisthatthecurveforJa=1468hasadecidedlyhighermaximathatCase2oftheReynoldsnumberstudy.ThisdierencecanbeattributedtothelowerWeintheJakobnumberstudy.Inthethirdsectionofthecurves,wherethesharpdropoccursattheliquidfrontandtheowtransitionstosingle-phase,NusseltnumberforJa=2122ishighest.Thisisbecausemorevolumeofthenewlygeneratedcoldervaporisavailabletochilldownthewallbyheattransfer.However,allthreecurvesapproachthesingle-phaseconstanttemperatureowvalueasymptotically. 116

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BV-velocity CTemperatureFigure6-28. FloweldofAratt=8:0forRe=2500 6-5 117

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and 6-7 .FortheWeandRestudy,theinitialwalltemperatureis319K.FortheJastudy,thevelocityis2.95cmandthediameteris1.64cmwhiletheinitialwalltemperatureisdierentforeachcase.Radiationeectsareincludedinthecomputations.Thewalltemperaturevariesduringthesimulations,asthewallgetscooleddownbylmboilingfromtheliquidphase.Figure 6-28 showsthevelocityeldforatypicalcaseofArgon.TheinterfaceshapeandthegeneralfeaturesoftheowaresimilartothosefoundwithOxygeninFigure 6-10 .Heretoo,avortexisseenattheliquidfront.Theinterfaceshowsthecharacteristicbulgingneartheend.Themaximumvaporvelocityoccursintheregionbetweenthisswellandthewall.TherecirculationzoneisformedatthetipoftheliquidandcanbeseenbythenegativeU-velocityinFigure 6-28A .IntheV-velocityplotitcanbelocatedbythecloselyspacedregionsofhighpositiveandnegativeV-velocitiesformedduetotheswirlingofvapor.Thetemperatureplotshowsalargerboundarylayerattheleadingedgeoftheliquidphaseduetothemixingofcoldandhotvaporinthisregion. Figure6-29. Developmentoftheoweldforatypicalcase 118

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Temperaturecontoursneartheliquidfront 6-5 .ThegeneralformatissimilartothatfollowedwithOxygen.ThevaluesorRe,WeandJahavebeenmatchedwithOxygen,onlythePebeingdierent.SoinadditiontoRe,theinuenceofPeonowcharacteristicswillalsobestudied. Table6-5. ParametersforargonReynoldsnumberstudy Diameter(m)Velocity(m/s)ReWeJaPe Case10.01643.54E-230002.2914687221Case20.01144.25E-225002.2914686018Case30.00735.31E-220002.2914684814 Thenon-dimensionalizedmassowrateatpipeexitatt=8:0isshowninFigure 6-31 .TheowrateisinverselyproportionaltotheRe,butislowerthanthat 119

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ComparisonofmassowrateforArandO2 ofoxygenforthesameReynoldsnumber.ThisdierencebetweenthetwogasesisduetotheeectofthePeasdemonstratedinEq. 6{2 .Figure 6-32A showsthewalltemperatureforthethreeRe.ThetemperaturesshowasimilartrendtoFigure 6-17 withrespecttotheReynoldsnumber.ThewallchilldowneectismoreforlowerReynoldsnumber. BComparisonwithO2,Re=2500Figure6-32. DependenceofwalltemperatureonReforAratt=8:0 120

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BComparisonwithO2,Re=2500Figure6-33. HeattransferatwallforAr,t=8:0 Figure 6-33 comparestheheattransfercharacteristicsofoxygenandargon.InFigure 6-32B ,thewallcoolingforoxygenissignicantlymore.ItshowsthatforthesameRe,WeandJa,oxygenisamoreeectivecoolantthanargonduetoitsuidproperties.TheNusseltnumberNu1isplottedinFigure 6-33B forthesamecase.Nu1forthetwouidsisaboutthesame.ThisisbecauseNusseltnumberdependsonthetemperaturegradient,TwandTv.AslongastheratioofthetemperaturegradientandthedrivingtemperaturedierenceTwTvisthesame,Nu1isnotaected. 6-34 showstheshapesobtainedforthreedierentcasesatt=1:0.Thedierence 121

