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Unsteady Forces on a Particle in Compressible Flows

Permanent Link: http://ufdc.ufl.edu/UFE0042516/00001

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Title: Unsteady Forces on a Particle in Compressible Flows
Physical Description: 1 online resource (207 p.)
Language: english
Creator: Parmar, Manoj
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: axisymmetric, basset, boussinesq, compressible, drag, force, gatignol, history, inhomogeneous, inviscid, maxey, multiphase, nonuniform, oseen, particle, quasisteady, riley, shock, steady, unsteady, viscous
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Compressible multiphase flows occur in a variety of industrial and environmental problems. The key to improved prediction and control of these flows at the macroscale depends on our understanding of the interaction of an individual particle in a compressible ambient flow at the microscale. For example, simulations of compressible multiphase flows at the macroscale require an accurate model for the time-dependent force on a particle. At present, due to lack of fundamental knowledge, no well-founded model exits for the evaluation of forces on a particle in a compressible flow. By contrast, the problem of determining forces on a particle in incompressible flows has been studied for more than 150 years and produced a comprehensive body of knowledge. Current understanding of the forces on a particle in a compressible flow is limited to the quasi-steady drag force, expressed in terms of a drag coefficient that is dependent on both the Reynolds and Mach numbers. In compressible flows, unsteady interactions between compressible waves and the particle gives rise to unsteady contributions to the force that can be very important but remain virtually unexplored. This dissertation attempts to lay the foundation for compressible multiphase flow by obtaining a rigorous equation of motion for an isolated particle that is applicable in complex compressible flows. Toward this end, we first rigorously derive the compressible extension to the celebrated Basset-Boussinesq-Oseen equation for the unsteady motion of a particle. We then rigorously derive the compressible extension of the Maxey-Riley-Gatignol equation that accounts for the inhomogeneity of the ambient compressible flow. Through carefully constructed simulations, finite Mach- and Reynolds- number extensions for the quasi-steady and unsteady forces on the particle are developed. The improved formulation is tested for shock-particle interaction.
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 Manoj Parmar.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Haselbacher, Andreas.
Local: Co-adviser: Balachandar, Sivaramakrishnan.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042516:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042516/00001

Material Information

Title: Unsteady Forces on a Particle in Compressible Flows
Physical Description: 1 online resource (207 p.)
Language: english
Creator: Parmar, Manoj
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: axisymmetric, basset, boussinesq, compressible, drag, force, gatignol, history, inhomogeneous, inviscid, maxey, multiphase, nonuniform, oseen, particle, quasisteady, riley, shock, steady, unsteady, viscous
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Compressible multiphase flows occur in a variety of industrial and environmental problems. The key to improved prediction and control of these flows at the macroscale depends on our understanding of the interaction of an individual particle in a compressible ambient flow at the microscale. For example, simulations of compressible multiphase flows at the macroscale require an accurate model for the time-dependent force on a particle. At present, due to lack of fundamental knowledge, no well-founded model exits for the evaluation of forces on a particle in a compressible flow. By contrast, the problem of determining forces on a particle in incompressible flows has been studied for more than 150 years and produced a comprehensive body of knowledge. Current understanding of the forces on a particle in a compressible flow is limited to the quasi-steady drag force, expressed in terms of a drag coefficient that is dependent on both the Reynolds and Mach numbers. In compressible flows, unsteady interactions between compressible waves and the particle gives rise to unsteady contributions to the force that can be very important but remain virtually unexplored. This dissertation attempts to lay the foundation for compressible multiphase flow by obtaining a rigorous equation of motion for an isolated particle that is applicable in complex compressible flows. Toward this end, we first rigorously derive the compressible extension to the celebrated Basset-Boussinesq-Oseen equation for the unsteady motion of a particle. We then rigorously derive the compressible extension of the Maxey-Riley-Gatignol equation that accounts for the inhomogeneity of the ambient compressible flow. Through carefully constructed simulations, finite Mach- and Reynolds- number extensions for the quasi-steady and unsteady forces on the particle are developed. The improved formulation is tested for shock-particle interaction.
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 Manoj Parmar.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Haselbacher, Andreas.
Local: Co-adviser: Balachandar, Sivaramakrishnan.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042516:00001


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UNSTEADYFORCESONAPARTICLEINCOMPRESSIBLEFLOWSByMANOJKUMARPARMARADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010ManojKumarParmar 2

