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Modeling and Simulation of Particle Dispersal by Shock and Detonation Waves

Permanent Link: http://ufdc.ufl.edu/UFE0042428/00001

Material Information

Title: Modeling and Simulation of Particle Dispersal by Shock and Detonation Waves
Physical Description: 1 online resource (193 p.)
Language: english
Creator: Ling, Yue
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: compressible, detonation, flow, fluid, multiphase, particle, shock
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: Particle dispersal by shock and detonation waves is an interesting and important phenomenon that can be observed in nature, such as in volcanic eruptions, and industrial applications, such as in detonation of multiphase explosives. This problem is difficult to investigate experimentally, because experiments are hazardous and expensive and the maximum velocities, pressures, and temperatures are very high. Therefore, the modeling and simulation approach employed in this dissertation is an attractive alternative. However, there are numerous challenges in modeling and simulating this problem. The problem involves multiple phases. Because shock and detonation waves and contact discontinuities exist in the flow, the phase interaction becomes much more complicated than in conventional multiphase flows because the unsteady mechanisms can become important. The problem also involves multiple scales. For the small scales that cannot be resolved, modeling techniques are required to capture the physics in these scales and their interactions with resolved scales. The modeling and simulation approach pursued in this dissertation is important because it can improve our understanding of how to predict particle dispersal by shock and detonation waves, and because it can lead to practical guidelines. In this dissertation, we develop a rigorous simulation approach for unsteady compressible multiphase flow involving shock and detonation waves, and apply this approach to investigate a sequence of problems of particle dispersal by shock and detonation waves. First, the particle dispersal in a one-dimensional shock-tube is considered. The primary focus is on how the existence of particles causes the flow to transition from the frozen to the equilibrium limit. The second problem entails the investigation and resolution of numerically induced particle-number-density fluctuations in Eulerian-Lagrangian simulations of multiphase flow. Guidelines to reduce the amplitude of the fluctuations are proposed that can be easily applied in practice. The third problem considered in the dissertation is the modeling and analysis of particle interaction with shock and blast waves. The study clearly demonstrates the importance of the unsteady force and heat transfer in the interaction of particles with shock and blast waves. Finally, we consider the problem of particle dispersal by blast waves. The focus is on investigating the interactions between particles with the complex wave system. The unsteady contributions to force and heat-transfer are again found to be substantial.
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 Yue Ling.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Balachandar, Sivaramakrishnan.
Local: Co-adviser: Haselbacher, Andreas.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042428:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042428/00001

Material Information

Title: Modeling and Simulation of Particle Dispersal by Shock and Detonation Waves
Physical Description: 1 online resource (193 p.)
Language: english
Creator: Ling, Yue
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: compressible, detonation, flow, fluid, multiphase, particle, shock
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: Particle dispersal by shock and detonation waves is an interesting and important phenomenon that can be observed in nature, such as in volcanic eruptions, and industrial applications, such as in detonation of multiphase explosives. This problem is difficult to investigate experimentally, because experiments are hazardous and expensive and the maximum velocities, pressures, and temperatures are very high. Therefore, the modeling and simulation approach employed in this dissertation is an attractive alternative. However, there are numerous challenges in modeling and simulating this problem. The problem involves multiple phases. Because shock and detonation waves and contact discontinuities exist in the flow, the phase interaction becomes much more complicated than in conventional multiphase flows because the unsteady mechanisms can become important. The problem also involves multiple scales. For the small scales that cannot be resolved, modeling techniques are required to capture the physics in these scales and their interactions with resolved scales. The modeling and simulation approach pursued in this dissertation is important because it can improve our understanding of how to predict particle dispersal by shock and detonation waves, and because it can lead to practical guidelines. In this dissertation, we develop a rigorous simulation approach for unsteady compressible multiphase flow involving shock and detonation waves, and apply this approach to investigate a sequence of problems of particle dispersal by shock and detonation waves. First, the particle dispersal in a one-dimensional shock-tube is considered. The primary focus is on how the existence of particles causes the flow to transition from the frozen to the equilibrium limit. The second problem entails the investigation and resolution of numerically induced particle-number-density fluctuations in Eulerian-Lagrangian simulations of multiphase flow. Guidelines to reduce the amplitude of the fluctuations are proposed that can be easily applied in practice. The third problem considered in the dissertation is the modeling and analysis of particle interaction with shock and blast waves. The study clearly demonstrates the importance of the unsteady force and heat transfer in the interaction of particles with shock and blast waves. Finally, we consider the problem of particle dispersal by blast waves. The focus is on investigating the interactions between particles with the complex wave system. The unsteady contributions to force and heat-transfer are again found to be substantial.
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 Yue Ling.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Balachandar, Sivaramakrishnan.
Local: Co-adviser: Haselbacher, Andreas.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042428:00001


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MODELINGANDSIMULATIONOFPARTICLEDISPERSALBYSHOCKANDDETONATIONWAVESByYUELINGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010YueLing 2

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TomywifeLaura 3

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ACKNOWLEDGMENTS IwouldliketoexpressmysinceregratitudetomyadvisorProf.S.Balachandarandmyco-advisorProf.AndreasHaselbacherfortheirsupportandguidancethroughoutmyPh.D.studies.Theirknowledgeandexpertiseinthiseldprovidedagoodfoundationforthiswork,andtheirencouragementandpatientinstructionmadethislearningexperienceexcitingandjoyful.Ifeelverygratefultohavetheopportunitytoworkwiththem.Theyaregreatrolemodelsasdedicatedandrespectfulscholarsformetofollow.Iwouldliketoacknowledgethemembersofmydissertationcommittee,Prof.JenniferSinclairCurtisandProf.RenweiMei.Thediscussionswiththemarealwayshelpfulandencouraging.Theirinsightsandcommentshaveinuencedmyviewonthetopicscontainedinthedissertation.IwouldalsoliketothankProf.B.J.FreglyforencouragingmetoapplyforthisPh.D.program.WithouthavingmethiminChina,IwouldneverhavethehonorofjoiningtheUniversityofFlorida.IalsowanttothankmyadvisorinBeihangUniversity,Prof.ZhigangShen,forhisinstruction.AlthoughIdidnotnishmyPh.D.degreeinBeihangUniversity,thegraduatestudythereintroducedmetheunexpectedcharmofresearchonuiddynamics.IappreciateverymuchProf.Shen'sencouragementandsupportwhenIdecidedtoswitchtotheUniversityofFloridaforPh.D.study.Inthepreparationofthisdissertation,IhavebeenassistedbymanypeoplewhoarecurrentlyorusedtobeintheComputationalMultiphysicsGroup.IparticularlywanttothankDr.SeungDoHong,Dr.JungwooKim,andProf.ThomasBonomettifortheirhelpandmentoringwhenIjuststartedmyresearch.IalsoacknowledgeDr.JianghuiChao,Dr.HyungooLee,andDr.BjornLandmannformanyhelpfuldiscussionswiththem.IamgratefulformycolleaguesManojParmar,YoshifumiNozaki,GordonTaub,andeachofwhomprovidedvaluablecommentsandsuggestionsforthiswork.TopursueaPh.D.degreeinaforeigncountry,thechallengesindailylifearenotnecessarilylessthaninresearch.Iamverygratefulforthehelpandlovefrommany 4

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closefriendsinGainesville.IparticularwanttothankmypastorTobySorrels,whohasalwaysbeenagreatfriend,teacher,mentor,andcouncilor.IalsowanttothankalltheAmericanfriendsinInternationalFriendshipwhohaveofferedcountlesshelpformyfamilyandme.IthankallthefriendsinCreeksideCommunityChurchandGainesvilleChineseChristianChurch,whohaveofferedhelpandencouragementduringmystudyhere.Ialsoneedtosendmythankstomyfamily.Ithankmyparentsandmyparents-in-lawfortheirsupportoveralltheyears.Finally,Iwouldliketosendmydeepestappreciationtomywife,Laura.Withoutherprayers,encouragement,andsupport,Iwouldnothavebeenbeabletoobtainthisdegree.WithgreatpleasureandgratitudeIdedicatethisdissertationtoher. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 16 CHAPTER 1INTRODUCTION ................................... 18 1.1BackgroundandMotivation .......................... 18 1.2ReviewofModelingandSimulationApproaches .............. 20 1.2.1MultiphaseFlowModeling ....................... 20 1.2.2Particle-FluidInteractionModeling .................. 22 1.2.3SimulationsofParticleInteractionwithShock/DetonationWaves 24 1.3GoalsandStructureoftheDissertation ................... 26 2TRANSIENTPHENOMENAINONE-DIMENSIONALCOMPRESSIBLEGAS-PARTICLEFLOWS ........................................ 28 2.1Introduction ................................... 28 2.2Theory ...................................... 29 2.2.1Assumptions .............................. 29 2.2.2GoverningEquations .......................... 30 2.2.3FrozenandEquilibriumSolutions ................... 32 2.2.3.1Frozenlimit .......................... 34 2.2.3.2Equilibriumlimit ....................... 36 2.2.4TimeScaleAnalysis .......................... 37 2.2.4.1Mechanicaltimescaleofparticlecontactsurface ..... 37 2.2.4.2Thermaltimescaleofparticlecontactsurface ...... 39 2.2.4.3Shocktimescale ....................... 41 2.2.4.4Expansionfantimescales ................. 42 2.3NumericalMethod ............................... 43 2.4Results ..................................... 44 2.4.1OverallBehavior ............................ 44 2.4.2EvolutionofExpansionWave ..................... 45 2.4.3EvolutionofParticleContactSurface ................. 50 2.4.4EvolutionofShockWave ........................ 56 2.5Conclusions ................................... 57 6

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3ANUMERICALSOURCEOFSMALL-SCALENUMBER-DENSITYFLUCTUATIONSINEULERIAN-LAGRANGIANSIMULATIONSOFMULTIPHASEFLOWS ... 59 3.1Introduction ................................... 59 3.2MathematicalFormulation ........................... 63 3.3ASimpleModelScenario ........................... 65 3.4Approach .................................... 68 3.4.1NumericalMethodsforGoverningEquations ............ 68 3.4.2InterpolationMethods ......................... 68 3.4.3ErrorDenitions ............................. 69 3.5Results ..................................... 72 3.5.1TestProblem1:SinusoidalProle .................. 72 3.5.1.1Particleredistribution .................... 73 3.5.1.2Globalerrornorms ...................... 77 3.5.2TestProblem2:Hyperbolic-TangentProle ............. 80 3.5.2.1Particleredistribution .................... 81 3.5.2.2Globalerrornorm ...................... 86 3.5.2.3Effectofdisturbancemagnitude .............. 86 3.5.2.4Effectofparticleinertia ................... 89 3.5.3TestProblem3:ExpansionWave ................... 91 3.5.4TestProblem4:ShockWave ..................... 97 3.6Conclusions ................................... 101 4MODELINGANDANALYSISOFPARTICLEINTERACTIONWITHSHOCKANDBLASTWAVES ................................. 105 4.1Introduction ................................... 105 4.2PhysicalModelingandGoverningEquations ................ 110 4.2.1GoverningEquations .......................... 111 4.2.2Momentum-TransferModel ...................... 112 4.2.3Energy-TransferModel ......................... 118 4.3Results ..................................... 120 4.3.1PlanarShockWave ........................... 120 4.3.1.1Peakvaluesofforceandheattransfer ........... 120 4.3.1.2Timescalesofquasi-steadyandunsteadyforceandheattransfer ......................... 125 4.3.1.3Neteffectofunsteadymechanismsonoverallforceandheattransfer ......................... 127 4.3.2SphericalShockWave ......................... 132 4.3.2.1Peakvaluesoftheforceandheattransfer ......... 135 4.3.2.2Particlevelocityandtemperature .............. 137 4.3.2.3Long-terminuencesofunsteadyforceandheattransfer 140 4.3.3Brode'sBlast-WaveProblem ...................... 141 4.4Discussion ................................... 146 4.5Conclusions ................................... 148 7

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5ANALYSISOFPARTICLEDISPERSALBYBLASTWAVES ........... 150 5.1Introduction ................................... 150 5.2GoverningEquationsandNumericalMethods ................ 155 5.2.1GoverningEquations .......................... 156 5.2.2NumericalApproach .......................... 158 5.3Results ..................................... 159 5.3.1EvolutionoftheGasFlow ....................... 160 5.3.2EvolutionofaSingleParticle ..................... 164 5.3.3EvolutionofAParticleCloud ..................... 170 5.3.4ParticleFront .............................. 174 5.3.5ImportanceofUnsteadyContributionstoForceandHeatTransfer 175 5.4Conclusions ................................... 179 6SUMMARY,CONCLUSIONS,ANDFUTUREWORK ............... 181 6.1Summary .................................... 181 6.2Conclusions ................................... 181 6.3FutureWork ................................... 183 APPENDIX ASURFACEANDVOLUMEAVERAGESOFTHEGASQUANTITIESFORTHEPLANARSHOCK-PARTICLEINTERACTION ................... 184 REFERENCES ....................................... 186 BIOGRAPHICALSKETCH ................................ 193 8

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LISTOFTABLES Table page 2-1Summaryofcomputationcases. .......................... 44 2-2Particlecontactsurfaceequilibrationtimescalesasgivenbytime-scaleanalysis. 56 2-3Shockequilibrationtimescalesasgivenbytime-scaleanalysis. ......... 57 3-1Summaryofcasesfortestproblem2. ....................... 81 9

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LISTOFFIGURES Figure page 1-1Snapshotsofparticledispersalinexplosions.FromZhangetal.[ 101 ]. ..... 18 1-2Lengthscalesinvolvedintheproblemconsideredinthisdissertation. ..... 20 2-1Schematicx-tdiagramshowingtheregionnumberingusedinthisarticle,whereH,T,PC,C,andSdenotetheheadandtailoftheexpansionfan,theparticlecontactsurface,thegascontactsurface,andtheshock,respectively;shadingindicatesthepresenceofparticles. ........................ 33 2-2VariationofdimensionlessmechanicalequilibrationtimescaleoftheparticlecontactsurfacewithRe0. .............................. 39 2-3VariationofequilibrationtimescaleratiooftheparticlecontactsurfacewithRe0. .......................................... 40 2-4Comparisonofcomputedandpuregassolutionsatt=210)]TJ /F5 7.97 Tf 6.59 0 Td[(7sforcase1,showingthatthetransientprocessstartsfromthepuregaslimit. ........ 45 2-5Comparisonofcomputedandequilibriumsolutionsatt=2.510)]TJ /F5 7.97 Tf 6.58 0 Td[(2sforcase1,showingthatthetransientprocessreachestheequilibriumlimit. .... 46 2-6Computedvelocityprolesofthetransientprocessforcase1. ......... 47 2-7Computedvelocityprolesofthetransientprocessforcase2. ......... 48 2-8Numericallygeneratedx)]TJ /F3 11.955 Tf 9.3 0 Td[(tdiagramforcase2,showingthetransientevolutionfrompuregaslimittoequilibriumlimit.Thecomputedresultisshowninsolidlines;thepuregasowresultisshownindashedlines.Atlongtimes,thecomputedspeedsofthewavesmatchestheequilibriumresults(showninboxes).Theshadingrepresentsthemagnitudeofthedensitygradient. ........... 49 2-9Computedvelocityprolesintheexpansionfanattheearlystagesofthetransientprocessfor110)]TJ /F5 7.97 Tf 6.59 0 Td[(66t6710)]TJ /F5 7.97 Tf 6.58 0 Td[(6s. ....................... 50 2-10Comparisonofcomputedevolutionofgas(dash-dottedlines)andparticle(solidlines)velocitiesatparticlecontactsurfacelocationasafunctionofYp4forvariousvaluesofdp. ..................................... 52 2-11VariationofYp4withdpforwhichapproachtoequilibriumofgasandparticlevelocitiesatparticlecontactsurfaceandshockvelocityisnon-monotonic. ... 53 2-12Comparisonofcomputedevolutionofgas(dash-dottedlines)andparticle(solidlines)velocitiesandtemperaturesatparticlecontactsurfacelocationasafunctionofdpforYp4=0.3. .................................. 54 10

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2-13Comparisonofcomputedevolutionofgas(dash-dottedlines)andparticle(solidlines)velocitiesandtemperaturesatparticlecontactsurfacelocationasafunctionofdpforYp4=0.8. .................................. 54 2-14EquilibriumtimescalesofshockandparticlecontactsurfaceasafunctionofdpforYp4=0.2and0.8fromcomputations,comparedwiththeresultfromthetimescaleanalysisindicatedinslopesymbol. .................. 55 2-15ComparisonofcomputedevolutionofshockspeedasafunctionofYp4,(shockspeedofequilibriumowsolutionisshowedasdashedlines). ......... 58 3-1Schematicofmodelprobleminwhichawavemovesthroughapacketofparticles.Inmodelproblem1(seeSection 3.5.1 ),asinusoidalwaveisconsidered.Inmodelproblem2(seeSection 3.5.2 ),ahyperbolic-tangentwaveisused. ... 66 3-2Exactandinterpolateduid-velocityprolesasafunctionoftimeforN=4andA=0.5.Thelledcirclesdenotelocationswithcollocatedvalues. .... 74 3-3Prolesofparticlepositionanddeviationfromtheexactsolutionatt=20asafunctionofinitialpositionforN=4,p=0,andA=0.5. ........... 76 3-4Prolesofparticlerelative-positionandlocalnumber-densityinsidetheparticlepacketwithinitialsizeunityafterthepassageofthesinusoidalwaveforN=4,p=0,andA=0.5. ............................... 77 3-5Prolesofparticlerelative-positionandspacingerrors(EposandEspac)insidetheparticlepacketwithinitialsizeofoneafterthepassageofthesinusoidalwaveforN=4,p=0,andA=0.5. ....................... 78 3-6Evolutionsoftherelative-position,spacing,mean,andspreaderrors(Epos,Espac,Emean,Esp)asfunctionsofNforp=0andA=0.5. ............ 79 3-7Prolesofrelativeparticlepositionandlocalnumber-densityoftheparticlepacketwithinitialsizeunityforcases2-1and2-2withN=4. .......... 83 3-8Prolesofparticlerelative-positionerrorandspacingerror(EposandEspac)insidetheparticlepacketwithinitialsizeunityforcases2-1and2-2withN=4. 84 3-9Schematicsforthegenerationofgrid-to-gridnumber-densityuctuationfromsubgridnumber-densityvariationforcases2-1,2-2,and2-3. .......... 85 3-10Behaviorofrelative-position,spacing,andmeanerrors(Epos,Espac,Emean)asfunctionsofNforcases2-1and2-3.Linesdenotevaluesofu2(solidlines:u2=1;dashedlines:u2=)]TJ /F4 11.955 Tf 9.29 0 Td[(0.4).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). ................... 87 11

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3-11Behaviorofnumber-densityerror(End)asafunctionofNforcases2-1and2-3.Linesdenotevaluesofu2(solidlines:u2=1;dashedlines:u2=)]TJ /F4 11.955 Tf 9.29 0 Td[(0.4).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). ................................. 88 3-12Behaviorofrelative-position,spacing,andnumber-densityerrors(Epos,Espac,End)asfunctionsofNandu2forcases2-1,2-4,and2-5.Linesdenotevaluesofu2(solidlines:u2=0.1;dashedlines:u2=1;dashed-dotlines:u2=10).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). .................................. 90 3-13Evolutionofrelative-position,spacing,andnumber-densityerrors(Epos,Espac,End)asfunctionsofNforcases2-1,2-6,and2-7.Linesdenotevaluesofp(solidlines:p=0;dashedlines:p=1;dashed-dotlines:p=10).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). 92 3-14Numberofparticlespercellnppcfromaone-way-coupledsimulationwithlteredHermiteinterpolation,andtheoreticallocationatwhichparticlesentertheexpansionfanandexpansion-fanthicknesswhileenteringasfunctionsoftheparticlelocationatt=136.1. ................................. 95 3-15Behaviorofparticlenumber-densityerrorEndatt=136.1foraone-waycoupledsimulationofanexpansionfanpropagatingintouniformlydistributedparticles.Solidlineswithsymbolsdenoteerrorestimatesforvariousinterpolationschemesbasedontheanalysisofthemodelproblemwiththehyperbolic-tangentprole(square:piecewiseconstant;delta:piecewiselinear;diamond:Hermite).Lineswithoutsymbolsdenotethenumericalresultsforvariousinterpolationschemes(dashedline:piecewiseconstant;dashed-dotline:piecewiselinear;longdashedlines:Hermite). .................................... 97 3-16Behaviorofparticlenumber-densityerrorEndatt=136.1foratwo-waycoupledsimulationofanexpansionfanpropagatingintouniformlydistributedparticlesasafunctionofthemassfractionYp. ....................... 98 3-17Velocityandnumberofparticlepercellprolesasfunctionsofspatiallocationattimet=130.Theresultsshownhereisfromthelteredone-waycoupledHermitesolution. ................................... 101 12

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3-18Behaviorofnumber-densityerrorEndatt=130foraone-waycoupledsimulationofashockwavepropagatingintouniformlydistributedparticles.Solidlineswithlledsymbolsdenoteerrorestimatesforvariousinterpolationschemesbasedontheanalysisofthemodelproblemwiththehyperbolic-tangentprole(square:piecewiseconstant;delta:piecewiselinear;diamond:Hermite).Lineswithoutsymbolsandwithunlledcirclesymbolsdenotenumericalresults.Thelineswithoutsymbolsdenotetheresultswith5000initialparticlespercell;whilethosewithunlledcirclesymbolsarefor1000initialparticlespercell.Thelinepatternsareusedtodistinguishdifferentinterpolationschemes(dashedline:piecewiseconstant;dashed-dotline:piecewiselinear;longdashedlines:Hermiteinterpolation). ............................ 102 4-1Evolutionofforceandheat-transfercontributionsinplanarshock-particleinteractionforMs=1.5,Repr=104,p=103,andCp=1. .................. 122 4-2Contoursoftheratiosofthepeakvaluesofforcestothepeakquasi-steadyforceasfunctionsofMsandReprinplanarshock-particleinteraction. ...... 125 4-3Contoursoftheratiosofthepeakvaluesofheat-transfercontributionstothepeakvaluesofquasi-steadyheat-transfercontributionsasfunctionsofMsandReprinplanarshock-particleinteraction. .................... 126 4-4Budgetsoftheparticlevelocityandtemperatureasfunctionsoftime,forMs=1.5,Repr=102,p=10,andCp=1inplanarshock-particleinteraction. .... 129 4-5Theparticlevelocityandtemperatureasfunctionsoftimewhenonlyoneforceorheattransferisactiveduringtheshock-particleinteractionforMs=1.5,Repr=102,p=10,andCp=1inplanarshock-particleinteraction. ....... 130 4-6TheterminalparticlevelocityasfunctionofMs,Repr,p,andthekernel,whenonlyOneforceisactiveduringtheplanarshock-particleinteraction. ...... 133 4-7TheterminalparticletemperatureasfunctionofMs,Repr,andpCpwhenonlythediffusive-unsteadyheat-transfercontributionisactiveduringtheplanarshock-particleinteraction. .............................. 134 4-8Evolutionofthetotalparticleforceandheattransferinsphericalshock-particleinteractionasafunctionofx0.OtherparametersarexedasMs=5,Repr=102,p=10,andCp=1. .............................. 135 4-9Thepeakvaluesoftheunsteadyforcesdividedbythepeakquasi-steadyforceasfunctionsofMsandReprforp=104andCp=1forsphericalshock-particleinteraction. ...................................... 136 4-10Thepeakvaluesoftheunsteadyheattransferdividedbythepeakquasi-steadyheattransferasfunctionsofMsandReprforp=104andCp=1forsphericalshock-particleinteraction. .............................. 137 13

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4-11Gassolutionofthesphericalshockwaveproblem.xmsdenotesthelocationofthemainshock. .................................... 138 4-12ParticlesolutionofthesphericalshockwaveproblemforMs=5,Repr=102,p=10,Cp=1,andx0=104. ........................... 139 4-13Themaximumvelocitydifference,max(up)]TJ /F3 11.955 Tf 9.3 0 Td[(upqs)=ug2,betweentheparticlevelocitycomputedwhenallforcesareactiveandthevelocitycomputedonlywiththequasi-steadyforce,forsphericalshock-particleinteraction. ........... 139 4-14Thetemperaturedifference,(Tp)]TJ /F3 11.955 Tf 13.43 0 Td[(Tpqs)jxs)]TJ /F8 7.97 Tf 6.59 0 Td[(xp=0.5=(Tg2)]TJ /F3 11.955 Tf 13.43 0 Td[(Tg1),betweentheparticletemperaturewhenallheat-transfercontributionsareactiveandthetemperaturecomputedonlywiththequasi-steadyheattransfer,atthetimewhenthesphericalshockjustpassestheparticle. ................ 140 4-15Evolutionofdifferentforcesandheattransferinthesphericalshock-particleinteractionforMs=5,Repr=102,p=10,Cp=1,andx0=104. ........ 141 4-16Evolutionoftheparticlevelocityandtemperatureincludingallforcesandheattransfer,comparedtotheresultswithonlyquasi-steadyforceandheattransferforMs=5,Repr=102,p=10,Cp=1,andx0=104insphericalshock-particleinteraction. ...................................... 142 4-17Schematicillustrationofparticleinteractionwithasphericalblastwavegeneratedbyacompressedgas. ................................ 144 4-18Trajectoriesofparticlesofdifferentdensityanddiameterin^x-^tdiagramfor^xp0=1.2inblastwave-particleinteraction. ..................... 145 4-19Timeevolutionofparticlevelocityandforcebudgetfor^dp=410)]TJ /F5 7.97 Tf 6.58 0 Td[(2and^xp0=1.2inblastwave-particleinteraction. .................... 146 4-20Timeevolutionofparticlevelocityandforcebudgetfor^dp=410)]TJ /F5 7.97 Tf 6.58 0 Td[(2and^xp0=1.2inblastwave-particleinteraction. ..................... 147 5-1Schematicillustrationofparticledispersalinasphericalblastgeneratedfromacompressedgas-particlemixture.Theshadedareaindicatestheexistenceoftheparticles. .................................... 151 5-2Thecharacteristicsofthegasowofthenite-sourcesphericalblastwaveinthex-tdiagramforairwithinitialconditions~pg4=~pg1=121,and~Tg4=~Tg1=1. ... 153 5-3Thetimeperiodandspatialregiondivisiononthex-tdiagram. ......... 160 5-4Temporalevolutionsofgasdensity,velocity,andtemperatureprolesfrom^t=0to8.19. .................................... 161 5-5Temporalevolutionsofgasdensity,velocity,andtemperatureprolesfrom^t=8.19to12.28. .................................. 162 14

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5-6Particlepositionevolutiononx-tdiagramasafunctionofparticledensity,diameterandinitialposition. .................................. 165 5-7Particlevelocityevolutionasafunctionofparticledensity,diameterandinitialposition. ....................................... 166 5-8Particletemperatureevolutionasafunctionofparticledensity,diameterandinitialposition. .................................... 168 5-9Temporalevolutionofthebudgetsofparticleforceandheattransferfor^p=^g4=10,^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3,and^xp0=0.6. .......................... 169 5-10Temporalevolutionsofparticleconcentration,velocity,andtemperaturefromt=0to6.82,for^p=^g4=10and^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3. ................ 172 5-11Temporalevolutionsofparticleconcentration,velocity,andtemperaturefromt=6.82to12.28,for^p=^g4=10and^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3. .............. 173 5-12Particlefrontevolutiononx-tdiagramasafunctionofparticledensityanddiameter. ....................................... 175 5-13Particlefrontpositionsasafunctionofparticledensity,diameterandtime. .. 176 5-14Errorinparticlefrontlocationiftheterms^fpg,^fam,and^fvuarenotincludedinEq.( 4 )asafunctionofparticledensityanddiameter,for^t=13.6,54.6and95.5. ......................................... 178 5-15Errorinparticlefrontlocationfordifferentforcemodelsasafunctionofparticledensityanddiameterfor^t=136.5. ........................ 180 15

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMODELINGANDSIMULATIONOFPARTICLEDISPERSALBYSHOCKANDDETONATIONWAVESByYueLingDec2010Chair:SivaramakrishnanBalanchandarCochair:AndreasHaselbacherMajor:MechanicalEngineering Particledispersalbyshockanddetonationwavesisaninterestingandimportantphenomenonthatcanbeobservedinnature,suchasinvolcaniceruptions,andindustrialapplications,suchasindetonationofmultiphaseexplosives.Thisproblemisdifculttoinvestigateexperimentally,becauseexperimentsarehazardousandexpensiveandthemaximumvelocities,pressures,andtemperaturesareveryhigh.Therefore,themodelingandsimulationapproachemployedinthisdissertationisanattractivealternative.However,therearenumerouschallengesinmodelingandsimulatingthisproblem.Theprobleminvolvesmultiplephases.Becauseshockanddetonationwavesandcontactdiscontinuitiesexistintheow,thephaseinteractionbecomesmuchmorecomplicatedthaninconventionalmultiphaseowsbecausetheunsteadymechanismscanbecomeimportant.Theproblemalsoinvolvesmultiplescales.Forthesmallscalesthatcannotberesolved,modelingtechniquesarerequiredtocapturethephysicsinthesescalesandtheirinteractionswithresolvedscales.Themodelingandsimulationapproachpursuedinthisdissertationisimportantbecauseitcanimproveourunderstandingofhowtopredictparticledispersalbyshockanddetonationwaves,andbecauseitcanleadtopracticalguidelines. Inthisdissertation,wedeveloparigoroussimulationapproachforunsteadycompressiblemultiphaseowinvolvingshockanddetonationwaves,andapplythis 16

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approachtoinvestigateasequenceofproblemsofparticledispersalbyshockanddetonationwaves.First,theparticledispersalinaone-dimensionalshock-tubeisconsidered.Theprimaryfocusisonhowtheexistenceofparticlescausestheowtotransitionfromthefrozentotheequilibriumlimit.Thesecondproblementailstheinvestigationandresolutionofnumericallyinducedparticle-number-densityuctuationsinEulerian-Lagrangiansimulationsofmultiphaseow.Guidelinestoreducetheamplitudeoftheuctuationsareproposedthatcanbeeasilyappliedinpractice.Thethirdproblemconsideredinthedissertationisthemodelingandanalysisofparticleinteractionwithshockandblastwaves.Thestudyclearlydemonstratestheimportanceoftheunsteadyforceandheattransferintheinteractionofparticleswithshockandblastwaves.Finally,weconsidertheproblemofparticledispersalbyblastwaves.Thefocusisoninvestigatingtheinteractionsbetweenparticleswiththecomplexwavesystem.Theunsteadycontributionstoforceandheat-transferareagainfoundtobesubstantial. 17

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CHAPTER1INTRODUCTION 1.1BackgroundandMotivation Particledispersalbyshockanddetonationwavesisaninterestingandimportantphenomenonthatcanbeobservedinnature,suchasinvolcaniceruptions,seeChojnickietal.[ 20 ],andinmanyindustrialapplications,suchasdetonationofmultiphaseexplosives,seeZhangetal.[ 101 ]andTanguayetal.[ 88 ],andsolid-propellantrocketpropulsion,seeNajjaretal.[ 65 ].Inthisdissertation,wefocusonthestudyofparticledispersalbyshockanddetonationwaves.Thereareseveralreasonswhytheextensionofconventionalexplosivetomultiphaseisofinterest.Onereasonisthatthetotalenergyreleaseisincreasedbyaddingreactivemetalparticles,seeTanguayetal.[ 88 ].Inamultiphaseexplosivedetonation,alargeamountofenergyissuddenlyreleasedfromamixtureofanenergeticmaterialandparticles.Therapidlyreleasedenergycausesthepressureofthegaseousreactionproductstorisetoverylargevaluesveryrapidly.Whenthehigh-pressuregasrushestowardtheambientenvironmentathighspeed,theparticlesinitiallymixedwiththeenergeticmaterialwillbepushedoutunderthedrivingforceofthegas,seeFig. 1-1 .Duetoinertia,theparticleswillingeneralmoveslowerthanthegasinitially. At=0ms Bt=2.5ms Figure1-1. Snapshotsofparticledispersalinexplosions.FromZhangetal.[ 101 ]. 18

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Particledispersalbyshockanddetonationwavesisadifcultproblemnotonlybecauseofthecomplexphysics,butalsobecauseitishardtoinvestigateexperimentally.First,experimentsarehazardousandexpensive.Furthermore,themaximumvelocities,pressures,andtemperaturesareveryhigh.Therefore,themodelingandsimulationapproachemployedinthisdissertationisanattractivealternative. Therearefourprimarychallengesthatmakemodelingandsimulationofparticledispersalbyshockanddetonationwavesdifcult.First,theprobleminvolvesmultiplephases:thegasphase(atmosphereandreactionproducts),thesolidphase(solidparticles),andliquidphase(completelyorpartiallymeltedparticlesbehavinglikedroplets).Second,becauseshockanddetonationwavesandcontactdiscontinuitiesexistintheow,theinteractionbetweenphasesbecomesmorecomplexthaninconventionalmultiphaseowsbecausetheunsteadymechanismscanbecomeimportant,seeParmaretal.[ 67 ].Third,theprobleminvolvesmultiplescales,seeFig. 1-2 .Thelengthscalesincludetheparticlesize,whichmayvaryfromsmallparticles(m)tofragments(cm);thecontainersize(cm)to(m);andthegasowlengthscales,whichmayvaryfromtheshockthickness(nm)totheshock-wavepropagationdistance(m).Thescaleseparationmakessimulationsverychallenging.Ifthereexistsmallscalesthatcannotberesolved,modelingtechniquesarerequiredtocapturethephysicsatthesescalesandtheirinteractionswithresolvedscales.Fourth,theprobleminvolvesmultiplephysical/chemicalmechanisms.Whentheenergeticmaterialreleasesalargeofamountenergy,thepressureandtemperatureofthegasisveryhigh,andthethermodynamicscannotbedescribedbytheidealgaslaw.Whentheshockanddetonationwavesinteractwiththeparticles,theimpactforcemaycauseparticledeformationorbreakage.Iftheparticlematerialisreactive,theintenseheatadditionmaycauseparticlemeltingorevenburning. 19

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Figure1-2. Lengthscalesinvolvedintheproblemconsideredinthisdissertation. Themodelingandsimulationstudypursuedinthisdissertationisimportantbecauseitcanimproveourunderstandingofhowtopredictparticledispersalbyshockanddetonationwaves,andbecauseitcanleadtopracticalguidelines.Forexample,intheapplicationtomultiphaseexplosives,addingreactivemetalparticlestotheexplosivehasbecomeacommonapproachtoincreasethetotalenergyrelease,seeTanguayetal.[ 88 ].Someimportantquestionsareunresolved.Forexample,whetherthereactivemetalparticlescanbeignitedorcontinuetoburnwhentheyaredispersedoutwardisaquestionofgreatconcern.Thechemicalreactionofaparticleinanexplosionowstronglydependsonitsinteractionswiththeshock/detonationwaves.Therefore,agoodunderstandingoftheparticleinteractionwithshockwavesiscrucialforpredictingtheoutcomeofaddingreactiveparticles. 1.2ReviewofModelingandSimulationApproaches 1.2.1MultiphaseFlowModeling Fourmainmodelingapproachesexistforcompressiblemultiphaseow:dusty-gas,Eulerian-Eulerian(E-E),Eulerian-Lagrangian(E-L),andfullresolution.MoredetailsabouttheseapproachescanbefoundinCrowe[ 23 ]andBalachandarandEaton 20

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[ 3 ].Alltheaboveapproacheshavebeenappliedtocompressiblemultiphaseowswithshock-particleinteractions.Thedusty-gasapproachconsiderstheparticlestobeinequilibriumwiththesurroundinggas,themixtureofgasandparticlescanthereforebeviewedasamodiedgas.Inasimulation,onlyonesetofequationsneedstobesolved,therebyloweringthecomputationalcost.Thedusty-gasapproachhasbeenusedtocomputetheasymptoticshock-propagationvelocitywhenashockmovesintoagas-particlemixture,seeforexampleRudinger[ 74 ]andLingetal.[ 52 ].However,astheparticlesareassumedtobeinequilibriumwiththegas,someimportantphysics,suchastherelaxationregionbehindtheshockcannotberesolvedbythedusty-gasapproach.TheEulerian-Eulerianapproachalsoviewstheparticlesasauid,butdoesnotrequiretheparticlestofollowthegasperfectly.Therefore,twosetsofequationsneedtobesolvedintheE-Eapproach.AsthecomputationalcostoftheE-Eapproachisstilllowcomparedtotheothertwoapproaches,itisthemostcommonlyusedapproachintheeld.ByusingtheE-Eapproach,therelaxationzone,thetransientbehaviorofashockthatmovesintogas-particlemixturecanberesolved,seeCarrier[ 17 ],Kriebel[ 48 ],andRudinger[ 74 ].Incontrasttotheprevioustwoapproaches,theEulerian-Lagrangianapproachconsiderstheparticlesaspointsandtracksindividualparticle.Whenthenumberofparticlesislarge,thecomputationalcostishigh.However,theE-LapproachnotonlycapturesallthephysicsthattheE-Eapproachcancapture,suchastherelaxationzonebehindashock,seeHaselbacheretal.[ 39 ],butcanalsocapturesomeadditionalphysicsthattheEulerian-Eulerianapproachmisses.TheE-Eapproachassumestheexistenceofauniqueeldrepresentationfortheparticlevelocityandtemperature.TheE-Lapproachisnotlimitedbythisassumptionandthusabletosolvethesituationswheretheeldisnotunique,suchasinterparticlecollision,seeBalachandarandEaton[ 3 ]formoreextensivediscussionofthisissue.Whenaparticleinteractswithashockwave,theforceexertedonaparticleisrelatedtoitsacceleration,i.e.,theBassethistoryforce,whichcannotberesolvedbyusingtheE-Eapproach. 21

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SomeadditionalexamplesofusingtheE-LapproachincompressiblemultiphaseowsareBalakrishnanandMenon[ 6 ]andLingetal.[ 51 53 ].Iftheparticlesizebecomeslargecomparedtotherelevantuidlengthscale,thepoint-particleassumptionintheE-EandE-Lapproachesbecomeinvalidandthefull-resolutionapproachmustbeused.Inthisapproach,theuidowaroundtheparticleiscompletelyresolvedandnomodelingisrequired.Therefore,itisthemostaccurate,butalsothemostexpensiveapproach.Duetoitshighcomputationalcost,thenumberofparticlesusedinthesimulationistypicallylimited.Thisapproachhasbeenusedinshock-particleinteraction,seeforexampleSunetal.[ 86 ],Milne[ 63 ],andZhangetal.[ 102 ].Ifnecessary,eventheparticledeformationcanbecaptured,seeZhangetal.[ 102 ]. Inthisdissertation,becausetheparticlemotioninvolvesunsteadyeffects,whichcannotbecapturedbythedusty-gasandE-Eapproach,theEulerian-Lagrangianapproachisadoptedasthemultiphaseowmodel.Theowscalesofinterestintheproblemconsideredinthedissertationareatthemacro-scale,whichareusuallymuchlargerthanthesizeofanindividualparticle.Itisimpracticaltoresolveallthemicro-scaledetailsoftheowaroundeachparticle,whilecomputingthecompressibleowatthemacro-scale.Thefull-resolutionapproachisthereforeimpractical. 1.2.2Particle-FluidInteractionModeling IntheE-Lapproach,physicalmodelsareneededtocalculatetheinstantaneousforceandheattransferbetweenthegasandtheparticle. Therearemanyparticleforceandheattransferformulasavailableintheliterature.Asummaryoftheseformulascanbefoundinmultiphasetextbooks,suchastheonebyCroweetal.[ 24 ].Inanincompressibleow,theavailablemodelsfortheuid-dynamicforceonaparticleandtheheattransferbetweengasandaparticleareonsolidtheoreticalfooting.MaxeyandRiley[ 59 ]andGatignol[ 35 ]havederivedrigorousexpressionsfortheuid-dynamicforceonaparticleundergoingarbitrarytime-dependentmotioninanunsteadyinhomogeneousambientowinthelimitof 22

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zeroReynoldsnumber.FollowingtheideaofMaxeyandRiley[ 59 ],MichaelidesandFeng[ 62 ]derivedtheanalogousexpressionforheattransfer.Inparticular,theoverallforceandheattransfercanbedividedintophysicallymeaningfulcontributions:thequasi-steadyandunsteadycontributions.ThislinearsuperpositionofthedifferentcontributionshasbeenrigorouslyestablishedonlyintheStokeslimitofzeroReynoldsnumber.FiniteReynolds-numberextensionsareinevitablyempiricalinnature.TheReynolds-numberdependenceofthequasi-steadyandviscousunsteadyforceshavebeenestablished,seeCliftandGauvin[ 21 ],MeiandAdrian[ 61 ],andKimetal.[ 47 ].Thepressure-gradientandadded-massforcesareprimarilyinviscidinnatureandarethereforeindependentoftheReynoldsnumber. Intheproblemconsideredinthisdissertation,theparticleswillinteractwithshock/detonationwaves,therefore,theparticleforceandheattransfermodelmustbeabletocapturetheessentialphysicsofshock-particleinteraction.Thisinteractionisstronglytime-dependent.Whenashockwavepassesoveraparticle,theparticleissubjectedtoaverystronggasacceleration.Recentcarefultime-resolvedmeasurementsoftheforceonastationaryparticlesubjectedtoashockwave,seeSunetal.[ 86 ]andBredinandSkews[ 12 ],haveshownthattheinstantaneousforceontheparticleundersuchhighlyunsteadyconditionscanbemuchlargerthanthecorrespondingquasi-steadyforce.Similarly,fullyresolvedsimulationsbyZhangetal.[ 102 ],Milneetal.[ 63 ],andRipleyetal.[ 72 ],haveshownarapidriseinparticlevelocityandtemperaturewhenashockordetonationwavepassesovertheparticle.Thisclearlyhighlightstheimportanceofunsteadycontributionstotheforceinshock-particleinteractionproblems.Nevertheless,theunsteadycontributionstoforceandheattransferarestillcommonlyneglectedevenincaseswherestronginteractionbetweencompressibleowfeaturesandparticlesareexpected,see,forexample,Najjaretal.[ 65 ]inthecontextofasolid-propellantrocketmotor,andLanovetsetal.[ 49 ]andZhangetal.[ 101 ]inthecontextofexplosivedispersalofparticles.Theneglectofunsteadycontributions 23