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6-6 Table6-6. ParametersforargonWebernumberstudy Diameter(m)Velocity(m/s)ReWeJaPe Case10.00825.90E-225003.1914686018Case20.01642.95E-225001.5914686018Case30.08225.90E-325000.31914686018 Figure6-34. InterfaceshapefordierentWeatt=1:0 6-7 liststheparametersforthethreecases.Figure 6-35 showstheinterfaceshapesobtainedwithargon,att=8:0.Inthisstudytoo,thehighestJahastheattestinterfaceprole.Thisisbecauseofhigherrateofphasechangeattheinterface. 122

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ParametersforargonJakobnumberstudy WallTemperature(K)ReWeJaPe Case1263.625001.599316018Case2319.025001.5914686018Case3373.425001.5921226018 Figure 6-36A takesthisargumentfurther,andshowsthatthemassowrateatthepipeexitisdirectlyproportionaltotheJa.Thisisadirectresultoflmboilingattheliquidsurface.InFigure 6-36B ,themassowrateiscomparedwithoxygen.Asisexpected,themassowrateofargonislower.Figure 6-37A showsthewallcoolingeect.Thegeneraltrendissimilartooxygen.TheeectofPecanbeseeninFigure 6-37B .Thecoolingeectismoreforthecaseofoxygen.TheNusseltnumberisplottedinFigure 6-38 .ThecaseofJa=931isexcludedbecauseitbecomesunstablebeforet=8:0isreached.TheJakobnumberdoeshaveaneectonNu1,unlikeRewhichonlyaectedthemassowrate.Inthiscase,itcanbeclearlyobservedthatJainuencesNu1inacomplexway.Intheregimeoftwo-phaseowparalleltothewall,Nu1ishigherforthehigherJa.Neartheliquidfront,thetrendisreversed.ThiscanbeseeninFigure 6-38A 7 ].Moreextensivestudywasdoneforoxygenandargon.Thedatabasewaschoseninsuchawayastovaryonlyonenon-dimensionalparameteroutofRe,JaandWe.The 123

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InterfaceshapefordierentJaatt=8:0 BComparisonwithO2,Ja=2122Figure6-36. MassowhistoryfordierentJaforAr 124

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BComparisonwithO2,Ja=2122Figure6-37. EectofJaonwalltemperatureofAratt=8:0 BComparisonwithO2,Ja=2122Figure6-38. EectofJaonNu1ofAratt=8:0 numberwasfoundtoaectthemassowrates,butdidnothaveasignicantinuenceonthewallcoolingortheNusseltnumber.Weaectedtheinterfaceshapeattheleadingedgeoftheliquidslug,alsoinuencingtheheattransferandvelocityeldthere.Jaaectsallthreequantitiesofinterest,i.e.,massowrate,wallcoolingandtheNusselt 125

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126

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25 30 31 ].Thisisbecausethewallsuperheatisquitehighevenwiththepipeatroomtemperature.IAFBowregimealsooccursinverticalquenchingowsinthepost-dryoutregion[ 19 ].IAFBinterrestrialgravityhasalsoreceivedattentionduetoitsroleintheLossofCoolantAccident(LOCA)failureofnuclearreactors[ 14 ].Controlledexperimentsarediculttoconductbecausethelower 127

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19 ].Theyuseatwo-phaseReynoldsnumberdenedintermsoftheequivalentvaporqualityxEofthetwo-phasemixtureandproposeamodiedformoftheDittus-Boelterequationforlmboiling 90 ]forowbetweentwoconcentricannularcylinders 1q00vi 17 ]haveusedtheKaysequationandgivethefollowing 128

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90 ]intheturbulentregion.[ 18 ]havealsoproposedatwo-uidmodelwithamorecomplicatedrelationshipwhichtakesintoaccounttheeectofRevandPr.ThisisbecausetheyassumethevaporowtobeturbulentandusetheDittus-Boeltercorrelationtopredictheattransfer. 6 ]alsomakeuseoftheKaysequationandfollowtheanalysisof[ 17 ]for<
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Figure7-1. TemperaturecontoursforRe=3000,O2 7-1 showstemperaturecontourplotatt=8:0forRel=3000.Theinterfaceislocatedatx=8:32.AlthoughtheliquidReynoldsnumberisconstant, 130