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idamhipumsastapasahsrutasyava svistasyasuktasyacabuddhi-dattayoh avicyuto0rthahkavibhirnirupito yad-uttamasloka-gunanuvarnanam Learnedcircleshavepositivelyconcludedthattheinfalliblepurposeoftheadvancementofknowledge,namelyausterities,studyoftheVedas,sacrice,chantingofhymnsandcharity,culminatesinthetranscendentaldescriptionsoftheLord,whoisdenedinchoicepoetry. AllparaphernaliaofthecosmicuniverseisbutanemanationfromtheLordoutofHisdifferentenergiesbecausetheLordhassetinmotion,byHisinconceivableenergy,theactionsandreactionsofthecreatedmanifestation.TheyhavecometobeoutofHisenergy,theyrestonHisenergy,andafterannihilationtheymergeintoHim.Nothingis,therefore,differentfromHim,butatthesametimetheLordisalwaysdifferentfromthem.WhenadvancementofknowledgeisappliedintheserviceoftheLord,thewholeprocessbecomesabsolute.Therefore,allthesagesanddevoteesoftheLordhaverecommendedthatthesubjectmatterofart,science,philosophy,physics,chemistry,psychologyandallotherbranchesofknowledgeshouldbewhollyandsolelyappliedintheserviceoftheLord. -SrimadBhagavatam,canto1,chapter5,text22.Source: http://vedabase.net/sb/1/5/22/en Thus,withthepermissionofmypreceptorsIdedicatethisworktooriginalscientist,thesupremepersonalityofGodheadSriKrishna. 3

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ACKNOWLEDGMENTS Iamverygratefultomyadvisors,Profs.HaselbacherandBalachandar,fortheircontinuedsupport,guidance,andencouragement.IwouldliketothankProfs.Mei,Klausner,andCurtis,forservingonmythesiscommittee.Inparticular,IwouldliketoacknowledgehelpfuldiscussionswithProf.Mei.IwouldalsoliketoextendmythankstolabmatesinComputationalMultiphysicsGroup.Particularly,IwouldliketoacknowledgeHyungooLee,JungwooKim,ThomasBonometti,YoshifumiNozaki,YueLing,andSubramanianAnnamalaiforhelpfuldiscussionsandtheirhelpduringmyPhDwork.IalsoacknowledgeRajeevJaimanforhissupportandinspirationduringmyPhD.Thanksarealsoduetomyfriendsfortheirconstantsupport,encouragement,andhelpinsomanywaysduringmyPhDwork.Ithankmybrother,Ravi,forhiscontinuouscareandsupport.Lastbutnottheleast,IthankmywifeSona,forherlove,care,andpleasantcompany,forbeingverypatientandunderstandingduringmyPhDwork.Ithankmyparentsforeverythingtheyhavedoneforme. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1MotivationandBackground .......................... 15 1.2ParticleEquationofMotioninIncompressibleFlows ............ 17 1.2.1CreepingMotioninaQuiescentFluid ................. 18 1.2.2UnsteadyAmbientFlow ........................ 19 1.2.3Non-UniformFlow ........................... 20 1.2.4Non-LinearRegime ........................... 21 1.2.5ForceSuperposition .......................... 22 1.3ParticleEquationofMotioninCompressibleFlows ............. 23 1.3.1Quasi-SteadyDrag ........................... 23 1.3.2Inviscid-UnsteadyForce ........................ 24 1.3.3Viscous-UnsteadyForce ........................ 25 1.4GoalofthePresentWork ........................... 26 1.5DissertationLayout ............................... 27 2GENERALIZEDBASSET-BOUSSINESQ-OSEENEQUATIONFORUNSTEADYFORCESONASPHEREINACOMPRESSIBLEFLOW ............. 30 2.1Introduction ................................... 30 2.2ProblemFormulation .............................. 32 2.3SolutionforImpulsiveMotion ......................... 33 2.4CompressibilityEffectonInviscidUnsteadyForce ............. 34 2.5AsymptoticBehaviorsofCompressibleViscousUnsteadyForce ..... 36 2.6NumericalEvaluationofViscousUnsteadyForce .............. 39 2.7InviscidandViscousUnsteadyForceKernelsandNumericalConrmation 42 2.8GeneralizationoftheBBOEquationtoCompressibleFlows ........ 45 2.9Conclusions ................................... 47 3EQUATIONOFMOTIONFORASPHEREINNON-UNIFORMCOMPRESSIBLEFLOWS ........................................ 49 3.1Introduction ................................... 49 3.2GoverningEquationsforFlowAroundaMovingParticle .......... 51 3.3MovingReferenceFrameandSeparationofDisturbanceFlow ...... 52 5