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totheforceandheattransferisoftenjustiedonthegroundsoftheparticle-to-uiddensityandheat-capacityratiosbeinglarge.Thisargumentisappropriateforunsteadycontributionsarisingfromparticleacceleration,asparticleaccelerationscalesasinverseofparticle-to-uiddensityratio.However,insituationssuchasshock-particleinteraction,asshownintheexperimentsofSunetal.[ 86 ]andBredinandSkews[ 12 ],thelargestcontributiontotheinstantaneousforceisfromtheambientuidaccelerationseenbytheparticle,whichisindependentoftheparticle-to-uiddensityratio,seeBagchiandBalachandar[ 1 ]. TheformulationofMaxeyandRiley[ 59 ]andGatignol[ 35 ]wasrecentlyextendedtocompressibleowsbyParmaretal.[ 69 ],whereanexpressionfortheforceonaparticleundergoingarbitrarytime-dependentmotioninanunsteadynon-uniformcompressibleambientowwasderived.TheresultingequationofmotionisasymptoticallyvalidonlyinthelimitofsmallMachandReynoldsnumbers.Nevertheless,itprovidesarmtheoreticalbasisforempiricalextensionstoniteMachandReynoldsnumbers.Inparticular,theeffectofcompressibilityontheunsteadycontributionstotheforcewasobservedtobesignicant.Inthisdissertation,theformulationofParmaretal.[ 69 ]willbeextendedempiricallytoniteMachandReynoldsnumber,andappliedtocomputetheforceexertedonaparticle.Forheattransfer,MichaelidesandFeng[ 62 ]andBalachandarandHa[ 4 ]havepresentedexpressionsvalidforincompressibleow.Toourknowledge,noextensionsoftheirresultstocompressibleowshavebeenpublished.Therefore,weusetheformulationsofMichaelidesandFeng[ 62 ]andBalachandarandHa[ 4 ]inthiswork. 1.2.3SimulationsofParticleInteractionwithShock/DetonationWaves Compressibleowinvolvingshockwavesgeneratedfromasuddenenergyreleasehasbeenintenselystudiedinthelastcentury.AgoodsummaryoftheevolutionofthestudyoftheshockandblastwavesinexplosionowscanbefoundinthebookbySachdev[ 76 ].TheoriginalcontributionsbyTaylor[ 91 92 ],vonNeumann[ 96 ],and 24

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Sedov[ 79 ]onintensepoint-sourceexplosionsaregenerallyviewedasabenchmarkinthestudyofshock/blastwaves,becausesimilaritysolutionscanbederivedanalytically.Brode[ 13 ]extendedthepoint-sourceblastwavetheorytothenite-size-sourcecase,wheretheblastwaveisgeneratedbyasphereofhigh-pressuregas.TheproblemconsideredbyBrodeisclosertotherealapplication,andhasbeenintenselystudiedsincethenbymanyothers,e.g.,Boyer[ 11 ]andFriedman[ 32 ]. Detonationhasbeenstudiedformorethanacentury.ThesummaryofthehistoricalevolutionofresearchondetonationcanbefoundinthebookbyBaueretal.[ 9 ].TheChapman-Jouguet(CJ)theoryisawell-knownbenchmarkfordetonationresearch.Chapman[ 19 ]andJouguet[ 44 45 ]werethersttoproposeformulationstoquantitativelypredictthedetonationwavepropagationspeed.IntheCJtheory,thedetonationwaveismodeledasadiscontinuity.TheinnerstructureofthedetonationwavewasresolvedbytheZNDmodelproposedindependentlybyZeldovich[ 100 ],vonNeumann[ 97 ],andDoring[ 25 ].ThegasowbehindadetonationwavewasrststudiedbyTaylor[ 90 ]forTNT.TheresultsofTaylorwereemployedbyBrodeasaninitialconditiontosimulatetheexplosionowafterthedetonation[ 15 ],whichcanbeviewedasanextensiononhis1955paper[ 13 ],asrealgaseffectsandignitionoftheexplosiveareincludedinthesimulation. Inmanypracticalapplications,theinteractionofparticleswithshockanddetonationwavesisofimportance,andasaresultthisproblemhasbeenconsideredbymanyresearchers.Carrier[ 17 ]isoneofthersttolookatparticleevolutionbehindashockwave.Sincethen,theso-calledrelaxationzonebehindaplanarshockhasbeenconsideredbyKriebel[ 48 ],Rudinger[ 74 ],MiuraandGlass[ 64 ],andmanyothers.Recently,progressontheparticleinteractionwithasphericalshockwavehasalsobeenachieved,seeforexample,Igraetal.[ 42 ],Lanovets[ 49 ],Zhangetal.[ 101 ].Thesphericalgeometryintroducesadditionalcomplexitytothegasowandtherebytotheparticlemotion,Lanovets[ 49 ]andZhangetal.[ 101 ]bothobservedparticlestocross 25

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eventhemainshockwaveinthedispersalprocess.Milneetal.[ 63 ],Zhangetal.[ 102 ],andRipleyetal.[ 72 ]carriedoutdirectnumericalsimulationsbyfullyresolvingtheowaroundasinglesphericalparticleorsmallnumberofsphericalparticlesduringtheinteractionwithashock/detonationwaveincludingtheeffectofparticledeformation.Tanguayetal.[ 88 ]proposedananalyticalmodeltopredictignitionwhenaparticleinteractswithadetonationwave. Onecommonissueinthesestudiesofmacro-scalesimulationsofparticledispersalbyshockanddetonationwavesisthattheunsteadycontributionstoforceandheattransferaregenerallyignoredintheparticle-uidinteractionmodels.Themicro-scalesimulations,suchasZhangetal.[ 102 ],areabletorecoverallthedetailsofshock-particleinteraction,however,itisimpracticaltocarryoutfully-resolvedsimulationsinapracticalproblemsinvolvingdistributionsofparticles.Asaresult,inthecurrentknowledgeofmodelingandsimulationforparticleinteractionwithshock/detonationwaves,agapexistsbetweenthemicro-scaleresearchresultsandthemacro-scaleapplications.Thedissertationtriestobridgethisgapbyimprovingtheparticle-uidinteractionmodel,sothatmacro-scalesimulationswillbeabletocapturetheessentialphysicsofshock-particleinteraction. 1.3GoalsandStructureoftheDissertation Theabovediscussionhasclearlyillustratedthecomplexitiesandchallengesoftheproblemofparticledispersalbyshockanddetonationwaves,andalsothelackofaccurateapproachestoinvestigatingtheproblem.Therefore,thegoalsofthisdissertationare:todeveloparigoroussimulationapproachforunsteadycompressiblemultiphaseowinvolvingshockanddetonationwaves,andapplythisapproachtoinvestigatetheproblemofparticledispersalbyshockanddetonationwaves. Toachievethisgoal,asequenceofproblemsarestudied: 1. Transientphenomenainone-dimensionalcompressiblegas-particleow(seeChapter 2 ). 26

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2. Numericalsourceofnumber-densityuctuationsinE-Lsimulationsofmultiphaseow(Chapter 3 ). 3. Modelingandanalysisofparticleinteractionwithshockandblastwaves(Chapter 4 ). 4. Analysisofparticledispersalbyblastwaves(Chapter 5 ). Afterthediscussionsoftheaboveproblems,summaryandconclusionswillbedrawnfromalltheworksinthisdissertation.Astheproblemconsideredhereisverycomplicated,thedissertationonlyaddressessomeoftheissues,andtherearestillmanypotentialdirectionstocontinuetheinvestigationoftheproblem.Thesummary,conclusions,andpotentialfutureresearchdirectionsarepresentedinChapter 6 27

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CHAPTER2TRANSIENTPHENOMENAINONE-DIMENSIONALCOMPRESSIBLEGAS-PARTICLEFLOWS 2.1Introduction Compressibleparticle-ladenowsareencounteredinmanyelds,suchassolid-propellantrocketmotorsanddetonationofmultiphaseexplosives.Thepresenceofinertparticlescansignicantlyaffecttheowthroughmomentumandenergyexchanges.Theseexchangescanintroduceinterestingnewphysicalphenomenathatarenotpresentinasingle-phasecompressibleow.Theclassicshock-tubeproblem,see[ 93 ],canbeemployedtostudycompressiblegas-particleowsatafundamentallevel. Theshock-tubeproblemisattractiveforseveralreasons.First,ananalyticalsolutionisavailableforthesingle-phasecase.Second,theshock-tubeproblemincludesthebasiccompressibleowfeaturessuchasanunsteadyexpansionfan,acontactdiscontinuity,andashockwave.Third,thenumericalsolutionoftheshock-tubeproblemisinexpensive,makingsystematicparametricstudiesfeasible.Paststudiesofthemultiphaseshock-tubeproblemcanbedividedintothreetypesdependingonwhetherparticlesarelocatedonlyinthedrivensection,onlyinthedriversection,orinbothsections.Studiesofthersttypemainlyfocusontheinteractionbetweentheparticlesandtheshockwave,whilestudiesofthesecondtypefocusontheinteractionbetweentheparticlesandtheexpansionwave. Themajorityofpaststudiesofgas-particleowintheshocktubehaveplacedtheparticlesinthedrivensectionandinvestigatedshock-particleinteraction,see,e.g.,[ 17 ],[ 83 ],[ 48 ],[ 74 ],[ 64 ],[ 82 ],and[ 77 ].Thissetupisrelevanttothestudyofthepropagationofshockwavesinadustyenvironmentsuchastheatmosphere.Becauseofthenitemechanicalandthermalinertiaofparticles,theparticlescannotadjustimmediatelytotheinstantaneouschangesofvelocityandtemperaturethroughtheshockwave.Asaresult,theinteractionbetweenashockwaveandparticlesleadstotheso-calledfrozen, 28

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relaxation,andequilibriumregionsthatarediscussedinmoredetailbelow.Carrier[ 17 ],Soo[ 83 ],Kriebel[ 48 ],andRudinger[ 74 ]studiedtherelaxationregionnumerically. Thecaseofparticlesinthedriversectiondoesnotappeartohavebeenstudiedintensivelydespiteitsimportanceinapplicationssuchasvolcaniceruptionsandparticulateexplosions.TwonotableeffortsarebyRudingerandChang[ 75 ],andElperinetal.[ 26 ].Ingeneral,theinteractionbetweenanexpansionwaveandparticlesisfarmorecomplexthanthatbetweenashockwaveandparticles.Frozenandequilibriumsolutionscanalsobedened,buttheapproachofthesolutiontoequilibriumisfarslowerthanfortheinteractionwithashockwave. Inthepresentwork,weinvestigatetheproblemofashock-tubewithparticlesinthedriversection.Theinuenceoftheparticlephaseisinvestigatedindetailthroughparameterssuchasmassloading,mechanical,andthermaltimescalesonthepropagationoftheshock,gasandparticlecontactsurfaces,andtheexpansionfan.Particularattentionispaidtothetimescaleonwhichtheequilibriumsolutionisapproached.Asimpleanalysisispresentedtoestimatethetimescaleonwhichthegasandparticleconstants,shockwaveandexpansionfanapproachtheirequilibriumstates.Theseestimatesarecomparedagainstthecorrespondingcomputedvalues. 2.2Theory 2.2.1Assumptions Wemakethefollowingassumptions:(1)Theuidmayberepresentedbyaperfectgas.(2)Theuidmotionmayberegardedasinviscid,sotheuidviscosityandconductivityareneglectedexceptintheinteractionwiththeparticles.(3)Theparticlesareinert,rigid,andspherical.(4)Theparticleshaveconstantheatcapacityanduniformtemperaturedistribution.(5)Thevolumefractionoftheparticlesisnegligible.(6)Theonlyforceactingontheparticlesistheviscousdragforce.(7)Theparticlesdonotcollidewitheachother. 29

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2.2.2GoverningEquations TheequationsgoverningthegasphaseareconvenientlyexpressedintheEulerianframeofreference.TheparticlephasecanbetreatedusingeithertheLagrangianapproach,whereeachindividualparticleistracked,orusingtheEulerianapproach,wheretheparticlephaseistreatedasacontinuum.Wehavecomputedsolutionsusingbothapproachesandobtainedresultsthatcomparedverywell.Forbrevity,wepresentonlytheresultsobtainedusingtheEulerianmethodfortheparticlephase. IntheEulerian-Eulerianframework,themass,momentumandenergyequationsofthegasphasecanbewrittenas@(Yg) @t+@(Ygug) @x=0, (2)@(Ygug) @t+@[Yg(ug)2] @x+@pg @x=)]TJ /F3 11.955 Tf 9.3 0 Td[(fp, (2)@(YgEg) @t+@(YgHgug) @x=)]TJ /F3 11.955 Tf 9.3 0 Td[(upfp)]TJ /F3 11.955 Tf 11.96 0 Td[(qp, (2) where,Y,u,p,E,andHrepresentthedensity,massfraction,velocity,pressure,totalenergy,andtotalenthalpy.Thesuperscriptsgandpindicatethegasandparticlephases,respectively.Quantitieswithoutsuperscriptdenotemixturevariables.ThetotalenthalpyofthegasHgisgivenbyHg=Eg+pg Yg. (2) Thecorrespondingmass,momentum,andenergyequationsoftheparticlephaseare:@(Yp) @t+@(Ypup) @x=0, (2)@(Ypup) @t+@[Yp(up)2] @x=fp, (2) 30

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@(YpEp) @t+@(YpEpup) @x=upfp+qp. (2) Thetermsfpandqpthatappearontheright-handsideoftheequationsrepresenttheviscousdragforceontheparticlesandheattransferredtotheparticlesperunitvolumeduetothemomentumandenergyexchangesbetweenthephases.Theexpressionsforfpandqparefp=Ypug)]TJ /F3 11.955 Tf 11.95 0 Td[(up pCDRe 24, (2) andqp=YpCpTg)]TJ /F3 11.955 Tf 11.96 0 Td[(Tp pNu 2, (2) whereCpisthespecicheatoftheparticles,andpandprepresentthemechanicalandthermalparticleresponsetimesgivenbyp=p(dp)2 18g, (2) andp=Cpp(dp)2 12g, (2) wheredpistheparticlediameterandgandgarethedynamicviscosityandthermalconductivityofthesurroundinggas. InEq.( 2 )and( 2 ),CD,Re,andNuaretheparticledragcoefcient,Reynoldsnumber,andNusseltnumber.TheReynoldsnumberisdenedintermsofrelativevelocitybetweenaparticleandthesurroundinggasasRe=gjug)]TJ /F3 11.955 Tf 11.95 0 Td[(upjdp g. (2) 31

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Inthepresentwork,wechoosethedragcoefcientandNusseltnumbertobegivenbythefollowingcorrelations,see[ 21 ]and[ 98 ],CD=24 Re(1+0.15Re0.687)+0.421+42500 Re1.16)]TJ /F5 7.97 Tf 6.59 0 Td[(1, (2) andNu=2+(0.4Re0.5+0.06Re2=3)Pr0.4, (2) wherePristhePrandtlnumber,Pr=gCgp g, (2) andCgprepresentsspecicheatatconstantpressureofthegas.WiththedenitionofthePrandtlnumber,theratioofparticlemechanicalandthermalresponsetimescanbeexpressedasp p=2 31 Pr, (2) where=Cp=Cgpisthespecicheatratiobetweentheparticlesandthegas. Assumingthatthegasfollowstheperfectgaslawandtheparticlesnottocontributetothepressure,theequationofstatecanbewrittenaspg=YgRgTg. (2) 2.2.3FrozenandEquilibriumSolutions Considertheproblemofashocktubewithparticlesinthedriversection.Figure 2-1 showsatypicalx-tdiagram,wherethenumbers4,3,2,and1representtheregionstotheleftsideoftheexpansionfan,betweentheexpansionfanandthegascontactsurface,betweenthegascontactsurfaceandtheshock,andtotherightsideoftheshock,respectively.Theshadedareaindicatesthepresenceofparticles.Thewaves 32

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Figure2-1. Schematicx-tdiagramshowingtheregionnumberingusedinthisarticle,whereH,T,PC,C,andSdenotetheheadandtailoftheexpansionfan,theparticlecontactsurface,thegascontactsurface,andtheshock,respectively;shadingindicatesthepresenceofparticles. inFigure 2-1 arerepresentativeofcaseswithverylowmassloading.Appropriatex-tdiagramsforlargermassloadingswillbeshownintheresultssection. Thedrivensectionisfullycharacterizedbythegaspressurepg1,temperatureTg1,gasconstantRg1,andratioofspecicheatsg1.Thedriversideissimilarlycharacterizedbythegaspressurepg4,temperatureTg4,gasconstantRg4,andratioofspecicheatsg4,butinadditionthesuspendedparticlesarecharacterizedbytheirdiameterdp,densityp,numberdensitynp4(numberofparticlesperunitvolume),andspecicheatCp.Assumingamono-disperseuniformdistributionofsphericalparticles,thevolumefractionp4andmassfractionYp4ofparticlesinthedriversectioncanbeexpressedas p4=np4(dp)3 6,(2) and Yp4=pp4 pp4+g4(1)]TJ /F9 11.955 Tf 11.95 0 Td[(p4).(2) Thegasinthedriveranddrivensectionsandtheparticlesareatrestinitially(ug1=ug4=up4=0)andtheparticlesinthedriversectionareinthermalequilibriumwiththegas 33

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(Tg4=Tp4).Inthepresentstudy,wefocusontheregimewherethemassfractionofparticlesislargeenoughsothatthebackeffectofparticlesontheowisimportant.However,weconsidertheparticledensitytobelargeenoughsothattheparticlevolumefractionissmalldespitethelargemassfraction.Asaresult,particle-particleinteractionsandothervolumetriceffectscanbeignored.Thegoverningequationspresentedaboveareconsistentwiththisregime. Oncetheshock-tubediaphragmisruptured,ashockwavepropagatesintothedrivensection,followedbythegascontactsurfaceseparatingthegasesoriginallycontainedindrivenanddriversections.Becauseoftheniteinertiaoftheparticles,theparticlecontactsurface(theinterfacebetweenparticle-ladengasandpuregas)willlagthegascontactsurface,asindicatedinFigure 2-1 .Simultaneously,anexpansionwavepropagatesintothedrivergas.Thepresenceoftheparticlesbreakstheself-similarityofthepuregascase,andhenceprecludesananalyticalsolution.Beforepresentingourresults,werstconsidersomelimitingcasesasguidelines. 2.2.3.1Frozenlimit Immediatelyaftertheruptureofthediaphragm,whilethegasrespondsinstantaneously,theparticlescanbetakentobestationary.AsconsideredbyRudingerandChang[ 75 ],ausefullimitwillbetoassumethattheparticlesarefrozenattheirinitialvelocityandtemperature,andtoconsiderthecorrespondinggasow.Inthisso-calledfrozenlimit,thegasowcanbestudiedassumingthattheparticlecontactsurfaceremainsxedsincetheparticlesarestationary.Accordingly,theparticlevolumefractionisequaltotheinitialvolumefraction. Ifwefurtherconsiderthevolumefractionofparticlestobenegligible,thefrozenowbehavesasapuregasow,i.e.,asifthedriversectioncontainsonlythehighpressuregaswithouttheparticles.Evenfornitebutsmallvolumefraction,thefrozenowbehaveslikepuregasowimmediatelyfollowingtheruptureofthediaphragm.Forthepresentstudy,astheparticlevolumefractioniskeptsmall,theanalyticalpure-gas 34

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solutionprovidesanadequateapproximationforthefrozenlimit.Therefore,weusethepure-gassolutiontorepresentthefrozenlimitinthisarticle. Itshouldalsobenotedherethatnearthehead,theexpansionwaveismovingintostationaryparticles.Sincetheexpansionwaveleadstocontinuousvelocitychanges,thegasvelocityneartheheadisquitesmall.Thusthepropagationoftheheadoftheexpansionwavecanbeconsideredtofollowthepuregasresultsatalltimes.Infact,theowneartheheadoftheexpansionwaveisgovernedbytheacousticequation,implyingthattheheadtravelsintothedriversectionatthespeedofsoundinpuregas,asobservedinthecomputationalresultsofRudingerandChang[ 75 ]andElperinetal.[ 26 ]. Thespeedsoftheshockwave,gascontactsurface,andtheheadandtailoftheexpansionfanin pg4 pg1=2g1M2s,f)]TJ /F4 11.955 Tf 11.95 0 Td[((g1)]TJ /F4 11.955 Tf 11.95 0 Td[(1) g1+11)]TJ /F3 11.955 Tf 13.15 8.09 Td[(ag1 ag4g4)]TJ /F4 11.955 Tf 11.96 0 Td[(1 g1)]TJ /F4 11.955 Tf 11.96 0 Td[(1M2s,f)]TJ /F4 11.955 Tf 11.95 0 Td[(1 Ms,f)]TJ /F17 5.978 Tf 10.66 5.56 Td[(2g4 g4)]TJ /F17 5.978 Tf 5.76 0 Td[(1,(2) whereMs,frepresentstheshockMachnumberinthepuregaslimit.Withthegiveninitialconditionsandgaspropertiesspeciedupstreamanddownstreamofthediaphragm,theaboveequationcanbeusedtoobtaintheshockMachnumber.Thespeedsofthewavescanthenbeexpressedasus,f=Ms,fag1, (2)uc,f=ug2=ug3=2ag1 g1+1Ms,f)]TJ /F4 11.955 Tf 21.05 8.09 Td[(1 Ms,f, (2)uh,f=)]TJ /F3 11.955 Tf 9.3 0 Td[(ag4, (2)ut,f=ug3)]TJ /F3 11.955 Tf 11.96 0 Td[(ag3, (2) whereus,f,uc,f,uh,f,andut,frepresentthespeedsoftheshock,contactsurface,head,andtailoftheexpansionfan,respectively.Thesubscriptfindicatesthepuregaslimit. 35

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2.2.3.2Equilibriumlimit Atverylatetimes,theparticles(exceptforasmallregionclosetotheheadoftheexpansionfan)canbeconsideredtobeinequilibriumwiththesurroundinggas,i.e.,up=ugandTp=Tg.Undertheseequilibriumconditions,themixtureofgasandparticlescanbeconsideredtobehavelikeagaswithmodiedproperties(aso-calleddustygas,see[ 57 ]).Governingequationsforthemixturecanbederivedbyaddingthoseforthegasandtheparticles.Theequilibriumspeedsoftheshockwave,gascontactsurface,andtheheadandtailoftheexpansionfancanthenbedeterminedfrom pg4 pg1=21M2s,e)]TJ /F4 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[(1) 1+11)]TJ /F3 11.955 Tf 13.15 8.09 Td[(a1 a44)]TJ /F4 11.955 Tf 11.96 0 Td[(1 1)]TJ /F4 11.955 Tf 11.96 0 Td[(1M2s,e)]TJ /F4 11.955 Tf 11.96 0 Td[(1 Ms,e)]TJ /F17 5.978 Tf 10.66 4.4 Td[(24 4)]TJ /F17 5.978 Tf 5.76 0 Td[(1,(2) whereMs,erepresentstheequilibriumshockMachnumber,andus,e=Ms,ea1,e, (2)uc,e=u2,e=u3,e=2a1,e 1+1Ms,e)]TJ /F4 11.955 Tf 21.03 8.09 Td[(1 Ms,e, (2)uh,e=)]TJ /F3 11.955 Tf 9.3 0 Td[(a4,e, (2)ut,e=u3,e)]TJ /F3 11.955 Tf 11.96 0 Td[(a3,e, (2) wherethesubscripteindicatestheequilibriumlimit.Theequilibriumspeedofsoundaofthemixtureis ae=p RTg,(2) whereandRrepresentthespecicheatratioandthegasconstantofthemixture, =g1+Yp 1)]TJ /F8 7.97 Tf 6.59 0 Td[(YpCp Cgp 1+gYp 1)]TJ /F8 7.97 Tf 6.59 0 Td[(YpCp Cgp,(2)R=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Yp)Rg. (2) BecauseMs,e
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thatasYp4increases,thepressureratioacrosstheexpansionfanbecomesstrongerwhilethatacrosstheshockdecreases. 2.2.4TimeScaleAnalysis Asdiscussedabove,immediatelyfollowingtheruptureofthediaphragm,thegas-particleowbehavesasapuregasowandatlargetimesthegas-particleowbehavesasanequilibriumow.Thechangefromtheinitialpuregasowtotheself-similarequilibriumstateoccursoveratransitionperiod.Herewewillexaminethetimescalesthatdictatethedurationofthistransitionphase. Becausepandprepresentthetimescalesonwhichtheparticlesadjusttotheambientow,fortimesmuchlargerthanbothpandp,theequilibriumstatecanbeconsideredtoexist.Sincetheowwithintheexpansionfanconstantlychangeswithtime,perfectequilibriumforparticlescanneverbeachieved.Nevertheless,astimeevolves,whentheexpansionregionissufcientlybroad,boththetemporalandspatialvariationoftheowwithinthefanbecomequitesmall.Sincethetimescaleoftheowcontinuestoincreasewiththeexpansion,theStokesnumberofagivenparticledecreasesovertimetoallowanapproximateequilibriumdescriptionofparticlesintermsofthelocaluidproperties(seeSection2.4.4). 2.2.4.1Mechanicaltimescaleofparticlecontactsurface Oncethediaphragmisruptured,thegascontactsurfacemovesinstantlyintothedrivensectionwithaspeedgivenbythepuregassolution.Incontrast,theparticlecontactsurfaceinitiallyhaszerovelocity.Onlyastheparticlesthatwereinitiallyatthediaphragmareacceleratedbythesurroundinggas,doestheparticlecontactsurfacemoveintothedrivensection.Thustheambientgasvelocityseenbytheparticlesattheparticlecontactsurfaceisug3,fatt=0+andeventuallychangestoug3,efortp.Thetimeittakesfortheparticlecontactsurfacetoreachtheuidvelocitycanbeestimated 37

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fromtheparticleequationofmotion,dup dt=ug3)]TJ /F3 11.955 Tf 11.96 0 Td[(up pCDRe 24, (2) whereCD,Re,andphavebeendenedinEqs.( 2 ),( 2 ),and( 2 )withug=ug3,andug3,eug3ug3,f.If,forsimplicity,weignorethetimevariationoftheambientgasvelocityseenbytheparticlecontactsurface,usetheparticletimescalepasthereferencetimescale,anddenet0=t=p,thenEq.( 2 )canbeexpressedasdRe dt0=)]TJ /F4 11.955 Tf 9.3 0 Td[(ReCDRe 24=)]TJ /F4 11.955 Tf 9.3 0 Td[(Re(1+0.15Re0.687). (2) Inderivingtheaboverelation,weonlyusethersttermofthedraglawgiveninEq.( 2 );thisapproximationisvalidforRe
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Figure2-2. VariationofdimensionlessmechanicalequilibrationtimescaleoftheparticlecontactsurfacewithRe0. 2.2.4.2Thermaltimescaleofparticlecontactsurface Thethermaltimescaleoftheparticlecontactsurfacecanbeestablishedinasimilarmanner.TheparticlesoriginallyatTp4attheparticlecontactsurfaceinitiallyseethegastemperaturetobethepuregastemperatureTg3,fandatlongtimestheequilibriumgastemperatureTg3,e.StartingfromEq.( 2 ),weobtainwithEq.( 2 ), dTp dt=Nu 2Tg)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp p.(2) Weintroduce=Tg)]TJ /F3 11.955 Tf 11.95 0 Td[(Tp,useEq.( 2 ),andignorethetimevariationinthegastemperatureseenbyparticlecontactsurface,torewritetheaboveequationas 1 d dt0=)]TJ /F9 11.955 Tf 10.5 8.09 Td[(p ph1+0.5(0.4Re0.5+0.06Re2=3)Pr0.4i.(2) Asbefore,thethermalequilibrationtimescaleoftheparticlecontactsurfacetpcanbedenedarbitrarilyasthetimeittakesforthetemperaturedifferencebetweentheparticlesattheparticlecontactsurfaceandthesurroundinggastodecreaseto1%of 39

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Figure2-3. VariationofequilibrationtimescaleratiooftheparticlecontactsurfacewithRe0. theinitialtemperaturedifference.Forverysmallparticles,Re1,Nu!2,andhence tp p=ln(100)p p,(2) wherep=pisgivenbyEq.( 2 ).ForlargerparticleswithniteReynoldsnumbers,Eq.( 2 )canbeintegratednumericallyandtheresultingtp=tpisplottedinFigure 2-3 asafunctionofRe0fordifferentvaluesofp=p.Clearly,overawiderangeofReynoldsnumbersandmechanicalandthermaltimescalesoftheparticles,theequilibrationtimescalesarecomparable.Together,theycharacterizethetimescaleonwhichtheparticlecontactsurfaceapproachesequilibrium. Inthepresentproblem,theapproachtoequilibriumisexpectedtohappeninthefollowingmanner.Ashorttimeafterthediaphragmisruptured,thegasvelocitiesandtemperaturesintheexpansionfanandbetweenthefanandtheshockinitiallyfollowtheirstandardevolutionasinpuregas.Astheheadoftheexpansionfanpropagatesintothedriversection,theparticlesbegintobeacceleratedbythesurroundinggas.Thevelocityoftheparticlecontactsurfaceincreasessteadily,andthetemperatureoftheparticlecontactsurfacedecreasessimilarly.Duringtheapproachtoequilibrium,owing 40

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toniteinertia,theparticlevelocityistypicallysmallerthanthelocaluidvelocityandtheparticletemperatureislargerthanthelocalgastemperature.Aswewillseelater,undercertaincircumstances,theparticlevelocitycanbrieyovershootthelocaluidvelocity.Theresultingmomentumandenergyexchangecouplesbacktothegas,whosevelocityandtemperatureevolvetotheequilibriumstateasaresult.Thetimescaleonwhichtheparticlesattheparticlecontactsurfaceapproachthelocalgaspropertiesisofthesameorderofmagnitudeasthetimescaleonwhichthegaspropertiesattheparticlecontactsurfaceapproachtheequilibriumstate. Atintermediatetimes,whentheowisawayfromequilibrium,thereisstrongmomentumandenergycouplingbetweenthegasandtheparticles,andasaresultboththegasandparticlepropertiesshowcontinuousvariationfromtheheadoftheexpansionfantotheparticlecontactsurface.Inotherwords,whiletheequilibriumstateisapproached,thetailoftheexpansionfancannotbediscernedveryclearly.Furthermore,onlyatequilibriumwillthegasandparticlecontactsurfacesmoveatthesamespeed.Untilequilibriumisreached,thegascontactsurfacemovesfasterthantheparticlecontactsurface.Theintegratedeffectisthatthegascontactsurfaceisanitedistanceaheadoftheparticlecontactsurfaceatequilibrium.Nevertheless,itcanbearguedthatsincethemechanismsdrivingthecontactsurfacestowardsequilibriumarethesame,wecanexpectthegascontactsurfaceandtheparticlecontactsurfacetoreachequilibriumonthesametimescaletc,whichcanbetakentobethelargeroftportp,i.e., tc=max(tp,tp).(2) 2.2.4.3Shocktimescale Althoughtheshockwavepropagatesintopuregasinthepresentproblem,andhencedoesnotcomeintodirectcontactwiththeparticles,theinteractionofparticleswiththeexpansionfancreatesdisturbancesthatpropagateforwardtodeceleratetheshockwave.Asaresult,theshockspeedstartsoutatthepure-gasvalueandevolvesto 41

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theequilibriumvalue.Thedecelerationoftheshockwaveinturnsendsoutdisturbancesthataffecttheinteractionbetweentheparticlesandtheexpansionfan.Basedonthissequenceofcauseandeffect,itcanbeanticipatedthatthetimescaleonwhichtheshockapproachesequilibrium,denedarbitrarilyasthetimeittakesfortheshockspeedtoreach1%ofitsequilibriumvalue,islongerthantc.Thedistancebetweenthecontactsurfaceandtheshocklocationincreasessteadilywithtime.Theposition(relativetothelocationofthediaphragm)atwhichthecontactsurfacesreachtheirequilibrationcanbeestimatedas(us,e)]TJ /F3 11.955 Tf 12.52 0 Td[(ug2,e)tc.Thespeedofpropagationofinformationfromthecontactsurfacetotheshockisgivenbya2,e+ug2,e)]TJ /F3 11.955 Tf 12.18 0 Td[(us,e.Thus,thetimescaleforshockequilibrationcanbecrudelyestimatedas ts p=tc p+M2,e(g+1)(M2s,e)]TJ /F4 11.955 Tf 11.95 0 Td[(1) 2gM2s,e)]TJ /F4 11.955 Tf 11.95 0 Td[((g+1).(2) 2.2.4.4Expansionfantimescales Althoughthetailoftheexpansionfanwasestimatedtoapproachequilibriumtogetherwithparticleandgascontactsurfaces,theexpansionfaninitsentiretycanbeexpectedtosettletotheequilibriumstateonamuchslowertimescale.Followingthearrivaloftheheadoftheexpansionfan,theparticleshavejuststartedtomove.Thusitcanbearguedthattheheadoftheexpansionfanalwaysmovesatthespeedofsoundofthepuregasinthedriversection.Aftertheexpansionfanhasbroadenedsufciently,thevelocityvariationwithinthefanbecomessmallenoughforparticlestofollowthegasclosely.Againwedeneequilibriumtoimplythattheparticlevelocityiswithin1%ofthelocalgasvelocity.Sincetheowwithintheexpansionfanisunsteady,weestimatethevelocitydifferencebetweentheparticlesandthegasusingtheequilibriumEulerianapproximationofFerryandBalachandar[ 30 ]as up)]TJ /F3 11.955 Tf 11.96 0 Td[(ugpDug Dt=pugdug dx,(2) 42

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whereupandugrepresentthelocalvelocitiesoftheparticlesandthegas,andDug=Dtanddug=dxrepresentthelocalmaterialderivativeandspatialderivativeofthegasvelocity,respectively. Asequilibriumisapproached,thegasvelocitygradientcanbeestimatedbasedontheequilibriumgasvelocitywithintheexpansionfan,wherethegasvelocitychangesfromzerotoug3overthefanthickness(a4,e)]TJ /F3 11.955 Tf 12.36 0 Td[(a3,e+ug3,e)t.Basedonthesearguments,thetimescaleonwhichthebulkoftheexpansionfanapproachesequilibriumcanbeestimatedas te p100ug3,e a4,e)]TJ /F3 11.955 Tf 11.96 0 Td[(a3,e+ug3,e.(2) Thusthetimeittakesfortheexpansionfantoreachequilibriumcanbemorethananorderofmagnitudelongerthanthatrequiredbythecontactsurfaceandtheshock. 2.3NumericalMethod Thesolutionmethodisbasedonthecell-centerednite-volumemethod.Thefollowingisonlyabriefoverview.Formoredetails,see[ 37 ],[ 38 ],[ 40 ],[ 41 ],and[ 39 ].TheinvisciduxforthegasiscalculatedbytheapproximateRiemannsolverofRoe[ 73 ].Theface-statesareobtainedbyasimpliedsecond-orderaccurateweightedessentiallynon-oscillatoryscheme(see,e.g.,[ 43 ]),whichmodiesthegradientscomputedusingtheleast-squaresreconstructionmethodofBarth[ 7 ]. ThemethodemployedtosolvetheEulerianequationsgoverningtheparticlephasevariablesisverysimilartothatusedforthegas.TheprimarydifferenceisthattheinvisciduxiscomputedusingthemethodofLarrouturou[ 50 ]toensurepositivityofYp. Boththegasandparticlevariablesareevolvedintimeusingthefourth-orderRunge-Kuttamethod. Thesolutionmethodhasbeenextensivelyveriedandvalidatedforgasandgas-particleowswithandwithoutshockwaves.Forbrevity,theresultsfromthesestudiesarenotreproducedhere.Thestudiesdemonstratedthattheexpectedorders 43

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Table2-1. Summaryofcomputationcases. CaseYp4dp(m)p4=p1T4=T1 10.2105120.8105130.8120140.81020150.8100201 ofconvergenceandgoodagreementwithexperimentaldatawereobtained.Detailedresultscanbefoundinthereferenceslistedabove. 2.4Results Awiderangeofparticlemassfractionsandparticlesizeswereconsideredtoobtainacompleteunderstandingoftheirinuence.Here,toillustratetheeffectoftheparticlemassfractionandparticlesize,weconsiderthevecasesshowninTable 3-1 .Gasandparticlepropertiescommontoallcasesare:Cp=1600J=(kgK),p=1800kg=m3,Cgp=1004.5J=(kgK),g=1.7510)]TJ /F5 7.97 Tf 6.58 0 Td[(5kg=(ms),Pr=0.72,andRg=287J=(kgK). Additionalcomputationsthatarenotreportedhereindicatethattheowbehaviorsdescribedinthisarticlearenotsignicantlyaffectedbyothertemperatureratiosanddragandheat-transferlaws. Withtwoexceptionsnotedbelow,themeshresolutionandthetimestepare510)]TJ /F5 7.97 Tf 6.59 0 Td[(4mand810)]TJ /F5 7.97 Tf 6.59 0 Td[(7s,respectively.Thesevaluesweredeterminedthroughrenementstudies. 2.4.1OverallBehavior Immediatelyafterthebreakingofthediaphragm,thepuregassolutionisestablished,wheretheparticlevelocityandtemperatureremainattheirinitialvaluesandthegasvariablesaregivenbythesolutionofthepure-gasshock-tubeproblem.Thisbehaviorisreproducedinthesimulations,ascanbeseeninFigure 2-4 forthevelocitiesandthetemperaturesforcase1.Here,toshowtheearlytimebehavior,wehaveusedameshresolutionandtimestepof110)]TJ /F5 7.97 Tf 6.59 0 Td[(7mand110)]TJ /F5 7.97 Tf 6.59 0 Td[(10s,respectively.Astp,theparticlesequilibratewiththegasandapproachtheequilibriumsolution,asshownin 44

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AVelocity BTemperature Figure2-4. Comparisonofcomputedandpuregassolutionsatt=210)]TJ /F5 7.97 Tf 6.59 0 Td[(7sforcase1,showingthatthetransientprocessstartsfromthepuregaslimit. Figure 2-5 .Inthefollowing,wefocusonthetransientbehaviorofthesolution,i.e.,thewayinwhichthesolutionevolvesfromthepuregastotheequilibriumstate. Theoveralltransientbehaviorforcase1isshowninFigure 2-6 andthatforcase2isshowninFigure 2-7 .Theoveralltransientprocessforcase2isalsoashowninanx)]TJ /F3 11.955 Tf 12 0 Td[(tdiagram,seeFigure 2-8 ,whichisgeneratednumericallyfromthemagnitudeofthedensitygradient.FromFigure 2-8A ,itcanbeseenclearlythatafterlongenoughtime,allthewavesanddiscontinuitiestravelatconstantspeeds,whichmatchtheequilibriumvalues.Figure 2-8B showstheearlystagesofthetransientprocess,fromwhichwecanobservehowthewavesmatchthefrozenowattheverybeginningandhowtheydeviatefromitlateron. Foreaseofexposition,wediscusstheevolutionoftheexpansionfan,particlecontactsurface,andtheshockwaveseparately. 2.4.2EvolutionofExpansionWave Threeprimaryobservationscanbemadeabouttheevolutionoftheexpansionfan.First,ascanbeseenfromFigs. 2-6A 2-7A ,and 2-8B ,andasdiscussedabove,theheadoftheexpansionfanpropagatesatconstantspeedag4intothedriversection. 45

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AVelocity BTemperature Figure2-5. Comparisonofcomputedandequilibriumsolutionsatt=2.510)]TJ /F5 7.97 Tf 6.58 0 Td[(2sforcase1,showingthatthetransientprocessreachestheequilibriumlimit. Second,andmostimportantgiventhepresentfocusontheowtransient,thestrengthoftheexpansionfanappearstochangerapidlyintheveryearlystages:Thevelocityandpressureofthegastotherightoftheexpansionfandecreaseveryquicklyfromthevaluesofthepuregaslimitduetothepresenceoftheparticles.Thismeansthatthetotalvelocitychangeacrosstheexpansionfandecreaseswhilethepressurechangeincreases.Inotherwords,boththeinternalenergyandkineticenergyofthegasphasedropthroughexpansionfan.Thisisbecausepartofgasenergyistransferredtotheparticles.Thetotalenergyofthemixtureacrosstheexpansionfanisstillconserved.ReferringtoFigure 2-7 ,thechangeinthestrengthoftheexpansionfanismostclearlyvisibleforlargevaluesofYp4.Itisthischangeintheexpansionfanstrengththatsendsoutdisturbancesintheformofexpansionwavesthattravelinthepositivex-directiontowardtheshockwave.Theyinteractwiththeparticlecontactsurface,gascontactsurface,andshockwave,causingthemtodecelerate.Inturn,theseinteractionsgeneratereectedexpansionwavesthataffecttheexpansionfan.Thethirdobservationconcernsthepositionoftheparticlecontactsurfacewithrespecttothevelocityandtemperatureprolesintheexpansionfan.Aswillbediscussedinmoredetailinthenext 46