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kv(7{10)ThisisthedenitionusedforalltheNusseltnumberguresinChapter6.Inlaminarsinglephasepipeow,TvistakenasthereferencetemperaturebecauseitwasfoundthatthisyieldedasimplerexpressionforNu,byvirtueofthefactthatthenon-dimensionaltemperatureproleremainsconstant,eventhoughtherealtemperaturechanges.ThiswastherationalebehindtryingTv,besidesthefactthatdownstreamoftheliquidplugtheowissinglephasepipeowwithalmostconstanttemperature.ThereforeNu1reducestothestandarddenitionofNuinthisregion.TheseconddenitionusedisbasedonTvand2.Thisdenitionistriedbecause2isthehydraulicdiameter.Thisisreasonablebecausetheheattransferisaectedbythedistanceoftheliquidphasefromthewall.ThereforeinFigure 7-2 131

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kv(7{11)ThethirddenitionofNusseltnumberisbasedonthetwo-phasecorrelationsassuminghomogeneousow(oralternativelyassumingnoknowledgeoftheowregime),whereitisdiculttodeterminethevaporphasetemperature.Becauseofthisdiculty,Tsatischosenasthereferencetemperature,anddiameterDisthelengthscale. kv(7{12)Figure 7-2 showstheplotsofNuforthecaseinFigure 7-1 basedonthesedenitions.Theyarecomparedthevaluecalculatedforsinglephasepipeowwithconstanttemperature,i.e.,Nu=3.657.Theinterfaceispresenttillx=8:0approximately.Beyondthatitisonlyvaporow.Nu3wasdenedbasedonTwTsatasistheconventionfor Figure7-2. ComparisonofdierentdenitionsofNusseltnumber two-phaseows.ItcanbeseenthatNu3ishighinthetwo-phasezone.Itisextremely 132

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7-2 itcanbeseenthatNu2isclosetothevaluepredictedforsinglephaselaminarowwithconstantwalltemperatureeverywhereexceptneartheliquidfront.Thiscanbeexplainedasfollows. D(7{13)Thevaporlmthicknessisdenedas=RoRint.staysnearlyconstantintheannulusregion.However,nearthetipoftheinterface,increasessharply,whileNu1issteadilydecreasing.Thisgivesapeakattheinterfacetip(atx=8:32)where=Ro.Physically,thismaybebecausethevaporowinthisregionisnotparallel,andthereisarecirculationzonewhichenhancestheheattransfersignicantlyneartheliquidfront.However,thefactthatNu2isalmostconstantinthetwo-phaseregion(excludingthelocalmaximum)showsthevalidityofpredictingtheheattransferforinvertedannularowbythesinglephasecorrelations.Thevaporowbetweenthewallandliquidplugcanbeapproximatedasowinanannuluswithdierentinnerandouterwalltemperatures.Inthesinglephaseregion(x>8:32)bothNu1andNu2arecloseto3.657.ThereforetheNusseltnumberobtainedfromthecurrentsimulationsseemsreasonableandaccordingtowhatonewouldexpectfromtheory.AmongthesedenitionsofNusseltnumber,Nu2showsthemostpromiseintermsofdevelopingacorrelationorcomparingwithotherresearchers,becauseofitsuseofTvandthehydraulicdiameter2asreference.IfweleaveasidethelocalpeakinNu2near 133

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7-3 showsNu2forallthecasesstudiedforoxygen.Inthisgure,therstcaseofReynoldsnumberstudyisdenotedasO2-Re-1,thesecondcaseasO2-Re-2andsoon.Forthelistofparameterscorrespondingtothesecases,refertoTable 6-2 6-3 and 6-4 .They-axishasbeenchangedtomakethedierencesbetweenthecasesclearer.TheNusseltnumbervaluesforpipeowwithconstanttemperature(NuT=3:657)andconstantheatux(NuH=4:364)arealsoshown.Sincetheliquid Figure7-3. speedatinletisconstant,andallthecasesareplottedatt=8:0,thepositionoftheinterfacetipisapproximatelythesameforallthecases,withminordierencesduetothedeformationnearthetip.Theoveralltrendisthesameforallthecases.ItisseenthattheNu2fromallthecasesofReandWestudyalmostoverlapintheannulusregion.ThecaseO2-Ja-3whichhasthehighestJa=2122isalittlehigherthantherest,andthecaseO2-Ja-1withJa=931isalittlesmallerthantherestofthecases.SoNu2isaectedbytheJaintheannulusregion,butnotbyReandWe.O2-Ja-3hasthehighestNu2notonlyintheannulusregionbutalsoneartheinterfacetipandintheregionjustaheadof 134