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3.4ScalingAnalysis ................................ 54 3.5Density-WeightedVelocityTransformation .................. 56 3.6HydrodynamicForceduetoUndisturbedFlow ................ 57 3.7ReciprocalTheoremforCompressiblePerturbationFlow .......... 58 3.8HydrodynamicForceduetotheDisturbanceow .............. 61 3.8.1ImportanceoftheDifferentTerms ................... 65 3.8.2InviscidandViscousKernels ..................... 66 3.9Discussion ................................... 72 4ONTHEUNSTEADYINVISCIDFORCEONCYLINDERSANDSPHERESINSUBCRITICALCOMPRESSIBLEFLOWS ................... 79 4.1Introduction ................................... 79 4.2NumericalMethod ............................... 81 4.3Results ..................................... 82 4.3.1EffectofMachNumber ......................... 84 4.3.1.1Cylinder ............................ 85 4.3.1.2Sphere ............................ 89 4.3.2Mach-NumberExpansion ....................... 91 4.4Discussion ................................... 95 5MODELINGOFTHEUNSTEADYFORCEFORSHOCK-PARTICLEINTERACTION 101 5.1Introduction ................................... 101 5.2ForceModel .................................. 105 5.2.1ForceParameterization ........................ 105 5.2.2ImportanceofInviscidUnsteadyContribution ............ 108 5.2.3EffectofFiniteMachNumber ..................... 112 5.2.4ApproximationofAmbientFlow .................... 113 5.3Results ..................................... 117 5.3.1StationarySphere ........................... 118 5.3.1.1ExperimentsofSunetal. .................. 118 5.3.1.2ExperimentsofSkewsetal. ................ 124 5.3.2MovingSphere ............................. 126 5.4Discussion ................................... 128 6ANIMPROVEDDRAGCORRELATIONFORSPHERESANDAPPLICATIONTOSHOCK-TUBEEXPERIMENTS ........................ 134 6.1Introduction ................................... 134 6.2ImprovedDrag-CoefcientCorrelation .................... 135 6.3Validation .................................... 141 6.4Discussion ................................... 145 6.5Conclusions ................................... 145 6

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7UNSTEADYFORCESONAPARTICLEINVISCOUSCOMPRESSIBLEFLOWSATFINITEMACHANDREYNOLDSNUMBERS ................. 147 7.1Introduction ................................... 147 7.2NumericalMethodology ............................ 151 7.3UnsteadyForcesOveraSphereatSmallMachNumbersandFiniteReynoldsNumbers .................................... 153 7.4UnsteadyForcesOveraSphereatSub-CriticalMachNumbersandFiniteReynoldsNumbers ............................... 156 7.5Conclusions ................................... 159 8SUMMARY,CONCLUSIONS,ANDFUTUREWORK ............... 161 8.1SummaryandConclusions .......................... 161 8.2FutureWork ................................... 164 APPENDIX:NUMERICALMETHODOLOGY ...................... 167 A.1GoverningEquations .............................. 168 A.1.1DimensionalForm ........................... 168 A.1.2Non-DimensionalForm ......................... 169 A.2SolutionMethods ................................ 170 A.2.1DissipativeSolver ............................ 170 A.2.2Non-DissipativeSolver ......................... 171 A.3DiscretizationofNon-DissipativeSolver ................... 172 A.3.1NotationandVariableArrangement .................. 172 A.3.2ContinuityEquation ........................... 172 A.3.3MomentumEquation .......................... 174 A.3.4EnergyEquation ............................ 177 A.4SolutionAlgorithmforNon-DissipativeSolver ................ 178 A.4.1ContinuityEquation ........................... 179 A.4.2MomentumEquation .......................... 180 A.4.3EnergyEquation ............................ 184 A.4.4Summary ................................ 189 A.5BoundaryConditionImplementationforNon-DissipativeSolver ...... 190 A.5.1SolidWalls ............................... 190 A.5.2Fareld .................................. 191 A.6AbsorbingBoundaryConditions ....................... 191 A.6.1CharacteristicBoundaryConditions .................. 192 A.6.2SpongeLayer .............................. 192 A.7MovingReferenceFrame ........................... 193 A.7.1EulerEquations ............................. 193 A.7.2CoordinateTransformation ....................... 193 A.7.3TransformationoftheEulerEquationstoMovingReferenceFrame 193 A.8AxisymmetricComputations .......................... 196 A.8.1EulerEquationsforAxisymmetricFlows ............... 196 7

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A.8.2VolumetricandSurfaceIntegration .................. 197 REFERENCES ....................................... 199 BIOGRAPHICALSKETCH ................................ 207 8

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LISTOFTABLES Table page 6-1SummaryofselectedtestcasestakenfromJourdanetal.[ 50 ]usedtocomparewithmodel.Thecolumnlabeled`gases'liststhegasesinthedriveranddrivensectionsoftheshocktube. ............................. 142 A-1Wallboundarycondition. .............................. 191 A-2Fareldboundarycondition. ............................. 191 9