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A110)]TJ /F22 6.974 Tf 6.23 0 Td[(66t6110)]TJ /F22 6.974 Tf 6.22 0 Td[(5s B110)]TJ /F22 6.974 Tf 6.23 0 Td[(36t6110)]TJ /F22 6.974 Tf 6.22 0 Td[(2s C1.510)]TJ /F22 6.974 Tf 6.23 0 Td[(26t62.510)]TJ /F22 6.974 Tf 6.23 0 Td[(2s Figure2-6. Computedvelocityprolesofthetransientprocessforcase1. section,theparticlecontactsurfacecoincideswiththepointatwhichtheslopeofthevelocityproleisdiscontinuous,ascanbeseenfromFigs. 2-6A and 2-7A .Thusitisimportanttonotethatthepointatwhichthevelocityproleisdiscontinuousdoesnotcoincidewiththetailoftheexpansionfan. TheinuenceoftheparticlediameterontheinitialphaseofthetransientbehavioroftheexpansionfanatahighpressureratiocanbeseeninFigure 2-9 .Forcomparison,thepuregassolutionisalsoshown.Here,toshowtheearlytimebehavior,wehave 47

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A110)]TJ /F22 6.974 Tf 6.23 0 Td[(66t6110)]TJ /F22 6.974 Tf 6.22 0 Td[(5s B110)]TJ /F22 6.974 Tf 6.23 0 Td[(36t6110)]TJ /F22 6.974 Tf 6.22 0 Td[(2s C1.510)]TJ /F22 6.974 Tf 6.23 0 Td[(26t62.510)]TJ /F22 6.974 Tf 6.23 0 Td[(2s Figure2-7. Computedvelocityprolesofthetransientprocessforcase2. usedameshresolutionandtimestepof110)]TJ /F5 7.97 Tf 6.58 0 Td[(6mand110)]TJ /F5 7.97 Tf 6.59 0 Td[(9s,respectively.Itisclearthatthesmallestparticlesleadtothelargestdeviationfromthepuregassolution.ThisiseasilyexplainedbecausethedragcoefcientCD/1=dpatlowparticleReynoldsnumbers,whileCDconstantathighparticleReynoldsnumbers.Thereforethetotalviscousdragforceperunitmassisproportionalto1=(dp)2atlowparticleReynoldsnumbers,andto1=dpathighparticleReynoldsnumbers.Thusforthesamemassfraction,largerparticlesexertasmallerdragforceonthegas.Asaresult,atthevery 48

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A06t60.75s B06t6110)]TJ /F22 6.974 Tf 6.22 0 Td[(3s Figure2-8. Numericallygeneratedx)]TJ /F3 11.955 Tf 11.95 0 Td[(tdiagramforcase2,showingthetransientevolutionfrompuregaslimittoequilibriumlimit.Thecomputedresultisshowninsolidlines;thepuregasowresultisshownindashedlines.Atlongtimes,thecomputedspeedsofthewavesmatchestheequilibriumresults(showninboxes).Theshadingrepresentsthemagnitudeofthedensitygradient. earlystagesofthetransient,thecaseswithlargerparticlesaremuchclosertothepuregassolution.NotealsoinFigure 2-9 B theveryclearlyvisiblelocationoftheparticlecontactsurfaceanditsinuenceonthevelocityproleintheexpansionfanasdiscussedabove. ItcanalsoobservedthatfromFigure 2-4 2-6C ,and 2-7C thatatt=25ms,boththeshockandparticlecontactsurfacehavereachedequilibrium,asthevelocitybetweentheshockandtailofexpansionfanisconstant,implyingthattheshockandparticlecontactsurfacetravelatconstantspeed.However,neartheheadandtailofexpansionfan,thewavesarestilltravelingwithspeedsdifferentfromtheirequilibriumvalues,whichmeansthattheexpansionfanhasnotfullyreachedequilibrium.Therefore,theequilibrationtimeforexpansionfanismuchlongerthanfortheshock,whichisconsistentwiththetimescaleanalysisintheprevioussection. 49

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ACase3 BCase4 CCase5 Figure2-9. Computedvelocityprolesintheexpansionfanattheearlystagesofthetransientprocessfor110)]TJ /F5 7.97 Tf 6.59 0 Td[(66t6710)]TJ /F5 7.97 Tf 6.58 0 Td[(6s. 2.4.3EvolutionofParticleContactSurface Becauseoftheabove-describedchangeofstrengthoftheexpansionfan,itisclearthatanydisturbancesintroducedbythischangewilltravelatthespeedu+aintheformofexpansionwavestowardtheparticlecontactsurface.Theinteractionwiththeexpansionwavescausestheparticlecontactsurfacetodecelerate. Effectofparticlemassfraction .Thetemporalevolutionofthegasandparticlevelocitiesattheparticle-contactsurfacelocationareplottedinFigure 2-10 asa 50

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functionofYp4forvariousdp.AcriticalvalueofYp4appearstoexist,abovewhichtheapproachoftheparticlevelocityattheparticlecontactsurfacetotheequilibriumvelocityisnon-monotonic;belowthatcriticalvalue,theapproachofthegasvelocityisnon-monotonic.Thenon-monotonicbehaviorofparticlevelocitywillbediscussedbelow,andsimilarbehaviorisseenforthespeedoftheshockwave.ThevariationofthecriticalvalueofYp4withdpdeterminednumerically,ispresentedinFigure 2-11 .Whentheparticlemassfractionisclosetothecriticalvalue,theamplitudesoftheover-orundershootsaresmallorbarelyvisible.Ontheotherhand,whentheparticlemassfractionisfarfromthecriticalvalue,suchasYp4=0.1andYp4=0.8inFigure 2-10B ,thenon-monotonicbehaviorcanbeobservedclearly. Apossibleexplanationforthenon-monotonicbehaviorisasfollows.Foraspecicparticlesize,ifthemassfractionisverylow,theequilibriumvalueofthecontactsurfacevelocityisveryclosetothepuregasvalue,thereforethetimeforthegasvelocityatthecontactsurfacetodecreasetoitsequilibriumvalueismuchshorterthanthatfortheparticlevelocitytoreachitsequilibriumvalue.Sowhenthegasvelocitydecreasestotheequilibriumvalue,theparticlevelocityisstillsmallerthantheequilibriumvalue,andthereforetheviscousdragforcewillcausethegasvelocitytocontinuetodecreaseandundershootitsequilibriumvalue.Ontheotherhand,whenthemassfractionisveryhigh,thedifferencebetweenequilibriumvalueandpuregasvalueofthegasvelocityatthecontactsurfaceismuchlarger,andthetimeforthegasvelocitytoreachequilibriumvalueincreases.Incontrasttothelowmassfractioncase,theparticlevelocityatthecontactsurfacereachestheequilibriumvaluebeforethegas,sotheviscousdragforceincreasestheparticlevelocitytoovershoottheequilibriumvalue.Itisinterestingtondthatexceptinthevicinityofthecriticalvalue,eitherthegasortheparticlevelocityhavetoexperiencenon-monotonicbehaviortoapproachequilibrium.Wehaveveriedthatthesebehaviorsarenotcausedbythespecicdragandheat-transfercorrelationortheinitialconditionsusedinthesimulations. 51

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Adp=1m Bdp=10m Cdp=100m Figure2-10. Comparisonofcomputedevolutionofgas(dash-dottedlines)andparticle(solidlines)velocitiesatparticlecontactsurfacelocationasafunctionofYp4forvariousvaluesofdp. Effectofparticlesize .ThetemporalevolutionofthegasandparticlevelocitiesandtemperaturesattheparticlecontactsurfacelocationareplottedinFigs. 2-12 and 2-13 asafunctionofdpforYp4=0.3and0.8,respectively.Thefollowingpointscanbemade.First,itisobservedthatifthetimeisscaledbyparticlemechanicalresponsetimep,boththemechanicalandthermalequilibrationtimesfordifferentsizeparticlesareofthesameorder,implyingthattheeffectoftheparticlesizeisreectedmainlythrough 52

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Figure2-11. VariationofYp4withdpforwhichapproachtoequilibriumofgasandparticlevelocitiesatparticlecontactsurfaceandshockvelocityisnon-monotonic. p.Second,itcanbefoundfromFigure 2-12A thatastheparticlesizeincreases,theevolutionofthegasvelocityattheparticlecontactsurfacelocationchangesfrommonotonictonon-monotonicforYp4=0.3.FromFigure 2-13A ,weobservethatthenon-monotonicbehavioroccursforallthreeparticlesizes,inwhichtheparticlevelocitiesriseabovethegasvelocitiesandtheequilibriumvaluesandthendropbacktothelatter.Third,inbothcases,thedimensionlessmechanicalandthermalequilibrationtimestc=pdecreaseasparticlesizeincreases.Finally,forYp4=0.3,thethermalequilibrationtimescaleisapproximatelyequaltothemechanicalequilibrationtimescaleforvariousparticlesizes;butforYp4=0.8,thethermalequilibrationtimescaleisobviouslysmallerthanthemechanicalequilibrationtimescaleforvariousparticlesizes.ItshouldalsobenotedthatthecomputedequilibriumtemperaturevaluesareslightlydifferentforthevariousvaluesofdpinbothFigs. 2-12B and 2-13B .Thereasonisthattheparticlecontactsurfaceisveryclosetothegascontactsurface,wherethenumericaltemperaturegradientisverylarge.Therefore,theslightdifferencesinFigs. 2-12B and 2-13B areduetoextractingthevaluesclosetoanear-discontinuity. 53

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AVelocity BTemperature Figure2-12. Comparisonofcomputedevolutionofgas(dash-dottedlines)andparticle(solidlines)velocitiesandtemperaturesatparticlecontactsurfacelocationasafunctionofdpforYp4=0.3. AVelocity BTemperature Figure2-13. Comparisonofcomputedevolutionofgas(dash-dottedlines)andparticle(solidlines)velocitiesandtemperaturesatparticlecontactsurfacelocationasafunctionofdpforYp4=0.8. 54

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Figure2-14. EquilibriumtimescalesofshockandparticlecontactsurfaceasafunctionofdpforYp4=0.2and0.8fromcomputations,comparedwiththeresultfromthetimescaleanalysisindicatedinslopesymbol. Comparisonwithtime-scaleanalysis .FromFigure 2-10 ,thetimescaletoreachequilibriumconditionscanbeobserved.TheresultsforthecomputedequilibrationtimescalesareplottedinFigure 2-14 asafunctionoftheparticlediameter.Forcomparison,wecanalsoestimatetheequilibrationtimescaleoftheparticlecontactsurfacefromEq.( 2 )forcaseswithdifferentmassfractionsandparticlesizes.TheresultsforYp4=0.2and0.8,dp=1mand100mareshowninTable 2-2 .Fromtheseresults,wecanestimatethevalueofd[log(tc)]=d[log(dp)],andcomparewiththecomputedresultspresentedinFigure 2-14 .ForYp4=0.2,overtherangedp=1mto100m,weobtaind[log(tc)]=d[log(dp)]=1.645;andforYp4=0.8,weobtaind[log(tc)]=d[log(dp)]=1.671.ThedependenceonYp4isweak,andthetimescaleanalysispredictsreasonablywellthecomputedpower-lawbehavioroftc/(dp)1.66. Itshouldalsobementionedherethattheparticlecontactsurfacealwaystravelsclosetothegascontactsurfacebecausetheparticlesstudiedhereareverysmall.Afterequilibriumisreached,theparticlesandgascontactsurfacetravelwiththesamespeed 55

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Table2-2. Particlecontactsurfaceequilibrationtimescalesasgivenbytime-scaleanalysis. Yp4dp(m)p4=p1T4=T1tplog(tp) 0.21511.846E)]TJ /F4 11.955 Tf 9.3 0 Td[(5)]TJ /F4 11.955 Tf 9.3 0 Td[(4.7340.2100513.602E)]TJ /F4 11.955 Tf 9.3 0 Td[(2)]TJ /F4 11.955 Tf 9.3 0 Td[(1.4430.81511.955E)]TJ /F4 11.955 Tf 9.3 0 Td[(5)]TJ /F4 11.955 Tf 9.3 0 Td[(4.7090.8100514.304E)]TJ /F4 11.955 Tf 9.3 0 Td[(2)]TJ /F4 11.955 Tf 9.3 0 Td[(1.366 (seeFigure 2-8B )andthereforethedistancebetweentheparticleandgascontactsurfacesbecomesconstant. 2.4.4EvolutionofShockWave Similartotheevolutionoftheparticlecontactsurface,theexpansionwavessentbythechangeofstrengthoftheexpansionfancausestheshockwavetodecelerateasalreadydiscussedabove.TheevolutionoftheshockspeedispresentedinFigure 2-15 asafunctionofdpandYp4.Aswiththecorrespondingevolutionofthevelocitiesattheparticlecontactsurface,wenotethattheasymptoticapproachoftheshockspeedtoitsequilibriumvalueisnotmonotonicifthemassfractionYp4isbelowacriticalvalueforagivenparticlediameterdp.Thecriticalvaluesfortheshockwavearequitesimilartothosefortheparticlecontactsurfaceoverarangeofdp,seeFigure 2-11 Itisinterestingtonotethat,whentheshockspeedisnon-monotonic,theshockisatrstmovingfaster,thenslowerthantheequilibriumvalue.ItisthereforepossibletopredicttheshocklocationtobeveryclosetotheequilibriumvalueforcertainYp4anddp.Thismeansthatevenfornite-sizedparticle,equilibriumtheorycanpredicttheshocklocationaccuratelyafterequilibriumconditionsarereached. ComparisonofFigs. 2-10 and 2-15 showsthatthetimefortheshockwavetoreachequilibriumislargerthanthatfortheparticlecontactsurface.ThiscanalsobeobservedfromFigure 2-14 ,whichshowstheequilibrationtimescalesoftheshockwaveandparticlecontactsurfacefordpvaryingfrom1mto100mforYp4=0.2and0.8. Forcomparison,wecanalsoestimatetheequilibrationtimescaleoftheshockfromEq.( 2 )fordifferentmassfractionsandparticlesizes.TheresultsforYp4=0.2 56

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Table2-3. Shockequilibrationtimescalesasgivenbytime-scaleanalysis. Yp4dp(m)p4=p1T4=T1tslog(ts) 0.21512.185E)]TJ /F4 11.955 Tf 9.3 0 Td[(5)]TJ /F4 11.955 Tf 9.3 0 Td[(4.6610.2100512.847E)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 9.3 0 Td[(0.5460.81511.685E)]TJ /F4 11.955 Tf 9.3 0 Td[(5)]TJ /F4 11.955 Tf 9.3 0 Td[(4.7730.8100512.134E)]TJ /F4 11.955 Tf 9.3 0 Td[(1)]TJ /F4 11.955 Tf 9.3 0 Td[(0.671 and0.8anddp=1mand100mareshowninTable 2-3 .Fromtheseresults,wecanestimatethevalueofd[log(ts)]=d[log(dp)],andcomparewiththecomputedresults(seeFigure 2-14 ).ForYp4=0.2,overtherangedp=1mto100m,weobtaind[log(ts)]=d[log(dp)]=2.057;andforYp4=0.8,weobtaind[log(ts)]=d[log(dp)]=2.051.Similartotheresultsfortheparticlecontactsurface,thedependenceonYp4isweakandthetimescaleanalysispredictsquiteaccuratelythecomputedpower-lawbehaviorofts/(dp)2.05. 2.5Conclusions Thepresentstudyfocusedontheshock-tubeproblemtostudythegas-particleowarisingfromparticle-ladendriversections.Theoryandcomputationwereusedtounderstandtheinteractionbetweentheexpansionfanandparticles.Thetransientprocessfromfrozentoequilibriumowwasinvestigatedindetail.Themechanismbywhichparticleschangethegasowwasanalyzedbystudyingthepropagationsofdisturbancewaves. Simpletheoreticalconsiderationsallowestimationofthetimescaleswithwhichtheexpansionfan,theparticlecontact,andtheshockwavereachequilibrium.Theseestimatesindicatedthatthelongestequilibrationtimescaleisassociatedwiththeexpansionfan.Numericalresultswereingoodagreementwiththetheoreticalestimates. Theparticlemassfractionwasfoundtodeterminewhethertheapproachtoequilibriumconditionsoftheshockandparticlecontact-surfacespeedsaremonotonicornot.Wehavefoundthatforvaluesofthemassfractionbelowadiameter-dependentcriticalvalue,theapproachtoequilibriumconditionsisnotmonotonic. 57

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Adp=1m Bdp=10m Cdp=100m Figure2-15. ComparisonofcomputedevolutionofshockspeedasafunctionofYp4,(shockspeedofequilibriumowsolutionisshowedasdashedlines). Theinitialtemperatureratioandthedragandheat-transferlawsdonotaffectourresultssignicantly. 58

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CHAPTER3ANUMERICALSOURCEOFSMALL-SCALENUMBER-DENSITYFLUCTUATIONSINEULERIAN-LAGRANGIANSIMULATIONSOFMULTIPHASEFLOWS 3.1Introduction InEulerian-Lagrangiansimulationsofmultiphaseows,thecarrier(uid)phaseisevolvedintheEulerianframeworkandthedispersed(uidorsolid)phaseistreatedbytheLagrangianframework.IntheEulerianapproach,conservationequationsaresolvedforthemass,momentum,andenergyperunitvolumeonaxedgrid.IntheLagrangianapproach,equationsaresolvedfortheposition,mass,momentum,andenergyofmaterialpointsthatmovethroughthexedEuleriangrid.(Forbrevity,wewillconsiderthedispersedphasetobesolidandhencerefertothematerialpointsasparticles.Theissuesdiscussedinthisarticleapplyequallytobubblesanddroplets,however.) IntheLagrangianequations,quantitiessuchastheundisturbeduidvelocity,uidacceleration,anduidtemperatureareneededattheparticlelocation.Becausetheparticlesdonot,ingeneral,coincidewiththelocationsatwhichtheuidsolutionisstored,someformofinterpolationisrequired.(Inthisarticle,weassumethattheuidsolutionisobtainedatgridpointsthroughanite-differencemethod.Therefore,theuidpropertiesmustbeinterpolatedfromgridpointstotheparticlelocations.)TheinterpolationissometimesreferredtoasforwardorEulerian-to-Lagrangiancoupling.Atsufcientlyhighmassloadings,theparticlesbegintoinuencetheuid.ThisbackeffectrepresentsLagrangian-to-Eulerianorbackwardcoupling. Todiscussthecouplinginmoredetail,considerthemomentumexchangebetweenthephases.AccordingtoNewton'sthirdlaw,thehydrodynamicforceexertedontheindividualparticlesisappliedbacktothecarrierphase.Inbackwardcoupling,theforcesfromparticlesmustbetransferredtothegridpointsandappliedassource/sinktermsinthemomentumequationsofthecarrierphase.Asmoothdistributionoftheparticlenumber-density,i.e.,thenumberofparticlesperunitvolume,onthegridisessentialforanaccurateaccountingofthebackeffectofthedispersedphaseonthe 59

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carrierphase.Largegrid-scaleuctuationintheparticlenumber-densitywillintroducespurioussmall-scaleoscillationsinthecarrierphasethroughbackwardcoupling.Theseoscillations,inturn,willthenbefedbacktotheparticlemotionanddistributionthroughforwardcoupling.Thisfeedbackmechanismcanleadtoaccumulationoferrors. Herewefocusonaclassofphysicalproblemswheretheparticlesaresmallandnumerous.Thiswillresultinalargenumberofparticleswithineachgridcell(inonedimension:theregionbetweentwoadjacentgridpoints),allowingaconceptuallysmoothrepresentationoftheparticlenumber-densityandthebackwardcoupling.BecausethecostofEulerian-Lagrangiansimulationsisproportionaltothenumberofparticles,itiscommontolimittheaveragenumberofparticlespercellanddeneso-calledcomputationalparticlesthatrepresentalargenumber(orcloud)ofrealparticles.Insuchsimulations,severalsourcesofnumericalerrorscanbeidentied: 1. TheEulerianDiscretizationError(EDE)thatarisesfromthespatialandtemporaldiscretizationoftheequationsgoverningthecarrierphase. 2. TheInterpolationError(IE)thatarisesfromtheinterpolationofthecarrier-phasesolutionfromtheEuleriangridtotheparticlelocations. 3. TheLagrangianIntegrationError(LIE)thatarisesfromthetemporaldiscretizationoftheLagrangianequationsfortheparticles. 4. TheBack-CouplingError(BCE)thatarisespartlyfromthediscretealgorithmthatapportionsthemomentumandenergycouplingoftheindividualparticlesbacktotheneighboringgridpoints(orcells)andpartlyfromthefactthatonlyalimitednumberofcomputationalparticlesareusedtorepresentthetruesystemwherespatialdistributionisrandom. Theseerrorsarecontrolledbythespatialdiscretization,thetemporaldiscretization,andtheLagrangiandiscretization(numberofcomputationalparticlespercell).Theerrorsfromthespatialandtemporaldiscretizationsareeasilycontrolled,atleastinprinciple,bychoosingappropriatemethodsandreducingthegridspacingxandtimestept.Forthisreason,weignoretheEDEandLIEinthefollowing.(AllnumericalresultspresentedbelowhavebeenveriedtohavenegligibleEDEandLIE.)Theother 60

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twoerrors(IEandBCE)arisefromthenumericalapproximationoftheforwardandbackwardcoupling.Severalinterpolationschemes(second-,fourth-,andsixth-orderLagrangeinterpolation;splineinterpolation;Hermiteinterpolation;etc.)havebeenusedinthepast.TheIEarisingfromvariousinterpolationschemeshasbeenconsideredby[ 5 ]and[ 94 ]forturbulentows.Althoughhigher-orderschemeshavebeenshowntoresultinsignicantlyreducedIE,lower-ordermethodssuchastrilinearinterpolationareoftenused,especiallyinthecontextofnite-volumeandnite-differenceapproaches(see,e.g.,[ 81 ],[ 80 ],[ 95 ]).Similarly,severaltechniqueshavebeenadvancedtoback-coupletheLagrangianparticlestothecarrierphase,suchastheparticle-in-cellapproach(e.g.,[ 27 ],[ 22 ],[ 36 ])andtheprojectionmethod(e.g.,[ 84 ],[ 87 ],[ 10 ]).Theaccuracyofbackward-couplingalgorithmshasrecentlybeenexaminedby[ 34 ]. Eulerian-Lagrangiansimulationsofmultiphaseowsoftenexhibitsmall-scaleuctuationsinthenumber-density.Here,thetermsmallscalesmeansscalesontheorderofthegridspacing.Traditionally,suchuctuationshavebeenascribedtotwoerrors.Ifparticlesarerandomlydistributedwithuniformprobabilityoveragivendomain,theinherentstochasticuctuationarisingfromtherandomdistributionwillcontributetogrid-celltogrid-cellvariationintheparticlenumber-density.Thisstochasticerrorscalesastheinversesquarerootofthemeannumberofparticlespergridcell.Afurthererrorisduetothenitenumberofcomputationalparticles,whichistypicallyfarsmallerthantheactualnumberofparticles.Thisdeterministicerrordecreasesastheinverseofthemeannumberofparticlespergridcell(see[ 71 ],[ 34 99 ]).Thus,whilethenatureofthesetwoerrorsisfundamentallydifferent,bothcanbecontrolledbyincreasingthenumberofcomputationalparticles. However,therearecases(see[ 51 ])wheresmall-scalenumber-densityuctuationscannotbereducedbyincreasingthenumberofcomputationalparticlesorbymaintainingthenumberofcomputationalparticlesineachcellconstant(see[ 71 ],[ 85 ],[ 46 ],[ 58 ]).Furthermore,thesesmall-scaleuctuationscannotbeexplainedonthebasisof 61

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stochasticerrorsarisingfromarandomparticledistribution.Itisthisadditionaloferrorinparticlenumber-densitythatisthefocusofthisarticle. Asimpleexplanationfortheaddedsourceofnumber-densityuctuationthatcannotbereducedbyincreasingthenumberofparticlesisasfollows.Considerthecarrierphasevelocitytoconsistofasimplepropagatingdisturbanceofxedshapeandsize.Lettheexactuidvelocitybeknownatthegridpointsatalltimeinstances(i.e.,EDEistakentobezeroatthediscretegridpoints).Eventhen,ifalowerorderinterpolationschemeisusedtoobtainuidvelocitybetweengridpointsforcomputingparticlemotion,theresultinginterpolateduiddisturbancewillchangeshapeasitpropagatesthroughthegrid.Thisinterpolationerrorinparticlemotioncontributestospuriousvariationsoftheparticlepositionatthesubgridscale.Thesevariationsmayaccumulateovertimeandbecomesignicant.Thus,particlenumber-densityerrorcanarisefrominterpolationerror(IE)evenintheabsenceofanyerrorintheuidsolution(EDE).However,inanytypicalnumericalimplementation,theuidvelocitywillincludeEuleriandiscretizationerrorandifthespatialdiscretizationoftheowsolverandtheinterpolationschemeareofthesameorderthenEDEandIEwillbecomparable.BothEDEandIEwillleadtoerrorsinparticlenumber-density. Aswillbeshownbelowthenatureofinterpolationerror'simpactonparticlemotionwillbesuchthatIEwillresultinspurioussmall-scaleuctuationinparticlenumber-density.Thiserrorwillbesubgridinoriginandappearasuctuatingclusteringofparticleswithinthegridcells.Therearesituationswhereparticlestendtopreferentiallyaccumulateandcreateregionsofenhancedconcentrationandregionsdevoidofparticles,eitherduetotheirinertialeffectorduetonon-zerodivergenceoftheunderlyinguidvelocity.SucheffectivedivergenceinparticlemotionspreadsthesubgriderrorduetoIEandgiverisetogrid-celltogrid-cellparticlenumber-densityuctuation. 62

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Thegoalsofthisarticleare(1)todeterminetheprecisesourceoftheerror,(2)todemonstratehowtheerrorcanbereduced,and(3)tocharacterizetheerrormagnitudessothatguidelinescanbeestablishedthatallowpractitionerstokeeptheerrorbelowspeciedthresholds.Theoverallapproachadoptedtoreachthesegoalsistosystematicallyeliminateothersourcesoferror,considerfoursimpletestproblemsinonedimension.Thersttwoaremodelproblemswherethepropagatinguiddisturbanceconsistsof(1)asinglesinusoidalwaveand(2)ahyperbolictangentdisturbance.Thesetwomodelsproblemsprovidetheidealtestbedtoillustratetheinterpolation-inducederrorinparticlenumber-density.Whiletheformerhasnonetuidvelocitydivergenceacrossthedisturbance,thelaterdoes.Thisallowsexaminationofhowsubgriderrortranslatestogrid-celltogrid-cellerror.Thelatertwotestcasesconsiderone-dimensionalcompressibleowsofpracticalinterest,namely(3)anexpansionfanand(4)ashockwave.Theresultsobtainedfromthemodelproblemswillbeusedtointerprettheresultsfortheselatertestcases. Theremainderofthisarticleisstructuredasfollows.ThegeneralmathematicalmodelofmultiphaseowinEulerian-LagrangianframeworkisdescribedinSection 3.2 .AsimpliedmodelisintroducedinSection 3.3 .ThenumericalapproachisoutlinedinSection 3.4 .ResultsarepresentedinSection 3.5 .ConclusionsareofferedinSection 3.6 3.2MathematicalFormulation TheequationsgoverningthecarrierphaseareconvenientlyexpressedintheEulerianframeofreference.Weconsideronlyone-dimensionalproblemsinthisarticlebecausethefundamentalnatureoftheerrorinvestigatedhereisnotdependentonthenumberofspatialdimensions.Themass,momentum,andenergyequationsofthe 63

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carrierphasecanthenbewrittenas@(g) @t+@(gug) @x=0, (3)@(gug) @t+@[g(ug)2] @x=)]TJ /F9 11.955 Tf 10.49 8.08 Td[(@pg @x)]TJ /F3 11.955 Tf 11.95 0 Td[(fp, (3)@(gEg) @t+@(gHgug) @x=)]TJ /F3 11.955 Tf 9.3 0 Td[(upfp)]TJ /F3 11.955 Tf 11.96 0 Td[(qp, (3) where,u,p,E,andHrepresentthedensity,velocity,pressure,totalenergy,andtotalenthalpy.Thesuperscriptsgandpdenotepropertiesassociatedwiththegasphaseandparticlephase,respectively.Thesuperscriptindicatesdimensionalquantities.ThetotalenthalpyofthegasHgisgivenbyHg=Eg+pg=g.Theideal-gaslawisassumedtoapply,i.e., pg=()]TJ /F4 11.955 Tf 11.96 0 Td[(1)gEg)]TJ /F4 11.955 Tf 13.16 8.09 Td[(1 2(ug)2,(3) whereistheratioofspecicheatsofthegas.Weuse=1.4throughout. TheLagrangianevolutionequationsfortheparticlepositionxp,velocityup,andtemperatureTparedxp dt=up, (3)dup dt=ug(xp,t))]TJ /F3 11.955 Tf 11.95 0 Td[(up p, (3)dTp dt=Tg(xp,t))]TJ /F3 11.955 Tf 11.96 0 Td[(Tp p, (3) whereug(xp,t)andTg(xp,t)arethegasvelocityandtemperatureattheparticlelocation,respectively,andtheparticlemechanicalandthermaltimescalespandparegivenby p=p(dp)2 18g,p=Cpp(dp)2 12g,(3) wherepistheparticledensity,dpistheparticlediameter,andCpistheparticlespecicheat,allofwhichareassumedtobeconstants.BecauseniteReynoldsnumbereffectsarenotimportantforthepurposeofthisstudy,nosuchcorrectionsareincorporated.Thevariationofthedynamicviscositygwithtemperatureisgivenby 64

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Sutherland'slaw.TheconductivitygandviscosityofthegasarerelatedbythePrandtlnumberPr=gCgp=g,whereCgpisthespecicheatatconstantpressureofthegas.Inthelimitp!0,theparticlebehavesasatracerandfollowsthelocalgas,i.e.,up!ug(xp,t).Whenp6=0,particleinertiabecomesimportantandtheparticlevelocitydiffersfromthatoflocalgasphasevelocity. Whentheparticlemassloadingisnotnegligible,theparticlesinuencethemomentumandenergyofthegasthroughfpandqpinEqs.( 3 )and( 3 )(backwardcoupling).Fortheithparticle,wehave fpi= 6p(dp)3 Vcellug(xpi,t))]TJ /F3 11.955 Tf 11.96 0 Td[(upi p,qpi= 6Cpp(dp)3 VcellTg(xpi,t))]TJ /F3 11.955 Tf 11.96 0 Td[(Tpi p,(3) whereVcellisequaltothegridspacingfortheone-dimensionalcomputationsconsideredhere.ThetermsfpandqpinEqs.( 3 )and( 3 )arecalculatedbysummingfpiandqpiofalltheparticlesinthevicinityofthegridpoint. 3.3ASimpleModelScenario ThoughresultsbasedonthemathematicalmodeldescribedinSection 3.2 willbeshowninSection 3.5 ,weconsideranevensimplermodelrsttoisolatetheprecisecauseforthesmall-scalenumber-densityuctuations.Inthissimplermodel,weassumethattheuidvelocityisaknownfunctionandthatheat-transfereffectsarenegligible.Thenonlythesolutionoftheparticlepositionandmomentumequationsisrequired. Forreasonsthatwillbecomeclearbelow,agasphasevelocitydisturbancethatisspatiallycompactisparticularlyusefulinthepresentstudy.Thenthegasphasevelocitycanbewrittenindimensionalformas ug(x,t)=8>>>>>><>>>>>>:0ifx+ust)]TJ /F3 11.955 Tf 21.91 0 Td[(N=2 (x+ust)if)]TJ /F3 11.955 Tf 9.29 0 Td[(N=2N=2,(3) 65

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Figure3-1. Schematicofmodelprobleminwhichawavemovesthroughapacketofparticles.Inmodelproblem1(seeSection 3.5.1 ),asinusoidalwaveisconsidered.Inmodelproblem2(seeSection 3.5.2 ),ahyperbolic-tangentwaveisused. whereNistheregionofsupport.Theshapeofthedisturbanceisdictatedbythechoiceofthefunction andisindependentoftime.Asdenedabove,thedisturbancetravelsleftwardataconstantvelocityofus>0.Thefunction isdenedtoguaranteeatleastC0-continuityatx+ust=N=2.Weassumethattheparticlesareinitiallystationary,uniformlydistributedwithinagivensub-domain,andtotheleftofthedisturbance.ThemodelproblemisshownschematicallyinFigure 3-1 .Ourprimaryinterestistodeterminehowtheparticlesareredistributedasthedisturbancepropagatesthroughtheparticles. Itisinstructivetorstconsiderthetheoreticalbehavioroftheparticlesinreactiontothedisturbance.Theparticlesremainstationaryuntilthearrivalofthedisturbance.Weplacearestrictiononthegasphasevelocitythat +us>0forallx.ThisensuresthattherearenoxedpointsinEqs.( 3 )and( 3 )whereparticlescanaccumulate.Thus,asthedisturbancemovestotheleft,theparticlesareguaranteedtoemergefromthedisturbanceontheright.Foratracerparticle,i.e.,forp=0,thevelocityinstantaneouslybecomesequaltou2assoonasthedisturbancemovespastit.Foraninertialparticle,i.e.,forp>0,theapproachtothevelocityisasymptotic.Intheregion 66

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ofthedisturbancewheretheparticlevelocityisvarying,thespacingbetweenparticlescanbenon-uniform.Buttothefarrightofthedisturbance,wheretheparticlevelocityhasreachedtheasymptoticvalueofu2,theparticlesareagainuniformlydistributed.Fromconservationofparticles,itcanbeshownreadilythattheratiooftheuniformspacingbetweentwoadjacentparticlesafterthepassageofthedisturbancetothatbeforethedisturbanceis1+u2=us. Inwhatfollows,itwillbeshownthattheabovetheoreticalresultsarenotrecoveredwhentheproblemissolvednumerically.Althoughalltheparticlesnumericallyreachthecorrectasymptoticvelocityofu2,theywillnotbeuniformlydistributed.Itisthisnon-uniformdistributionofparticlesthatcangiverisetothegridscaleuctuationsdescribedintheintroduction.Therootcauseoftheerrorliesintheprecisewayinwhichtheuidvelocityisinterpolatedtotheparticlelocation.Inthenumericalsolution,theuidvelocitythatiscollocatedatthegridpointsisinterpolatedtoobtainanumericalapproximationto .AsthedisturbancemovesthroughthexedEuleriangrid,theinterpolateduidvelocityvariesovertime(i.e.,thenumericalapproximationto varies).Thisvariationistime-periodic,sincetheuidvelocityatthegridpointsanditsinterpolationareidenticalafterthedisturbancehasmovedoveranentiregridcell. Inessence,althoughtheexactdisturbanceiscollocatedatthegridpointsasitpropagatestotheleft,itsnumericalapproximationobtainedfrominterpolationchangesasthedisturbancemoveswithinagridcell,andthisvariationcyclicallyrepeatsoveradjacentgridcells.Consequently,particlesthatareinitiallylocatedatdifferentpositionswithinagridcellseeslightlydifferenthistoriesofthecarrier-phasevelocityasthedisturbancepropagatesthroughthegridcell.Thisresultsinanon-uniformparticledistributionastheyemergeoutofthedisturbance.Again,thenon-uniformityiswithinapacketofparticlesthatwasbetweengridpointstotheleftofthedisturbancebeforeitsarrival.Bycontrast,intheexactsolution,thereisnogriddependenceandeachparticle 67

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seesthesamehistoryofgasphasevelocityandthusauniformspacingispreservedevenafterparticlesemergefromthedisturbance. Akeyparameterthatcontrolsthelevelofnon-uniformityinthenumericalsolutionisthegridsizex,ormorepreciselythenumberofgridpointsusedtoresolvethedisturbance(N=N=x).Hereweusethegridsizexandthevelocityofpropagationofthedisturbanceusasthelengthandvelocityscales.Thenthegasphasevelocityandtheequationsofmotionfortheparticlecanbewrittenindimensionlessformasug=ug(x,t), (3)dxp dt=up, (3)dup dt=ug(xp,t))]TJ /F3 11.955 Tf 11.96 0 Td[(up p, (3) wherethevariableswithoutsuperscriptsaredimensionless. 3.4Approach 3.4.1NumericalMethodsforGoverningEquations Forthemodelproblem,thegassolutionisassumedtobeknownandisnotinuencedbytheparticles,i.e.,weonlyconsiderone-waycoupling.Fortherealmultiphase-owproblems,boththegasandtheparticlephasesneedtobesolved.Inthiswork,Eqs.( 3 )-( 3 )aresolvedbyafth-orderaccuratehybridcompact-WENOschemewithRK4timeintegrationdescribedby[ 18 ].Equations( 3 )-( 3 )areintegratedintimewithRK4also.Thegridspacingandtimesteparechosensmallenoughtoensurethattheassociatederrorsarenegligible.Theimplementationoftwo-waycouplingisdescribedby[ 51 ]and[ 52 ]. 3.4.2InterpolationMethods InthenumericalintegrationofEqs.( 3 )and( 3 )orEq.( 3 ),uidquantitiesattheparticleposition,e.g.,ug(xp,t)orug(xp,t),areneeded.Theymustbeobtained 68

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throughinterpolationfromgrid-pointvaluesoftheuidvelocity.Inthepresentstudy,weemployfourinterpolationmethods: 1. Piecewiseconstantinterpolation:Theinterpolateduidvelocityisapproximatedasaconstantbetweengridpointsandisthereforediscontinuousatthegridpoints.Theuidvelocityattheparticlelocationisobtainedbyaveragingthevaluesatthetwogridpointsstraddlingthecellinwhichtheparticleislocated.Thisinterpolationschemeisnotofpracticalinterestduetoitsloworderofaccuracy. 2. Piecewiselinearinterpolation:Thisinterpolationmethodcanbeviewedasrst-ordersplineinterpolation.Theuid-velocitydistributioninsideacellisapproximatedaslinear.TheinterpolateduidvelocityisonlyC0-continuousacrossthegridpoints.Intwoandthreedimensions,thisschemeleadstothewidelyusedbilinearandtrilinearinterpolationschemes. 3. Naturalcubic-splineinterpolation:Incubic-splineinterpolation,theuidvelocityisconstructedsuchthattheinterpolateduidvelocityissmoothandcontinuousintherstandsecondderivativesatthegridpoints.Naturalcubicsplines,whichassumethatthesecondderivativesoftheinterpolantatthedomainboundariesarezero,arethemostcommonlyusedcubicspline. 4. Hermiteinterpolation:Inthismethod,theuidvelocityinsideacellisinterpolatedbyHermitepolynomials.Therefore,notonlythefunctionvaluesbutalsotherstderivativeareneededatthegridpoints.TheinterpolateduidvelocityisC1-continuous.Inthemodelproblems,therstderivativeatthegridpointsisavailableanalytically.Inarealcomputation,therstderivativeoftheuidvelocityatthegridpointsmustbecomputednumerically.Inmultipledimensions,thecompleteimplementationofHermiteinterpolationcanbetedious,andinstead,asimplershape-functionmethodcanbeusedsee[ 5 ]. 3.4.3ErrorDenitions Toevaluatetheinuenceoftheinterpolationerrorandtheresultingnon-uniformityintheparticledistribution,weonlyneedtostudyasinglepacketofparticles,denedasagroupofparticleslocatedwithinagridcellpriortothearrivalofthedisturbance.Asdiscussedabove,andtobeillustratedinmoredetailbelow,thebehaviorofeachpacketofparticlesisidentical. WedenetherightandleftboundsofthepacketasxpR(t)andxpL(t),respectively.Thedistributionofparticlesinthepacketatanygiventimeisafunctionoftheinitialdistribution.Thusthedistributionofparticleswithinthepacketcanbedenedintermsof 69

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theirrelative-positionasfollows (0,t)=xp(t))]TJ /F3 11.955 Tf 11.95 0 Td[(xpL(t) xpR(t))]TJ /F3 11.955 Tf 11.96 0 Td[(xpL(t),(3) suchthat01atalltimes.Similarly,theinitialdistributionofparticleswithinthepacketisgivenby 0=xp(0))]TJ /F3 11.955 Tf 11.96 0 Td[(xpL(0) xpR(0))]TJ /F3 11.955 Tf 11.96 0 Td[(xpL(0).(3) Therefore,=(0,t)givestheparticledistributioninsidethepacketandservesasamappingfromtheinitialtothecurrentparticledistribution. Thederivatived=d0measurestheparticlespacing,i.e.,thedistancebetweentwoneighboringparticleswithinthepacket.Theinverseoftheparticlespacing,i.e.,d0=d,providesameasureofthelocalparticlenumber-density.Theparticlenumber-densitywithinagridcellofunitwidthcenteredaroundcanthenbeobtainedas np()=1 Z+=2)]TJ /F15 7.97 Tf 6.59 0 Td[(=2d0 dd,(3) where=1=(xpR(t))]TJ /F3 11.955 Tf 12.45 0 Td[(xpL(t))istheunitgridspacingnormalizedbythecurrentpacketsizexpR(t))]TJ /F3 11.955 Tf 11.96 0 Td[(xpL(t). Wedeneseveralerrorstomeasurethesmall-scaleuctuationsarisingfromtheinterpolation.Inalldenitions,wecomparethenumericalresults(subscriptnum)withthecorrespondingexactsolution(subscriptex).Fortheexactsolution,theasymptoticparticledistributionremainsuniformasdiscussedabove,andwehavetheresultsex(0,t)=0and(d=d0)ex=1ast!1.However,thewidthofthepacketofparticleschangesandtheratioofnaltoinitialnumber-densityisgivenby1=(1+u2).Theerrordenitionsare: Meanerror:Measurestheerrorinthemeanlocationofthepacket, Emean=1 LxpL,ex+xpR,ex 2)]TJ /F3 11.955 Tf 13.15 8.88 Td[(xpL,num+xpR,num 2.(3) 70