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theinterface.Itdoeshoweversettledowntothevalueof3:657likealltheothercases.Figure 7-4 showsalltheO2andArcasesinthisresearch.Thenamingconventionforthecasesissameasforoxygen.TheparameterscanbelookedupinTable 6-5 6-6 and 6-7 .AlltheNusseltnumbersfallinanarrowbandirrespectiveofthecryogen,withthehighestNu2occuringforthecaseswiththehighestJa.Intheannulusregion(awayfromtheentranceandinterfacetip),alltheresultsliebetweenthetwolimitsofNuHandNuT.NextitwasdecidedtocompareNu2fortherepresentativecaseO2-Re-1withsomecorrelationsfromliterature.Figure 7-5 showsNu2plottedalongsidethevaluespredictedby[ 17 ]andtheColburnequation(Eq. 7{5 )whichhasbeenusedby[ 6 ]forR.TocalculatetheNusseltnumberfromEq. 7{3 ,wetaketheTv,TsatandTwfromthesimulationsanduseNuoo=5:385and=0:346forlaminarow (10:3462)10:346(TwTsat) (TwTv)(7{14) 135

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ComparisonofNusseltnumberwithotherresearchers Similarly,tocalculateNusseltnumberfromEq. 7{5 17 ]isverygood.Neartheentranceregion(x<1:5)howevertherearedierencesbetweentheresultsfromthenumericalsimulationandthepredictionof[ 17 ].[ 17 ]predictamuchmoresmoothlyvaryingNuwhereasinthesimulationsthereissharpdropinNu2neartheentrance.Inthecoreannulusregionawayfromtheentranceandawayfromtheinterfacetip,thetworesultscloselymatch.Takingtheaverageoferrorfor1:5
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17 ]isnotabletocapturetheeectofsharpchangesinontheNusseltnumber.Attheinterfacetip(x=8:32),thedierencebetweenthetwoismaximumat53percent.ItshouldbenotedthatEq. 7{3 wasderivedassuminganannuluswithconstantinnerandouterradiiandfullydevelopedow.Thereforeitshouldbeusedwithcautionwherevertheowisnotfullydeveloped,forexampleneartheentranceandwheretherearesharpchangesin.Also,theresultfrom[ 17 ]isnotvalidintheregionbeyondtheinterfaceasthereisnoannulusandonlyvaporowthere,sothesepointsshouldbeignored.AlthoughtheColburnequationismainlyvalidforturbulentow,itwaschosenbecauseitcanbeusedovertheentireowregime.ItisseenthatColburnequationpredictsamuchlowervalueofNusseltnumberthroughoutandtheagreementwithpresentresultsisnotclose,especiallyintheannulusregion.From[ 91 ],therangeofapplicationforthisequationis0:7Pr160andRe>4000.Forthiscase,theReynoldsnumberofthevaporismuchlower;Rev=280inthevaporphasenearthepipeexit.ThediscrepancyreducesastheReynoldsnumberofthevaporowincreasesalongthex-direction.Inthesingle-phaseowregimetheresultsarethesameorderofmagnitude,buttheerrorintroducedbyusingColburnequationevenforfullydevelopedvaporowwherethedierenceisminimum,wouldbealmostftypercent. 137

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17 ].Itwasfoundthatthepresentresultsmatchcloselywiththelatter'sprediction(within3percent)inthemiddlesectionoftheinterfacewheretheowisfullydeveloped,awayfromtheentranceandinterfacetip.Theerrorneartheentranceandtheliquidfrontwasmuchhigher,sotheequationby[ 17 ]shouldbeusedwithcautionintheseregions.ThepredictionfromColburnequationwhichisprimarilymeantforturbulentowwasthesameorderofmagnitude,buttheerrorintroducedbyusingthisequationwouldbeveryhigh. 138