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LISTOFFIGURES Figure page 1-1Examplesofcompressiblemultiphaseows.Inthespaceshuttle,multiphaseowarisesduetothesolid-propellantrocketmotorsbecausethepropellantisenrichedwithaluminumparticles. .......................... 16 1-2State-of-the-artfortheforcesonaparticle. .................... 26 2-1ThebehaviorofthecorrectionfunctionC()thataccountsforthecompressibilityeffectontheviscousunsteadyforce(seeEq.( 2 )).Resultsareplottedforb=0andKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,10)]TJ /F10 7.97 Tf 6.59 0 Td[(5,10)]TJ /F10 7.97 Tf 6.59 0 Td[(8,10)]TJ /F10 7.97 Tf 6.59 0 Td[(11g.Theopencirclesrepresentthecurve-tgivenbyEq.( 2 )evaluatedforKn0=10)]TJ /F10 7.97 Tf 6.59 0 Td[(5. .............. 41 2-2ThedependenceofthecorrectionfunctionC()onthebulkviscosity.Resultsareplottedforb==f0,1,59=12,10gandKn0=f10)]TJ /F10 7.97 Tf 6.58 0 Td[(2,10)]TJ /F10 7.97 Tf 6.58 0 Td[(8g. ......... 42 2-3Timeevolutionofthenormalizedunsteadyforce.Theoreticalpredictions(lasttwotermsofEq.( 2 ))areplottedassolidlinesforKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,10)]TJ /F10 7.97 Tf 6.59 0 Td[(3,10)]TJ /F10 7.97 Tf 6.59 0 Td[(4gandb=0.Inviscidunsteadykernel(secondlastterminEq.( 2 ))isshownasdashesline.Correspondingsimulationresultsforfourdifferentcasesareshownassymbols. .................................. 44 2-4lowanduppforb==f0,1,59=12,10g. ...................... 45 3-1ThebehaviorofthecorrectionfunctionCv()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==0andKn0=f10)]TJ /F10 7.97 Tf 6.58 0 Td[(2,10)]TJ /F10 7.97 Tf 6.58 0 Td[(5,10)]TJ /F10 7.97 Tf 6.58 0 Td[(8,10)]TJ /F10 7.97 Tf 6.58 0 Td[(11g. ........ 67 3-2ThebehaviorofthecorrectionfunctionCs()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==0andKn0=f10)]TJ /F10 7.97 Tf 6.58 0 Td[(2,10)]TJ /F10 7.97 Tf 6.58 0 Td[(5,10)]TJ /F10 7.97 Tf 6.58 0 Td[(8,10)]TJ /F10 7.97 Tf 6.58 0 Td[(11g. ........ 68 3-3ThebehaviorofthecorrectionfunctionCv()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==f0,1,59=12,10gandKn0=f10)]TJ /F10 7.97 Tf 6.58 0 Td[(2,10)]TJ /F10 7.97 Tf 6.58 0 Td[(5g. ...... 69 3-4ThebehaviorofthecorrectionfunctionCs()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==f0,59=12,10gandKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2g. ........... 70 4-1SchematicdepictionofvariationoffreestreamMachnumberduringcomputations. 84 4-2EffectofaccelerationparameterontheunsteadyforcecoefcientoncylinderforM1,0=0.3and=1.4 ............................. 85 4-3ComparisonofcomputedresultsforcylinderwiththeoreticalresultsofMiles(1951)for=1.4 .................................. 86 10

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4-4Evolutionofnon-dimensionalperturbationpressure(scaledtorangebetweenminusandplusoneateachinstant)forcylinderatM1=0.2and=1.4 ... 87 4-5Computedbehaviorofpeakandsteady-statevaluesofeFfor=1.4 ...... 90 4-6ComparisonofcomputedresultsforspherewiththeoreticalresultsofLonghorn(1952)for=1.4 .................................. 92 4-7Effectofonunsteadyforcecoefcientoncylinder ................ 95 4-8Behaviorof(t!1,t)denedbyEq.( 4 ) ................. 99 5-1ResponsekernelsofParmaretal.[ 79 ] ....................... 109 5-2Schematicofshockpositioninthesymmetryplaneofasphericalparticleanddenitionofvariables ................................. 114 5-3Comparisonofmodelwithcomputationsforspherewithdiameter8mofSunetal.[ 102 ] ....................................... 120 5-4Comparisonofmodelwithcomputationsforspherewithdiameter80mofSunetal.[ 102 ] .................................... 121 5-5Comparisonofmodelwithexperimentforspherewithdiameter80mmandcomputationsfor8mmofSunetal.[ 102 ] ..................... 122 5-6ComparisonofmodelwithexperimentsofSkewsetal.[ 97 ] ........... 125 5-7ComparisonofmodelwithexperimentsandcomputationsofBritanetal.[ 14 ] 129 5-8BreakdownofdragforceforexperimentalconditionsconsideredbyBritanetal.[ 14 ] ......................................... 130 5-9ResultsforshockinteractionwithcylinderatMs=1.22 .............. 133 6-1ComparisonofdragcorrelationswithdataofBaileyandStarr[ 5 ]assumingthat=1.4andthattheparticletemperatureisequaltothesurroundinggastemperature. ..................................... 137 6-2ComparisonofnewdragcorrelationwithdataofBaileyandStarrBaileyandStarr[ 5 ]. ........................................ 140 6-3Comparisonofnewdrag-coefcientcorrelationwithdataofGoinandLawrence[ 38 ]andMayandWitt[ 64 ]. ............................. 141 6-4ComparisonofmodelwithexperimentaldataofJourdanetal.[ 50 ]Notetheabscissascaling. ................................... 144 11