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Spreaderror:Measurestheerrorinthewidthofthepacket, Espr=(xpR,ex)]TJ /F3 11.955 Tf 11.95 0 Td[(xpL,ex))]TJ /F4 11.955 Tf 11.95 0 Td[((xpR,num)]TJ /F3 11.955 Tf 11.96 0 Td[(xpL,num) (xpR,ex)]TJ /F3 11.955 Tf 11.95 0 Td[(xpL,ex).(3) Relative-positionerror:Evaluatestheerroroftherelativeparticlepositionwithinthepacketcomparedtotheexactsolution, Epos()=j)]TJ /F9 11.955 Tf 11.95 0 Td[(0j.(3) Spacingerror:Evaluatestheerrorinadjacentparticlespacing, Espac()=d d0)]TJ /F4 11.955 Tf 11.95 0 Td[(1.(3) Number-densityerror:Evaluatestheerrorinparticlenumberdensitywithineachgridcellarisingfromthenumericalsolution, End()=np())]TJ /F4 11.955 Tf 11.96 0 Td[(1.(3) Notethattherelative-position,spacing,andnumber-densityerrorsaredenedasafunctionofthecurrentrelativeparticleposition.Tomeasuretheerrorfortheentirepacketofparticles,theL2-andL1-normsareused,L2(E)=Z10(E())2d, (3)L1(E)=maxjE()j, (3) whereEstandsforEpos,Espac,orEnd.Intheabovedenitions,theparticledistributioninapacketisassumedtobecontinuous.Therefore,theerrordenitionsareindependentofthenumberofparticleswithinthepacket,andhencetheLagrangiandiscretizationdoesnotplayarole.Whencalculatingtheaboveerrornormsnumerically,nitenumbersofparticlesareused.Thenumbersofcomputationalparticlesusedinallthecalculationsreportedbelowhavebeenchosenlargeenoughtoresolvethecontinuouserrorproleinsidethepacket.Asaresult,anyerrorsduetonitenumbersofparticlesarenegligiblysmallcomparedtotheinterpolationerror. 71

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3.5Results Inthissection,resultsforfourtestproblemsarepresented.ThersttwoproblemsaremodelproblemsofthekinddescribedinSection 3.3 withsinusoidalandhyperbolic-tangentproles,respectively.Theprimarydifferencebetweenthetwoprolesisthatthelatterhasnon-zeronetuidvelocitydivergenceandthereforerepresentsasimplemodelforcompressionandexpansionwaves.ThesecondtwoproblemsarebasedonthefullEulerequationsforthegasphaselistedinSection 3.2 andconsiderproperexpansionandshockwaves. 3.5.1TestProblem1:SinusoidalProle Theuidvelocityistakentobeasinglesinusoidalwavethatmovestotheleftatunitspeedusintoaparticlepacket,asshownschematicallyinFigure 3-1 .Thenon-dimensionaluidvelocityis ug(x,t)=8>><>>:Asinh2(x+t) Niif)]TJ /F3 11.955 Tf 9.3 0 Td[(t)]TJ /F3 11.955 Tf 11.95 0 Td[(N=2x)]TJ /F3 11.955 Tf 21.92 0 Td[(t+N=2,0else,(3) whereNdenotesthenumberofgridpointsacrosstheunitwavelengthofthesinusoidalwaveandA=A=usisthedimensionlessamplitude.NotethatAmustbesmallerthanunitytoavoidparticlesettlingatxedpoints.Nprovidesameasureofhowwellthecarrier-phasevelocityisresolvedbythecomputationalgrid.TheuidvelocitydenedbyEq.( 3 )isonlyC0-continuousbecauseofslopediscontinuitiesattheheadandtailofthewave.Toreduceinterpolationerrorsnearthesediscontinuities,itisassumedthattheheadandtailpositionsareknowntosubgridresolution.Therefore,theuidvelocityisonlyinterpolatedoverthedomain)]TJ /F3 11.955 Tf 9.3 0 Td[(t)]TJ /F3 11.955 Tf 12.44 0 Td[(N=2x)]TJ /F3 11.955 Tf 23.13 0 Td[(t+N=2andtakentobezeroelsewhere.Animportantcharacteristicofthismodelproblemisthattheparticlevelocityreturnstozeroafterthepassageofthedisturbance,i.e.,u2=u2=us=0.Thereforethenetdivergenceofuidvelocityiszero,andthenon-dimensionalpacketwidthreturnstoitsstartingvalueofunity.Theresultspresentedbelowcanbeinterpretedasa 72

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von-Neumann-typeerroranalysisofparticleredistributionwithinthepacketandusedtoevaluatetheaccuracyofinterpolationschemes. Inthismodelproblem,thekeyparametersareN,p,andA.Weonlyconsidertracerparticles,i.e.,p=0.Werstdemonstratetheeffectofinterpolationerrorwithacaseofverypoorspatialresolution,namelyN=4andA=0.5. 3.5.1.1Particleredistribution Thefundamentalsourceofnon-uniformparticleredistributionwithinapacketisduetothemannerinwhichtheinterpolationschemeapproximatesthesinusoidaluid-velocitydisturbanceasthelatterpropagatesthroughthexedEuleriangrid.Figure 3-2 showstheexactvelocityproleanditsapproximationbythefourinterpolationschemes.Innon-dimensionalterms,thewavepropagatestotheleftbyonegridspacinginunittime.Foreaseofinterpretation,eachgureiscenteredaboutthewaveandthegridlocationsaremarkedbyblacklledcircles.Severalobservationscanbemade.First,asexpected,theinterpolationerrordecreasesastheorderofinterpolationincreases,withthelargesterrorobservedforpiecewiseconstantinterpolation.TheerrorisparticularlylargeinFigure 3-2 sinceonlyfourpointsareusedtoresolvethewave.Moresignicantthantheerroritselfishowtheerrorchangesintime.Asthesinusoidaldisturbancemovesthroughthegrid,thecollocatedvaluesatthegridpointsvaryintime.Thenumericalrepresentationofthedisturbanceobtainedfromtheinterpolationchangescorrespondingly.Thischangecanbeobservedreadilyinthedifferentshapesofthelower-orderinterpolants.Althoughnotasreadilyobservable,theapproximationschangeswithtimeevenforthehigher-ordersplineandHermiteinterpolations.Thisvariationintheapproximationtotheuidvelocityrepeatsafterthewavehasmovedbyonegridspacing. Theadverseimpactoftheperiodicvariationintheinterpolateduidvelocityonparticlemotionandredistributioncannowbeaddressed.Whentheleadingedgeofthewavereachesagivenparticle,itstartstomoveduetothenon-zerorelativevelocity. 73

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At=0 Bt=0.25 Ct=0.5 Dt=0.75 Figure3-2. Exactandinterpolateduid-velocityprolesasafunctionoftimeforN=4andA=0.5.Thelledcirclesdenotelocationswithcollocatedvalues. Whenagivenparticleexitsthewave,itdeceleratesbacktozerovelocity.Thetracerparticlesconsideredherestopmovingoncetheyreachthetailofthesinusoidalwave.Thus,eachparticleseesahistoryofuidvelocitythatdictatesitsmotionandnalrestingposition.Duetovariationintheinterpolatedsinusoidalwave,asillustratedinFigure 3-2 ,thehistoryofuidvelocityseenbyagivenparticlewilldependonits 74

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initialpositionrelativetothegrid.Asaresult,aninitiallyuniformdistributionofparticlesbetweentwogridpointswillnotremainuniformlydistributed. Ininterpretingtheresults,itisimportanttodistinguishbetweentwocomponentsoftheinterpolationerror.Therstisthemeaninterpolationerrorthataffectsallparticlesirrespectiveoftheirinitiallocationwithinthepacket.Theeffectofthiserroristhatthenalnumericallypredictedpositionofaparticlewithinthepacketisdifferentfromitsexactposition.Becausethiserroristhesameforallparticleswithinapacket,itdoesnotaffecttheuniformityofparticledistribution.Theseconderror,whichisthatofprimaryconcernhere,isthatcomponentoftheinterpolationerrorthatchangesasthewavepassesthroughthexedgrid.Sincethemeanerrorhasbeentakenintoconsideration,thiserrorcanbethoughtofastheuctuatinginterpolationerror.Theuctuatinginterpolationerroristhesolecauseofthenon-uniformparticledistribution. Figure 3-3A showsthenalparticlepositionanditserrorcomparedtotheexactsolutionasafunctionoftheinitialparticleposition.Fromthisgure,boththemeanandtherelative-positionerroroftheparticlescanbediscerned.Accordingtotheexactsolution,everyparticleshiftstotheleftby0.6188becauseofthepassageofthesinusoidalwave.Bycomparison,theaverageleft-shiftsoftheparticlesinthenumericalsolutionsare0.1910(69.14%),0.3690(40.36%),0.5983(3.31%)and0.6070(1.90%)respectivelyforthepiecewiseconstant,piecewiselinear,spline,andHermiteinterpolations(themean-positionpercentageerrorsaregiveninparentheses).ThevariationinthepositionerrorasafunctionoftheinitialparticlelocationpresentedinFigure 3-3 bisresponsibleforthenon-uniformdistributionofparticles.Notethattheerrorrepeatsforeverypacketofparticlesinitiallylocatedbetweengridpoints.Therefore,wewillfocusonthebehaviorofasinglepacketfortherestofthearticle.Theresultspresentedbelowcanbeextendedtodistributionsstretchingovermultiplegridspacingsbyapplyingperiodicity. 75

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AParticleposition BParticlepositiondeviation Figure3-3. Prolesofparticlepositionanddeviationfromtheexactsolutionatt=20asafunctionofinitialpositionforN=4,p=0,andA=0.5. Figure 3-4A givestherelative-positionofaparticleinsidethepacketasafunctionofitsinitialrelative-position0andthevariousinterpolationmethods.Theexactsolutionissimply=0.Theerrorintherelativeparticlepositiontranslatesintoanerrorinspacingbetweenadjacentparticlesd=d0andhenceintoanerrorinlocalparticlenumber-densityd0=d.Thelocalparticlenumber-densityispresentedinFigure 3-4B forthevariousinterpolationschemes.Ascanbeexpected,theerrorsassociatedwithpiecewise-constantinterpolationarelarge.Forinstance,inFigure 3-4A ,itcanbeseenthattherelative-positionremainsnearlyconstantat0.92for0.500.95.Inotherwords,particlesthatwereoriginallydistributedovernearlyhalfthepacketareconcentratedinanarrowregionaround0.92afterthepassageofthewave.Thecorrespondingsharpincreaseinlocalnumber-densitycanbeobservedinFigure 3-4B .Therelative-positionerrorforthepiecewiselinearinterpolationisnotnearlyaslarge.Nevertheless,itcanbeseeninFigure 3-4B thatthelocalnumber-densityvariesfromabout0.7to1.5.ThecorrespondingerrorsforthesplineandHermiteinterpolationaresmall.Figure 3-5 showstherelative-positionandspacingerrorsofparticlesinsidethe 76

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ARelative-position BLocalnumber-density Figure3-4. Prolesofparticlerelative-positionandlocalnumber-densityinsidetheparticlepacketwithinitialsizeunityafterthepassageofthesinusoidalwaveforN=4,p=0,andA=0.5. packetforthedifferentinterpolationschemes.Notethatthescalingsoftheordinatesaredifferent.ItisobservedthattheerrorofthepiecewiselinearschemeisaboutanorderofmagnitudelowerthanthepiecewiseconstantinterpolationandthecorrespondingerrorsofthenaturalcubicsplineandHermiteinterpolationmethodsarethreeordersofmagnitudelower. 3.5.1.2Globalerrornorms TheglobalerrorsintegratedacrosstheentirepacketasdenedbyEqs.( 3 )and( 3 )arepresentedinFigure 3-6 .WeobservethattherateofdecayoftheL2-normoftherelative-positionandspacingerrorsisconsistentingeneralwiththeorderofaccuracyoftheinterpolation.Theonlyexceptionistheresultforpiecewiseconstantinterpolation,forwhichtherateofdecayseemssmallerthanO(1).Thelowererrorandthefasterdecaymakeshigher-orderinterpolationschemesattractive.Foradesiredleveloferror,higher-orderschemestypicallyrequiremuchlowergridresolution.Forexample,ifitisdesiredthatL1(Espac)10)]TJ /F5 7.97 Tf 6.59 0 Td[(4,Figure 3-6B indicatesthatmorethan100 77

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APiecewiseconstant BPiecewiselinear CNaturalcubicspline DHermite Figure3-5. Prolesofparticlerelative-positionandspacingerrors(EposandEspac)insidetheparticlepacketwithinitialsizeofoneafterthepassageofthesinusoidalwaveforN=4,p=0,andA=0.5. 78

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AL2(Epos) BL2(Espac) CEmean DEspr Figure3-6. Evolutionsoftherelative-position,spacing,mean,andspreaderrors(Epos,Espac,Emean,Esp)asfunctionsofNforp=0andA=0.5. pointsperwavelengthareneededforpiecewiselinearinterpolation,butabout10pointsaresufcientforcubicsplineorHermiteinterpolation.ThebehaviorofthemeanerrorpresentedinFigure 3-6C issimilartothatofthespacingerror. Thespreaderror,showninFigure 3-6D ,isgenerallymuchsmallerthantheothererrorsforallinterpolationschemes,suggestingthatthenon-dimensionalpacketwidthremainsclosetounitydespitetheinterpolationerror.Thisresulthasimportant 79

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ramications.Becausethepacketsizedoesnotchangeappreciably,theuctuationoftheparticlepositionandthespacingerrorwithinthepacketremaincontainedwithinagridcell.Theparticlenumber-densityorconcentrationofparticlesmeasuredwithineachgridcellshownovariationandthustheerrorsaretrulysubgrid.Inproblemswherethenon-dimensionalwidthofthepacketeitherincreasesordecreasesinresponsetoadisturbancewithu26=0,thesub-packetuctuationwillmanifestitselfasconcentrationuctuationonthescaleofthegridspacing.Thesecondmodelproblemisdesignedtoillustratethiseffect. 3.5.2TestProblem2:Hyperbolic-TangentProle Inthissection,weexploretheinterpolationerrorwithamodelproblemforwhichu26=0.Asaresult,theuidvelocityhasanon-zeronetdivergence.Asaresultparticlespacingafterthepassageofthedisturbanceisdifferentfromitsinitialvalue.Theobjectiveistoshowthatbecauseofthisnon-zeroparticlevelocitydivergencethesub-packetuctuationofrelative-positionerrorwillresultinsubstantialgrid-to-griductuationinthenumber-density.Non-zerodivergenceoftheuidvelocityiscommonincompressibleows.Forexample,expansionandcompressionwaveshavepositiveandnegativedivergence,respectively.Therefore,whenanexpansionfanorashockwavepassthrougharegionofuniformlydistributedparticles,signicantconcentrationuctuationcanbeexpectedbasedontheresultsoftherstmodelproblem. Herewetaketheuidvelocityproletobeahyperbolictangentfunction.Withthesamelengthandvelocityscalesasinthepreviousproblem,thedimensionlessuidvelocitycanbewrittenas ug(x,t)=u2 21+tanh2(x+t) N,(3) whereu2denotesthevelocityjumpandNrepresentsthenumberofgridpointsacrossthethicknessoftheprole.Notethattheuidvelocityhasasmoothbutsharptransitionaroundx+t=0andthatug(x!)=0andug(x!+1)=u2,i.e.,theuid 80

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Table3-1. Summaryofcasesfortestproblem2. Casepu2 2-101.02-20)]TJ /F4 11.955 Tf 9.29 0 Td[(0.52-30)]TJ /F4 11.955 Tf 9.29 0 Td[(0.42-400.12-5010.02-611.02-7101.0 velocityonbothsidesisconstantfarawayfromthejump.Whenu2>0,Eq.( 3 )canbeusedtomodelanexpansionfanwithconstant(orfrozen)thickness.Conversely,whenu2<0,Eq.( 3 )modelsashockwave.Ofcourse,itisphysicallyinconsistenttohaveanexpansionfanofconstantthickness,sincethewidthofarealexpansionfanincreasesintime.Nevertheless,thismodelproblemisusefulinprovidinginsightintohowanexpansionfancancontributetoconcentrationuctuations.Theconstantthicknessoftheexpansionfansimpliestheinterpretationoftheresults.Ashockwaveofconstantthicknesspropagatingthroughadistributionofparticlesisrealizable,however,becauseoftheopposingmechanismsofsmoothingthroughdiffusionandsteepeningthroughnon-linearity. 3.5.2.1Particleredistribution Whenthewavereachesaparticle,thelatterstartstomovetotherightorleftdependingonthesignofu2.Aftersufcientlylongtime,thehyperbolic-tangentwavehaspassedcompletelyandtheparticlereachestheterminalvelocityofu2.Aftertheparticlepacketreachesthesteadystate,itcanbeshownthatthesizeofparticlepacketchangesto1+u2.Therefore,thepacketexpandsorcontractsdependingonwhetheru2ispositiveornegative.However,whentheinterpolateduidvelocityisused,theparticlesareredistributedinanon-uniformmannerasforthesinusoidal-waveproblem.ThekeyparametersforthisproblemareN,p,andu2.Sevencasesarepresentedforthevaluesofpandu2showninTable 3-1 81

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Figure 3-7 showstherelative-positionandlocalnumber-densityd0=dvariationinsideapacketofparticlesforcases2-1and2-2.HereN=4,i.e.,thethicknessofthehyperbolic-tangentproleisresolvedwithonlyfourgridpoints.Similartothepreviousproblem,forbothexpansionandcompression,theinterpolationerrorredistributestheparticlesandgenerateserrorsintherelativeparticlepositionandhenceinthenumber-density.Figure 3-8 presentstherelativeparticle-positionandspacingerrorsforcases2-1and2-2.Againthepiecewise-constantinterpolationyieldsthelargesterror,theerrorofthepiecewise-linearinterpolationisanorderofmagnitudelower,andtheerrorsofthenaturalcubicsplineandHermiteinterpolationsaretwoordersofmagnitudelowerstill. Incase2-1,1+u2=2,thereforetheparticlesthatwereinitiallylocatedinonegridcellwillbedistributedovertwocellsafterthepassageofthewave.Thenthesub-packetvariationofthelocalnumber-densitygeneratesgrid-to-gridvariationinparticlenumber-densitywithawavelengthoftwogridcells.AschematicofthissituationisshowninFigure 3-9A todemonstratehowthegrid-to-gridnumber-densityuctuationisgenerated.Incase2-2,1+u2=0.5,hencetheparticlesthatwereinitiallyinonecellwillbedistributedoverhalfagridcell.Sodespitethesub-packetuctuationduetotheinterpolationerror,therewillbenogrid-to-gridnumber-densityuctuation,becausetheuctuationswillbecontainedatthesubgridlevelafterthecompressionoftheparticles(seeFigure 3-9B ).Ontheotherhand,incase2-3,where1+u2=0.6,thepacketofparticlesthatwereinitiallydistributedoveronegridcellgetcompressedto3=5ofacell,resultinginagrid-to-griductuationinparticlenumber-densitythatisperiodicoverthreegridcellsasshowninFigure 3-9C .Therefore,wecanstatethefollowingrule:If1+u2isarationalnumberp=q,thesub-packetvariationinparticledistributionafterthepassageoftheuid-velocitydisturbancewillresultinagrid-to-gridnumber-densityvariationthatisperiodicoverpgridcells.If1+u2isirrational,thegrid-to-gridnumber-densityvariationwillhavenoperiodicstructure. 82

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ACase2-1:Relative-position BCase2-1:Localnumber-density CCase2-2:Relative-position DCase2-2:Localnumber-density Figure3-7. Prolesofrelativeparticlepositionandlocalnumber-densityoftheparticlepacketwithinitialsizeunityforcases2-1and2-2withN=4. 83

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APiecewiseconstant BPiecewiselinear CNaturalcubicspline DHermite Figure3-8. Prolesofparticlerelative-positionerrorandspacingerror(EposandEspac)insidetheparticlepacketwithinitialsizeunityforcases2-1and2-2withN=4. 84

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Acase2-1:1+u2=2 Bcase2-2:1+u2=0.5 Ccase2-3:1+u2=0.6 Figure3-9. Schematicsforthegenerationofgrid-to-gridnumber-densityuctuationfromsubgridnumber-densityvariationforcases2-1,2-2,and2-3. 85

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3.5.2.2Globalerrornorm Figure 3-10 showstherelative-position,spacing,andmeanerrorsforthedifferentinterpolationschemesforcases2-1and2-3.Ingeneral,theerrorsdecreasewithincreasingNandtherateofdecayisconsistentwiththeorderofaccuracyoftheinterpolationscheme.Theonlyexceptionisthemeanerror,whereN)]TJ /F5 7.97 Tf 6.59 0 Td[(2behaviorisobservedforthepiecewiseconstantinterpolation.However,recallthatthemeanerrorhasnoinuenceonthenumber-densityuctuation.ExceptforsmallN,theerrorsforcases2-1and2-3casesarenearlyidentical.Unlikefortestproblem1,herewecancomputethenumber-densityerror.Figure 3-11 showsthattheerrorsforCase2-1(theexpansionwave,u2=1)areconsistentlylargerthanthoseforCase2-3(theshockwaveu2=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.4). ItcanbeobservedthattheerrorofHermiteinterpolationissmallerthanthatofthenaturalcubicsplineinterpolationwhenNissmall.Notethatthenaturalcubicsplineinterpolationbehavesevenworsethanthepiecewiselinearinterpolation.WhenNiscloseto1,thediscretizedhyperbolic-tangentproleappearsasadiscontinuity.Asiswellknown,higher-orderinterpolationschemesgeneratespuriousoscillationsintheinterpolantneardiscontinuities.Thesespuriousoscillationsalsovaryintimeandcontributetotheredistributionofparticles.Comparedtothenaturalcubicspline,Hermiteinterpolationtakesadvantageoftheexactrstderivativesinconstructingtheinterpolant,andhenceperformsmuchbetterforsmallN.Furthermore,Hermiteinterpolationbetterlocalizestheinuenceofthediscontinuity,whereasthecubicsplinetendstospreadtheinterpolationerrortoalargerdomain. 3.5.2.3Effectofdisturbancemagnitude Thevelocityjumpordisturbancemagnitudeu2acrossthehyperbolictangentisanimportantparameter.ForgivenvaluesofN,especiallysmallones,thelargeru2,thelargerthevelocitygradient,andhenceinterpolationismorechallenging.Figure 3-12 showsthevariationoftheerrormeasuresasafunctionofNforthedisturbance 86

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AL2(Epos) BL2(Espac) CEmean Figure3-10. Behaviorofrelative-position,spacing,andmeanerrors(Epos,Espac,Emean)asfunctionsofNforcases2-1and2-3.Linesdenotevaluesofu2(solidlines:u2=1;dashedlines:u2=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.4).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). 87

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AL2(End) BL1(End) Figure3-11. Behaviorofnumber-densityerror(End)asafunctionofNforcases2-1and2-3.Linesdenotevaluesofu2(solidlines:u2=1;dashedlines:u2=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.4).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). magnitudesu2=f0.1,1,10g.AlltheerrorspresentedinFigure 3-12 decaywithincreasingNattheasymptoticratedictatedbytherespectiveordersoftheinterpolationschemes.Fortherelative-positionandspacingerrors,theeffectofu2issignicantonlyatsmallvaluesofN,i.e.,whenthehyperbolictangentproleisnotwellresolved.AtlargevaluesofN,theerrorsbecomeindependentofu2.Thenumber-densityerror,however,seemstoincreasewithincreasingu2.Forsmallvaluesofu2,e.g.,u2=0.1,theerrorsaretypicallysmallerforsmallvaluesNandtheapproachtotheasymptoticpower-lawbehavioroccursonlyforN'30.Thesuperiorperformanceofhigher-orderinterpolationisclear.Forinstance,ifthegrid-to-gridnumber-densityuctuationneedstobecontainedbelow10)]TJ /F5 7.97 Tf 6.59 0 Td[(4,lessthan10gridpointsarerequiredtoresolvethehyperbolic-tangentprolewiththetwohigher-orderschemes,while70to100gridpointsarerequiredwithpiecewise-linearinterpolation. 88

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Thenaturalcubic-splineinterpolationdoesnotworkwellforlargeu2andsmallN.ItcanbeseenfromFigure 3-12C ,thatforu2=10andsmallN,naturalcubic-splineinterpolationintroducesanerrorcomparabletoorevenlargerthanthatforpiecewise-linearinterpolation.Theoscillationsintheinterpolantadverselyaffecttheparticleredistribution,causinglargeuctuationsintheparticleposition. Hermiteinterpolationseemstobethebestinterpolationschemeforthecurrentproblemamongthemethodstested.ItbehaveswellevenfortheextremecaseofsmallNandlargeu2.However,itshouldbekeptinmindthatinthepresentimplementation,theanalyticalrstderivativeisusedbytakingadvantageoftheknownuid-velocityprole.Inarealcomputation,therstderivativeisnotknownandmustbecalculatedfromthefunctionvalues.Furthermore,theproblemobservedwiththecubic-splineinterpolationcanalsoaffecttheHermiteinterpolation.[ 33 ]developedamethodofcomputingtherstderivativethatcanguaranteemonotonicinterpolants.Withthismethod,numericaloscillationsintheinterpolantupstreamanddownstreamofthewavecanbeeliminated,guaranteeingthatHermiteinterpolationworkswellevenforpoorresolution. 3.5.2.4Effectofparticleinertia Wenowconsiderinertialparticles.TheequationofmotionfortheparticlesonlyincludestheStokes-dragtermandthusthedimensionlessparticleresponsetimepistheonlyparameterrepresentingtheparticle-inertiaeffect.Wechoosecases2-6and2-7(p=f1,10g),andcomparethemwithcase2-5(p=0). Theresultsfortheposition,spacing,andnumber-densityerrorsintegratedacrossthepacketareshowninFigure 3-13 asafunctionofNandp.ForlargeN,theerrorsseemtobethelowestforthetracerparticlesandincreasewithincreasingparticleinertia.ForverylargeN,theinuenceofpissmallerthanthatoftheinterpolationscheme. 89

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AL2(Epos) BL2(Espac) CL2(End) Figure3-12. Behaviorofrelative-position,spacing,andnumber-densityerrors(Epos,Espac,End)asfunctionsofNandu2forcases2-1,2-4,and2-5.Linesdenotevaluesofu2(solidlines:u2=0.1;dashedlines:u2=1;dashed-dotlines:u2=10).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). 90

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TheerrorbehaviorreversesforsmallNandasaresultthedeviationoftheinertialparticlesfromthetracerparticlesisnotmonotonic.ItcanbeobservedclearlythattheerrorfortheinertialparticlescrossesthatfortracerparticlesatacertainvalueofN=^N.WhenN<^N,theerrorsforinertialparticlesaresmallerthanfortracerparticles.Conversely,whenN>^N,theoppositetrendisobserved.Itisalsointerestingtonotethat^Ndecreasesastheorderofinterpolationschemegoesdown.Infact,forpiecewise-constantinterpolationtheerrorforinertialparticlesisalwayslargerthanfortracerparticles(i.e.,^Ndoesnotexist).Itseemsforthehigher-orderinterpolationschemestheinertialeffectofparticleshelpsreducetheparticlenumber-densityuctuationwhenthegridresolutionispoor. 3.5.3TestProblem3:ExpansionWave Wenowextendtheanalysisfromthemodelproblemswithimposeduid-velocityprolestoproblemsinwhichtheuidvelocityisobtainedfromnumericalsolutionsoftheEulerequations.First,weconsiderthepropagationofaone-dimensionalexpansionfanintoaparticle-gasmixture.IncontrasttothefrozenexpansionwaveconsideredinSection 3.5.2 ,heretheexpansion-wavewidthincreaseswithtime.Oneparticularfocusofthefollowingistoestablishthedegreetowhichtheresultsobtainedforthemodelproblemapplytotherealproblemconsideredhere. Theinteractionofparticleswithexpansionwavesisoffundamentalimportanceandhasbeeninvestigatedby[ 75 ]and[ 52 ].WeinitializetheowasaRiemannproblemwiththeleftandrightstates(denotedbysubscripts1and2,respectively)chosensuchthatanexpansionfanwithapressureratioofpg1=pg2=3.5andatemperatureratioofTg1=Tg2=1.4results.Theexpansionregionisinitiallyofzerothickness,i.e.,thepressure,density,andvelocityexhibitdiscontinuitiesatx=0.Theheadoftheexpansionfanpropagatestotheleftatthespeedofsoundag1.Thegasvelocityincreasesacrosstheexpansionfanandreachesaconstantvalueug2atthetrailing 91

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AL2(Epos) BL2(Espac) CL2(End) Figure3-13. Evolutionofrelative-position,spacing,andnumber-densityerrors(Epos,Espac,End)asfunctionsofNforcases2-1,2-6,and2-7.Linesdenotevaluesofp(solidlines:p=0;dashedlines:p=1;dashed-dotlines:p=10).Symbolsdenoteinterpolationschemes(lledsquare:piecewiseconstant;unlleddelta:piecewiselinear;unlleddiamond:naturalcubicspline;lledgradient:Hermite). 92

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edge,where ug2 a1=2 )]TJ /F4 11.955 Tf 11.96 0 Td[(1"1)]TJ /F16 11.955 Tf 11.96 16.86 Td[(pg2 pg1)]TJ /F17 5.978 Tf 5.76 0 Td[(1 2#.(3) Withtheabove-statedconditions,weobtainug2=a1=0.82.Thevelocityofthetailoftheexpansionfanisgivenbyug2)]TJ /F3 11.955 Tf 12.81 0 Td[(a2,wherethespeedofsounda2followsfromthetemperatureratioandhence(ug2)]TJ /F3 11.955 Tf 13.21 0 Td[(a2)=a1=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.02.Therefore,theowoutoftheexpansionfanisnearlysonic.Thewidthoftheexpansionfanincreasesas(ug2)]TJ /F3 11.955 Tf 12.26 0 Td[(a2+a1)t.Here,wechoosea1tobethevelocityscaleandthegridspacingxtobethelengthscale.Thus,theheadoftheexpansionfantravelsthroughonegridcellineachunittimeinterval.Startingfromzerothickness,theexpansionfanexpandsbyug2=a1)]TJ /F3 11.955 Tf 12.66 0 Td[(a2=a1+1=0.98gridcellsineachunittimeinterval.Therefore,thespatialresolutionoftheexpansionfanimproveswithtime. Theparticlesareassumedtobeuniformlydistributedinx<0att=0.Theparticlesandthegasareinitiallystationaryandareacceleratedbytheexpansionfan.Aftertheparticlestravelthroughtheexpansionfanandexittotherightofthetailendofthefan,theyreachthepost-expansionnon-dimensionalvelocityofu2=a1.Likeinthehyperbolictangentproblem,theparticlesstartfromzerovelocityandreachanalsteadystateagain.Hereweconsiderbothtracerparticles(p=0)andinertialparticlesdenedintermsoftheirnon-dimensionaltimeconstant(p=3.89).Asdiscussedabove,theapproachtosteadystateisasymptoticforinertialparticles.Asaresult,arelaxationzoneexistsbehindtheexpansionfan. Intheresultspresentedbelow,weusedauniformdistributionof5000particlespercell(nppc=5000)asaninitialconditionintheregiontotheleftoftheexpansionwave.Thislargenumberofparticlespercellprovidesasmoothrepresentationofthecontinuousparticlenumber-densityeld.Increasingthedensityofparticlesfurtherdidnotchangetheresultspresentedbelow.Althoughananalyticalsolutionfortheparticleconcentrationeldcanbecomputedusingthesolutionforthevelocityeldofacenteredexpansionfan,hereweusedthenumericalresultsobtainedwithHermiteinterpolation 93

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todenethebaselinesolution(ornear-exactsolution)fromwhichtheerrorscanbecomputed.Thereasonforthisapproximateapproachisasfollows. Atveryearlytimes,duetothepoorresolutionoftheexpansionwavethenumericalsolutionforthegasvelocitywithintheexpansionfandeviatesfromtheanalyticalsolution.Whiletheanalyticsolutionpresentsdiscontinuitiesinthevelocitygradientatboththeheadandtailoftheexpansionfan,thesediscontinuitiesaresmoothedoutinthenumericalsolution.Thesedifferencesintroducedeviationsinthecomputedparticlepositionfromitsexactposition.Butthesesystematicdifferencesintheparticlepositiondonotcontributetogrid-to-gridparticlenumber-densityuctuationandthereforearenotthefocusofthecurrentdiscussion. Toisolateinterpolation-induceductuationsintheparticlenumber-density,hereweextractthebaselinesolutionfromtheHermite-interpolationsolution.Thissolutionistakentobeasuperpositionofthebaselinesolutionplusauctuatingcomponent.Aeighth-ordercompactspatiallterisusedtoextractthebaselineportion,whichthencanbesubtractedfromthedifferentinterpolatedresultstoobtaintherespectiveerrors.Hereitshouldbepointedoutthatunlikeinthepreviousexamples,thecarrier-phasevelocityiscomputednumericallyatthegridpointsandthereforethederivativesatthegridpointsforHermiteinterpolationmustbeevaluatednumerically.Tomaintainthehigherorderaccuracyofinterpolationweemployafth-ordercompactschemetocomputetherstderivativesatthegridpoints. ThenumberofparticlespercellnppcthusobtainedfromthelteredHermitesolutionforthepresentcaseisshowninFigure 3-14A .Thenon-dimensionaltimeist=136.1andaccordinglytheexpansionfanatthistimespreadsover134.1gridpoints.Theresultsforboththetracerandinertialparticlesarequitesimilar.Itcanbeseenthatnppcdecreasesfromitsinitialvalueof5000percelltoaround2000percelltotherightofthefan. 94

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ANumberofparticlespercellprole BParticleentrylocationandthicknessofexpansionfan Figure3-14. Numberofparticlespercellnppcfromaone-way-coupledsimulationwithlteredHermiteinterpolation,andtheoreticallocationatwhichparticlesentertheexpansionfanandexpansion-fanthicknesswhileenteringasfunctionsoftheparticlelocationatt=136.1. Figure 3-15 showsthenumber-densityerroratt=136.1.Atthistime,theheadandtailoftheexpansionfanarelocatedatx=)]TJ /F4 11.955 Tf 9.3 0 Td[(136.1and)]TJ /F4 11.955 Tf 9.3 0 Td[(1.98,respectively.Also,thelocationofthetracerparticlethatinitiallywasattheoriginisnowlocatedatx=110.69;thisparticlerepresentstheparticlecontactsurface.Notethatparticlesthatareimmediatelybehindtheparticlecontactpassedthroughtheexpansionfanwhenitwasverynarrowandresolvedbyveryfewgridpoints.Asaresult,theinterpolationerrorfortheseparticlesisexpectedtobequitelarge.InFigure 3-15A thetracerparticlesbetweenthecontactandthetailoftheexpansionfan(i.e.,for)]TJ /F4 11.955 Tf 9.3 0 Td[(1.98x110.69)havepassedthroughtheexpansionfan.Particleslocatedbetween)]TJ /F4 11.955 Tf 9.3 0 Td[(136.1x)]TJ /F4 11.955 Tf 22.2 0 Td[(1.98arestillwithintheexpansionfan,andparticlestotheleftoftheheadofthefanremainunaffected.Figure 3-14B presentsthetheoreticallocationxhatwhichaparticleenterstheexpansionfanandthethicknesshoftheexpansionfanwhentheparticleentersasafunctionofaparticle'slocationatt=136.1. 95

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Withthecurrentnon-dimensionalization,thethicknessofthefanisequaltothenumberofgridpointsusedtoresolveit.Itisexpectedthattheexpansionfanseenbyaparticleisthickerwhentheparticleexitsthanwhenitenters.However,thelargestnumber-densityuctuationisrelatedtothesmallestthicknessoftheexpansionfanwhichaparticleencounters.Therefore,onecouldchoosethethicknessoftheexpansionfanwhenaparticleentersastheexpansionfanthicknessseenbytheparticle.Basedonthisthickness,theresultsforthehyperbolic-tangentprolepresentedinFigure 3-10 canbeusedtoestimatethenumber-densityerrorsexpectedforthepresentproblemofanunsteadyexpansionfanatt=136.1.TheestimatesforthedifferentinterpolationmethodsareplottedassymbolsinFigure 3-15 .Therapiddecreaseinthenumber-densityerrorupstreamoftheparticlecontactobservedinthenumericalresultsisinexcellentagreementwiththeestimatesbasedonthefrozendisturbance.Thenumber-densityerrorforthepiecewiseconstantinterpolationisthelargestanddecaysfromaboutO(1)closetotheparticlecontacttoaboutO(10)]TJ /F5 7.97 Tf 6.58 0 Td[(2)intheexpansionfan.Thenumber-densityerrorsforthepiecewiselinearandHermiteinterpolationsareordersofmagnitudelower.Substantiallyhighererrorsareobservedclosetotheparticlecontactforallinterpolationmethods.Theresultsforinertialparticlesarequitesimilartothosefortracerparticles.Thusparticleinertiadoesnotappeartogreatlyinuenceinterpolationerrors. Theaboveresultswereobtainedwithone-waycoupling,wherethegasowisunaffectedbytheparticles.Toassesstheinuenceoftwo-waycouplingonnumber-densityerrors,weconsidertwocaseswithparticlemassfractionsofYp=f0.1,0.5g.Numericalsimulationswereusedtocomputenumber-densityerrorsinthesamewayasdescribedabove.TheresultsareshowninFigure 3-16 .Itisobservedthatthebehaviorofthenumber-densityerrorwithtwo-waycouplingisqualitativelysimilartothatforone-waycoupling. 96

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Ap=0 Bp=3.89 Figure3-15. Behaviorofparticlenumber-densityerrorEndatt=136.1foraone-waycoupledsimulationofanexpansionfanpropagatingintouniformlydistributedparticles.Solidlineswithsymbolsdenoteerrorestimatesforvariousinterpolationschemesbasedontheanalysisofthemodelproblemwiththehyperbolic-tangentprole(square:piecewiseconstant;delta:piecewiselinear;diamond:Hermite).Lineswithoutsymbolsdenotethenumericalresultsforvariousinterpolationschemes(dashedline:piecewiseconstant;dashed-dotline:piecewiselinear;longdashedlines:Hermite). AfurtherobservationcanbemadebycomparingFigs. 3-15 and 3-16 .Itisseenthatthenumber-densityuctuationwavelengthchangeswiththemassfraction.Whenthemassfractionincreases,thevelocityjumpacrosstheexpansionfandecreases,andthereforethenumber-densityratioacrosstheexpansionfanchangesalso.AsshowninSection 3.5.2 andFigure 3-9 ,theuctuationwavelengthdependsonthenumeratorpifthenumber-densityratiocanbewrittenasp=q.Forthepresentproblem,asthemassloadingincreases,thenumber-densityratio1=(1+ug2)decreases,andthenumber-densityuctuationwavelengthalsodecreases. 3.5.4TestProblem4:ShockWave Thesecondcasewithauidvelocitythatiscomputedratherthanimposedinvolvesaplanarshockwavepropagatingtotherightintoadilutedistributionofparticles.TheinitialconditioncorrespondstoaRiemannproblemleadingtoashockwavewitha 97

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AYp=0.1 BYp=0.5 Figure3-16. Behaviorofparticlenumber-densityerrorEndatt=136.1foratwo-waycoupledsimulationofanexpansionfanpropagatingintouniformlydistributedparticlesasafunctionofthemassfractionYp. pressureratioofpg2=pg1=2.08andatemperatureratioofTg2=Tg1=1.25.Theparticlestotherightoftheshockareinitiallystationary.Thevelocityoftracerparticlesequalsthegasvelocityug2behindtheshock.Forinertialparticles,arelaxationregionexistsbehindtheshockwheretheparticlevelocitychangesfromthefrozenvaluetothenalpost-shockgasvelocity,see,e.g.,[ 75 ]. Theshockvelocitycanbeobtainedfromtheshock-waveMachnumber, Ms=us a1=+1 2p2 p1+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 21=2,(3) wherea1isthespeedofsoundaheadoftheshock.Wetakeustobethevelocityscaleandthegridspacingxtobethelengthscale.Thenon-dimensionalpost-shockgasvelocityisthengivenby ug2=ug2 us=2 +11)]TJ /F4 11.955 Tf 18.22 8.08 Td[(1 M2s.(3) Fortheconditionsconsideredhere,thenon-dimensionalpost-shockgasvelocityisug2=0.4.Asinthepreviousproblem,theparticlesareinitiallyuniformlydistributedin 98