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8-1 .IthasbeendemonstratedthattheSIMCCisaccurateandfullycapableofsimulatingtwo-phaseowswithphasechange. Table8-1. VericationandvalidationstudyforSIMCC TestCaseAspectofcodetestedRangeofCriticalParameter(s) Liddrivencavityow2dNavierStokesSolver100Re4000FlowoversolidsphereSharpInterfaceCut-cellalgorithms1Re100DeformablebubbleingravityMovingInterfacealgorithm1Re100,1We10LiquidplugowInternaltwo-phaseowsRe=10,0:05Ca0:25 However,itshouldbeemphasizedherethateventhoughthenumericalmethoddevelopedbythepreviousresearcherslike[ 7 54 59 64 66 ]oersmanyadvantagesforsimulatingmultiphaseows,theinherentcomplexitiesofcryogenicchilldownprocessposeseveralnumericalchallenges.Thereforethetechniquesdevelopedbythemarenotsucienttosimulatetheinvertedannularowproblemindepthformorechallengingcases.Thelocalizedhighvelocitiesandsteepgradientsproducedbecauseoftheunique 139

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70 ]fortheconvectiveterm,havebeenproposedandimplemented.TheQUICKschemewasdemonstratedtoeectivelyalleviatethestabilityproblemsassociatedwiththediscretizationoftheconvectivetermforPecell>2.Themultigridmethodwastestedforthepressure-Poissonequationwithoutthecut-cellsrst.Itwasshownthattheperformancewithmultigridisimproved.Next,atestcasewithanarticialinterfaceembeddedinliddrivensquarecavitywasusedtotesttheaccuracyandspeedenhancementbyusingmultigrid.Itwasshownthatmultigridshowsdecreasesthecomputationtimebyafactorofmorethan2.5,whilemaintainingthesamelevelofaccuracy.Thisisverysignicantimprovementinspeed.ThusoneofthecontributionsmadebythisresearchistoextendthecapabilityofSIMCCtohandleinternaltwo-phaseowswithphasechange.TheimprovedandenhancedSIMCCwasusedtosimulatechilldownofthreeimportantcryogens:nitrogen,oxygenandargon.Bystudyingthemomentumandheattransfercharacteristicsofnitrogen,oxygenandargoninthiswork,liquidairwhichiscomposedofmainlythesethreeisalsocovered.Thenitrogenchilldownresultswerefoundtobeingoodagreementwith[ 7 ].Moreextensivestudywasdoneforoxygenandargon.Thedatabasewaschoseninsuchawayastovaryonlyonenon-dimensionalparameteroutofRe,JaandWe.TheeectofPewasstudiedbycomparingtheresultsofoxygenandargon.Itwasfoundthateachoftheparametersaectsthethermal-uidcharacteristicsoftheowsquantitatively.Howeverqualitativelyalltheresultsshowanunderlyingsimilarityintheowpatternsandheattransfertrends.Thereforeitcanbe 140

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88 89 ].Theassumption 141

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52 92 { 95 ]haveimplementedthree-dimensionaltechniquesforinterfacetracking.However,mostoftheseaddressisothermalows.Infuture,thesetechniquescanbedevelopedtohandlemorecomplexowswithphasechange.SimulationofLowBoilingCryogens 142

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AlpanaAgarwalwasborninthecityofLucknow,India.ShewenttoschoolatSt.AgnesLoretoHighSchooluntil1997.ThereaftersheattendedLaMartiniereGirlsCollege.ShewasadmittedtotheprestigiousIndianInsituteofTechnology(IIT),Kanpurintheyear2000.ShereceivedherBachelorinTechnology(B.Tech.)inMechanicalEngineeringin2004.SheworkedatGEGlobalResearch,Bangaloreinthematerials'mechanicsgroupforayearbeforedecidingtopursuehigherstudies.ShereceivedtheAlumniFellowshipatUFandjoinedthedirectPh.Dprograminfallof2005withDr.J.N.Chungasheradvisor.Sincethenshehasbeenengagedinresearchaboutcomputationaluidmechanicsandheattransferformultiphaseowsusingthemovinginterfacetechniques. 150