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7-1TimeevolutionofnormalizedunsteadyforceforM=0.01,Re=1.Bassethistoryforce(Eq.( 7 )),modiedhistoryforceduetoMeiandAdrian[ 68 ](Eq.( 7 )),inviscidunsteadyforceincompressibleowsduetoLonghorn[ 58 ](Eq.( 7 )),andinviscidandviscousunsteadyforceincompressibleowsduetoParmaretal.[ 81 ](Eq.( 7 ))areplotted. ................. 150 7-2Meshquality. ..................................... 152 7-3SchematicdepictionofvariationoffreestreamMachandReynoldsnumberduringcomputations. ................................. 153 7-4TimeevolutionofnormalizedunsteadyforceforM=0.01,Re=1.Bassethistoryforce(Eq.( 7 )),modiedhistoryforceduetoMeiandAdrian[ 68 ](Eq.( 7 )),andinviscidandviscousunsteadyforceincompressibleowsduetoParmaretal.[ 81 ](Eq.( 7 ))areplotted.Correspondingsimulationresultsareshownasopencirclesymbols. ..................... 154 7-5ComparisonofnormalizedunsteadyforceobtainedbynumericalsimulationandthatgivenbythelasttwotermsofEq.( 7 ).Simulationresultsareshownasopencirclesymbols. ............................... 155 7-6Comparisonofnormalizedunsteadyforceobtainedbynumericalsimulationandthatgivenbynewmodel(thelasttwotermsofEq.( 7 )). ......... 157 7-7Comparisonofnormalizedunsteadyforceobtainedbynumericalsimulationandthatgivenbynewmodel(thelasttwotermsofEq.( 7 )). ......... 158 A-1Variablearrangement. ................................ 173 A-2Timediscretization. ................................. 173 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyUNSTEADYFORCESONAPARTICLEINCOMPRESSIBLEFLOWSByManojKumarParmarDecember2010Chair:AndreasHaselbacherCochair:S.BalachandarMajor:AerospaceEngineering Compressiblemultiphaseowsoccurinavarietyofindustrialandenvironmentalproblems.Thekeytoimprovedpredictionandcontroloftheseowsatthemacroscaledependsonourunderstandingoftheinteractionofanindividualparticleinacompressibleambientowatthemicroscale.Forexample,simulationsofcompressiblemultiphaseowsatthemacroscalerequireanaccuratemodelforthetime-dependentforceonaparticle.Atpresent,duetolackoffundamentalknowledge,nowell-foundedmodelexitsfortheevaluationofforcesonaparticleinacompressibleow.Bycontrast,theproblemofdeterminingforcesonaparticleinincompressibleowshasbeenstudiedformorethan150yearsandproducedacomprehensivebodyofknowledge.Currentunderstandingoftheforcesonaparticleinacompressibleowislimitedtothequasi-steadydragforce,expressedintermsofadragcoefcientthatisdependentonboththeReynoldsandMachnumbers.Incompressibleows,unsteadyinteractionsbetweencompressiblewavesandtheparticlegivesrisetounsteadycontributionstotheforcethatcanbeveryimportantbutremainvirtuallyunexplored.Thisdissertationattemptstolaythefoundationforcompressiblemultiphaseowbyobtainingarigorousequationofmotionforanisolatedparticlethatisapplicableincomplexcompressibleows.Towardthisend,werstrigorouslyderivethecompressibleextensiontothecelebratedBasset-Boussinesq-Oseenequationfortheunsteadymotionofaparticle.WethenrigorouslyderivethecompressibleextensionoftheMaxey-Riley-Gatignol 13

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equationthataccountsfortheinhomogeneityoftheambientcompressibleow.Throughcarefullyconstructedsimulations,niteMach-andReynolds-numberextensionsforthequasi-steadyandunsteadyforcesontheparticlearedeveloped.Theimprovedformulationistestedforshock-particleinteraction. 14