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theundisturbedregionwithnppc=5000.Afterthepassageoftheshock,thenumberofparticlepercellincreasesto5000=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ug2)=8333. Thenumber-densityuctuationamplitudeisexpectedtobedirectlyrelatedtothenumberofgridpointsusedtorepresenttheshock.Fromtheanalysisofthemodelproblemwiththehyperbolic-tangentprole,itisanticipatedthatthenumber-densityerrorsincurredbythevariousinterpolationschemesaresimilarwhenthegridresolutionisverypoor(seeFigure 3-11 ).Therefore,weintroduceextraviscosityinthepresentproblemtoarticiallyincreasetheshockthickness.Thejusticationfordoingsoisdiscussedbelow.AsshowninFigure 3-17A ,theshockthicknessisaround10cellsifmeasuredbyasimilarapproachusedforthemeasuringthehyperbolic-tangentprole. Similartotheproblemoftheexpansionfan,thelteredHermiteinterpolationresultsareusedtodeneanexactsolution.TheprolesofthenumberofparticlespercellforthelteredHermitesolutionforbothtracerparticlesandinertialparticles(p=6.25)atthenon-dimensionaltimet=130areshowninFigure 3-17B .Figure 3-18 showsthenumber-densityerrorplottedasafunctionofspatiallocation.Withthecurrentnon-dimensionalization,theshocktravelsthroughonegridcellperunittimeinterval.Theparticlecontactsurfaceislocatedatx=51.6forthetracerparticlesandx=47.9fortheinertialparticles.Oncetheparticlescompletelyexittheshockwave,thenumber-densityuctuationismaintainedatthesamelevel(seetheregion50x125inFigure 3-18A andtheregion50x90inFigure 3-18B ),anddonotdecayawayfromtheparticlecontactasintheexpansionfan.Wheretheparticleshavecompletelycrossedtherelaxationzone,thenumber-densityuctuationwavelengthisexactlythreecells.Becausethenumber-densityratioacrosstheshockis1)]TJ /F3 11.955 Tf 12.84 0 Td[(u2=3=5,andthenumber-densityuctuationwavelengthisdeterminedbythenumerator,thecomputednumber-densityerroructuationproleagreeswiththetheory. Thenumber-densityerrorforboth5000initialparticlespergridcelland1000initialparticlespercellareshown.Theresultisclearlyinvariantanddoesnotexhibitone 99

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oversquarerootdecay.Thisestablishestheindependenceoftheinterpolation-inducednumber-densityuctuationfromthemeannumber-density.Basedontheestimatedshockthicknessof10,theresultsofthehyperbolic-tangentprolepresentedinFigure 3-10 canbeusedtoestimatethenumber-densityerrorsforthepresentproblem.ThisestimateisalsoshowninFigure 3-18 andtheagreementisquitegood.Therefore,itisseenthatevenfordifferentgridresolution,thesimpleanalysisofthehyperbolic-tangentprolecanbeusedtoestimatethenumber-densityuctuationamplitudeencounteredinrealcomputations. Theadvantageofhigher-orderinterpolationschemesisagainnoticeableinFigure 3-18 .Thenumber-densityerrorforthepiecewiseconstantinterpolationisthelargestataboutO(10)]TJ /F5 7.97 Tf 6.59 0 Td[(2).LinearinterpolationgiveserrorsmorethanoneorderofmagnitudelowerataboutO(10)]TJ /F5 7.97 Tf 6.59 0 Td[(3),whileHermiteinterpolationyieldsthesmallesterrorataboutO(10)]TJ /F5 7.97 Tf 6.58 0 Td[(5).Iftheresolutionusedtoresolvetheshockismuchsmaller,forexampleifthedimensionlessshockthicknessistwo,theerrorsfromtheinterpolationschemesareofapproximatelythesameorderofmagnitude.Thisisnotsurprising,asitiswell-knownthathigher-ordermethodscanperformworsethanlower-ordermethodswhenthegridresolutionispoor.Incomplexthree-dimensionalsimulations,itisunlikelythatonecanaffordtoresolveashockwavewithtenormoregridpoints.Thisislikelytoposeproblemsinaone-waycoupledsimulation,astheshockgenerallytendstoremainthin.However,iftheparticlemassfractionisnotsmall,sothattwo-way-coupledsimulationsarerequired,anite-sizerelaxationzoneexistsbehindtheshockwave.Therelaxationzone,togetherwiththeactualshockwave,canbeviewedasathickenedmultiphaseshock.Usingtenormorepointstoresolvetherelaxationzoneisabsolutelypractical. Finally,itshouldberemindedthattheresultsshowninsection 3.5.3 and 3.5.4 areindependentofthenumberofparticlesusedinthecomputation,namelythenumber-densityuctuationobservedherewillnotdecreaseifoneincreasesthenumberofparticles.ThiscanbeconrmedbyFigure 3-18A ,whereresultswithdifferentparticle 100

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AVelocity BNumberofparticlespercell Figure3-17. Velocityandnumberofparticlepercellprolesasfunctionsofspatiallocationattimet=130.Theresultsshownhereisfromthelteredone-waycoupledHermitesolution. numberareshown.Thelineswithoutsymbolsdenotethecasesinwhichtheinitialnumberofparticlepercellis5000;whilethelineswithunlledcirclesymbolsdenotethecasewithhalfnumberofparticles.Itcanbeseenthatthesethesetwosetsoflinesalmostperfectlyoverlapwitheachother.Italsoshowsthatthenumberofparticlesusedhereislargeenoughtoscreenouttheinuenceofcomputationalparticlenumber.Thisalsomeansthat,theothertwocommonlyknownsourcesofnumber-densityuctuationhavebeenperfectlyexcluded. 3.6Conclusions Thepresentstudyexploresasourceofparticlenumber-density,orconcentrationuctuationsinEulerian-Lagrangiansimulationsofmultiphaseows.Theconventionalunderstandingofnumber-densityuctuationsisthattheyaregeneratedbyLagrangiandiscretization(i.e.,becauseofalimitednumberofparticles).Therefore,increasingthenumberofparticlesisusuallyviewedassufcienttolimittheuctuationamplitudes.Wehavefoundthatanothersourceofparticlenumber-densityuctuationexiststhatisnot 101

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Ap=0 Bp=6.25 Figure3-18. Behaviorofnumber-densityerrorEndatt=130foraone-waycoupledsimulationofashockwavepropagatingintouniformlydistributedparticles.Solidlineswithlledsymbolsdenoteerrorestimatesforvariousinterpolationschemesbasedontheanalysisofthemodelproblemwiththehyperbolic-tangentprole(square:piecewiseconstant;delta:piecewiselinear;diamond:Hermite).Lineswithoutsymbolsandwithunlledcirclesymbolsdenotenumericalresults.Thelineswithoutsymbolsdenotetheresultswith5000initialparticlespercell;whilethosewithunlledcirclesymbolsarefor1000initialparticlespercell.Thelinepatternsareusedtodistinguishdifferentinterpolationschemes(dashedline:piecewiseconstant;dashed-dotline:piecewiselinear;longdashedlines:Hermiteinterpolation). relatedtoLagrangiandiscretization.Thiserrorsourceisfundamentallydifferentbecausetheuctuationscannotbereducedbyincreasingthenumberofparticles.Asshowninthisarticlethroughasequenceofproblems,thistypeofuctuationiscausedbytheerrorintheinterpolationoftheuidvelocitytotheparticlelocation. Thecrucialpointisthatwhenafrozenuidvelocityvariationpropagatesthroughaxedgrid,theinterpolantoftheuidvelocityvariationchangesnotonlyinspacebutalsointime.Inotherwords,eveniftheexactuidvelocityisknownatthegridpoints,theinterpolatedapproximationtotheuidvelocityvariationwillchangeovertimeinaperiodicmanner(seegure2).Asaconsequence,whentheuidvelocityvariation 102

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propagatesthroughthexedgrid,thetimehistoryofuidvelocityseenbyaparticlewilldependonitslocationwithrespecttothegrid,affectingitsmotionandnalposition.Thustheinterpolationerrorgeneratesasubgridvariationintheparticlepositionafterthepassageoftheuidvelocityperturbation. NotethatinanactualsimulationwheretheuidowisalsonumericallycomputedtherewillbeEuleriandiscretizationerrorintheuidvelocityevaluatedatthegridpoints.Thiserrorinthecomputeduidvelocitywillcontributetoanerrorinparticlepositionandnumber-density.Butasshownhereanerrorininterpolatingtheuidvelocitytooff-gridlocationswillleadtoaspecialkindofnumber-densityerror,whichcanbesubstantialandsubgridinnature.Inotherwords,aninitiallyuniformlyspacedparticledistributionbetweentwogridpoints,afterthepassageoftheuidvelocityperturbation,willbenon-uniformlydistributedwithlargevariationintheinter-particlespacing.Ifeithertheuidvelocityhasnon-zerodivergenceortheparticleshavenon-zeroinertia,theparticledistributioncanundergopreferentialaccumulation.Inwhichcasetheinterpolation-inducedsubgriductuationscanmanifestasgrid-to-griductuations.Notethatherewehavefocusedonthespatial(grid-to-grid)natureofthenumber-densityuctuation.Insteadifoneconsidersmeasurementovertimeasobservedbyaxedprobe,thespatialuctuationwillresultinacorrespondingspurioustemporaluctuationinthemeasuredparticlenumber-density.Theshortspatialwavelengthoftheuctuationwilltranslatetoahighfrequencynoiseinthemeasurednumber-density.Theseuctuationsaregenerallysmall,whichmayexplainwhytheyhavenotbeennoticedsofar. However,thereexistssomesituationsinwhichtheuctuationscanbecomelargeandthereforemustbedealtwith.Thecaseofanexpansionfanpropagatinginaparticle-ladenuidconsideredinSection 3.5.3 isagoodexample.Inthatcase,alltheinterpolationmethodstestedhereintroducesignicantnumber-densityuctuationsfortheparticlesthatwereacceleratedbytheexpansionfanatearlytimeswhenthefanwas 103

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verythin.Therefore,thissourceofnumber-densityuctuationsmaynotbenegligibleincompressiblemultiphaseowinwhichthinwavesexist.Thebasicnatureofthissourceoferrorisnot,however,dependentoncompressibility,andthuslargeerrorscaninprinciplebeencounteredinincompressibleowsalso. Thisstudyemployedtwomodelproblemswithimposeduid-velocityprolestoshowhowthenumber-densityuctuationsaregeneratedbytheinterpolationoftheuidvelocitytotheparticlelocation.Differenttypesoferrorhavebeenplottedasafunctionofthegridresolution,withthegoalofservingasguidelinesforchoosingthegridresolutionforagivenerrorlevel.TwophysicalproblemsinwhichtheuidvelocitywasobtainedfromnumericalsolutionsoftheEulerequations,namelyofanexpansionandashockwave,werealsoconsidered.Thenumber-densityuctuationsobservedinthosecomputationsmatchtheresultsfromthemodelproblem.Ingeneral,higher-orderinterpolationmethods,suchasHermiteinterpolation,resultinreducednumber-densityuctuationscomparedtolower-orderschemes.Finallyitmustbepointedouttheseveralsimplicationshavebeenmadeinthepresentstudy.Forexample,wehaveusedonlyone-dimensionalexampleswithparticlemotioninuencedbyonlytheStokesdrag.However,thephenomenondescribedherewillbevalidinmorecomplex3Dtime-dependentowsthatusemoreaccuratedragandliftforcesforparticlemotion.Wehopethepresentstudywillhelpbetterexplain(atleastqualitatively)number-densityuctuationsseeninsuchcomplexows. 104

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CHAPTER4MODELINGANDANALYSISOFPARTICLEINTERACTIONWITHSHOCKANDBLASTWAVES 4.1Introduction Theinteractionbetweenshockwavesandparticlesisacommonandimportantphenomenonincompressibleparticle-ladenows.Whenashockwavepropagatesaroundanobstacle,theoweldbecomescomplicated.Eveniftheshapeoftheobstacleissimple,suchasforasphere,acomplexsystemofregularandirregularshock-wavereection,diffraction,andfocusingmayexist,seeBrysonandGross[ 16 ],Sunetal.[ 86 ],andTannoetal.[ 89 ].Theowfeaturesarisingfromtheinteractionbetweenashockwaveandasphericalparticlecanbeaccuratelycapturedinfullyresolveddirectnumericalsimulations,see,forexample,Sunetal.[ 86 ].However,inmanypracticalapplicationsofcompressibleparticle-ladenows,shockwavesinteractwithaverylargenumberofparticles.Thescalesofprimaryinterestaretypicallyatthemacro-scale,i.e.,oftheorderofthesizeofphysicalsystem,andareusuallymuchlargerthanthesizeofanindividualparticle.Therefore,itisimpracticaltoresolveallthemicro-scaledetailsoftheowaroundeachparticle,andtosimultaneouslycomputetheowatthemacro-scale.Instead,thepoint-particleapproachisoftenused,whereparticlesaremodeledaspointmasses,seeBalachandarandEaton[ 3 ].Inthisapproach,physicalmodelsareneededtocalculatetheinstantaneousforceandheattransferbetweenthegasandtheparticle,becausetheowaroundtheparticleisnotresolved. Inthepoint-particleapproach,theuid-dynamicforcemodelmustrepresentthenetmomentumexchangebetweentheparticleandthesurroundinguidandtherebyaccountsfortheeffectofthemicro-scaleow.Thustheuid-dynamicforcemustbeexpressedintermsofallthekeyparametersthatcharacterizethegasowaroundtheparticleatthemicro-scale.Understeadyconditions,thedragforceontheparticlewilldependonlyontherelativevelocitybetweentheparticleandtheambientow 105

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andonthethermodynamicstateandtransportpropertiesoftheambientow.Innondimensionalterms,thisdependenceisexpressedasadragcoefcientthatisafunctionoftheparticleMachandReynoldsnumbers(basedontherelativevelocity).Underunsteadyconditions,additionalcontributionstothedragforcearise,thatdependonthehistoryoftheparticleaccelerationandthetotalaccelerationoftheambientow. Byitsverynature,shock-particleinteractionisstronglytime-dependent.Theparticleissubjectedtoaverystronggasacceleration,astheshockwavepassesoverit.Recently,Sunetal.[ 86 ]andBredinandSkews[ 12 ]presentedtime-resolvedmeasurementsoftheforceonastationaryparticlesubjectedtoashockwave.Theinstantaneousforceontheparticleundersuchhighlyunsteadyconditionswasshowntobemuchlargerthanthecorrespondingquasi-steadyforcethatwouldhaveresultedifthechangefromthequiescentpre-shockstatetotheuniformpost-shockstateweretohappenveryslowly,seealsoParmaretal.[ 67 ].Inparticular,theinstantaneousforceduringthepassageoftheshockwavewasobservedtobeanorderofmagnitudelargerthanthesteadydragforceresultingfromthepost-shockgasvelocity,althoughforastationaryparticletherelativevelocitymonotonicallyincreasesfromzerobeforethearrivaloftheshockwavetoitsmaximumvalueafterthepassageoftheshockwave.Thisclearlyhighlightstheimportanceofunsteadycontributionstotheforceinshock-particleinteractionproblems.Similarly,fully-resolvedsimulationsbyZhangetal.[ 102 ]andRipleyetal.[ 72 ]indicatearapidriseinparticlevelocity,whenadetonationwavepassesoveraparticle.Unsteadyeffectsalsoenhanceheattransferandcontributetoarapidchangeinthetemperatureoftheparticle,seeRipleyetal.[ 72 ].Thesesharpincreasesinvelocityandtemperaturecannotbeexplainediftheunsteadyeffectsarenotincludedinthemodelsofparticleforceandheattransfer.Thus,weanticipatethatignoringunsteadyforcesandheattransferinmacro-scalesimulationsofcompressibleparticle-ladenowinvolvingshockwavesmayleadtosignicanterrors. 106

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Insomeapplications,suchasdetonation,thelargeunsteadyforcesexertedontheparticlecancausedeformationandbreakage.Similarly,intenseunsteadyheatadditioncancausemeltingorinitiatechemicalreactions.Therefore,thepeakvaluesofthedifferentcontributionstotheforceandheattransferareofimportanceandofpracticalinterest.Nevertheless,theunsteadyforcesandheattransferareusuallyneglectedevenifstronginteractionsbetweencompressibleowfeaturesandparticlesaretobeexpected;seeforexample,Najjaretal.[ 65 ]forasolid-propellantrocketmotors,andLanovetsetal.[ 49 ]andZhangetal.[ 101 ]fortheexplosivedispersalofparticles.Theneglectofunsteadycontributionstotheforceandheattransferisoftenjustiedonthegroundsoftheparticle-to-uiddensityandheat-capacityratiosbeinglarge.Thisargumentisappropriateonlyforunsteadycontributionsarisingfromparticleacceleration,becausethelatterscaleswiththeinverseoftheparticle-to-uiddensityratio.However,insituationssuchasshock-particleinteraction,asshownclearlyintheexperimentsofSunetal.[ 86 ]andBredinandSkews[ 12 ],thelargestcontributiontotheinstantaneousforceisfromtheambientuidaccelerationseenbytheparticle,whichisindependentoftheparticle-to-uiddensityratio,seeBagchiandBalachandar[ 1 ]foranextensivediscussionofthisissue. Theoverallgoalofthispaperistore-considertheproblemofshock-particleinteractionandtoinvestigatetheimportanceofunsteadycontributionstotheuid-dynamicforceoverarangeofshock-waveMachnumbers,particleReynoldsnumberandparticle-to-uiddensityratios. Itishelpfulatthisstagetobrieyreviewthestateoftheartofthemodelingforforceandheat-transferinthepoint-particleapproach.Inanincompressibleowourabilitytomodeltheuid-dynamicforceonaparticleisonsolidtheoreticalfooting.MaxeyandRiley[ 59 ]andGatignol[ 35 ]obtainedrigorousexpressionsfortheuid-dynamicforceonaparticleundergoingarbitrarytime-dependentmotioninanunsteadyinhomogeneousambientow.Inparticular,theywereabletoseparatetheoverall 107

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forceintophysicallymeaningfulcontributions:thequasi-steadyforceFqs,thepressuregradientforceFpg,theadded-massforceFam,andtheviscous-unsteadyforceFvu.ThelatterisoftencalledtheBassethistoryforce.ThislinearsuperpositionofthedifferentcontributionshasbeenrigorouslyestablishedonlyintheStokeslimitofzeroReynoldsnumber.Finite-Reynolds-numberextensionsareinevitablyempiricalinnature.TheReynolds-numberdependenceofthequasi-steadyandviscous-unsteadyforceswasestablishedbyCliftandGauvin[ 21 ]andMeiandAdrian[ 61 ]andKimetal.[ 47 ],respectively.Thepressure-gradientandadded-massforcesareprimarilyinviscidinnatureandarethereforeindependentofReynoldsnumber. TheformulationofMaxeyandRiley[ 59 ]andGatignol[ 35 ]wasrecentlyextendedtocompressibleowsbyParmaretal.[ 68 69 ],whereanexpressionfortheuid-dynamicforceonaparticleundergoingarbitrarytime-dependentmotioninanunsteadynon-uniformcompressibleambientowwasobtained.TheresultingequationofmotionisasymptoticallyvalidonlyinthelimitofsmallMachandReynoldsnumbers.Nevertheless,itprovidesarmtheoreticalbasisforempiricalextensionstoniteMachandReynoldsnumbers.Inparticular,theeffectofcompressibilityontheunsteadycontributionstotheforcewasobservedtobesignicant.Here,theformulationofParmaretal.[ 68 69 ]willbeempiricallyextendedtoniteMachandReynoldsnumber,andappliedtocomputetheforceduringthepassageofashockwaveoveraparticle.Forheattransfer,MichaelidesandFeng[ 62 ]andBalachandarandHa[ 4 ]havepresentedexpressionsvalidforincompressibleow.Toourknowledge,noextensionsoftheirresultstocompressibleowshavebeenpublished.Therefore,weusetheformulationsofMichaelidesandFeng[ 62 ]andBalachandarandHa[ 4 ]inthiswork. Thegoalofthispaperisthustopresentasystematicanalysisoftheunsteadyforceandheattransfer,andtoevaluatetheirimportanceintheproblemofshock-particleinteraction.Fromapracticalpointofview,theresultscanbeusedtoquantitativelyevaluatetherelativeimportanceoftheunsteadycontributions.Foragivenproblem,i.e., 108

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foragivenshockMachnumber,particlesize,andparticledensity,theresultscanbeusedtodeterminewhethertheunsteadycontributionsoughttobeincludedintheforceandheat-transfermodels. Inpursuitofthisgoal,westudythreecanonicalproblemsinwhichshock-particleinteractionsoccur.Intherstproblem,weconsideraplanarshockwavepassingoveraninitiallystationaryparticle.SincethegassolutionisgivenanalyticallybytheRankine-Hugoniotconditions,thevariouscontributionstotheoverallforceandheattransfercanbesolvedanalytically,providedthatthedetailsoftheinteractionsasdescribedaboveareneglected.Twomeasuresareusedtoevaluatetheimportanceoftheunsteadycontributions.First,thepeakvaluesarisingfrompressure-gradient,added-mass,andviscous-unsteadycontributionsarecomparedagainstthecorrespondingpeakvalueofthequasi-steadycontribution.Second,theterminalvelocityoftheparticle(i.e.,thatattainedlongafterthepassageoftheshock)thatwouldresultfromtheactionofpressure-gradient,added-mass,orviscous-unsteadycontributionsinisolationarecomparedagainstthepost-shockgasvelocity.Similarmeasuresareusedtoevaluatetheimportanceofunsteadycontributionstoheattransfer.Therationaleforchoosingthesemeasuresisasfollows.Processessuchasdeformationandfragmentationdependonpeakstressesactingontheparticleandthustherstmeasurewillbeusefulinjudgingtheimportanceofunsteadycontributionstosuchevents.Althoughtheunsteadycontributionstoforceandheattransfercanbeveryimportant,theyremainsignicantonlyduringthebriefperiodwhentheshockwavepassesovertheparticle.Bycontrast,thequasi-steadydrag(whichdependsonrelativevelocity)actspersistentlyuntiltheparticlevelocityapproachesthepost-shockuidvelocity.Thusthesecondmeasureprovidesquantitativeinformationontheimportanceoftheintegratedeffectoftheunsteadycontributions. Thesecondproblemconsideredisthatofasphericalshockwavepassingoveraparticle.Theshockwaveisassumedtobegeneratedbyapoint-sourceandto 109

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propagateoutwardinasphericalsymmetricmanner.Theblast-wavetheoryofTaylor[ 91 ]isemployedandtheanalyticalsimilaritysolutionisusedfortheowbehindtheshockwave.Theequationsofmotionfortheparticlecannotbeintegratedanalyticallyasintherstproblem,andhenceareintegratednumerically.Thesphericalshock-waveproblementailsadditionalcomplexitiesinthattheshockspeeddecayswithtime,andtheowbehindtheshockisnon-uniformandtime-dependent.Asintherstproblem,twodifferentmeasureswillbeusedtoevaluatetheimportanceofunsteadycontributionstotheforceandheattransfer. Finally,weinvestigatetheblast-waveproblemconsideredbyBrode[ 13 ].Thesphericalshockwaveisgeneratedbythesuddenreleaseofcompressedgasinitiallycontainedwithinasphereofniteradius.Incontrasttothesecondproblem,anexpansionfan,asecondarysequenceofshockwaves,andacontactdiscontinuityaregeneratedinadditiontothemainshockwave.Theseadditionalwavescaninteractwiththeparticle.Numericalsimulationsareperformedtoaccesstheimportanceoftheunsteadycontributionstoforceandheattransfer.Thetimeevolutionofparticlepositionandvelocitywillbecomparedtoevaluatetheimportanceoftheunsteadycontributions. 4.2PhysicalModelingandGoverningEquations Intheshock-particleinteractionproblemsconsideredhere,theparticleistakentobespherical,rigid,andinert.Itisalsostationaryandinthermalequilibriumwiththeambientuidbeforeencounteringtheshock.WeassumethattheBiotnumberisverysmall,sothattheparticlecanbeconsideredtobeisothermal.Whentheshockwavepropagatesovertheparticle,thevelocityandtemperatureoftheparticlestarttochangeduetomomentumandenergyexchange.Inaframeattachedtotheparticle,theapproachingshockwaveisaxisymmetricanditwillbeassumedthattheowremainsaxisymmetricevenastheshockwavepropagatespasttheparticle.Asaresult,theforceexertedontheparticleisalignedwiththedirectionofshockmotion.Weconsidergravitytobeunimportantcomparedtotheuid-dynamicforcesforthetimescalesof 110

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interest.Thus,inallthecasestobeconsideredhere,boththeuidandparticlemotionwillremainonedimensional(takentobethex-coordinatedirection).Thegaswillbetakentoobeytheideal-gaslaw.Theuidatthemacro-scalewillbeconsideredtobeinviscid,exceptinthemicro-scaleinteractionwiththeparticle.Furthermore,thethermalandtransportpropertiesofgasareassumedtobeconstant. Throughoutthisarticle,thesubscripts1and2denotepre-andpost-shockquantities,respectively.Thesuperscriptsgandpindicatesquantitiesassociatedwiththegasandtheparticles,respectively.Thetildeindicatesdimensionalquantities.Theparticlediameter~dpandthepre-shockspeedofsound~ag1aretakentobethecharacteristiclengthandvelocityscales.Thecharacteristictimescaleisthen~dp=~ag1.Thepre-shockgasdensity~g1andtemperature~Tg1areemployedasreferencescales. 4.2.1GoverningEquations Asdiscussedabove,theowisassumedtobeinviscidandone-dimensional,sothegoverningequationscanbewritteninnondimensionalformas@g @t+@gug @x+jgug x=0, (4)@gug @t+@g(ug)2 @x+jg(ug)2 x=)]TJ /F9 11.955 Tf 10.49 8.09 Td[(@pg @x, (4)@gEg @t+@gHgug @x+jgHgug x=0, (4) whereg,ug,pg,Eg,andHgrepresentthedensity,velocity,pressure,totalenergy,andtotalenthalpyofgasindimensionlessform,andj=0,1,and2correspondtoplanar,cylindrical,andsphericalcases,respectively. 111

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Theequationsgoverningtheparticleposition,velocity,andtemperatureindimensionlessformaredxp dt=up, (4)dup dt=fp, (4)dTp dt=qp, (4) wherexp,upandTprepresentthedimensionlessparticleposition,velocity,andtemperature. 4.2.2Momentum-TransferModel Equation( 4 )canbewrittenasdup dt=fp=fqs+fpg+fam+fvu, (4) wherefqs,fpg,fam,andfvudenotethenondimensionalquasi-steady,pressure-gradient,added-mass,andviscous-unsteadyforces,respectively.Traditionally,theparticlemotioniscomputedbasedonlyonthequasi-steadyforce.Implicitinthispracticeistheassumptionthatthethreeunsteadycontributionscanbeignored.Here,weincludethesetermsandestablishtheirimportanceforshock-particleinteraction.Todoso,itisessentialtoemployanaccurateequationofmotionthatpreciselycapturesallcontributions. Inanincompressibleow,preciseexpressionsforfqs,fpg,fam,andfvuhavebeenderivedrigorouslybyMaxeyandRiley[ 59 ]andGatignol[ 35 ].However,theMaxey-Riley-Gatignol(MRG)equationofmotionisasymptoticinnatureandisstrictlyvalidonlyinthelimitofRep!0,where Rep=~gj~up)]TJ /F4 11.955 Tf 12.2 0 Td[(~ugj~dp ~g,(4) where~gisthedynamicviscosityofthegas.Furthermore,theasymptoticresultisbasedontheassumptionthatthespatialvariationoftheambientoweldonthescale 112

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oftheparticlediameterissmallcomparedtotherelativevelocity,andthustheMRGequationofmotionrequiresweakambientowgradients.TheReynoldsnumberRepcanbeexpressedintermsofdimensionlessvariablesasRep=Reprgjug)]TJ /F3 11.955 Tf 11.96 0 Td[(upj, (4) whereRepristhereferenceparticleReynoldsnumber,denedasRepr=~g1~ag1~dp ~g. (4) ThereferenceparticleReynoldsnumberisafunctionofparticlediameterandpre-shockstagnantambientuidpropertiesonly,andhenceisindependentofthedetailsoftheshockstrengthorthepost-shockow. EmpiricalextensionsoftheMRGequationforniteRephavebeenproposedbyMagnaudetandEames[ 56 ]andBagchiandBalachandar[ 1 ].IntheMRGequation,fqsisgivenbytheStokes-dragexpressionwithFaxen-correctiontoaccountforinhomogeneousambientow.AtniteRep,theStokes-dragexpressionwillbereplacedwiththestandard-dragrelationthatisappropriateforawiderrangeofReynoldsnumber,seeSchillerandNaumann[ 78 ]andCliftandGauvin[ 21 ].Similarly,atniteRep,thekernelinfvumustbereplacedbyanappropriateempiricalnite-Reynolds-numberextension,suchasthosedevelopedbyMeiandAdrian[ 61 ]andKimetal.[ 47 ].Theinviscidpressure-gradientandadded-masscontributionsareindependentofRep. ThetheoreticalformulationoftheforcecontributionshasbeenextendedtocompressibleowrecentlybyParmaretal.[ 68 69 ].Theresultingasymptoticequationofmotion,liketheMRGequationofmotion,isrigorousinthelimitofRep!0andMp!0,whereMpistheMachnumberbasedonrelativevelocityandspeedofsoundoftheambientgas, Mp=j~up)]TJ /F4 11.955 Tf 12.2 0 Td[(~ugj ~ag.(4) 113

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Theformulationassumesthatthevariationoftheambientowvelocityanddensityonthescaleoftheparticlediametertobesmallcomparedtotherelativevelocityandreferenceuiddensity.Inspiteoftheselimitations,theformulationofParmaretal.[ 68 69 ]providesarigorousfoundationforconsideringarbitrarytime-dependentparticlemotioninanunsteadycompressibleinhomogeneousambientowthatisvalidatniteRepandMp.Theexpressionsfortheforcecontributionsconsideredinthisworkarefqs=3 4g pCD(ug)]TJ /F3 11.955 Tf 11.95 0 Td[(up)2, (4)fpg=g pDug Dt, (4)fam=1 pZtKam(t)]TJ /F9 11.955 Tf 11.95 0 Td[()D(gug) Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(d(gup) dtt=d, (4)fvu=9 pp gRerZtKvu(t)]TJ /F9 11.955 Tf 11.96 0 Td[()D(gug) Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(d(gup) dtt=d. (4) Thecontributionsrequireclearexplanations.Inthequasi-steadyforce,CDisthedragcoefcientthatisafunctionofRepandMp.Inthiswork,theempiricalcorrelationofParmaratal.[ 70 ]willbeusedforthedragcoefcient. Thepressure-gradientforcefpgistheforceexperiencedintheabsenceoftheparticlebytheuidthatwouldoccupythevolumeoftheparticleandthusisthesameasthelefthandsideofthecompressibleNavier-Stokesequationsappropriatelynondimensionalizedforthepresentpurposes. Intheadded-massforcefam,KamistheinviscidunsteadykernelthatisafunctionofdimensionlesstimeandMp.Inanincompressibleow,i.e.,ifMp=0,thekernelbecomesKam()=()=2.Asaresult,werecovertheinviscidunsteadyforcetobeproportionaltoinstantaneousrelativeaccelerationwiththeadded-masscoefcientCMequaltoone-half,asCM=RtKam(t)]TJ /F9 11.955 Tf 12.75 0 Td[()d.Notethatincompressibleows,theadded-massforceinvolvesahistoryintegralliketheviscous-unsteadyforce.Therefore,theaddedmassconceptinincompressibleows,strictlyspeaking,doesnotexist.Forthisreason,itismorepropertocombinethisforcetogetherwiththe 114

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pressure-gradientforceintoaninviscidunsteadyforceassuggestedbyParmaretal.[ 67 ].Forconvenience,wediscussthesetwoinviscidunsteadyforcesseparatelyandusetheconventionalnamesinthispaper.ForMp!0,Longhorn[ 55 ]derivedKam()=exp()]TJ /F9 11.955 Tf 9.29 0 Td[()cosforasphere.AtniteMachnumber,thekernelcannotbederivedanalytically,butnumericalsolutionswerepresentedbyParmaretal.[ 66 ]forthesub-criticalMach-numberregime. Notethattimehasbeennondimensionalizedbytheacoustictimescale~dp=~ag1.Asaresult,theinviscidunsteadykernelbecomesnegligiblysmallafterafewacoustictimescales.AtniteMachnumbers,Parmaretal.[ 66 ]observedthedecaytoslowdownslightly,butKamwasstillnegligiblefor10.IfD(gug)=Dt)]TJ /F3 11.955 Tf 12.71 0 Td[(d(gup)=dtchangesslowlyontheacoustictimescale,itcanbetakenoutsideoftheintegral.Theintegralofthekernelfunctionwillthenyieldaneffectiveadded-masscoefcientthatwillbeafunctionofMponly.Thentheadded-massforcecanbesimpliedtofam=CM pD(gug) Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(d(gup) dt (4) eveninacompressibleow.ForMp!0,Longhorn'skernelcanbeintegrated,yieldingCM=1=2.ItwasshownbyParmaretal.[ 66 ]thatCMincreaseswithMp.Insection 4.3 ,wewilltakeCM=1=2forsimplicity.Insection 4.4 ,weassesstheeffectofMachnumberontheresults. Intheviscous-unsteadyforcefvu,Kvuistheviscous-unsteadyforcekernelthatisafunctionoftime,particleReynoldsnumber,andparticleMachnumber.Intheincompressiblezero-Reynolds-numberlimit,theviscous-unsteadykernelwasderivedanalyticallybyBasset[ 8 ]tobeKvu,B()=)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2.Atlongtimes,thekernelgenerallydecaysatafasterratethantheBassetkernel,butitisalsoobservedtobedependentonthenatureofaccelerationordeceleration.Atniteparticle-Reynolds-number,empiricalkernelsKvu,inc(,Rep)havebeenformulatedbyMeiandAdrian[ 61 ]andKim 115

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etal.[ 47 ]thatgivemoreaccuraterepresentationsoftheviscous-unsteadyforceforlongtimes. Theeffectofcompressibilityontheviscous-unsteadyforcehasrecentlybeenestablishedbyParmaretal.[ 69 ].ForRep!0andMp!0,theeffectofcompressibilitycanbereducedtoacorrectionfunctionandthekernelcanbeexpressedasKvu()=Kvu,B()C().ThepreciseformofthecorrectionfunctionC(),whichwasfoundtobeboundedbetween0.44and1.45,hasbeenshowntobedependentonboththeKnudsennumber(whichisproportionaltoMp=Rep)andtheratioofthebulkviscositytothedynamicviscosity.ExtensionsoftheviscousunsteadykerneltoniteRepandMpremaintobeexplored.AsimpleproposalistoapplythecompressibilitycorrectiontotheniteReynoldsnumberformulationoftheviscousunsteadykernel,anddeneKvu()=Kvu,inc(,Rep)C(),butthisassumestheReynoldsnumbercorrectiontobeindependentofMachnumber.Nevertheless,Parmaretal.[ 69 ]observedthatC()approachesunityrapidlyontheacoustictimescaleandtheBassetkernelwasrecoveredafterafewacoustictimescales.Furthermore,ifD(gug)=Dt)]TJ /F3 11.955 Tf 12.53 0 Td[(d(gup)=dtchangesslowlyontheacoustictimescale,KvucanbereplacedbyKvu,incsinceRtKvud!RtKvu,incdfort~dp=~ag1.Asaresult,consistentwithourearlieruseofEq.( 4 )fortheadded-massforce,insection 4.3 wewillinitiallyuseKvu,inc(,Rep)byMeiandAdrian[ 61 ]fortheviscousunsteadyforce.Insection 4.4 ,wewilldiscusstheeffectofincludingtheinuenceofcompressibilitythroughC(). TheexpressionsgiveninEq.( 4 )to( 4 )areappropriateforhomogeneousambientows.Intheseequations,gasquantitiessuchasug,g,Dug=Dt,andD(gug)=Dt,arethoseoftheundisturbedambientow.Inthecaseofaplanarorsphericalshockwaveapproachingaparticle,theundisturbedambientowisknown.Longafterthepassageoftheshockwave,theambientowaroundtheparticlecanbeconsideredtobespatiallyuniformalso.However,astheshockwaveinteractswiththeparticle,theowisingeneralhighlynon-uniform,andthereforesomeadditional 116

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modelingisrequired.Basedontheobservationthattheshockthicknessisoftheorderof0.1m(atstandardconditionsinair)andthusmuchsmallerthanatypicalparticlesize,itisreasonabletoassumetheshocktobeaperfectdiscontinuity.Thus,astheshockispassingovertheparticle,theambientowseenbytheparticlecannotbeconsideredspatiallyhomogeneousanditisnotappropriatetotakegasquantitiessuchasugtobesimplygivenbytheirvaluesatthecenteroftheparticle.Faxen[ 28 ]wasthersttoconsiderspatialvariationsoftheambientowontheforceactingonaparticle.IntheStokes-owlimit,heobtainedtheelegantresultthatinaninhomogeneousambientowtheeffectiveuidvelocityseenbytheparticleisgivenbyanaverageoverthesurfaceoftheparticle.Faxen'sresulthasbeenextendedtotime-dependentincompressibleowsbyMaxeyandRiley[ 35 59 ]andtotime-dependentcompressibleowsbyParmaretal.[ 68 ].BasedontheworkbyParmaretal.[ 68 ]andwiththeabovementionedsimplicationtotheadded-massforce,wecalculatetheparticleforceasfollowsinthecaseofshock-particleinteraction,fqs=3 4CD gs p( ugs)]TJ /F3 11.955 Tf 11.95 0 Td[(up)2, (4)fpg=1 p gDug Dtv, (4)fam=CM p D Dtv (gug)v)]TJ /F3 11.955 Tf 15.3 8.09 Td[(d dt (gup)v!, (4)fvu=9 pp gsReprZtKvu(t)]TJ /F9 11.955 Tf 11.96 0 Td[() D Dts (gug)s)]TJ /F3 11.955 Tf 15.3 8.09 Td[(d dt (gup)s!t=d. (4) Barswithasuperscriptsorvdenotegasquantitiesaveragedovertheparticlesurfaceorvolume,respectively.Forexample, ugs=1 SpISpugds, (4) ugv=1 VpIVpugdv, (4) 117

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whereSpandVparetheparticlevolumeandsurface,respectively.Theaveragedsubstantialderivatives D=Dtsand D=Dtvaredenedas D=Dts=@ @t+ ugs@ @x(4) and D=Dtv=@ @t+ ugv@ @x,(4) respectively. 4.2.3Energy-TransferModel ThenondimensionalparticlethermalequationcanbewrittenasdTp dt=qqs+quu+qdu, (4) whereqqs,quu,andqduarethequasi-steady,undisturbed-unsteady,anddiffusive-unsteadycontributionstotheparticlethermalacceleration.Therstcontributionaccountsforthesteadythermaldiffusionfromtheparticletothesurroundinggasduetothetemperaturedifference.Thelastcontributionaccountsfortheunsteadythermaldiffusionduetothetemporaldevelopmentofthethermalboundarylayeraroundtheparticle.Theheattransferduetotheunsteadinessoftheundisturbedtemperatureeldisdenotedbyquu,whichaccountsfortheheattransferthatwouldhaveoccurredintheabsenceoftheparticlebetweenthevolumeofgasoccupiedbytheparticleandthesurrounding.Theundisturbed-unsteadycontributionquuisanalogoustothepressure-gradientforcefpg.Notethattheadded-massforceisinviscidinnatureandarisesfromtheno-penetrationboundaryconditionontheparticle.Thereforethereisnoanaloginheattransfer.Thediffusive-unsteadycontributionqduarisesfromtheunsteadydevelopmentofthethermalboundarylayeranddependsonthepasthistoryofrelativethermalacceleration.MichaelidesandFeng[ 62 ]obtainedathermalequationfortheparticleanalogoustotheMRGequationofparticlemotion.FengandMichaelides[ 29 ]andBalachandarandHa[ 4 ]haveconsiderednite-Reynolds-numberextensionsofthethermalequations.The 118

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quasi-steady,undisturbed-unsteady,anddiffusive-unsteadycontributionstoheattransfercanbeexpressedasqqs=6Nu( Tgs)]TJ /F3 11.955 Tf 11.96 0 Td[(Tp) pCpPrRer, (4)quu=1 pCp gDTg Dtv, (4)qdu=6 pCpp gsPrReprZtKdu(t)]TJ /F9 11.955 Tf 11.95 0 Td[() D Dts (gTg)s)]TJ /F3 11.955 Tf 15.3 8.09 Td[(d dt (gTp)s!t=d. (4) Intheseequations,NuandPraretheNusseltnumberandPrandtlnumber,denedas Nu=~g~dp ~kg(4) and Pr=~g ~g(4) where~kg,~g,and~garethethermalconductivity,thermaldiffusivity,andkinematicviscosityofthegas,respectively.Thequasi-steadycontributionqqstotheoverallheattransferisparameterizedintermsoftheNusseltnumberasafunctionofRep,Mp,andPr.HereweignoredtheweakMach-numberdependence,andemploythecorrelationfortheNusseltnumberbyWhitaker[ 98 ]. ThekernelfunctionKduatzeroparticleReynoldsnumberandMachnumberisequaltoKvu.AtniteReynoldsnumber,thethermalkernelfunctionismorecomplicated,seeMichaelidesandFeng[ 62 ].Toourknowledge,thereisnopreviousworkonthekernelfunctionforunsteadyheattransferincompressibleows.Forsimplicity,thepresentstudyonlyconsiderstheBassetkernelforheattransfer.Theimplicationsofthisassumptionwillbediscussedinsection 4.4 .Alsonotethat,asinEqs.( 4 )to( 4 ),weemploysurfaceandvolumeaveragesoftheowquantitiestoaccountfortheinhomogeneousnatureoftheambientowduringtheshock-particleinteraction. 119

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4.3Results 4.3.1PlanarShockWave Thepre-andpost-shockstatesforaplanarshockwavemovingatconstantspeedare(g,ug,pg)=8><>:(g2,ug2,pg2)ifxxs,(g1,0,pg1)ifx>xs. (4) Thepost-shockpropertiesareug2=2 +1us)]TJ /F4 11.955 Tf 15.38 8.09 Td[(1 us, (4)g2=(+1)u2s ()]TJ /F4 11.955 Tf 11.96 0 Td[(1)u2s+2, (4)pg2=~pg2 ~g1(~ag1)2=2u2s)]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F4 11.955 Tf 11.96 0 Td[(1) (+1), (4) whereistheratioofspecicheats,andus=Ms=~us=~ag1isthenondimensionalshockspeed.FromEqs.( 4 )to( 4 ),explicitexpressionsfortheaveragedgasquantitiesthatappearinEqs.( 4 )to( 4 )andEqs.( 4 )to( 4 )canbeobtained,seeAppendix A 4.3.1.1Peakvaluesofforceandheattransfer Whentheshockpassesoveraparticle,theinstantaneousforceandheattransfercanbeverylarge.Thetimeduringwhichtheinteractiontakesplaceisgenerallyveryshort,andthemotionoftheparticleisgenerallylimited.Therefore,herewerstmaketheapproximationthattheparticleisfrozenastheshockpassesoverit.Inotherwords,weassumeup0,dup=dt0,anddTp=dt0.Thisassumptionallowstheforceandheattransfercontributionstobedeterminedanalytically.Basedonthis 120

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approximation,Eqs.( 4 )to( 4 )becomefqs=3 4CD p(ug2)2y2hyg2+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y)i, (4)fpg=6 pug2usy(1)]TJ /F3 11.955 Tf 11.96 0 Td[(y), (4)fam=6CMg2 pug2us1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(ug2 usy2(3)]TJ /F4 11.955 Tf 11.96 0 Td[(2y)y(1)]TJ /F3 11.955 Tf 11.96 0 Td[(y), (4)fvu=9p usyg2ug22)]TJ /F5 7.97 Tf 13.15 4.71 Td[(4 3ug2y us pr h(g2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)y+1iRepr, (4) andEqs.( 4 )to( 4 )becomeqqs=6 pCpNu PrRery(Tg2)]TJ /F4 11.955 Tf 11.95 0 Td[(1), (4)quu=6 pCp(Tg2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)us(y)]TJ /F3 11.955 Tf 11.95 0 Td[(y2), (4)qdu=6p usy"2g2(Tg2)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[((g2Tg2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)4 3ug2y us# pCpr h(g2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)y+1iPrRepr, (4) wherey=ustisthedistancethattheshockwavehastraveledovertheparticle.Equations( 4 )to( 4 )canbeexpectedtobeverygoodapproximations,eveniftheparticleisallowedtomove,providedthattheparticle-to-uiddensityratioislarge.Onlyiftheparticle-to-uiddensityratioisO(1)orsmaller,willEqs.( 4 )to( 4 )beinaccurate.ThenitcanbeshownthattheactualcontributionstotheforceandheattransferwillbereducedandthefrozenresultsgiveninEqs.( 4 )to( 4 )canbeconsideredasupperbounds.Sincetheviscous-unsteadyforcereachesitspeakvalueataveryearlytime,thekernelbyMeiandAdrian[ 61 ]differsonlynegligiblyfromtheBassetkernel.Therefore,forsimplicitytheBassetkernelwasusedinobtainingEq.( 4 ).Figure 4-1 showstheevolutionofthedifferentcontributionstotheforceandheattransferasfunctionsofthedistancesweptbytheshockforMs=1.5,Repr=104,p=103,andCp=1.Alsoitcanbeseenthat,theshocktouchesthefrontofthe 121

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AForce BHeattransfer Figure4-1. Evolutionofforceandheat-transfercontributionsinplanarshock-particleinteractionforMs=1.5,Repr=104,p=103,andCp=1. particleatt=0andleavestherearatt=2=3.Itcanbeobservedthatfpg,fam,andfvureachtheirpeakvalueswhen0
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Thesearegivenbyfqs,max=3 4CDjy=1g2 p(ug2)2wheny=1, (4)fpg,max=3 2usug2 pwheny=1=2, (4)fam,max=3 2CMg2 pug2usmax0y11)]TJ /F3 11.955 Tf 13.16 8.09 Td[(ug2 usy2(3)]TJ /F4 11.955 Tf 11.95 0 Td[(2y)y(1)]TJ /F3 11.955 Tf 11.96 0 Td[(y) (4)fvu,max=9 p g2 pug2 p Rermax0y124p usy2)]TJ /F5 7.97 Tf 13.15 4.7 Td[(4 3ug2y us p (g2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)y+135, (4) andqqs,max=6 pCpNu PrRer(Tg2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)wheny=1, (4)quu,max=3 2(Tg2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)us pCpwheny=1=2, (4)qdu,max=6 p g2Tg2)]TJ /F4 11.955 Tf 11.96 0 Td[(1 pCpp PrRermax0y124p usy2g2(Tg2)]TJ /F4 11.955 Tf 11.96 0 Td[(1))]TJ /F5 7.97 Tf 13.15 4.71 Td[(4 3ug2y us(g2Tg2)]TJ /F4 11.955 Tf 11.95 0 Td[(1) p (g2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)y+135. (4) ThepeakvaluesofthevariouscontributionstotheforceandheattransferarefunctionsofgasphasepropertiesPrand,particleparametersp,Cp,andRepr,andtheshockstrengththroughMs.Notethatallthepost-shockquantitiescanbefullydeterminedbyus=Msand,seeEqs.( 4 )to( 4 ).Iftheunsteadycontributionstotheforcearenormalizedbyfqs,maxandtheunsteadyheattransfercontributionsbyqqs,max,theparticleparameterspandCpbecomeirrelevant.Furthermore,wechoosetheambientuidtobeairatstandardconditionsandset=1.4andPr=0.72.Thustheratiosofpeakvaluesdependonlyonthetwoparameters:theshockMachnumberMsandthereferenceparticleReynoldsnumberRepr,whichisaproxyfortheparticlediametersincethereferenceambientairpropertiesarexed.Forairatstandardconditions,thecorrespondingparticlediameterforreferenceparticleReynoldsnumberRepr=103is42.7m. 123

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Figures 4-2 and 4-3 showtheratiosofthepeakforcesandpeakheat-transfercontributionsrelativetotheirrespectivequasi-steadycontributionsasfunctionsofMsandReprfor1.1Ms10and1Rer105.Notethattheabscissasareplottedintermsoflog(Ms)]TJ /F4 11.955 Tf 12.23 0 Td[(1).Severalimportantobservationscanbemade.Firstandforemost,unsteadyforcesandheattransfercontributionsareimportantcomparedtothequasi-steadycontribution.ItisobservedthatintherangesofMsandReprconsideredhere,fpg,maxandfam,maxcanbemorethananorderofmagnitudelargerthanfqs,max,whilefvu,maxcanbefactortwoormorelarger.Theseratiosbecomeevenlargerforweakshocks,e.g.,forMs1.1,butforsuchweakshockstheratioslosetheirsignicance,sincetheindividualforcecontributionsthemselvesarenegligiblyweak.TherelativecontributionsoftheunsteadymechanismstoforceandheattransferincreasewithdecreasingMs.ForlargeMsandsmallRepr,theunsteadycontributionstothetotalforceandheattransferdecreases. Theinviscid-unsteadyforcesfpgandfamshowsimilarbehaviorandareofthesameorderofmagnitude.Evenwhentheparticleissufcientlysmall,i.e.,ifRepr=O(1),fpg,maxandfam,maxarenon-negligiblecomparedtothequasi-steadyforce,andfvu,maxisstillabout75%ofthepeakquasi-steadyforce.Theorderofmagnitudeofthepeakheat-transferratiosquu=qqsandqdu=qqsaregenerallysimilartotheirforcecounterparts.AtsmallRepr,theratiosquu=qqsandfpg=fqsareveryclose,butduetodifferencesinthenon-linearadvectionofvorticityandheat,quu=qqsincreasesmorestronglythanfpg=fqsatlargeRepr. Inmostshock-particleinteractionproblems,theassumptionofafrozenparticlewillnotbevalidbecausetheparticlewillmoveandbeheated.ThenthepeakvaluesofforceandheattransferneedtobecomputednumericallyandtheparticleparameterspandCpbecomerelevant.Asdiscussedabove,whenp1,theresultsforamovingparticlewillbeverysimilartothosepresentedinFigs. 4-2 and 4-3 forafrozenparticle.Whenp=O(1)orsmaller,theresultsofamovingparticleareexpectedtobedifferent, 124

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Afpg,max=fqs,max Bfam,max=fqs,max Cfvu,max=fqs,max Figure4-2. Contoursoftheratiosofthepeakvaluesofforcestothepeakquasi-steadyforceasfunctionsofMsandReprinplanarshock-particleinteraction. buteventhenweobservethedifferencestobeonlyquantitative,andthefrozenparticleresultsofFigs. 4-2 and 4-3 arequalitativelycorrect. 4.3.1.2Timescalesofquasi-steadyandunsteadyforceandheattransfer Thetimescalesforthequasi-steadyforceandheattransferaretheparticlemechanicalandthermalresponsetimes.Ifnite-Reynolds-numbereffectsaretaken 125

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Aquu,max=qqs,max Bqdu,max=qqs,max Figure4-3. Contoursoftheratiosofthepeakvaluesofheat-transfercontributionstothepeakvaluesofquasi-steadyheat-transfercontributionsasfunctionsofMsandReprinplanarshock-particleinteraction. intoaccount,themechanicalandthermaltimescalescanbewrittenas~pf,qs=~p(~dp)2 18~g24 CDRe, (4)~pq,qs=~Cp~p(~dp)2 12~kg2 Nu. (4) Thesearethetimescalesonwhichthevelocityandtemperatureofaparticleintroducedintoasteadyuniformisothermalowwillapproachtheambientconditions.Thetwotimescaleestimatesareappropriateforniteparticle-Reynolds-numbersprovidedthatp1.Thetimescalefortheunsteadycontributionscanbetakenasthetimefortheshocktomovepasttheparticle,i.e.,~ps=~dp=~us.Thepressure-gradientandadded-massforceandundisturbed-unsteadyheattransfer,arenon-zeroonlyfor0~t~ps.Theviscous-unsteadyforceanddiffusive-unsteadyheattransferdecayonalongertimescale,butthetimescaleofitsdominantcontributionisalsoO(~ps).Thethreetimescales 126

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canbenondimensionalized,pf,qs=4p 3CDgug, (4)pq,qs=pCpPrRer 6Nu, (4)ps=1 Ms. (4) Thuswehave ps pf,qs=18 pRerMs,(4) and ps pq,qs=12 pCpPrRerMs.(4) Generally,thetimescalesofquasi-steadyforceandheattransferaremuchlargerthanthetimescaleofunsteadyforceandheattransfercontributions.Therefore,althoughthemagnitudesofthepeakunsteadycontributionstoforceandheattransferarelargerthanthecorrespondingpeakquasi-steadycontributions,thedifferenceintimescalesmustbetakenintoaccountwheninvestigatingtheneteffectoftheunsteadyforcesonparticlevelocityandtheunsteadyheattransfertotheparticletemperature.Thisisaddressednext. 4.3.1.3Neteffectofunsteadymechanismsonoverallforceandheattransfer Theimportanceofunsteadycontributionstopeakvaluesofforceandheattransferencounteredduringshock-particleinteractionhasbeenrmlyestablished.However,itisofinteresttoexaminehowsignicanttheunsteadymechanismsareinaccountingforthechangeinparticlevelocityandtemperatureafterthepassageoftheshock. AccordingtointegralformofEq.( 4 ) up=Zfqsdt+Zfpgdt+Zfamdt+Zfvudt,(4) wherethetermsontherighthandsidearetheparticlevelocitybudgetsforquasi-steady,pressure-gradient,added-mass,andviscous-unsteadyforcecontributions,respectively. 127

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Similarly,theintegralformofEq.( 4 )canbewrittenas Tp)]TJ /F3 11.955 Tf 11.95 0 Td[(Tg1=Zqqsdt+Zquudt+Zqdudt,(4) wherethetermsontherighthandsidearetheparticletemperaturebudgetsforquasi-steady,undisturbed-unsteady,anddiffusive-unsteadyforcecontributions,respectively.Figure 4-4 showstheparticlevelocityandtemperaturebudgetsduetotheforceandheat-transfercontributionsasfunctionsoftimeforMs=1.5,Repr=102,p=10,andCp=1.Itisobservedthatthevelocityandtemperaturebudgetsforunsteadycontributionsareinitiallysignicantinthetotalvelocityandtemperature,butsoonafterthepassageoftheshock,thequasi-steadycontributionbecomesdominant.ThiscanbeexplainedbytheforceevolutiondescribedinFig. 4-1 .Althoughthemagnitudeoftheunsteadyforceandheattransferscontributionsarequitelargeduringthepassageoftheshock,theyquicklydecay.Bycontrast,thequasi-steadyforceandheattransfermakesustainedcontributions.Wealsonotethat,thevelocitybudgetforfvuandthetemperaturebudgetforqdureachmaximaatt=3.96andt=5.41,respectively.Afterwards,fvuandqdubecomenegative,andhencecontributetodecelerationandcooling.Theexplanationforthisisasfollows.Whiletheuidaccelerationasseenbytheparticleisnon-zeroonlyduringthepassageoftheshock,theparticleaccelerationislong-lived,i.e.,untiltheparticlereachesasymptoticallytheconstantpost-shockstate.Thus,shortlyafterthepassageoftheshock,thehistoryeffectofuidaccelerationdominatesandcontributespositivelytofvuandqdu.Bycontrast,overlongtime,theeffectofuidaccelerationhasdecayed,andtheeffectofparticleaccelerationcontributesnegativelytofvuandqdu,seeEqs.( 4 )and( 4 ).Therefore,itisimportanttoassesstheintegratedeffectofthequasi-steadyandunsteadycontributions. Anotherapproachofevaluatingtherelativeimportanceoftheunsteadyforceandheat-transfercontributionsistoevolvetheparticlemotionandtemperaturewithonlyoneforceorheat-transfercontributionactiveatatime.Forexample,whenevaluating 128

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AVelocity BTemperature Figure4-4. Budgetsoftheparticlevelocityandtemperatureasfunctionsoftime,forMs=1.5,Repr=102,p=10,andCp=1inplanarshock-particleinteraction. theimportanceofthepressure-gradientforce,wecomputetheparticlemotionduetotheshockonlyusingfpg,withallotherforcesarticiallydeactivatedbysettingthemtozero.Ifthecontributionofanindividualmechanismisnotsignicant,itneednotbeconsideredincomputingthemotionorthetemperatureoftheparticle.Tojustifytheneglectofallunsteadymechanismsintheclassicalapproachestoshock-particleinteraction,itisnecessarytoestablishthedominanceofthequasi-steadymechanismoverallothers. Duetothepersistentnatureofthequasi-steadyforce,aslongasitisactive,theparticlewillalwaysreachthepost-shockgasvelocityug2,althoughtherateofapproachwillbedifferentdependingonwhichothercontributionsareincludedandhowsignicanttheyare.Itisimportanttonotethat,thenatureoftheunsteadyforcesissuchthatwhentheyareactiveintheabsenceofquasi-steadyforce,theparticlevelocityneednotasymptoticallyapproachthepost-shockgasvelocityofug2.Figure 4-5 showstheparticlevelocityandtemperatureevolutionswhenonlyoneforceorheat-transfercontributionisactiveforMs=1.5,Repr=102,p=10,andCp=1.Here,weuseuptodenotethe 129

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AVelocity BTemperature Figure4-5. Theparticlevelocityandtemperatureasfunctionsoftimewhenonlyoneforceorheattransferisactiveduringtheshock-particleinteractionforMs=1.5,Repr=102,p=10,andCp=1inplanarshock-particleinteraction. actualparticlevelocitycomputedbyEq.( 4 ),andupqs,uppg,upam,andupamtodenotetheparticlevelocitycomputedwithonlyonecorrespondingforcecontributioninEq.( 4 )active.Similarly,TpdenotestheactualparticletemperaturecomputedbyEq.( 4 ),andTpqs,Tpuu,andTpdudenotetheparticlevelocitycomputedwithonlycorrespondingheat-transfercontributioninEq.( 4 )active.AscanbeseeninFig. 4-5A ,whileupaswellasupqsapproachug2,theterminalvaluesofup=ug2forallotherforcesreachvaluessmallerthanunityasjustexplained.Fortheparticularcaseconsideredinthisgure,up(t!1)=ug2takesvaluesof0.10,0.066,and0.13forthepressure-gradient,added-massandviscous-unsteadyforces,thusestablishingtheirrelativeimportanceintheoverallmotionoftheparticle. InFig. 4-5B ,theterminalvalueof(Tp(t!1))]TJ /F3 11.955 Tf 12.37 0 Td[(Tg1)=(Tg2)]TJ /F3 11.955 Tf 12.37 0 Td[(Tg1)forquuis0.075.(Tp(t!1))]TJ /F3 11.955 Tf 12.12 0 Td[(Tg1)=(Tg2)]TJ /F3 11.955 Tf 12.12 0 Td[(Tg1)forqduisequalto0.25att=50,whichhasnotreacheditsterminalvalueyet.Thus,itcanbeseenthatdiffusive-unsteadytermcontributessignicantlytotheheattransfer. 130

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Ifthepressure-gradientcontributionistheonlyforcetoactontheparticleduringitsinteractionwiththeshockwave,thesimpliedequationofmotiondup=dt=fpgcanbeanalyticallyintegratedtoobtainuppg,t us=1)]TJ /F16 11.955 Tf 11.95 21.96 Td[(s 1)]TJ /F4 11.955 Tf 15.61 8.09 Td[(2 pug2 us, (4) whereuppg,trepresentstheterminalparticlevelocitycomputedwithonlythepressure-gradientforceactive.Generally,thepost-shockvelocityug2issmallerthantheshockvelocityus,butofthesameorderofmagnitude.Thusifp1,theaboveexpressioncanbesimpliedtoobtainuppg,tu2=p.Inotherwords,thevelocitygainedbyaparticlefromthepressure-gradientforceisinverselyproportionaltotheparticle-to-uiddensityratioandindependentofRePrandMs.EveninEq.( 4 ),thedependenceofuppg,tonMsisgenerallysmall. Numericalintegrationisneededtoobtainupam,tandupvu,t,whicharedenedanalogouslytouppg,t.Thevelocitiesuppg,tandupam,taregeneratedbyinviscidforcesandareindependentofRepr.Thevelocityupvu,tisrelatedtoviscouseffects,andisthereforeafunctionofRepr,Ms,andp.Similaranalysiscanbecarriedoutfortheparticletemperature,whereTpqs,t,Tppg,t,andTpdu,tdenotetheparticletemperatureast!1ifonlyqqs,quu,andqduareactiveduringtheshock-particleinteraction.ForTppg,t,wearriveatTppg,t)]TJ /F4 11.955 Tf 11.95 0 Td[(1 Tg2)]TJ /F4 11.955 Tf 11.96 0 Td[(1=1 pCp, (4) whereasTpdu,tneedstobeintegratednumerically.Itshouldbenotedthat,Tppg,tandTpdu,twilldependalsoonCp.Inparticular,thetemperaturechangeduetoquuisinverselyproportionaltopCpandduetoitsinviscidnatureisindependentofRepr. Thenumericalresultsforupam,t=ug2,upvu,t=ug2and(Tpdu,t)]TJ /F4 11.955 Tf 12.72 0 Td[(1)=(Tg2)]TJ /F4 11.955 Tf 12.71 0 Td[(1)areshowninFigs. 4-6 and 4-7 .ItcanbeobservedthattheunsteadycontributionstoforceandheattransferaredependentmainlyonpandpCp,respectively.Ingeneral,the 131

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terminalparticlevelocityisinverselyproportionaltopwhenonlyoneunsteadyforceisactive.TheterminalparticletemperatureisinverselyproportionaltopCpwhenonlyoneunsteadyheat-transfercontributionisactive.HerewehaveusedboththeclassicalBassetkernelandtheniteReynoldsnumberviscous-unsteadykernelbyMeiandAdrian[ 61 ].ComparingFigs. 4-6C and 4-6D ,itcanbeobservedthatupvu,tisoverpredictedbyafactoroffourtovebytheBassetkernelonaccountofitsslowert)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2decaycomparedtothet)]TJ /F5 7.97 Tf 6.58 0 Td[(2decayofthekernelofMeiandAdrian[ 61 ]forlongtimes.BecausewehaveusedtheBassetkernelforqdu(seeremarksinsection 4.2.3 ),itcanbeconjecturedthattheresultsinFig. 4-7 likelyoverpredicttheimportanceofviscousunsteadyheattransfer.Insummary,Eqs.( 4 )and( 4 ),alongwithFigs. 4-6 and 4-7 ,providegoodestimatesoftheimportanceoftheunsteadycontributionstoforceandheattransferincomputingparticlemotionandtemperatureduringplanarshock-particleinteraction. 4.3.2SphericalShockWave Intheprevioussection,theshockwaveisplanar,movingatconstantspeed,andthepropertiesbehindtheshockareuniform.Bycontrast,forthesphericalshockwavegeneratedbyapointsource,theshockshapeisnotplanar,theshockspeedreduceswithtime,andtheowbehindasphericalshockisnon-uniformandunsteady.Toconsidertheinteractionofaparticlewithasphericalshockwave,weemploytheblast-wavetheorydevelopedbyTaylor[ 91 ],andintegratenumericallytheequationsforparticlevelocityandtemperature. Intheplanarshockcase,theonlyparameterforthegassolutionistheshockMachnumberMs.Forthesphericalshockwave,twoparametersarenecessarytodenetheoweld.ThetwoparameterscanbechosenastimeandthetotalenergyreleaseEtotassuggestedbyTaylor.Inthepresentstudy,tobeconsistentwiththeplanarcase,wepickinsteadtheshockMachnumberMsandtheshockradiuswhenithitstheparticle,denotedby~x0.WithMsand~x0,thetimewhenshockhitstheparticle~t0andthetotal 132

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Aupam,t=ug2 Bupvu,t=ug2forMs=2andMeiandAdrian'sker-nel Cupvu,t=ug2forRepr=103andBasset'skernel Dupvu,t=ug2forRepr=103andMeiandAdrian'skernel Figure4-6. TheterminalparticlevelocityasfunctionofMs,Repr,p,andthekernel,whenonlyOneforceisactiveduringtheplanarshock-particleinteraction. 133

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A(Tpdu,t)]TJ /F14 9.963 Tf 9.96 0 Td[(1)=(Tg2)]TJ /F14 9.963 Tf 9.96 0 Td[(1)forMs=2 B(Tpdu,t)]TJ /F14 9.963 Tf 9.96 0 Td[(1)=(Tg2)]TJ /F14 9.963 Tf 9.96 0 Td[(1)forRepr=103 Figure4-7. TheterminalparticletemperatureasfunctionofMs,Repr,andpCpwhenonlythediffusive-unsteadyheat-transfercontributionisactiveduringtheplanarshock-particleinteraction. energyEtotcanbecalculatedinastraightforwardmanner.Thenondimensionalformof~x0isx0=~x0=~dp.Generally,theshockradiuswhenitreachestheparticleismuchlargerthantheparticlesize,sox01.Thentheimpactsoftheshockcurvatureandthechangeinshockspeedduringitsinteractionwiththeparticlearenegligible.Figure 4-8 showsthetimeevolutionofthetotalforceandheattransferwhenx0variesbetween102to104forMs=5,Repr=102,p=10,andCp=1.Itcanbeobservedthatthedifferencesbetweentheresultsaresmall.Iftheradiusoftheshockwhenithitstheparticleislargerthan103timestheparticlesize,theforceandheattransferareindependentofx0. Inthepresentstudy,weassumex01,andthereforeneglectitsinuenceinthefollowing.Asaresult,themaindifferenceofthesphericalshockcasefromtheplanarcaseisthenon-uniformandunsteadynatureoftheowbehindtheshock.Similartotheplanarshockproblem,therearefourdimensionlessparameters:Ms,Repr,p,andCp.AsinTaylor'sblast-wavetheory,theshockwaveisassumedtobeinnitelystrongto 134

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AForce BHeattransfer Figure4-8. Evolutionofthetotalparticleforceandheattransferinsphericalshock-particleinteractionasafunctionofx0.OtherparametersarexedasMs=5,Repr=102,p=10,andCp=1. simplifyshockjumpconditions.Tobeconsistentwiththestrongshockassumption,wehereonlyconsidertherangeof3Ms11. 4.3.2.1Peakvaluesoftheforceandheattransfer Similartotheplanarcase,thepeakvaluesofthedifferentcontributionstotheforceandheattransferareusedtomeasuretheimportanceofunsteadyforceandheattransfer.Theunsteadycontributionstotheoverallforceareagainnormalizedbythequasi-steadycontributionsandplottedinFig. 4-9 asfunctionsofMsandReprforp=104andCp=1.Forsuchlargedensityratio,theparticlecaneffectivelytakentobefrozenduringthepassageoftheshock.Andthus,similartotheplanarcase,theresultsdependweaklyonpandCpprovidedp1.ItcanbeobservedthatintherangeofMsconsideredhere,Repristhedominantparameter.Generally,fpg,max=fqs,maxandfam,max=fqs,maxincrease,andfvu,max=fqs,maxdecreaseswithincreasingRepr.Thetrendsarethussimilartotheplanarshockproblemasexpected.Anditcanbeseenthattheunsteadyforcesarealsosignicantinthisproblemalso;fpg,max=fqs,maxvaries 135

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Afpg,max=fqs,max Bfam,max=fqs,max Cfvu,max=fqs,max Figure4-9. Thepeakvaluesoftheunsteadyforcesdividedbythepeakquasi-steadyforceasfunctionsofMsandReprforp=104andCp=1forsphericalshock-particleinteraction. between0.1to0.45,whilefam,max=fqs,maxvariesbetween0.4to0.9.WhenReprissmall,fvu,max=fqs,maxcanbeaslargeas0.7. Figure 4-10 showsthepeakvaluesoftheunsteadycontributionstotheheattransfernormalizedbythepeakquasi-steadyheattransferasfunctionsofMsandReprforp=104andCp=1.Againforlargepquu,max=qqs,maxandqdu,max=qqs,maxareobservedtobeindependentofpandCp.Itcanbeseenthatquu,max=qqs,maxincreases 136

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Aquu,max=qqs,max Bqdu,max=qqs,max Figure4-10. Thepeakvaluesoftheunsteadyheattransferdividedbythepeakquasi-steadyheattransferasfunctionsofMsandReprforp=104andCp=1forsphericalshock-particleinteraction. withincreasingRepr,butatamuchfasterratethanfpg,max=fqs,max.WhenReprisO(105),quu,max=qqs,maxcanbeashighas50.Bycontrast,qdu,max=qqs,maxincreasestoreachapeakandthendecreaseswithRepr,withamaximumlocatedataroundRepr=312. Ingeneral,weobservethatthepeakvaluesofforceandheattransferinthesphericalshockproblemarequitesimilartothosefoundintheplanarshockproblem.Thisisaconsequenceoftheassumptionthatx01 4.3.2.2Particlevelocityandtemperature ThesimilarityproleofthegasvelocityandtemperaturebehindtheshockasdeterminedbyTaylor'stheoryisshowninFig. 4-11 .Thegasvelocitydecreases,whilethegastemperatureincreasestowardtheorigin.Theunphysciallylargetemperaturestowardtheoriginarecausedbytheuseoftheidealgaslaw.Forthepresentpurpose,thisbehaviorisimmaterialbecauseourprimaryinterestisnotontheregionneartheorigin,andthebehaviorneartheorigindoesnotinuencetheevolutionofparticlesforlargerradii.TheparticlemechanicalandthermalresponsetotheowarepresentedinFig. 4-12 .Boththeparticlevelocityandtemperatureincreasewhentheshockreaches 137

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Figure4-11. Gassolutionofthesphericalshockwaveproblem.xmsdenotesthelocationofthemainshock. theparticle.Ifonlythequasi-steadyforceandheattransferareactive,theparticlevelocityandtemperaturewillincreaseataslowerrate.Aftertheshockpassesovertheparticle,thedifferencebetweenupandupqswillgenerallydecrease,whilethedifferencebetweenTpandTpqsincreases.Themaximumdifferencebetweenupandupqscanbeusedtomeasurethecontributionsoftheunsteadyforcestotheparticlevelocity,andareplottedinFig. 4-13 asfunctionsofMs,Repr,andp.ThedifferencebetweenTpandTpqsincreaseswithtime,thusthemaximumvaluecannotbeusedasameasure.Instead,wetakethevalueofTp)]TJ /F3 11.955 Tf 12.23 0 Td[(Tpqswhentheshockjustpassedtheparticleasameasureofthecontributionoftheunsteadyheattransfertotheparticletemperature.TheresultsareplottedasafunctionofMs,Repr,andpCpinFig. 4-14 ItcanbeobservedfromFig. 4-13 thatphasthekeyinuenceonmax(up)]TJ /F3 11.955 Tf 12.01 0 Td[(upqs)=ug2,consistentwiththeresultsoftheplanarcase.Itisseenthatmax(up)]TJ /F3 11.955 Tf 11.41 0 Td[(upqs)=ug2isinverselyproportionaltop.FortherangeofMsconsideredhere,Mshaslittleinuenceonmax(up)]TJ /F3 11.955 Tf 12.53 0 Td[(upqs)=ug2.WhenReprissmall,max(up)]TJ /F3 11.955 Tf 12.54 0 Td[(upqs)=ug2decreasesasReprincreases.ButasReprbecomeslarge,thedependenceofmax(up)]TJ /F3 11.955 Tf 12.35 0 Td[(upqs)=ug2onReprvanishes.Thetemperaturedifferenceduetoneglectingunsteadyheattransferissimilar.Wendthat(Tp)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpqs)jxs)]TJ /F8 7.97 Tf 6.59 0 Td[(xp=0.5=(Tg2)]TJ /F3 11.955 Tf 11.96 0 Td[(Tg1)isgenerallyinverselyproportionaltopCp. 138

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AVelocity BTemperature Figure4-12. ParticlesolutionofthesphericalshockwaveproblemforMs=5,Repr=102,p=10,Cp=1,andx0=104. ARepr=103 BMs=5 Figure4-13. Themaximumvelocitydifference,max(up)]TJ /F3 11.955 Tf 11.95 0 Td[(upqs)=ug2,betweentheparticlevelocitycomputedwhenallforcesareactiveandthevelocitycomputedonlywiththequasi-steadyforce,forsphericalshock-particleinteraction. 139

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ARepr=103 BMs=5 Figure4-14. Thetemperaturedifference,(Tp)]TJ /F3 11.955 Tf 11.95 0 Td[(Tpqs)jxs)]TJ /F8 7.97 Tf 6.59 0 Td[(xp=0.5=(Tg2)]TJ /F3 11.955 Tf 11.96 0 Td[(Tg1),betweentheparticletemperaturewhenallheat-transfercontributionsareactiveandthetemperaturecomputedonlywiththequasi-steadyheattransfer,atthetimewhenthesphericalshockjustpassestheparticle. 4.3.2.3Long-terminuencesofunsteadyforceandheattransfer Figure 4-15 showstheevolutionsofthedifferentcontributionstoforceandheattransferoverlongtimeforMs=5,Repr=102,p=10,Cp=1andx0=104.Figure 4-16 showsthecorrespondingtimeevolutionoftheparticlevelocityandtemperature,comparedwiththeresultswhenonlythequasi-steadymechanismsareactive.Therefore,thedifferencebetweenthetwovelocity/temperatureprolescanbeviewedasanerrorifunsteadycontributionsareignored.Severalobservationscanbemadefromthesegures.First,similartotheplanar-shockcase,theunsteadycontributionsaremuchlargerthanthequasi-steadycontributionstotheforceandheattransferwhentheshockispassingovertheparticle.Asaresult,theparticlevelocityandtemperatureincreasesmuchfasteratearlytimescomparedtotheresultswithonlyquasi-steadyforceandheattransferareactive.Second,theunsteadycontributionstotheforceandheattransferbecomenegativeafterthepassageoftheshock.Asaresult,theparticlevelocityandtemperaturewithunsteadycontributionsisatearlytimelarger 140

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AForces BHeattransfer Figure4-15. Evolutionofdifferentforcesandheattransferinthesphericalshock-particleinteractionforMs=5,Repr=102,p=10,Cp=1,andx0=104. thanthatwithoutunsteadycontributions,buteventuallybecomessmaller.Theparticletemperatureexhibitsasimilartrend.Third,incontrasttotheplanarcase,theparticlewillnotreachthepost-shockvelocityasthesurroundinggasvelocitydecreaseswhentheparticleismovingawayfromtheshockfront.Theparticlewillreachamaximumvelocitywhenitovershootsthesurroundinggasvelocity.Becauseofthedecelerationcausedbytheunsteadymechanisms,thismaximumvelocityoftheparticlewithunsteadycontributionsissmallerthanthatwithout. 4.3.3Brode'sBlast-WaveProblem Thethirdcaseconsideredhereisthesphericalblastwavegeneratedbyasuddenreleaseofcompressedgasinitiallycontainedinasphereofniteradius.ThisproblemwasrstconsideredbyBrode[ 13 ].Itcanbeinterpretedasanextensionofthepoint-sourceproblemconsideredbyTaylor[ 91 ]tonite-size-sourcecase.Attemptstosolvethisproblemanalytically,seee.g.,McFadden[ 60 ]andFriedman[ 32 ]wereonlypartiallysuccessfulbecauseapproximationshadtobemadetomaketheproblemtractable.HereweuseanumericalapproachdescribedinHaselbacher[ 37 ],Haselbacheretal.[ 41 ]andLingetal.[ 51 ]tosolveEqs.( 4 )to( 4 )forthegasow. 141

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AVelocity BTemperature Figure4-16. Evolutionoftheparticlevelocityandtemperatureincludingallforcesandheattransfer,comparedtotheresultswithonlyquasi-steadyforceandheattransferforMs=5,Repr=102,p=10,Cp=1,andx0=104insphericalshock-particleinteraction. Asashock-capturingschemeisusedhere,theshockwillbesmearedduetonumericaldissipation.Therefore,negridresolutionisnecessarytoguaranteethattheshockthicknessismuchsmallerthantheparticlesize.Thus,thegridsizeisdeterminednotonlybythegasowbutalsobytheparticlesize.Hereweuse~dp=~x20.Itshouldbenotedherethatwedonotadvocateusingsuchasmallgridspacinginpracticalcomputations.Herewedesiretoavoidnumericalissuespertainingtoresolutionbyemployingverynegrid,andtherebyfocusattentiononthequestionofunsteadycontributiontoforceandheattransfer.Theappropriatemodelingofforcesandheattransferforpracticalgridresolutionisatopicforfutureinvestigation.Equations( 4 )to( 4 )andEqs.( 4 )to( 4 )areintegratednumericallysimilartothepreviousproblems. Comparedtothepoint-sourceprobleminsection 4.3.2 ,theblastwavefromanite-size-sourceismuchmorecomplicated.Whereasthegasowofthepoint-sourceproblemcanbefullydeterminedbytwoparameters,thenite-size-sourceproblem 142

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requiresthreeparameters(time,initialpressureratio,andinitialtemperatureratio).Furthermore,althoughtheowbehindtheshockinthepoint-sourceblastwaveisnon-uniformandunsteady,theproleissimpleandself-similar.Butinthenite-size-sourceproblem,amainshock,anexpansionwave,acontactdiscontinuity,andasecondaryshockwaveexist.Thesewavesinteractwitheachother,leadingtoreectionsandacomplicatedunsteadyow,seetheschematicoftheprobleminFig. 4-17 .Thenumbers0to4inthegureindicateregionsseparatedbywavesandareusedassubscriptsinthefollowing.Asthefocusofthispaperisontheparticleforceandheattransfer,wepickjustonesetofparametersforthegassolution,andvaryonlytheparticleparameters.TheinitialpressureandtemperatureratioaretakentobethoseconsideredbyBrode,i.e.,~pg4=~pg0=121and~Tg4=~Tg0=1.Thegaspropertiesinregion0correspondtoairatstandardconditions. Thegassolutionoftheproblemcanbeperfectlyscaledbythecontainerradius~Rsc.However,whenparticlesareadded,~Rscbecomesrelevanttotheparticlemotion.Forthenite-size-sourceproblem,thecontainerradiusisactuallyabetterscaletocharacterizetheowthantheparticlediameter.Therefore,forconvenienceinpresentingtheresults,wedeneanewsetofnondimensionalvariablesherethataredenotedbyhat.Comparedtothenondimensionalizationinsection 4.2 ,theonlychangeisthattheparticlediameter~dpisreplacedby~Rscasthecharacteristiclengthscale.Thespeedofsound~ag0,density~g0,andtemperature~Tg0oftheambientairareusedasthevelocity,density,andtemperaturescales.Basedonthenewnondimensionalization,^x=~x=~Rsc=x=Rscand^t=~t=(~Rsc=~ag0)=t=Rsc.Thenondimensionalvariablesthatdonotinvolvelengthscaleareidenticaltothosedenedpreviously,i.e.,^T=~T=~Tg0and^u=~u=~ag0. WefollowBrode[ 13 ]andset~Rsc=0.0254m.Inaddition,wechooseg=1.4,Rg=287J=(kgK),Pr=0.72,and~Cp=1600J=(kgK).Theparticleparametersinclude 143

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Figure4-17. Schematicillustrationofparticleinteractionwithasphericalblastwavegeneratedbyacompressedgas. thediameter~dp,density~p,andspecicheat~Cp.Wevaried~dpfrom100mto1mm(^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3)]TJ /F4 11.955 Tf 11.95 0 Td[(410)]TJ /F5 7.97 Tf 6.59 0 Td[(2)and~pfrom141to1410kg=m3(^p=^g4=1)]TJ /F4 11.955 Tf 11.96 0 Td[(10). Aparticleisinitiallylocatedjustoutsideofthecontainer.Beforebeingreachedbythemainsphericalshock,itisstationaryandinequilibriumwiththeambientgas.Figure 4-18 showsthetrajectoriesofparticlesofdifferentdensityanddiameterfor^xp0=1.2alongwiththemainshock,thecontactdiscontinuity,andthesecondaryshockwaveinan^x-^tdiagram.Duetoinertia,theparticlesinitiallymoveslowly,andarecaughtupbytheoutgoingcontactdiscontinuityandsecondaryshockwave.Theparticlescontinuestoacceleratebehindthesecondaryshock,whileboththecontactdiscontinuityandsecondaryshockwavearedeceleratingduetotheradialeffect.Asaresult,theparticlesareabletocatchupwiththecontactdiscontinuity.Thelargerparticle(^dp=410)]TJ /F5 7.97 Tf 6.58 0 Td[(2)andtheheavierparticle(^p=^g4=10)areobservedtocrossthesecondaryshock.Theoretically,itispossibleforaparticletocrossthemainshock,asobservedbyLanovetsetal.[ 49 ]andZhangetal.[ 101 ].Thisphenomenadoesnotoccurfortheparameterrangeweconsiderhere. 144

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Figure4-18. Trajectoriesofparticlesofdifferentdensityanddiameterin^x-^tdiagramfor^xp0=1.2inblastwave-particleinteraction. Astheparticlesinteractinacomplexmannerwiththewavesinthegasow,theparticle-waveinteractionscannotbecapturedcorrectlyiftheunsteadycontributionsareignored.Figure 4-19A showstwosetsofresultsofthetemporalevolutionoftheparticlevelocityfor^xp0=1.2,^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(2,and^p=^g4=f1,10g.Unsteadycontributionsareincludedintherstsetandneglectedinthesecond.Itcanbeobservedthatneglectingtheunsteadyforceleadstosignicanterrorsfortheparticlewith^p=^g4=1.Whentheparticledensityincreases,theerrordecreases,whichisconsistentwithourobservationfortheplanarandsphericalshockproblems.Figure 4-19B showsthetemporalevolutionoftheforcebudget.Itcanbeseenthattheunsteadycontributionsaredominantwhentheparticleinteractswiththeshockwave.Becausethedurationoftheinteractionissoshort,thequasi-steadyforcedominatesmostofthetime.However,theimpactoftheunsteadyforceduringtheseshortperiodssignicantlyinuencestheoverallmotionoftheparticle.Amongtheunsteadycontributions,theinviscidunsteadycontributionsaregenerallymuchlargerthantheviscousunsteadycontribution. Figure 4-20 showsthetemporalevolutionofparticletemperatureandtheheat-transferbudget.Again,weobservethatunsteadyheattransferdominatestheoverallheat-transfer 145