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CHAPTER1INTRODUCTION 1.1MotivationandBackground Multiphaseowsareamongthemostinterestingandchallengingproblemsinuidmechanics.Thestudyofmultiphaseowsisimportantbecauseofitsapplicationstogeophysicalandengineeringowssuchasvolcaniceruptionsandsolid-propellantrocketmotors(seeg. 1-1 ).GeneralinformationonmultiphaseowscanbefoundinCroweetal.[ 25 ],Brennen[ 12 ],andBalachandarandProsperetti[ 7 ].Thepresentworkfocusesonarelativelylittleexploredbranchofmultiphaseow,namelythoseinwhichcompressibleeffectsareimportant.Relevantexamplesaretheowofaluminum-oxideparticlesthroughanimperfectlyexpandednozzle(seeg. 1-1B andNajjaretal.[ 75 ]),shock-particleinteractions(seeThomas[ 111 ]andTedeschietal.[ 108 ]),particleremovalfromsurfaces(seeSmedleyetal.[ 98 ]andLeeandWatkins[ 56 ]),needle-freedrugdelivery(seeMenezesetal.[ 69 ]),anddetonationofmultiphaseexplosives(seeLanovetsetal.[ 55 ],Zhangetal.[ 118 ],andRipleyetal.[ 86 ]).Theoverarchinggoalofthiswork,tobediscussedinmoredetailbelow,istoputthesimulationofcompressiblemultiphaseowsonamoresolidtheoreticalfooting.Beforereviewingrelevantpriorwork,somebackgroundonmultiphaseowswillbediscussed. Werestrictourattentiontodispersedtwo-phaseowsinwhichonephaseismateriallydisconnected,theso-calleddispersedphase,andtheotherismateriallyconnected,theso-calledcontinuousphase.Thebehaviorofdispersedtwo-phaseowsisdeterminedbytheinteractionbetweenthephases.Ingeneral,theinteractionsoccurintheformofmass,momentum,andenergyexchangesatthephaseboundaries.Inthepresentwork,attentionisfocusedontwo-phaseowsinwhichthedispersedphaseconsistsofsphericalparticlesandthecontinuousphaseisauid.Theparticlesareassumedtoberigidandinert,sothemasstransferisneglected.Inadditiontotheinteractionsbetweenthetwophases,complexitiescanariseduetotheinteractions 15

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AMountSt.Helens.Source: http://logancullen.files.wordpress.com/2010/05/mt-st-helens-erupting.jpg BSpaceshuttleplume.Source: http://www.nasa.gov/ Figure1-1. Examplesofcompressiblemultiphaseows.Inthespaceshuttle,multiphaseowarisesduetothesolid-propellantrocketmotorsbecausethepropellantisenrichedwithaluminumparticles. withinthedispersedphase,i.e.,intra-phaseinteractions,suchascollision,agglomeration,andbreakup.Atwo-phaseowiscalleddilutewhenonlyinter-phaseinteractionsareimportant.Otherwise,itiscalledadenseow.Throughoutthisdissertation,weconsideronlydilutetwo-phaseows.Theunderstandingoftheinteractionsbetweenthephasesiscomplicatedbydisparatelengthandtimescalesinvolved,unsteadiness,non-linearity,inhomogeneity,andcompressibility.Inpreviousinvestigationsofmultiphaseows,littleattentionhasbeenfocusedontheeffectsofcompressibilitycomparedtootherfeaturesinmultiphaseows.Compressibleowsarecharacterizedbyvariabledensity,non-negligibleenergytransfer,andwavepropagation,whichfurtheraddtothedifculties. Analyticalsolutionsfortheinteractionsbetweentheparticulateandtheuidphasesarepossibleonlyinthelinearizedregimeoflow-speedowsandsmallperturbationspropagation.Inthenon-linearregimeofhigh-speedowsandlarge 16

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perturbations,analyticalsolutionsareseldomavailable,andthecomplexinteractionsmustbestudiedexperimentallyand/orcomputationally.Thepresentworkfocusesonstudyingadilute,non-reactingsuspensionofrigidsphericalparticlesinacompressibleuidusingtheoreticalanalysisinthelinearizedregimeandcomputationaltoolsinthenon-linearregime.Throughthelinearityassumption,energyexchangesarenegligible,sotheoreticalanalysisconsiderstheeffectsofmomentumexchangesonly. Inprinciple,anumericalsimulationofanunsteadythree-dimensionalowaroundaparticlecanprovideaccurateinformationabouttheoweld.Theforcesontheparticlecanthenbecomputedbyintegratingthepressureandstressdistributionaroundtheparticle.Inatypicalmultiphaseow,however,thenumberofparticlescanrangefrommillionstobillions.Withcurrentcomputationalcapabilities,itisimpossibletoresolvetheowaroundeveryparticle.Asanalternativeapproach,aparticlecanbetreatedasapointparticleinwhichthedetailsoftheoweldaroundtheparticlearenotresolvedandmodeledinstead,seeBalachandarandEaton[ 6 ].Theaccuracyofthepoint-particleapproachdependsstronglyontheaccuracyoftheforcemodelusedintheparticleequationofmotion.Forcemodelshavebeenstudiedextensivelyforincompressibleows,aswillbedescribedinSection 1.2 .Therearemanyapplicationsoftwo-phaseowswherecompressibilityisnotnegligibleaslistedabove.Thedevelopmentofthepointparticleapproachforcompressibleowshasbeenlimited.Becauseofthelackofunderstandingoftheforcesonaparticleincompressibleows,anincompressibleforcemodelisoftenused,seeSaito[ 93 ]andJourdanetal.[ 50 ]. 1.2ParticleEquationofMotioninIncompressibleFlows Inthepoint-particleapproach,thevelocityofaparticleisobtainedfromNewton'ssecondlaw, mpdv dt=F(t),(1) wherempisthemassoftheparticle(hereassumedtobeconstant),tistime,vistheparticlevelocity,andFistheforceontheparticle.Theforceonaparticlecanbedivided 17