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AParticlevelocity BParticleforcebudgetfor^p=^g4=1 Figure4-19. Timeevolutionofparticlevelocityandforcebudgetfor^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(2and^xp0=1.2inblastwave-particleinteraction. whentheparticleinteractswiththeshockwave.Therefore,neglectingunsteadyeffectsleadstosignicanterrorintheparticletemperature.Similartotheparticlevelocity,theerrordecreaseswiththeparticledensity.Incontrasttotheparticlevelocity,whentheparticlecrossesthecontactdiscontinuity,thegastemperatureincreasesatamuchfasterratewithunsteadyheat-transferincluded.Asthegasthermalaccelerationisverylargewhentheparticlecrossesthecontactdiscontinuity,theunsteadyheattransferbecomesverylarge.Tosummarize,unsteadymechanismsarenotonlyimportantincapturingtheparticleinteractionwithshockwaves,butalsowithcontactdiscontinuities. 4.4Discussion Compressibilityeffectonadded-massforce .Equation( 4 )withCM=1=2isbasicallytheincompressibleformoftheadded-massforce,withtheexceptionofthedensityappearinginsidethederivatives.Usingthisadded-massforceformulaignorestwocompressibleeffects:thehistoryeffectandtheMach-numbereffect.Ignoringthehistoryeffect,namelyusingEq.( 4 )toapproximateEq.( 4 ),willoverpredictthepeakvalueoftheadded-massforce,asshowninParmaretal.[ 67 ],buttheneteffectontheparticlevelocitywillnotchange.Furthermore,Parmaretal.[ 66 ]also 146

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AParticletemperature BParticleheat-transferbudgetfor^p=^g4=1 Figure4-20. Timeevolutionofparticlevelocityandforcebudgetfor^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(2and^xp0=1.2inblastwave-particleinteraction. showedthatthekernelfunctionvaluesincreaseastheparticleMachnumberincreases,resultinginincreasesintheeffectiveadded-masscoefcient.Forexample,theeffectiveadded-masscoefcientforaparticleMachnumberof0.5canbetwicethevalueofCMatzeroMachnumber.Unfortunately,theadded-massforceforMachnumbershigherthanthecriticalMachnumberisnotavailableintheliterature. Compressibilityandnite-Reynolds-numbereffectsonviscous-unsteadyforceanddiffusive-unsteadyheattransfer .Theincompressibleformofthekernelisusedintheviscous-unsteadyforceanddiffusive-unsteadyheat-transfer.IthasbeenshownbyParmaretal.[ 69 ]thatthecorrectionfunctionC()fortheviscous-unsteadyforcekernel,whichisdenedasC()=Kvu()=Kvu,B(),wasboundedbetween0.44and1.45.Therefore,theerrorofthepeakvaluesoftheviscous-unsteadyforcepresentedhereisboundedtolessthan45%.Furthermore,theBassetkernelisusedtocomputethediffusive-unsteadycontributiontoheattransfer,whichbasicallyignoresboththecompressibilityandnite-Reynolds-numbereffect.Accordingtothenite-Reynolds-numbereffectonviscous-unsteadyforce,itisexpectedthatthediffusive-unsteadycontributiontoheattransfertodecayfasterinrealitydueto 147

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non-lineareffects.Asaresult,theneteffectofthediffusive-unsteadycontributionisoverpredicted.However,asthereisnosolidknowledgeonniteReynolds-numberkernelfordiffusive-unsteadyheattransferincompressibleows,herewehaveusedtheBassetkernelasanapproximation. Effectofdifferentformsofsurfaceandvolumeintegralsofgasquantities .InEqs.( 4 )to( 4 ),andEqs.( 4 )to( 4 ),surfaceorvolumeintegralsareneededtoevaluatetheeffectivegasquantitiesandtheirsubstantialderivatives.Therearemultiplewaystocomputetheseintegrals,forexample,therstterm( D Dtv (gug)v)inEq.( 4 )canbealsocalculatedas D(gug)=Dtv.Foraplanarshockpassingoveraxedparticle,theexpressionoftherstformisgiveninAppendix A .Bysimilarapproach,theexpressionofthesecondformcanalsobederivedanalyticallyas D Dt(gug)v=g2ug2(us)]TJ /F3 11.955 Tf 12.96 0 Td[(ug2=2)6(y)]TJ /F3 11.955 Tf 12.95 0 Td[(y2).ItcanbeprovedeasilythatthedifferencebetweenthesetwointegralformsvanisheswhenshockMachnumberapproachesunity;whentheshockMachnumbergoestoinnity,thesecondformhasonlymaximum16%decitcomparedtotherstform. 4.5Conclusions Theaimofthisarticleistoanalyzesystematicallytheimportanceoftheunsteadycontributionstoforceandheattransferinshock-particleinteractions.Weconsideredrstthesimplestcase:aplanarshockpassingoveraninitiallystationaryparticle.Theresultsshowthatthepeakvaluesofunsteadycontributionsnormalizedbythequasi-steadycontributiontobothforceandheattransferareonlyfunctionsofshockMachnumberMsandReynoldsnumberRepr.ForawiderangeofMsandRepr,unsteadycontributionscanbemuchlargerthanquasi-steadycontributions.Astheunsteadycontributionstobothforceandheattransferareactiveforshorttimesonlycomparedtothequasi-steadycontributions,theirneteffectontheparticlevelocityandtemperatureissmall.Thesecondproblemconsideredisthatofasphericalshockwavegeneratedbyapoint-sourceblast.Theadditionalcomplexitiesaretheshockdecayandthe 148

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unsteadysolutionbehindtheshock.Theresultsofthepeakvaluesoftheunsteadycontributionstoforceandheattransferaresimilartotheplanarshockproblem.Asthepost-shockgasowisnon-uniformandunsteady,theunsteadymechanismscontinuetoinuencetheparticlemotionaftertheinteractionwiththeshock.Theresultswithonlythequasi-steadycontributionsactiveshowthattheerrorduetoneglectingunsteadycontributionsarelonglastingandevenmagnied.Finally,weconsidertheblastwavegeneratedbyanite-size-source.Thegasowinthisproblemiscomplicatedbythepresenceofmultiplewavesandreections.Theparticleafterinteractingwiththemainshockmayinteractwithotherwavesmultipletimes.Theresultsshowthattheinteractionsbetweentheparticleandthesewaveswillnotbecapturedcorrectly,andcanintroducesignicanterrors,ifunsteadycontributionsareignored.Themainmessageofthisarticleisthusthat,althoughunsteadycontributionstoforceandheattransferareonlyactiveforshorttimes,theireffectsonthelongtermparticlemotioncanstillbesignicant.Theresultspresentedinthispapercanbeusedtoestimatetheerrorifoneignoresunsteadycontributionstoforceandheattransferinthesimulations.Ithasbeenveriedthattheseconclusionsarenotaffectedqualitativelybyapproximationsinthiswork. 149

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CHAPTER5ANALYSISOFPARTICLEDISPERSALBYBLASTWAVES 5.1Introduction Particledispersalbyblastwavesisaninterestingphenomenonthatcanbeobservedinnature,suchasinvolcaniceruptions,seeChojnickietal.[ 20 ],andinindustrialapplications,suchasindetonationsofmultiphaseexplosives,seeZhangetal.[ 101 ]andTanguayetal.[ 88 ].Asasimpleidealizationoftheseproblems,wecanconsiderthesuddenreleaseofahighlycompressedsphericalgas-particlemixture.Whenthemixtureisexposedtoalowerpressureenvironment,astrongblastwaveisgeneratedandtheparticlesaredispersedoutwardathighspeed.Thetimeevolutionofthegas-particlemixtureisveryrapid,makingmeasurementsthroughexperimentaltechniquesverydifcult.Therefore,themodelingandsimulationapproachisanimportanttoolininvestigatingthedispersalofparticlesbysphericalblastwaves. Tofocusonthefundamentalphysicsofparticledispersalbyblastwaves,inthisworkweemployaspecicproblemofsimplegeometry.AschematicoftheproblemtobeconsideredhereisshowninFigure 5-1 .Thegas-particlemixtureisinitiallystoredinasphericalcontainerofradius~Rscandsurroundedbyairatstandardatmosphericconditions.At~t=0,thecontainerisremovedandinstantaneouslythemixtureisexposedtothesurroundings,leadingtoablastwaveandtheoutwardmotionoftheparticles. Theblastwavegeneratedbyasphereofpressurizedgaswithoutparticlesisaclassicalproblem.Theblast-wavetheorydevelopedbyTaylor[ 91 92 ],vonNeumann[ 96 ],andSedov[ 79 ]givesananalyticalsolutionforasphericalblastwavegeneratedbyapointsourceofenergy.Brode[ 13 ]extendedtheinvestigationfromthepoint-sourceproblemtoanite-sizesourceofthekindshowninFig. 5-1 usingnumericalsimulations.Theresultswerecomparedwiththepoint-sourcetheoryanditwasfoundthattheblastwavegeneratedbyanite-size-sourceismorecomplicated.Otherstudiesof 150

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Figure5-1. Schematicillustrationofparticledispersalinasphericalblastgeneratedfromacompressedgas-particlemixture.Theshadedareaindicatestheexistenceoftheparticles. thesphericalblast-waveproblemincludetheexperimentsofBoyer[ 11 ]andBaker[ 2 ],simulationsbyBrode[ 13 15 ]andLiuetal.[ 54 ]),andthe(approximate)analyticalinvestigationsbyFriedman[ 32 ]andMcFadden[ 60 ].Agoodsummaryoftheevolutionofblast-wavestudiesisgiveninSachdev[ 76 ]. Thegenerationofsphericalblastwavesbyanite-size-sourcewascalledthesphericalshock-tubeblastbyBrode[ 14 ],astheproblemsettingandtheowatearlystagesaresimilartotheplanarshock-tubeproblem.However,theradialeffecthasastronginuenceandeventuallymakesthesphericalblastwavesubstantiallydifferent.Assumingsphericalsymmetry,asimpledescriptionoftheowisasfollows.ItcanbeseenfromFig. 5-1 that,whenthecompressedgasisreleased,ashockwave(theso-calledmainshock),anexpansionfan,andacontactdiscontinuityaregenerated,similartotheplanarshock-tubeproblem.Theshockwaveandthecontactdiscontinuitytraveloutward,butincontrasttotheplanarcase,theirspeedsdonotremainconstant.Instead,thespeedsdecreasewithtime.Here,wedenotetheinnerandouterboundariesoftheexpansionfanastheheadandtail,respectively.Duetotheover-expansionarisingfromtheradialeffect,thepressuretotheleftofthetailofthe 151

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expansionfanislowerthanontheright,resultingintheformationofasecondaryshockwave.Asthepressuregradientacrossitispositive,thesecondaryshockisanimplodingshock.Butduetothelargeinitialgasvelocity,thesecondaryshockisinitiallysweptoutward,beforeeventuallyturnsinward.Similarly,thecontactdiscontinuityalsoreachesamaximumradialpositionandturnsinward. Theformationofthesphericalblastwavescanbefurtherclariedbylookingatthecharacteristicsoftheowinthe^x-^tdiagram,seeFig. 5-2 forairwithinitialconditionsas~pg4=~pg1=121and~Tg4=~Tg1=1.Severalobservationscanbemade.First,duetotheradialeffect,thenegativecharacteristicsintheexpansionfancurvetowardthepositive^x-direction.Themainshockbecomesweaker,sothenegativecharacteristicsfromthemainshockinclinetowardthenegative^x-direction.Therefore,thenegativecharacteristicsarisingfromdifferentsourceswilltendtointersectatthetailoftheexpansionfan,whereasecondaryshockwaveforms.Second,theslopesofthecharacteristicsintheregionbetweenthesecondaryshockwaveandthecontactdiscontinuity,andtheregionbetweenthecontactdiscontinuityandthemainshockchangeintimeandspace.Therefore,theowisnon-uniformandunsteadyintheseregions,whichisdifferentfromtheplanarcase.Third,aftertheexpansionwavesreectfromtheorigin,thenegativecharacteristicsinsidetheexpansionfancurvebacktowardthenegative^x-directionagain.Therefore,thespeedofthesecondaryshockwavedecreasesandtheshockmovesinward.Itshouldbenotedherethat,theassumptionofsphericalsymmetryisvalidonlyforshorttimes.Forlongtimes,theeffectsofgravityanddiffusionbecomeimportant,whicharenotconsideredinthiswork. Hereweconsidertheextensionofthenite-sourcesphericalblastproblemtomultiphaseow.Thus,asshowninFig. 5-1 thenite-sizedsphericalcontainerisinitiallylledwithacompressedgas-particlemixture.Withtheadditionofparticles,theproblembecomesmorecomplex.Oncethecompressedgas-particlemixtureisreleased,theparticlesaredrivenbythegasowandmoveoutward.Duetoinertia,theparticlefront 152

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Figure5-2. Thecharacteristicsofthegasowofthenite-sourcesphericalblastwaveinthex-tdiagramforairwithinitialconditions~pg4=~pg1=121,and~Tg4=~Tg1=1. (theboundaryoftheparticlecloud)initiallylagsbehindthecontactdiscontinuity.Intheplanarshock-tubeproblem,theparticlefrontwillnevercatchupwiththecontactdiscontinuity,seeLingetal.[ 51 ].Butinthesphericalcase,asthecontactdiscontinuitydeceleratesandreversesdirection,theparticlefrontmaycrossthecontactdiscontinuityandmoveintotheshock-compressedambientair.Infact,ithasbeenobservedthatundersomeconditions,theparticlefrontcanevencrossthemainshock,seeLanovetsetal.[ 49 ]andZhangetal.[ 101 ]. Inmanyapplications,thedispersalofparticlesresultingfromthesphericalblastisofinterest.Inparticular,wewanttoanswerquestionssuchasunderwhatconditionswillparticlesbepropelledaheadofthemainshock?Whatfractionofparticleswillbepropelledaheadofthemainshock?Similarly,underwhatconditionwillparticlesbepropelledaheadofthecontactdiscontinuityandwhatfractionofparticleswillgetaheadofthecontactdiscontinuity?Forexample,incaseofdispersalofreactivemetalparticlesinamultiphaseexplosion,particlesthatremainwithinthecontactdiscontinuitycanburnonlyanaerobically,whileparticlesoutsidethecontactdiscontinuitycanburnaerobically. 153

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Furthermore,particlesaheadoftheshockaresubjectedtoambientair,whilethosejustbehindtheshockaresubjectedtoshockheatedambientair. Inordertoreliablycomputeparticledispersalresultingfrommultiphaseexplosions,itisessentialtoaccuratelyaccountforthemomentumexchangebetweenanindividualparticleandthesurroundinggas.Similarly,tocapturethethermalevolutionoftheparticledistribution,itisimportanttoaccuratelyaccountforthethermalexchange.Inthepresentapplication,particlesinteractwiththemainandsecondaryshockwaves,thecontactdiscontinuity,andexpansionwavesastheydisperseoutward.Therefore,toyieldaccuratepredictionofparticledispersal,theinteractionsbetweenparticlesandunsteadycompressibleowfeaturesmustbeaccuratelycapturedintheparticleforceandheat-transfermodels.InChapter4,weestablisheddrag-forceandheat-transfermodelsthataccountfortheinteractionswithshockwavesandcontactdiscontinuities.Themodelswererstappliedtotheproblemofaparticleinteractingwithaplanarshockwave.Analyticalresultsestablishedthesignicanceoftheunsteadyforceandheat-transfercontributionstoshock-particleinteractionoverwiderangesofshockMachnumber,particleReynoldsnumber,andparticledensityratio.Themodelswerealsoappliedtotheproblemofparticlesinteractingwithasphericalblastwavegeneratedbyanite-sizesource.Theresultsshowedclearlythatneglectingunsteadyforceandheat-transfercontributionsintroducesignicanterrors.Theseresultsaresignicantbecausethemajorityofsimulationsofcompressiblemultiphaseowsonlytakeintoaccountquasi-steadycontributions.Here,themultiphaseowmodelofChapterisappliedtotheproblemofparticledispersalbyasphericalblastwavepicturedinFig. 5-1 .Asystematicstudywillbecarriedouttodeterminetheinuenceofrelevantparameters,suchasparticledensityratio,ratiosoftheparticlediameterandparticleinitiallocationtothecontainerradius,ratiosoftheinitialpressureanddensityofthecompressedgastothoseoftheambientair.Asthefocusofthisworkisontheparticlebehavior,wekeepthepressureratioanddensityratioxed,andvarytheparticleparameters. 154

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5.2GoverningEquationsandNumericalMethods Inthepresentproblem,theparticlesareassumedtobespherical,rigid,andinert.Theparticlesareinitiallyuniformlydistributedinsidethesphericalcontainer,andareinthermalequilibriumwiththecompressedgas.Thesolutiondomainislargeenoughtoeliminatetheinuenceoftheboundaryconditionsandcanthusbeconsideredtobeunbounded.Asaresult,groundinteractionsandothershockreectionsarenotconsidered.Becausewealsoneglectgravity,themotionoftheparticlesandthegasisalwayssphericallysymmetric.Thegasobeystheidealgaslawandtheuidatthemacro-scalewillbeconsideredtobeinviscid.Atthemicro-scale,theinteractionoftheparticleswiththesurroundinggaswillincludeviscouseffectsthroughappropriatemodels.Furthermore,thethermalandtransportpropertiesofthegasareassumedtobeconstant.Thegoverningequationswillbebrieypresentedbelow.ThedetailsoftheforceandheattransfermodelsandquantitativeinformationontheimportanceofthedifferentcontributionstothemodelscanbefoundinChapter4.Akeyassumptioninthepresentworkisthatthemassandvolumefractionsoftheparticlesaresmall,sothatthemultiphaseowcanbeassumedtobeone-waycoupledandparticlecolllitionsareignored.Asaresult,theparticlemotionisdictatedonlybytheow,buttheparticlesareassumednottoinuencethegasow.Insomepracticalapplications,theassumptionofone-waycouplingmaybeinvalid.However,two-waycoupledsimulationsintroduceadditionalcomplexities.Therefore,itisimportanttoestablishthebasicphysicsofmultiphaseowarisingfromsphericalblastblastinaone-waycoupledcontextbeforeconsideringtwo-waycoupledsimulations. Throughout,thesuperscriptsgandpindicatesquantitiesassociatedwiththegasandtheparticles,respectively.Thetildeindicatesdimensionalquantities,andthehatrepresentsnondimensionalquantitiesdenedasfollows.AssuggestedinChapter4,fortheblastwavefromanite-sizesource,thesphericalcontainerradius~Rscandthepre-shockspeedofsound~ag0aretakentobethetypicallengthandvelocityscales.The 155

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timescalecanbewrittenas~Rsc=~ag0.Thepre-shockgasthermalproperties,suchasthedensity~g0,andtemperature~Tg0areemployedasreferencescales.Thesubscript0denotesthequantitiesofambientairasshowninFig. 5-1 5.2.1GoverningEquations ThegasowisgovernedbytheEulerequations.Indimensionlessform,theycanbewrittenas@^g @^t+@^g^ug @^x+2^g^ug ^x=0, (5)@^g^ug @^t+@^g(^ug)2 @^x+2^g(^ug)2 ^x=)]TJ /F9 11.955 Tf 10.5 8.08 Td[(@^pg @^x, (5)@^g^Eg @t+@^g^Hg^ug @^x+2^g^Hg^ug ^x=0, (5) where^g,^ug,^pg,^Eg,and^Hgrepresentdensity,velocity,pressure,totalenergy,andtotalenthalpyofgas. Thecorrespondingposition,momentum,andenergyequationsoftheparticleinnondimensionalformared^xp d^t=^up, (5)d^up d^t=^fp=^fqs+^fpg+^fam+^fvu, (5)d^Tp d^t=^qp=^qqs+^quu+^qdu. (5) where^xp,^up,and^Tprepresentposition,velocity,andtemperatureoftheparticle.Theforceperunitmass(particleacceleration)andheattransferperunitheatcapacity(particlethermalacceleration)innondimensionalformsaredenotedby^fpand^qp,respectively.Intheaboveexpressions,^fqs,^fpg,^fam,and^fvudenotethequasi-steady,pressure-gradient,added-mass,andviscous-unsteadycontributionstotheforce,respectively.Similarly,qqs,quu,andqdudenotethequasi-steady,undisturbed-unsteady,anddiffusiveunsteadycontributionstotheoverallheattransfer. 156

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Theexpressionsfortheforcecontributionsare^fqs= ^ugs)]TJ /F4 11.955 Tf 12.19 0 Td[(^up ^pCDRep 24, (5)^fpg=1 ^p ^gD^ug D^tv, (5)^fam=CM ^p D D^t(^g^ug)v)]TJ ET q .478 w 216.19 -98.54 m 264.32 -98.54 l S Q BT /F3 11.955 Tf 220.11 -108.52 Td[(d d^t(^g^up)v!, (5)^fvu=1 ^g^pZ^tKvu(^t)]TJ /F4 11.955 Tf 13.13 0 Td[(^) D D^t(^g^ug)s)]TJ ET q .478 w 300.05 -138.39 m 348.18 -138.39 l S Q BT /F3 11.955 Tf 303.97 -148.37 Td[(d d^t(^g^up)s!d^, (5) whereCDisthequasi-steadydragcoefcientforwhichthecorrelationofParmaretal.[ 70 ]isemployed.ThedenitionsoftheparticleReynoldsnumberRepandMachnumberMpare Rep=~gj~up)]TJ /F4 11.955 Tf 12.2 0 Td[(~ugj~dp ~g,(5) and Mp=j~up)]TJ /F4 11.955 Tf 12.2 0 Td[(~ugj ~ag.(5) Theeffectiveadded-masscoefcientisdenotedbyCM.ForasphericalparticleatMp=0,CMtakesthevalueof1=2.ThecompressibilityeffectonCMisnotconsideredhereforsimplicity.AjusticationofthischoiceisgiveninChapter4.Theviscous-unsteadyforcekernelisdenotedbyKvu,theformulationofwhichforniteReynoldsnumberisgivenindimensionlessformbyMeiandAdrian[ 61 ].InEq.( 5 )and( 5 ),^pisthenondimensionalizedparticlemechanicalresponsetime ^p=~p(~dp)2 18~g~ag0 ~Rsc.(5) 157

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Theexpressionsoftheheat-transfercontributionsare^qqs= ^Tgs)]TJ /F4 11.955 Tf 13.95 2.66 Td[(^Tp ^pNu 2, (5)^quu=1 ^p^Cp ^gD^Tg D^t!v, (5)^qdu=p Pr ^g^pZ^tKdu(^t)]TJ /F4 11.955 Tf 13.14 0 Td[(^) D D^t(^g^Tg)s)]TJ ET q .478 w 301.71 -110.67 m 353.34 -110.67 l S Q BT /F3 11.955 Tf 305.64 -120.65 Td[(d d^t(^g^Tp)s!d^. (5) whereNuistheNusseltnumber.TheheattransfercorrelationofFoxetal.[ 31 ]isemployedtocalculatetheNusseltnumberatniteMachandReynoldsnumbers.NotethatthecorrelationsofCDandNuusedherearefunctionsoftheparticleMachandReynoldsnumbers.Thediffusive-unsteadyforcekernelindimensionlessformisdenotedbyKdu,theformulationofwhichforzeroReynoldsnumberisgivenbyBasset[ 8 ].InEq.( 5 )and( 5 ),^pisthenondimensionalizedparticlethermalresponsetime, ^p=~p~Cp(~dp)2 12~kg~ag0 ~Rsc.(5) BarswithasuperscriptsorvdenotegasquantitiesaveragedovertheparticlesurfaceSporvolumeVp,respectively,i.e., ugs=1 SpISpugds, (5) ugv=1 VpIVpugdv. (5) 5.2.2NumericalApproach ThenumericalapproachfortheEulerequationisbasedonthecell-centerednite-volumemethodology.TheinvisciduxforthegasiscalculatedbytheapproximateRiemannsolverofRoe[ 73 ].Theface-statesareobtainedbyasimpliedsecond-orderaccurateweightedessentiallynon-oscillatory(WENO)scheme,see,e.g.,JiangandShu[ 43 ],whichmodiesthegradientscomputedusingtheleast-squaresreconstructionmethodofBarth[ 7 ].Thetemporalintegrationmethodforbothgasandparticle 158

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equationsisthefourth-orderRunge-Kuttamethod.Thesolutionmethodhasbeenextensivelyveriedandvalidatedforgasandgas-particleowswithandwithoutshockwaves.Forbrevity,theresultsfromthesestudiesarenotreproducedhere.Formoredetails,seeHaselbacher[ 37 ]andHaselbacheretal.[ 41 ]. 5.3Results Asdescribedpreviously,theblast-wavestructuregeneratedbyasphereofcompressedgasisquitecomplicated,andinvolvesthemainshock,thegascontactdiscontinuity,thesecondaryshock,andtheexpansionfan.Forconvenienceinpresentingtheresults,thedifferentowregionsarenumberedasinFig. 5-1 .Thus,thenumbers4,3,2,1,and0representtheregionsinsidetheexpansionfan,betweentheexpansionfanandthesecondaryshock,betweenthesecondaryshockandthecontactdiscontinuity,betweenthecontactdiscontinuityandthemainshock,andaheadofthemainshock,respectively.Theaboveseparationintovedifferentregionsremainsvalidonlyatearlytimes.Oncetheheadoftheexpansionfanreachesthecenterofthesphereandlaterasthesecondaryshockwavereectsfromtheoriginandinteractswiththecontactdiscontinuity,theresultingwavestructureofthegasowbecomesmorecomplex. Inthecomputations,thecompressedgasandtheambientgasarebothtakentobeairwithg=1.4,~Rg=287J=(kgK),andPr=0.72.Inthepresentone-waycoupledlimit,thegassolutionisfullydeterminedbytheinitialconditionsofthecompressedgas(pg4,g4),andtheambientair(pg0,g0).Intheabsenceofparticles,thesphericalcontainerradius~Rscistheonlylengthscale.Gassolutionsofdifferentcontainersizescanbecollapsedbyscalingby~Rsc.Onceparticlesareintroduced,anadditionallengthscaleisintroduced,andtheparticlesolutionwillnotbeperfectlyscaledby~Rsc.Givenaspecicgasow,theparticlesolutioncanbefullydeterminedby~dp,~p,and~Cp.Asthefocusofthisworkconcernsthedispersalofparticlesbyblastwaves,weconsideronlyonesetofinitialconditionsforthegas,andvaryparticleparameters.Theparametersfor 159

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Figure5-3. Thetimeperiodandspatialregiondivisiononthex-tdiagram. thegasowarethoseconsideredbyBrode[ 13 ],i.e.,~pg4=~pg0=121,~Tg4=~Tg0=1,andRsc=0.0254m.Theparticlediameterisvariedfrom^dp=~dp=~Rsc=410)]TJ /F5 7.97 Tf 6.59 0 Td[(4to410)]TJ /F5 7.97 Tf 6.59 0 Td[(2(~dp=10mto1mm).Theparticledensityisvariedfrom^p=~p=~g0=121to12100(~p=141to14100kg=m3or^p=^g4=1to100).Wetake^Cp=~Cp=~Cgp=1.6. 5.3.1EvolutionoftheGasFlow Figures 5-4 and 5-5 showthetemporalevolutionofthegasdensity,velocity,andtemperatureproles.Tohelpinterprettheresultstobepresented,wedividethetimeevolutionintothreeperiodsasshowninFig. 5-3 .Therstperiodstartsat^t=0andlastsuntilthesecondaryshockreversesdirection.Thesecondperiodstartswhenthesecondaryshockreversesdirectionandlastsuntilitreaches^x=0.Thethirdperiodstartsattheendofthesecondperiodandlastsuntilthesecondaryshockwaveinteractswiththecontactdiscontinuity.Fortheparameterschosenhere,theperiodshavethefollowingdivision:0^t4.1forperiod1,4.1^t8.6forperiod2,and8.6^t12.7forperiod3. 160

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ADensity BVelocity CTemperature Figure5-4. Temporalevolutionsofgasdensity,velocity,andtemperatureprolesfrom^t=0to8.19. Intherstperiod,thecompressedgasexpandsrapidly.InFig. 5-4A ,itcanbeobservedthatthegasdensityinregion3rapidlydecreasesduringthistimeperiod.At^t=4.09,thegasdensityinregion3reducestonearlyzero,indicatingthatalmostallofthegasinitiallystoredinthecontaineriscompressedintoregion2.Thegasvelocityinregion3decreasewhentheheadoftheexpansionfanreachestheorigin.Duetothestrongexpansion,thetemperaturealsodecreasesinregion1.Atthetailofthe 161

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ADensity BVelocity CTemperature Figure5-5. Temporalevolutionsofgasdensity,velocity,andtemperatureprolesfrom^t=8.19to12.28. expansionfan(therightboundaryofregion1),thesecondaryshockwaveforms.Itcanbeobservedthatthestrengthofthesecondaryshockincreaseswithtime,seeFig. 5-4B Inthesecondperiod,thesecondaryshockwavereversesandmovestowardtheorigin.Thestrengthofthesecondshockwaveincreasesasitapproachestheorigin.Therefore,itisseeninFig. 5-4B and 5-4B that,thepost-shockvelocity(totherightofthesecondaryshockwave)decreasesfasterwithtime.At^t=5.46,itisseenthatthepost-shockvelocitybecomesnegative.Thepost-shockvelocitywillcontinuetodecrease 162

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untiltheshockreectsfromtheorigin.Thegaswithnegativevelocityreversesdirectionandmovesbacktowardtheorigin.Therefore,thedensityintheregionssweptbythesecondaryshockincreases.Whenthevelocityatthecontactdiscontinuityalsobecomesnegative,thecontactdiscontinuitystartstopropagatebackward.Thetemperatureofthegassweptbythesecondaryshockrisesduetoshockheating. Inthethirdperiod,thesecondaryshockreectsfromtheoriginandmovesoutwardagain.ItcanbeobservedfromFig. 5-5B thatthevelocityinmostoftheregionbehindthemainshockwave(1^x8)isnegativeat^t=8.19.Thatismostofthegasmovesinwardduetothesecondaryshockwavewhenitismovingtowardtheorigin.Thegascontinuestomoveinwardwhenthereectedsecondaryshockdrivesthegasbehindittomoveoutward,see^t=9.55inFig. 5-5B .Asaresult,thegasishighlycompressedatthesecondaryshock,whichcausesthedensityofgasnearthesecondaryshocktoincreaserapidly.PeaksareobservedatthesecondaryshocklocationinthedensityprolesinFig. 5-5A for^t>9.55.Asthereectedsecondaryshockcontinuestomoveoutward,thevelocityofthegasbehinditincreases. Itshouldbenotedherethatwhenthesecondaryshockreachestheorigin,theassumptionsunderlyingtheideal-gasmodelusedhereareinvalid.Weforegotheuseofamorecomplexequationofstatebecause(i)ourprimaryinterestisontheparticledispersalawayfromtheorigin,and(ii)fewparticlesarelocatedneartheorigin. Thereectedsecondaryshockwillinteractwiththecontactdiscontinuityat^t=12.7asshowninFig. 5-3 .Throughthisinteraction,theinward-propagatingcontactdiscontinuityispushedoutwards.Areectedandatransmittedshockwavewillbegeneratedintheinteraction.Thenasimilarcyclestarts,withthetransmittedshockplayingtheroleofthemainshock,andthereectedshockplayingtheroleofthesecondaryshock.Thisinteractionprocesscontinuestocreateasystemofreectedandtransmittedshocksthatbecomeweakerwitheachinteraction.Throughthese 163

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interactions,thegasboundedbythecontactdiscontinuitycontinuestoexpandandcompressesuntilitreturnstoastationarystate. 5.3.2EvolutionofaSingleParticle Werstexaminethemotionofasingleparticleinthisow.Figure 5-6 showsthetrajectoriesofisolatedparticlesonthe^x-^tdiagramalongwiththetrajectoriesofthemainshock,thecontactdiscontinuity,thesecondaryshockwave,andtheheadoftheexpansionfan.InFig. 5-6A ,threeparticle-to-uiddensityratiosareconsidered;inFig. 5-6B ,theparticlediametercomparedtothecontainersizeisvaried;andinFig. 5-6C theinitiallocationoftheparticlewithinthecompressedgas-particlemixtureisvaried.Theparticleremainsstationaryuntiltheinward-propagatingheadoftheexpansionfanarrivesattheparticlelocation.Theparticlebeginstoaccelerateandmoveoutwardastheexpansionfansweepspast.Theparticleisnextsubjectedtothesecondaryshockwave.Becausethesecondaryshockeventuallyreversesdirectionandmovesinward,allparticleswillinteractwithit.Dependingontheparticleparameters,theparticlemaycrossthecontactdiscontinuity.ForthecasesshowninFig. 5-6 theparticledoesnotcrossthemainshock. Figures 5-6A and 5-6B clearlydemonstratethenon-monotonicnatureoftheparticlemotionasafunctionofparticledensityanddiameter.Forexample,inFig. 5-6A theparticlewith^p=^g4=1initiallymovesoutwardfasterthantheparticlewith^p=^g4=10asexpected.Butafter^t&8theparticlewith^p=^g4=1reversesdirection,becauseitrespondstotheinwardowaheadofthecontactdiscontinuity.However,theparticlewith^p=^g4=10continuestomoveoutwardduetoitslargerinertiaandcrossesthetrajectoryoftheparticlewithunitdensityratioat^t=13.3.Thisnon-monotonicbehaviorcanbeevenbetterseeningure 5-6B ,wherethesmallestparticleinitiallymovesoutwardfasterthantheintermediate-sizeparticle.Buttheirtrajectoriescrossat^t9and,infact,itappearsthatthetrajectoryofthesmallestparticleconsideredmayevenintersectthatofthelargestparticle.Thepositionsoftheparticlesdecreasewiththeirinitialpositions 164

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A^dp=410)]TJ /F22 6.974 Tf 6.22 0 Td[(3,^xp0=0.6 B^p=^g4=10,^xp0=0.6 C^p=^g4=10,^dp=410)]TJ /F22 6.974 Tf 6.22 0 Td[(3 Figure5-6. Particlepositionevolutiononx-tdiagramasafunctionofparticledensity,diameterandinitialposition. inthecontainerasshowninFig. 5-6C .Theparticlewith^xp0=0.9isalwaysleadingthosewithsmaller^xp0.Itisalsoobservedthattheparticlewith^xp0=0.3doesnotcrossthecontactdiscontinuityandstaysintheneareld.Thisindicatesthatparticlesinitiallydistributedneartheoriginmayneverbedispersedoutbytheblastwave. ThechangeinparticlevelocityasitinteractswiththewavescanbeseeninFig. 5-7 ,whichshowstheparticlevelocityevolutionasafunctionofparticledensity,diameter,andinitialposition.Theparticleexperienceshighaccelerationasitenterstheexpansion 165

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A^dp=410)]TJ /F22 6.974 Tf 6.22 0 Td[(3,^xp0=0.6 B^p=^g4=10,^xp0=0.6 C^p=^g4=10,^dp=410)]TJ /F22 6.974 Tf 6.22 0 Td[(3 Figure5-7. Particlevelocityevolutionasafunctionofparticledensity,diameterandinitialposition. fan.Theaccelerationrapidlydecreasesandtheparticlereachesanear-constantvelocitybeforethearrivaloftheinward-propagatingsecondaryshockwave.Astheparticlecrossesthesecondaryshockwave,itsvelocitydropsrapidlytoapproachthepost-shockgasvelocity.Theparticledecelerationdecreaseswhentheparticlecrossescontactdiscontinuity.Thisobservationisimportantfortheevolutionoftheparticleconcentrationeldaswediscussedinsection 5.3.3 .Theparticleresponsetimeisproportionaltothedensityratioandquadraticallydependentontheparticlesize, 166

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seeEqs.(??)and(??).Thus,itcanbeobservedthatthesmallerandthelightertheparticle,thebetteritrespondstotheambientgasow.Thevelocitiesofthesmallerandlighterparticlescanbeobservedtobecomenegativeafterthepassageoftheinward-propagatingsecondaryshockwave.InFig. 5-6 theheaviest,largestandmostinnerparticle,respectively,aresubjectedtothereectedsecondaryshock,assecondaryshockwavenowpropagatesoutwardandcatchesupwiththeparticle.Itseffectistoacceleratetheoutwardmotionoftheparticle,butthechangeinaccelerationisnormallytooweaktobeclearlyseeninthevelocityplots. TheevolutionoftheparticletemperatureisshowninFig. 5-8 asafunctionofparticledensity,diameter,andinitialposition.Insidetheexpansionfan,theparticletemperaturedecreasesasthesurroundinggascools.Thethermaldecelerationoftheparticlerstdecreasesandthenincreases.Similartotheparticle-velocityevolution,itcanbeobservedthatthesmallerandthelightertheparticle,thebetteritrespondstotheambientgastemperaturechange,anditstemperatureisobservedtodecreasemorequicklyinsidetheexpansionfan.Thetemperatureoftheheavierandlargerparticlevariesonlyslightly(lessthan5%intotal)whenitinteractswiththegasow.Thethermaldecelerationdecreaseswhentheparticlecrossesthesecondaryshock,asthesurroundinggastemperaturerises. Figure 5-9A showstimeevolutionoftheforcebudgetfortheparticlewith^p=^g4=10,^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3,and^xp0=0.6.Itcanbeobservedthat,attheearlytimewhentheparticletravelswithintheexpansionfan,theunsteadyforcecontributions^fpg,^fam,and^fvuarelargerthanthequasi-steadyforcefqsduetothehighgasacceleration.However,thedensityofthegassurroundingtheparticledecreasesrapidlyasthegasexpands.Whentheparticlemovestowardthesecondaryshockwave,theunsteadycontributionstotheforcebecomequiteweak.Notethatthe^fvureachesasmallerpeakcomparedtotheinviscid-unsteadyforcecontributions^fpgand^fam,butduetotheslowdecayoftheviscous-unsteadykernel,^fvupersistsslightlylonger.Thequasi-steady 167

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A^dp=410)]TJ /F22 6.974 Tf 6.22 0 Td[(3,^xp0=0.6 B^p=^g4=10,^xp0=0.6 C^p=^g4=10,^dp=410)]TJ /F22 6.974 Tf 6.22 0 Td[(3 Figure5-8. Particletemperatureevolutionasafunctionofparticledensity,diameterandinitialposition. forcecontinuestoincreaseandbecomethedominantcontributiontotheoverallforcebalance.Thequasi-steadyforcereachesitspeakvaluewhenthevelocityofthegassurroundingtheparticlestartstodecreaseduetotheradialeffect.Byabout^t3,therelativevelocitybetweentheparticleandsurroundinggashasdecreasedandallcontributionstotheuid-dynamicforceontheparticlearequitesmall.Whentheparticlecrossesthesecondaryshockwave,thesurroundinggasvelocitydropssuddenlyandthequasi-steadyforcejumpstoanegativevalueinresponsetothenegativerelative 168

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AForce BHeattransfer Figure5-9. Temporalevolutionofthebudgetsofparticleforceandheattransferfor^p=^g4=10,^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3,and^xp0=0.6. velocity.Undertheinuenceofthequasi-steadyforce,therelativevelocityoftheparticledecreasesafteritcrossesthesecondaryshockwave.However,thelocalgastemperaturealsodecreasesasshowninFig. 5-4C .Asthespeedofsoundincreaseswithgastemperature,theMachnumberoftherelativeowincreases.Therefore,thequasi-steadyforce,althoughitremainsnegative,continuestoincreaseinmagnitude.Astheparticlecontinuestomoveoutwardandcrossthecontactdiscontinuity,thequasi-steadyforceremainsnegative,butitsmagnitudeundergoesasuddendecrease.ThisisduetothesuddenincreaseingastemperatureandsothusasuddendecreaseofparticleMachnumberwhentheparticlecrossesthecontactdiscontinuityintotheshockheatedambientair.Asaresult,thequasi-steadydragcoefcientdropsaccordingly.AsdiscussedinChapter4,theunsteadycontributionstotheoverallforcearesmallfortheparticularparticleconsideredinFig. 5-9A ,sincethelocalgasdensityisatleastanorderofmagnitudesmallerthantheparticledensityandcontinuestofurtherdecreaseasthegasrapidlyexpands.Iftheparticle-to-uiddensityratiodecreases,theinuenceofunsteadyforcesincreases. 169

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Figure 5-9B showsthetimeevolutionoftheheat-transferbudgetfortheparticlewith^p=^g4=10,^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3,and^xp0=0.6.Similartotheforcebudget,attheearlytimewhentheparticletravelswithintheexpansionfan,theunsteadycontributionstoheattransfer,i.e.,^quuand^qdu,arelargerthanthequasi-steadycontribution^qqs.Theparticletemperaturedecreases,asallheat-transfercontributionsarenegative.Thedensityoftheexpandinggasdecreasesrapidlyintheexpansionfan,theunsteadycontributionsdecay,and^qqsdominates.Itisinterestingtonotethatthediffusive-unsteadyheat-transferreachesasmallerpeakcomparedtotheundisturbed-unsteadyheat-transfercontribution^quu.Butnotealsothatduetotheslowdecayofthediffusive-unsteadykernel,thethermaldecelerationinsidetheexpansionfankeepsaccumulatingin^qdu,making^qduincreaseagainafterashortdecay.Asaresult,incontrasttotheforcebudget,thediffusive-unsteadyheattransferismostlythedominantcontribution.Here,itmustberememberedthatthediffusive-unsteadykernelusedhereisstrictlyvalidonlyforRep=0.Forthecasesconsideredhere,Rep1ingeneral,andweexpectthereforethatthediffusive-unsteadycontributiontoheattransfershownhereisoverpredicted.InFig. 5-8B ,thetemperatureoftheparticlewithdiameter^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(4decreasesandundershootsthegastemperaturetobecomenegative.Thisisphysicallyimpossible,ofcourse,andismainlyduetotheoverpredictionofthediffusive-unsteadyheattransfer.Nevertheless,thediffusive-unsteadycontributionstillappearstobeimportantintheparticlethermalbehaviorwhendispersedbyablastwave.ThisindicatesthattheniteReynolds-numberandMach-numberextensionofthediffusive-unsteadyheattransferkernelisanimportantareaofresearch. 5.3.3EvolutionofAParticleCloud Wenowcontinuetoshowtheresultsforthedispersalofacloudofparticles.Inthepresentproblem,theparticlesareassumedtobeinitiallyuniformlydistributedinsidethesphericalcontainer.However,duetotheinteractionwiththeow,theparticleconcentrationwillnotremainuniform,butratherevolvewithtimeinacomplexmanner. 170

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Theevolutionoftheparticlecloudisinuencedbytheevolutionoftheblast-wavestructure.Theparticle-concentrationdistribution,inturn,isdeterminedbytheevolutionoftheparticle-velocityprole.Thevariationofparticlequantitiesisthereforedependentonthetimeperiodandtheregioninwhichtheparticlesarelocated,seeFig. 5-3 .Figures 5-10 and 5-11 showthetemporalevolutionsofparticleconcentration,velocity,andtemperaturefor^p=^g4=10and^dp=410)]TJ /F5 7.97 Tf 6.58 0 Td[(3.Theresultsforthisspecicparticledensityanddiameteraretakenasanexampletoexplainthedispersaloftheentireparticlecloudintheblastwave. Intherstperiod,theparticlecloudisgenerallyexpandingwiththegas.Whentheparticlesareacceleratinginsideregion3,theparticle-velocitygradientispositive,andhencetheparticleconcentrationmonotonicallydecreasesinthisregion.Astheparticleskeepmovingoutward,theparticleconcentrationrapidlydecreaseswithtime.At^t=1.36,alltheparticlesarebehindthesecondaryshock.At^t=4.09,itisobservedthatsomeparticleshavecrossedthesecondaryshockwaveandenteredregion2.Theparticlevelocitydoesnotdropasquicklyasthegasvelocityduetoinertia;instead,itdecreasesgraduallyinregion2.Itcanbeseenthattheparticle-velocitygradientisstillpositive,butthemagnitudedecreases,see^t4.09inFig. 5-10B Inthesecondperiod,thesecondaryshockreversesdirectionandpropagatesinward.Theparticlessweptupbytheimplodingsecondaryshockdonotreversedirectionimmediatelyduetoinertia,buttheydeceleraterapidly.Theparticle-velocitygradientinregion3isstillpositive,butitsmagnitudecontinuestodecrease.Asthebackward-propagatingsecondaryshockwavecompressestheparticlesinregion3,thedecreaseinconcentrationreduces.Inregion2,theparticle-velocitygradientcontinuestodecrease.Neartherightendofregion2(closetothecontactdiscontinuity),theparticle-velocitygradientbecomesnegative.Itisalsoobservedthatsomeparticlespenetratethecontactdiscontinuityandenterregion1duringthisperiod.AsshowninFig. 5-7 ,therateatwhichparticlesdeceleratedecreaseswhentheycrossthecontact 171

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AConcentration BVelocity CTemperature Figure5-10. Temporalevolutionsofparticleconcentration,velocity,andtemperaturefromt=0to6.82,for^p=^g4=10and^dp=410)]TJ /F5 7.97 Tf 6.59 0 Td[(3. discontinuity.Asaresult,theparticle-velocitygradientchangessignatthecontactdiscontinuity.Correspondingtothislocalminimumatthecontactdiscontinuityintheparticle-velocityprole,theconcentrationalsoexhibitsakinkthere.Itisinterestingtonotethatthereectedexpansionwavesgenerateawaveintheparticlevelocitynear^x=1.6and2.3,thatpropagatesoutwardwiththeparticles,seeFig. 5-11B at^t=6.82and8.19,respectively.Thiswaveintheparticlevelocitybehaveslikeacompression 172

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AConcentration BVelocity CTemperature Figure5-11. Temporalevolutionsofparticleconcentration,velocity,andtemperaturefromt=6.82to12.28,for^p=^g4=10and^dp=410)]TJ /F5 7.97 Tf 6.58 0 Td[(3. wave,whichintroducesabumpintheparticleconcentrationproleandthendevelopsintoaspike,seeFig. 5-11A ). Inthethirdperiod,thesecondaryshockreectsfromtheoriginandpropagatesoutward.Theinteractionbetweentheparticlesandthesecondaryshockwavemakestheparticledistributioninregion3complex.Whenthereectedsecondaryshockwavepropagatesovertheparticles,theparticlevelocityincreases.Asaresult,particlesthataremovingtowardtheorigindecelerateandmaystarttomoveoutwardagain.Particles 173

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thathavenotyetinteractedwiththereectedsecondshockmaystillbemovingtowardtheorigin.Thiscausesparticleaccumulationbehindthesecondaryshock,seeFig. 5-11A forthepeakintheconcentrationproleat^x=1.5for^t=12.28.Thecontactdiscontinuityinthisperiodpropagatestowardtheoriginuntilitishitbythereectedsecondaryshockwave.Undertheinuenceofthenegativegasvelocity,theparticlevelocitycontinuestodecreasesinregion2.At^t10.92,theparticlevelocityisnegativefor1.3.^x.3.5.Fortheparticlesinregion1,thetrendismuchsimpler.Theparticlevelocityincreaseswith^xtowardtheparticlefront,andthemagnitudeofthevelocitydecreaseswithtime. Fortheparticleparametersconsideredhere,thethermalresponseoftheparticlecloudtothegasowsislesssensitivetotheunsteadyeffectscausedbytheinteractionwithdiscontinuitiesandexpansionwaves.Intherstperiod,theparticletemperatureinregion3hasnegativegradientsanddecreaseswithtime.Thedecreasingtemperaturesuddenlyriseswhenparticlescrossthegasdiscontinuityandenterregion2.Asaresult,thetemperaturegradientbecomespositiveattheleftendoftheregion2.Thetemperaturegradientdecreasesinregion2asthethermaldecelerationdecreaseswhenparticlescrossthesecondaryshock(asshowninFig. 5-8 ).Thetemperaturegradientneartheparticlefrontinregion2decreasestobenegativeastimeevolvestothesecondtimeperiod,seeFig. 5-10C at^t=6.82.Afterthat,temperatureprolemaintainsasimilaroverallshape,butcontinuestodecrease. 5.3.4ParticleFront Theevolutionoftheparticlefront(theouterboundaryoftheparticlecloud)isoftenofgreatinterest.Thisisbecausetheparticlefrontindicatesthemaximumdistanceparticlescanreachwhentheyaredispersedbytheblastwave.Figure 5-12 showsthetimeevolutionoftheparticlefrontonx-tdiagramasafunctionofparticledensityanddiameter.Theparticlefrontinitiallylagsbehindtheoutgoingsecondaryshockwaveandcontactdiscontinuity,butitisobservedthatitcancatchupwiththemlaterasthe 174

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A^dp=410)]TJ /F22 6.974 Tf 6.23 0 Td[(3 B^p=^g4=10 Figure5-12. Particlefrontevolutiononx-tdiagramasafunctionofparticledensityanddiameter. outgoingwavesareslowingdown.Similartothesingle-particlebehaviordiscussedinsection 5.3.2 ,thenon-monotoniccomplexmotionoftheparticlefrontisobservedagain.Foraspecicparticledensityordiameter,itcanbeseenthatthepositionofparticlefrontisnotamonotonicfunctionofdiameterordensityforalltime,althoughinitiallythesmallerandlighterparticlesarealwaysmovingfaster.Figure 5-13 showstheparticlefrontposition^xpfasafunctionofparticlediameter,density,andtime.Forparticledensities^p=^g4=10and100,itcanbeobservedthatateachtime,thereexitsanintermediatesizeparticlethatreachesamaximumradialposition.Furthermore,thisdiameterincreaseswithtimeifthedensityisxed.Converselyiftimeisxed,thenthisdiameterdecreaseswithdensity.Themaximumparticle-frontpositionasafunctionofparticlediameterisnotobservedfor^p=^g4=1.Fortherangeofparticlediametersconsideredhere,thepositionoftheparticlefrontfor^p=^g4=1increaseswithdiameter. 5.3.5ImportanceofUnsteadyContributionstoForceandHeatTransfer Finally,wehighlighttheimportanceoftheunsteadycontributionstotheforceandheattransfer.AlthoughthesignicanceoftheseunsteadycontributionshasalreadybeenpartiallyrevealedinFig. 5-9 ,aclearerillustrationisobtainedifwemeasurethe 175

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A^p=^g4=1 B^p=^g4=10 C^p=^g4=100 Figure5-13. Particlefrontpositionsasafunctionofparticledensity,diameterandtime. errorbynotincludingtheunsteadycontributionsintheforceandheat-transfermodel.Here,theparticlefrontpositionistakenastheparameterforwhicherrorsareevaluated.ThesolutioncalculatedfromEq.( 4 )isconsideredastheexactsolution,whichisusedtocomparewiththesolutionscomputedbyotherthreeforcemodels: 1. Equation( 4 )withouttheterms^fpg,^fam,and^fvu,and^fqscomputedfromthecorrelationofCliftandGauvin[ 21 ] 2. Equation( 4 )withouttheterms^fpg,^fam,and^fvu,and^fqscomputedfromthecorrelationofParmaretal.[ 70 ] 176

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3. Equation( 4 )withouttheterm^fvu Therstmodelistheconventionalincompressiblenite-Reynolds-numberdraglaw,thatignorescompressibilityandunsteadyeffects.ThesecondmodelignoresallunsteadytermsinEq.( 4 )andcompute^fqsbythecorrelationofParmaretal.[ 70 ].Thedifferencebetweentheresultsobtainedwiththismodelandtheexactsolutioncanbeviewedastheerrorofneglectingallunsteadyforces.Thethirdmodelisconsideredheretoevaluatetheimportanceoftheviscous-unsteadyforce.Thecomputationoftheviscous-unsteadyforceisfairlyexpensivebecausethemotionhistoryofeachparticleneedstobeintegrated.Ifconsiderationsofcomputationalcostindicatethatviscous-unsteadyforcecannotbecomputed,theresultsforthethirdmodelshownherecanbeusedasanestimatefortheincurrederror. Figure 5-14 showsthetemporalevolutionoftheerrorsintheparticlefrontlocationcomputedbythesecondmodelasafunctionoftheparticledensityanddiameter.Figure 5-15 showstheerrorsintheparticlefrontscomputedbyallthethreemodelsat^t=136.5asfunctionsofparticledensityanddiameter.At^t=136.5,thegasaroundtheentireparticlecloudisalmoststationary.Itcanbeobservedthat,allthreemodelsintroducenegativeerrorsintheparticlefrontposition,i.e.,theresultsunderestimatethemaximumradiusoftheparticlecloud.FortheresultsobtainedwiththesecondmodelinFig. 5-14 ,themaximumerrorkeepsincreasingwithtime,from3to8containerradii.Thefrontofthesmallandlightparticlesreachestheasymptoticvalueearly,sothattheerrornolongervarieswithtime,seeforexample,log(^dp))]TJ /F4 11.955 Tf 22.64 0 Td[(2.5andlog(^p=^g4)1inFigs. 5-14B and 5-14C .Butforlargeandheavyparticles,theerrorcontinuestoincreaseupto^t=136.5,whichimpliesthatthesmallerrorintroducedbyneglectingtheunsteadytermswillmagnifywithtime. Generally,theerrorsincurredbythesecondmodelaresmallerthanthoseobtainedwiththerst,andthoseobtainedwiththethirdmodelaresmallerthanthoseobtainedwiththesecond.Theerrorofusingonlythestandarddraglawcanbeupto20container 177

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A^t=13.6 B^t=54.6 C^t=95.5 Figure5-14. Errorinparticlefrontlocationiftheterms^fpg,^fam,and^fvuarenotincludedinEq.( 4 )asafunctionofparticledensityanddiameter,for^t=13.6,54.6and95.5. radii;whiletheerrorofignoringallunsteadyforcescanbeaslargeas10containerradii.Theseresultsclearlyshowthatincludingunsteadyforcesisnecessaryforawiderangeofparticledensitiesanddiameters.Theerrorsincreasewithparticlediameterandparticledensity.Thetrendfortheparticledensityisnotasclearasthatfortheparticlediameter.Especiallyforthelargeparticles,theerrorseemstoincreaseandthendecreasewithparticledensity.Forexample,fortheparticlediameterlog(^dp)=)]TJ /F4 11.955 Tf 9.29 0 Td[(1.4, 178

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themaximumerrorsofallofthethreemodelslocatehappenfortheintermediatedensitylog(^p=^g4)=1.Nevertheless,onlywhentheparticleisverylightandsmall,theunsteadyforcescanbeignoredwithoutintroducingsignicanterrorsintheparticlefrontlocation.InFig. 5-15C ,theerrorintheparticlefrontwhentheviscous-unsteadyforceisignoredseemstobesmall(lessthanabout1.6containerradii).Therefore,iftheparticlefrontlocationisofinterest,andiftheerrorshowninFig. 5-15C isacceptable,theviscous-unsteadyforcemaybeexcludedinthemodeltoreducethecomputationexpense. 5.4Conclusions AphysicalmodelforunsteadycompressibleowwithshockwaveshasbeenproposedandtestedbysomesimplecasesinChapter4.Theunsteadycontributionstotheoverallforceandheattransferareshowntobeimportantifthephysicalproblemincludesshockwavesorcontactdiscontinuities.Inthispaper,themultiphaseowmodelisappliedtoinvestigateafundamentalproblemofparticledispersalbyblastwaves.Theblastwaveconsideredinthispaperisgeneratedbyaspheicalcompressedgas-particlemixture.Whenthegas-particlemixtureissuddenlyreleased,amainshock,acontactdiscontinuity,asecondaryshockwave,andanexpansionfanaregenerated.Theinteractionsofparticleswiththediscontinuitiesandthestrongexpansionwavesdeterminethemechanicalandthermalmotionoftheparticles.Theresultsshowthatunsteadycontributionstoforceandheattransferaresignicant.Atearlytimeswhenparticlesaretravelingintheexpansionfan,thedensityofthegassurroundingtheparticlesislarge,andthustheunsteadycontributionstoforceandheattransferaremuchlargerthanthequasi-steadycontributions.Wendthattheparticlemotionisanon-monotonicfunctionoftheparticledensityordiameter.Thelighterorsmallerparticlesmovefasterinitially,butalsoslowdownfasterasthecontactdiscontinuityslowsdown,andeventuallymaybeovertakenbytheheavierorlargerparticles.Complexwaveinteractionsredistributetheparticlesinsidetheparticlecloud.Thefrontsof 179

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AStandarddraglawbyCliftandGauvin BEquation( 4 )withouttheterms^fpg,^fam,and^fvu CEquation 4 withouttheterm^fvu Figure5-15. Errorinparticlefrontlocationfordifferentforcemodelsasafunctionofparticledensityanddiameterfor^t=136.5. theparticlecloudsarecomputedwithdifferentforcemodelsthatignoredunsteadyforcesorcompressibilityeffectsonquasi-steadyforce.Theresultsofthesemodelsarecomparedwiththatobtainedwiththepresentmodel.Theerrorintheparticlefrontduetoneglectingunsteadyforceandcompressibilityeffectsonquasi-steadyforceislargeforawiderangeofparticledensitiesanddiameters,demonstratingtheimportanceoftheunsteadymechanismstoparticledispersalinblastwaves. 180

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CHAPTER6SUMMARY,CONCLUSIONS,ANDFUTUREWORK 6.1Summary Inthisdissertation,theproblemofparticledispersalbyshockanddetonationwaveswasinvestigatedthroughnumericalsimulations.Thisprobleminvolvescomplexphysicalmechanismsandchallengingnumericalissues.Thegoalsofthisdissertationweretodeveloparigoroussimulationapproachforunsteadycompressiblemultiphaseowsinvolvingshockanddetonationwaves,andapplythisapproachtotheproblemofparticledispersalbyshockanddetonationwaves.Toachievethesegoals,asequenceofproblemswerestudiedinthisdissertation.First,particledispersalinaone-dimensionalshock-tubewasconsidered(Chapter 2 ).Theprimaryfocusisonhowtheexistenceofparticlescausestheowtotransitionfromthefrozentotheequilibriumlimit.Thesecondproblementailedtheinvestigationandresolutionofnumericallyinducedparticle-number-densityuctuationsinEulerian-Lagrangian(E-L)simulationsofmultiphaseows(Chapter 3 ).Guidelinestoreducetheamplitudeoftheuctuationswereproposedthatcanbeappliedeasilyinpractice.Thethirdproblemconsideredinthedissertationwasthemodelingandanalysisofparticleinteractionwithshockandblastwaves(Chapter 4 ).Thestudyclearlydemonstratedtheimportanceoftheunsteadyforceandheattransferintheinteractionofparticleswithshockandblastwaves.Finally,weconsideredtheproblemofparticledispersalbyblastwaves(Chapter 5 ).Thefocuswasoninvestigatingtheinteractionsbetweenparticleswiththecomplexwavesystem.Theunsteadycontributionstoforceandheat-transferwereagainfoundtobesubstantial. 6.2Conclusions Themainresultsandconclusionsoftheproblemsstudiedinthisdissertationcanbesummarizedasfollows: 1. Transientphenomenainone-dimensionalcompressiblegas-particleow. 181

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(a) Theinteractionbetweenparticlesandtheexpansionfangeneratesdisturbancewavesthatpropagateintothecontactdiscontinuityandtheshockwave,therebycausingthemtoslowdownuntiltheyreachtheirequilibriumspeeds. (b) Theoreticalestimatesofthetimescaleswithwhichtheexpansionfan,theparticlefront,andtheshockwavereachequilibriumwereproposedandfoundtobeingoodagreementwiththenumericalresults. (c) Theparticlemassfractiondetermineswhetherthedecelerationtotheirequilibriumspeedsoftheshockwaveandparticlefrontismonotonic.Aparticle-diameter-dependentcriticalmassfractionwasfound,belowwhichtheapproachisnon-monotonic. 2. Numericalsourceofnumber-densityuctuationsinE-Lsimulationsofmultiphaseow. (a) Anewnumericalsourceofparticlenumber-densityuctuationwasfound.Whenauid-velocityprolepropagatesthroughaxedgrid,aperiodicvariationintheinterpolantofthisprolegeneratessubgridvariationsintheparticleposition. (b) Thesubgridvariationsintheparticlepositionresultinsubgriductuationsofthelocalnumberdensity.Thesubgriductuationscanmanifestthemselvesasuctuationsonthegridscaleifthedivergenceoftheparticlevelocityisnotzero. (c) Theamplitudeofuctuationdecreaseswithgridspacingatarategivenbytheorderoftheinterpolationmethod.Higher-orderinterpolationmethodssuchasHermiteinterpolationwerefoundtoreducethenumber-densityuctuations. (d) Theanalysisofvariousinterpolationmethodsresultedinquantitativeguidelinesthatprovideestimatesforgridresolutionasafunctionofdesiredmaximumerrorintheparticledistribution. 3. Modelingandanalysisofparticleinteractionwithshockandblastwaves. (a) Thepeakvaluesofunsteadycontributionstoforceandheattransferwerefoundtobemuchlargerthanthequasi-steadycontributionsforwiderangesofMachandReynoldsnumbers.Theratiosofunsteadyforceandheat-transfercontributionstoquasi-steadycontributionsdidnotdependstronglyontheparticledensityandheatcapacity. (b) Theneteffectofunsteadycontributionsonparticlevelocityandtemperatureinshock-particleinteractiondependsmainlyontheparticle-to-uiddensityandheat-capacityratios. (c) Iftheowbehindtheshockwaveisunsteadyandnon-uniform,suchasforasphericalshockwaveanda(point-source)blastwave,theunsteady 182

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mechanismscontinuetoinuencetheparticlemotionaftertheinteractionwiththemainshockwave.Theerrorsduetoneglectingunsteadycontributionsarelonglastingandcanevenbemagnied. 4. Analysisofparticledispersalbyblastwaves. (a) Theparticlemotionisanon-monotonicfunctionoftheparticledensityordiameter.Thelighterorsmallerparticlesinitiallymovefasterthantheheavierorlargerparticles,butwillslowdownandeventuallybeovertakenbytheheavierorlargerparticles. (b) Signicanterrorsintheparticlefrontpositionareobservedatlongtimesifunsteadyforcesorcompressibilityeffectsonthequasi-steadyforceareneglected. 6.3FutureWork Potentialfuturedirectionsforcontinuingthestudiesreportedinthisdissertationare: 1. Scalinganalysisfortheparticlemotioninblastwave:Fortheblastwavegeneratedbyasphereofcompressedgas,thegasmotionswithdifferentcontainerradiiandinitialpressureratioscollapseifscaledbythepropercharacteristiclengthandtimescale,seeLingetal.[ 53 ].Similarscalesmayexistforparticles. 2. Two-waycoupledsimulationofparticledispersalbydetonations:Inrealmultiphasedetonations,theparticlemassloadingisnotnegligiblysmall.Thepresenceofparticleswillinuencethegasowsignicantly.Therefore,theassumptionofone-waycouplingmaybeunjustied.Two-waycoupledsimulationsareneededtorigorouslyassesstheaccuracyofthisassumption.Two-waycoupledsimulationswiththeEulerian-Lagrangianapproachpursuedinthisdissertationwillintroduceadditionalchallenges.Forexample,thecomputationofunsteadyforcesintwo-waycoupledsimulationsisanunresolvedproblem. 3. Dense-oweffectsonparticledispersalbydetonations:Attheearlystagesofamultiphasedetonation,thevolumefractionofparticlesishigh,sothatsomedense-oweffects,suchasparticlecollisionmaybeimportant. 183

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APPENDIXASURFACEANDVOLUMEAVERAGESOFTHEGASQUANTITIESFORTHEPLANARSHOCK-PARTICLEINTERACTION ThesurfaceandvolumeaveragedquantitiesappearinginEqs.( 4 )to( 4 )andEqs.( 4 )to( 4 )are ugs=yug2, (A) Tgs=y(Tg2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+1, (A) gs=y(g2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+1, (A) gugs=yg2ug2, (A) gTgs=y(g2Tg2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+1, (A) and ugv=y2(3)]TJ /F4 11.955 Tf 11.95 0 Td[(2y)ug2, (A) gv=y2(3)]TJ /F4 11.955 Tf 11.95 0 Td[(2y)(g2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)+1, (A) gugv=y2(3)]TJ /F4 11.955 Tf 11.95 0 Td[(2y)g2ug2, (A) whereyisthedistancesweptthroughbytheshock,whichcanbedenedasy=ust)]TJ /F4 11.955 Tf 11.99 0 Td[((xp)]TJ /F4 11.955 Tf 11.98 0 Td[(1=2).Whent=0,y=0andxp=1=2.Allthevariablesthatappearherearenondimensional. Averagedtermsinvolvingtimederivativesare gDug Dtv=6usug2y(1)]TJ /F3 11.955 Tf 11.96 0 Td[(y), (A) gDTg Dtv=6us(Tg2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)y(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y), (A) D Dtv (gug)v=6g2ug2(us)]TJ ET q .478 w 227.67 -555.34 m 239.68 -555.34 l S Q BT /F3 11.955 Tf 227.67 -564.23 Td[(ugv)y(1)]TJ /F3 11.955 Tf 11.96 0 Td[(y), (A)d dt (gup)v= gvdup dt+6(g2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)up(us)]TJ /F3 11.955 Tf 11.95 0 Td[(up)y(1)]TJ /F3 11.955 Tf 11.95 0 Td[(y), (A) 184

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and D Dts (gug)s=g2ug2(us)]TJ ET q .478 w 244.58 -38.93 m 256.6 -38.93 l S Q BT /F3 11.955 Tf 244.58 -47.82 Td[(ugs), (A)d dt (gup)s= gsdup dt+(g2)]TJ /F4 11.955 Tf 11.96 0 Td[(1)up(us)]TJ /F3 11.955 Tf 11.96 0 Td[(up), (A) D Dts (gTg)s=(g2Tg2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)(us)]TJ ET q .478 w 278.73 -99.42 m 290.75 -99.42 l S Q BT /F3 11.955 Tf 278.73 -108.31 Td[(ugs), (A)d dt (gTp)s= gsdTp dt+(g2)]TJ /F4 11.955 Tf 11.95 0 Td[(1)Tp(us)]TJ /F3 11.955 Tf 11.96 0 Td[(up). (A) 185

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REFERENCES [1] Bagchi,P.andBalachandar,S.InertialandviscousforcesonarigidsphereinstrainingowsatmoderateReynoldsnumbers.J.FluidMech.481(2003):105. [2] Baker,W.E.ExplosionsinAir.Austin,TX:UniversityofTexasPress,1973. [3] Balachandar,S.andEaton,J.K.Turbulentdispersedmultiphaseow.Annu.Rev.FluidMech.42(2010):111. [4] Balachandar,S.andHa,M.Y.Unsteadyheattransferfromasphereinauniformcross-ow.Phys.Fluids13(2001):3714. [5] Balachandar,S.andMaxey,M.R.Methodsforevaluatinguidvelocitiesinspectralsimulationsfoturbulence.J.Comput.Phys.83(1989):96. [6] Balakrishnan,K.andMenon,S.Ontheroleofambientreactiveparticlesinthemixingandafterburnbehindexplosiveblastwaves.Combust.Sci.Technol.182(2010):186. [7] Barth,T.J.A3DupwindEulersolverforunstructuredmeshes.,1991.AIAAPaper91-1548. [8] Basset,AB.Onthemotionofasphereinaviscousliquid.Phil.Trans.R.Soc.A179(1888):43. [9] Bauer,P.,Dabora,E.K.,andManson,N.Chronologyofearlyresearchondetonationwave.AIAA,1991. [10] Boivin,M.,Simonin,O.,andSquires,K.D.Directnumericalsimulationofturbulencemodulationbyparticlesinisotropicturbulence.J.FluidMech.375(1998):235. [11] Boyer,D.W.Anexperimentalstudyoftheexplosiongeneratedbyapressurizedsphere.J.FluidMech.9(1960):401. [12] Bredin,M.S.andSkews,B.W.Dragmeasurementsinunsteadycompressibleow.Part1:Anunsteadyowfacilityandstresswavebalance.R&DJ.SouthAfr.Inst.Mech.Eng.23(2007):3. [13] Brode,H.L.NumericalSolutionsofSphericalBlastWaves.J.Appl.Phys.26(1955):766. [14] .Theoreticalsolutionsofsphericalshocktubeblasts.Tech.Rep.RM-1974,RandCorporationReport,1957. [15] .Blastwavefromasphericalcharge.Phys.Fluids2(1959):217. 186

PAGE 187

[16] Bryson,AEandGross,RWF.Diffractionofstrongshocksbycones,cylinders,andspheres.J.FluidMech.10(1961):1. [17] Carrier,G.F.Shockwavesinadustygas.J.FluidMech.4(1958):376. [18] Chao,J.,Haselbacher,A.,andBalachandar,S.Amassivelyparallelmulti-blockhybridcompact-WENOschemeforcompressibleows.J.Comput.Phys.228(2009):7473. [19] Chapman,D.L.Ontherateofexplosioningases.Phil.Mag.47(1889):90. [20] Chojnicki,K.,Clarke,A.B.,andPhillips,J.C.Ashock-tubeinvestigationofthedynamicsofgas-particlemixtures:Implicationsforexplosivevolcaniceruptions.Geophys.Res.Lett.33(2006):L15309. [21] Clift,R.andGauvin,W.H.Themotionofparticlesinturbulentgasstreams.Proc.Chemeca1(1970):14. [22] Crowe,C.T.Numerical-modelsfordilutegas-particleows-review.J.FluidEng.104(1982):297. [23] Crowe,C.T.,ed.Multiphaseowhandbook.CRCPress,2006. [24] Crowe,C.T.,Sommereld,M.,andTsuji,Y.Multiphaseowswithdropletsandparticles.CRCPress,1998. [25] Doring,W.UberdenDetonationsvorganginGasen.Ann.DerPhysik43(1943):421. [26] Elperin,T.,Igra,O.,andBen-Dor,G.Rarefactionwavesindustygases.FluidDyn.Res.4(1988):229. [27] Evans,M.W.,Harlow,F.H.,andBromberg,E.Theparticle-in-cellmethodforhydrodynamiccalculations.,1957.LosAlamosNationalLabNM. [28] Faxen,H.DerWiderstandgegendieBewegungeinerstarrenKugelineinerzahenFlussigkeit,diezwischenzweiparallelenebenenWandeneingeschlossenist.Ann.DerPhysik373(1922):89. [29] Feng,Z.G.andMichaelides,E.E.AnumericalstudyonthetransientheattransferfromasphereathighReynoldsandPecletnumbers.Int.J.HeatMassTransfer43(2000):219. [30] Ferry,J.andBalachandar,S.AfastEulerianmethodfordispersetwo-phaseow.Int.J.MultiphaseFlow27(2001):1199. [31] Fox,TW,Rackett,CW,andNicholls,JA.Shockwaveignitionofmagnesiumpowders.Proc.11thInt.Symp.ShockTubesandWaves.Seattle,WA:UniversityofWashingtonPress,1978,262. 187

PAGE 188

[32] Friedman,M.P.Asimpliedanalysisofsphericalandcylindricalblastsaves.J.FluidMech.11(1961):1. [33] Fritsch,F.N.andCarlson,R.E.Monotonepiecewisecubicinterpolation.SIAMJ.Numer.Anal.17(1980):238. [34] Garg,R.,Narayanan,C.,Lakehal,D.,andSubramaniam,S.AccuratenumericalestimationofinterphasemomentumtransferinLagrangian-Euleriansimulationsofdispersedtwo-phaseows.Int.J.MultiphaseFlow33(2007):1337. [35] Gatignol,R.TheFaxenformulaeforarigidparticleinanunsteadynon-uniformStokesow.J.Mec.Theor.Appl1(1983):143. [36] Harlow,F.H.PICanditsprogeny.Comput.Phys.Comm.48(1988):1. [37] Haselbacher,A.AWENOreconstructionalgorithmforunstructuredgridsbasedonexplicitstencilconstruction.,2005.AIAAPaper2005-0879. [38] .Onconstrainedreconstructionoperators.,2006.AIAAPaper2006-1274. [39] Haselbacher,A.,Najjar,F.M.,Balachandar,S.,andLing,Y.Lagrangiansimulationsofshock-wavediffractionataright-angledcornerinaparticle-ladengas.Proc.6thInternationalConferenceonMultiphaseFlow.Leipzig,Germany,2007. [40] Haselbacher,A.andNajjar,F.M.Multiphaseowsimulationsofsolid-propellantrocketmotorsonunstructuredgrids.,2006.AIAAPaper2006-1292. [41] Haselbacher,A.,Najjar,F.M.,andFerry,J.P.Anefcientandrobustparticle-localizationalgorithmforunstructuredgrids.J.Comput.Phys.225(2007):2198. [42] Igra,O.,Elperin,T.,andBen-Dor,G.Blastwavesindustygases.Proc.R.Soc.Lond.AMat.414(1987):197. [43] Jiang,G.S.andShu,C.W.EfcientimplementationofweightedENOschemes.J.Comput.Phys.126(1996):202. [44] Jouguet,E.Surl'ondeexplosive.C.R.Acad.Sci.Paris140(1904):1211. [45] .Surlapropagationdesreactionschimiquesdanslesgaz.J.Math.PuresAppl.6thSeri.1(1905):347. [46] Kannenberg,K.C.andBoyd,I.D.StrategiesforefcientparticleresolutioninthedirectsimulationMonteCarlomethod.J.Comput.Phys.157(2000):727. [47] Kim,I.,Elghobashi,S.,andSirignano,W.A.Ontheequationforspherical-particlemotion:effectofReynoldsandaccelerationnumbers.J.FluidMech.367(1998):221. 188

PAGE 189

[48] Kriebel,A.R.Analysisofnormalshockwavesinparticleladengas.J.BasicEng-T.ASME(1964):655. [49] Lanovets,V.S.,Levich,V.A.,Rogov,N.K.,Tunik,Yu.V.,andShamshev,K.N.Dispersionofthedetonationproductsofacondensedexplosivewithsolidinclusions.Combust.Explo.Shock+29(1993):638. [50] Larrouturou,B.Howtopreservethemassfractionpositivitywhencomputingcompressiblemulti-componentows.J.Comput.Phys.95(1991):59. [51] Ling,Y.,Haselbacher,A.,andBalachandar,S.Modelingandsimulationofexplosivedispersalofparticlesinamultiphaseexplosion.,2009.AIAAPaper2009-1532. [52] .Transientphenomenainone-dimensionalcompressiblegas-particleows.ShockWaves19(2009):67. [53] .Numericalinvestigationofparticledispersalinmultiphaseexplosions.,2010.AIAAPaper2010-768. [54] Liu,T.G.,Khoo,B.C.,andYeo,K.S.Thenumericalsimulationsofexplosionandimplosioninair:UseofamodiedHarten'sTVDscheme.Int.J.Numer.Meth.Fl.31(1999):661. [55] Longhorn,A.L.Theunsteady,subsonicmotionofasphereinacompressibleinvisciduid.Quart.J.Mech.Appl.Math.5(1952):64. [56] Magnaudet,J.andEames,I.Themotionofhigh-Reynolds-numberbubblesininhomogeneousows.Annu.Rev.FluidMech.32(2000):659. [57] Marble,F.E.Dynamicsofadustygas.Annu.Rev.FluidMech.2(1970):397. [58] Mashayek,F.andJaberi,F.A.Particledispersioninforcedisotropiclow-Mach-numberturbulence.Int.J.HeatMassTransfer42(1999):2823. [59] Maxey,M.R.andRiley,J.J.Equationofmotionforasmallrigidsphereinanonuniformow.Phys.Fluids26(1983):883. [60] McFadden,JA.Initialbehaviorofasphericalblast.J.Appl.Phys.23(1952):1269. [61] Mei,R.andAdrian,R.J.Flowpastaspherewithanoscillationinthefree-streamvelocityandunsteadydragatniteReynoldsnumber.J.FluidMech.237(1992):323. [62] Michaelides,E.E.andFeng,Z.Heattransferfromarigidsphereinanonuniformowandtemperatureeld.Int.J.HeatMassTransfer37(1994):2069. 189

PAGE 190

[63] Milne,A.M.Detonationinheterogeneousmixturesofliquidsandparticles.ShockWaves10(2000):351. [64] Miura,H.andGlass,I.I.Onadusty-gasesshocktube.Proc.R.Soc.Lond.AMat.382(1982):373. [65] Najjar,F.M.,Ferry,J.P.,Haselbacher,A.,andBalachandar,S.Simulationsofsolid-propellantrockets:effectsofaluminumdropletsizedistribution.J.Space-craftRockets43(2006):1258. [66] Parmar,M.,Haselbacher,A.,andBalachandar,S.Ontheunsteadyinviscidforceoncylindersandspheresinsubcriticalcompressibleow.Phil.Trans.R.Soc.A366(2008):2161. [67] .Modelingoftheunsteadyforceforshockparticleinteraction.ShockWaves19(2009):317. [68] .EquationofMotionforaSphereinEquationofmotionforasphereinnon-uniformcompressibleows.,2010.Inpreparation. [69] .GeneralizedBasset-Boussinesq-Oseenequationforunsteadyforcesonasphereinacompressibleow.,2010.SubmittedtoPhys.Rev.Lett. [70] .ImprovedDragCorrelationforSpheresandApplicationtoShock-TubeExperiments.AIAAJournal48(2010):1273. [71] Pope,S.B.Particlemethodforturbulentows:Integrationofstochasticmodelequations.J.Comput.Phys.117(1995):332. [72] Ripley,R.C.,Zhang,F.,andLien,F.-S.Shockinteractionofmetalparticlesincondensedexplosivedetonation.AIPConf.Proc.845(2006):499. [73] Roe,P.L.ApproximateRiemannsolver,parametervectors,anddifferenceschemes.J.Comput.Phys.43(1981):357. [74] Rudinger,G.Somepropertiesofshockrelaxationingasowscarryingsmallparticles.Phys.Fluids7(1964):658. [75] Rudinger,G.andChang,A.Analysisofnonsteadytwo-phaseow.Phys.Fluids7(1964):1747. [76] Sachdev,P.L.Shockwavesandexplosions.Chapman&Hall/CRC,2004. [77] Saito,T.Numericalanalysisofdusty-gasows.J.Comput.Phys.176(2002):129. [78] Schiller,L.andNaumann,Z.UberdiegrundlegendenKraftebeiderSchwerkraftaufbereitung.Z.Ver.Dtsch.Ing.77(1933):318. 190

PAGE 191

[79] Sedov,L.I.Similarityanddimensionalmethodsinmechanics.NewYork:AcademicPress,1959. [80] Snider,D.M.Ancompressiblethree-dimensionalmultiphaseparticle-in-cellmodelfordenseparticleows.J.Comput.Phys.170(2001):523. [81] Snider,D.M.,O'Rourke,P.J.,andAndrews,M.J.Sedimentowininclinedvesselscalculatedusingamultiphaseparticle-in-cellmodelfordenseparticleows.Int.J.MultiphaseFlow24(1998):1359. [82] Sommerfeld,M.Theunsteadinessofshockwavespropagatingthroughgas-particlemixture.Exp.Fluids3(1985):197. [83] Soo,S.L.Gasdynamicprocessesinvolvingsuspendedsolids.AIChEJ.3(1961):384. [84] Squires,K.D.andEaton,J.K.Particleresponseandturbulencemodicationinisotropicturbulence.Phys.Fluids2(1990):1191. [85] Subramaniam,S.Statisticalrepresentationofasprayasapointprocess.Phys.Fluids12(2000):2413. [86] Sun,M.,Saito,T.,Takayama,K.,andTanno,H.Unsteadydragonaspherebyshockwaveloading.ShockWaves14(2005):3. [87] Sundaram,S.andCollins,L.R.Numericalconsiderationsinsimulatingaturbulentsuspensionofnite-volumeparticles.J.Comput.Phys.124(1996):337. [88] Tanguay,V.,Higgins,A.J.,andZhang,F.Asimpleanalyticalmodelforreactiveparticleignitioninexplosives.Propell.Explos.Pyrot.32(2007):371. [89] Tanno,H.,Itoh,K.,Saito,T.,Abe,A.,andTakayama,K.Interactionofashockwithaspheresuspendedinaverticalshocktube.ShockWaves13(2003):191. [90] Taylor,G.Thedynamicsofthecombustionproductsbehindplaneandsphericaldetonationfrontsinexplosives.Proc.R.Soc.Lond.AMat.200(1950):235. [91] Taylor,G.I.Theformationofablastwavebyaveryintenseexplosion.I.Theoreticaldiscussion.Proc.R.Soc.Lond.AMat.201(1950):159. [92] .Theformationofablastwavebyaveryintenseexplosion.II.Theatomicexplosionof1945.Proc.R.Soc.Lond.AMat.201(1950):175. [93] Thompson,P.A.Compressible-uiddynamics.McGraw-Hill,Inc.,1972. [94] Vedula,P.andYeung,P.K.Similarityscalingofaccelerationandpressurestatisticsinnumericalsimulationsofisotropicturbulence.Phys.Fluids11(1999):1208. 191

PAGE 192

[95] Verevkin,A.A.andTsirkunov,Yu.M.FlowofadispersedphaseintheLavalnozzleandinthetestsectionofatwo-phasehypersonicshocktunnel.J.Appl.Mech.Tech.Phys.49(2008):789. [96] vonNeumann,J.Thepointsourcesolution.Tech.Rep.Div.BReportAM-9,NationalDefenseResearchCommittee,1941. [97] .Theoryofdetonationwaves.Tech.Rep.Div.BO.S.R.D.Rept.549,NationalDefenseResearchCommittee,1942. [98] Whitaker,S.Forcedconvectionheattransfercorrelationsforowinpipes,pastatplates,singlespheres,andforowinpackedbedsandtubebundles.AIChEJ.18(1972):361. [99] Xu,J.andPope,S.B.AssessmentofnumericalaccuracyofPDFMonteCarlomethodsforturbulentreactingows.J.Comput.Phys.152(1999):192. [100] Zeldovich,Y.B.Onthetheoryofthepropagationofdetonationingaseoussystems.Zh.Exp.Teor.Fiz10(1940):542. [101] Zhang,F.,Frost,D.L.,Thibault,P.A.,andMurray,S.B.Explosivedispersalofsolidparticles.ShockWaves10(2001):431. [102] Zhang,F.,Thibault,P.A.,andLink,R.Shockinteractionwithsolidparticlesincondensedmatterandrelatedmomentumtransfer.Proc.R.Soc.Lond.AMat.459(2003):705. 192

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BIOGRAPHICALSKETCH YueLingwasborninJiangmen,China.HereceivedhisBachelorofEngineeringfromBeihangUniversity(theformerBejingUniversityofAeronauticsandAstronautics)in2004.In2003,beforegettinghisBachelordegree,hewasaccepteddirectlyasagraduatestudentbyBeihangUniversitybecauseofhisoutstandingaccomplishments.HisresearchatBeihangUniversitywasonexperimentalandnumericalstudiesofparticlesizereductionbyimpinging-jetmills.HetransferredtotheUniversityofFloridatocontinuehisPh.D.studiesin2006.HejoinedtheComputationalMultiphysicsGroupintheDepartmentofMechanicalandAerospaceEngineeringunderthesupervisionofProf.Balachandar.HisresearchattheUniversityofFloridaisonthemodelingandsimulationofcompressiblemultiphaseowsinvolvingshockanddetonationwaveswithfocusontheimportanceofunsteadyforceandheattransfer. 193