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intosurfaceandbodyforces.Examplesofbodyforcesaregravity,electrostatic,andmagneticforces.Examplesofsurfaceforcesarepressureandviscousforces.Inthecurrentwork,bodyforcesareneglected. Theproblemofdeterminingforcesonrigidbodiesimmersedinauidisafundamentalandclassicalprobleminuidmechanics. 1.2.1CreepingMotioninaQuiescentFluid OneoftheearliestcontributionstothedeterminationofforcesisduetoStokes[ 101 ].StokesinvestigatedthesteadyrectilinearmotionofanisolatedsphereinaquiescentviscousuidbasedonthelinearizedincompressibleNavier-Stokesequationsandobtainedananalyticalexpressionforthedragforceas F=6av,(1) whereisthedynamicviscosityoftheuidandaistheradiusofthesphere.Stokesalsoextendedhisanalysistotheunsteadycase.Expressedinfrequencydomain,theresultforthetotalforceonasinusoidallyoscillatingsphereinaquiescentuidis F=6av0ei!t"1+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(i)!a2 21=2)]TJ /F3 11.955 Tf 11.96 0 Td[(i!a2 9#,(1) wherev0and!aretheamplitudeandfrequencyoftheoscillatingvelocity,respectively.==isthekinematicviscosity,andistheuiddensity.TherstterminsidethesquarebracketscorrespondstotheinstantaneousStokesdraggivenbyEq.( 1 ).ThesecondtermiswhatisnowcalledtheBassetforce.Thethirdtermistheforceduetotheaddedmass. Subsequently,Basset[ 8 ],Boussinesq[ 10 ],andOseen[ 77 ]independentlyexaminedthetime-dependentforceonasphereduetorectilinearmotioninaquiescentviscousincompressibleuid.TheyalsobasedtheiranalysesonthelinearizedunsteadyincompressibleNavier-Stokesequationsvalidforcreepingmotion,i.e.,inthelimitofvanishingReynoldsnumber.Theresultingequationofmotionforasphericalparticle, 18

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theso-calledBBOequation,canbewrittenas mpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2mfdv dt)]TJ /F4 11.955 Tf 11.95 0 Td[(6a2p Zt1 p t)]TJ /F8 11.955 Tf 11.96 0 Td[(dv dtt=d,(1) wheremfisthemassoftheuiddisplacedbytheparticle.Thethreetermsontheright-handsidearethequasi-steady(Stokes)drag,inviscidunsteady(addedmass),andviscousunsteady(Bassethistory)forces,respectively. Theadded-massforceappearsduetotheno-penetrationconditionontheparticlesurfaceandisrealizedinstantlyinthepresenceofparticleacceleration.Ingeneral,theaddedmassforcecanbederivedanalyticallyinthelimitofpotentialandStokesow,seeLandauandLifshitz[ 54 ].Theadded-massforceFamistypicallyexpressedas Fam=Cammfdv dt,(1) whereCamistheadded-masscoefcient.Theadded-masscoefcientisxedforagivenparticleshape.Forasphericalparticle,Cam=1=2. TheBassethistoryforcearisesduetotheredistributionofvorticitygeneratedattheparticlesurface.Forimpulsiveacceleration,thehistoryforcedecaysmonotonically. TheBBOequationofparticlemotionisstrictlyvalidonlyinthelinearizedincompressibleregimeforamovingparticleinaquiescentuid.Anaturalquestionishowtheforcecomponentsandthereforetheequationofmotionchangesfornon-quiescentuids,non-uniformows,andwhennonlinearitiesandcompressibilitybecomeimportant. 1.2.2UnsteadyAmbientFlow Thequasi-steadycontributiontothehydrodynamicforceonamovingparticleinanon-stationaryambientowistypicallyparameterizedintermsoftherelativevelocityoftheuidwithrespecttotheparticle,urel=u)]TJ /F15 11.955 Tf 11.96 0 Td[(v,whereuistheundisturbeduidvelocity(intheabsenceoftheparticle)atthecenteroftheparticle.Thequasi-steadydragisfoundtobegivenby Fqs=6aurel.(1) 19

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Foraparticlemovinginanon-stationaryuid,theinviscidunsteadyforceconsistsofthepressure-gradientforceinadditiontotheadded-massforce.Theadded-massforcenowdependsontherelativeaccelerationoftheparticlewithrespecttotheow.Thepressure-gradientforceisduetotheowaccelerationintheabsenceoftheparticle.Theadded-massforceFamandthepressure-gradientforceFpgareexpressedas Fiu=Fam+Fpg=CammfDu Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dv dt+mfDu Dt.(1) whereD=Dtisthesubstantialtimederivativefollowingtheuidparticle. Theviscousunsteadyforceforthecaseofamovinguiddependsontherelativeaccelerationoftheuidwithrespecttotheparticle.Theviscousunsteadyforcecanbeexpressedas Fvu=6a2p Zt1 p t)]TJ /F8 11.955 Tf 11.96 0 Td[( Du Dtt=)]TJ /F3 11.955 Tf 14.35 8.08 Td[(dv dtt=!d.(1) 1.2.3Non-UniformFlow Faxen[ 31 ]rstconsideredtheeffectofnon-uniformambientowsontheforceonaparticle.ForsteadyStokesow,Faxenobtainedtheremarkableresultthat Fqs=6a( uS)]TJ /F15 11.955 Tf 11.96 0 Td[(v),(1) where ()Sdenotesanaverageoverthesurfaceofthesphericalparticle,i.e., fS=1 4a2ISfdS.(1) MazurandBedeaux[ 65 ]generalizedFaxen'sresultstounsteadymotionofaparticleinanarbitraryincompressibleStokesow.Theypresentedtheirresultinthefrequencydomain.Gatignol[ 36 ]derivedFaxen'sformulaforunsteadyforcesinthetimedomain.Independently,MaxeyandRiley[ 63 ]derivedfromrstprinciplestheparticleequationofmotionfornon-uniformcreepingows.TheMaxey-Riley-Gatignol(MRG) 20

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equationofmotioncanbewrittenas(neglectingbodyforces)mpdv dt=6a)]TJ ET q .478 w 106.2 -40.5 m 112.91 -40.5 l S Q BT /F15 11.955 Tf 106.2 -47.82 Td[(uS)]TJ /F15 11.955 Tf 11.96 0 Td[(v+1 2mfD uV Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dv dt+mfD uV Dt+6a2p Zt1 p t)]TJ /F8 11.955 Tf 11.96 0 Td[( D uS Dtt=)]TJ /F3 11.955 Tf 14.34 8.09 Td[(dv dtt=!d, (1) where ()Vdenotesanaverageoverthevolumeofthesphericalparticleas fV=3 4a3ZVfdV.(1) 1.2.4Non-LinearRegime Non-lineareffectsarequantiedthroughtheparticleReynoldsnumberRe,denedas Re=2urela=.(1) AtniteReynoldsnumbers,non-lineareffectsonthequasi-steady,inviscidunsteady,andviscousunsteadyforcesmustbetakenintoaccount. Becauseoftheabsenceofananalyticalsolutionforthedragforce,empiricalcorrelationshavebeenproposedforthequasi-steadydrag.Thequasi-steadydragisgenerallyexpressedinanon-dimensionalformasadragcoefcientCD,denedas CD=Fqs 1 2u2rela2,(1) whereFqs=kFqsk.Awealthofexperimentaldatahasbeengatheredandcompiledintheformofthestandarddragcurve,seeCliftandGauvin[ 21 ]andCliftetal.[ 22 ]forreferences.Thestandarddragcurverepresentsthedragcoefcientonasphericalparticleas CD=24 Re)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1+0.15Re0.687+0.421+42500 Re1.16)]TJ /F10 7.97 Tf 6.59 0 Td[(1.(1) TheabovecorrelationisvalidforRe/2105. 21

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ForniteReynolds-numberows,ithasbeenfoundthattheadded-massforceisidenticaltothatinpotentialandcreepingows,seeMeietal.[ 66 ],Riveroetal.[ 87 ],ChangandMaxey[ 19 ],andWakabaandBalachandar[ 114 ]. TheBassethistoryforcehasalsobeenextendedtoniteReynoldsnumbers.OdarandHamilton[ 76 ]suggestedacomplexnonlineardependenceofthehistoryforceontherelativeparticleacceleration.ThisapproachwaslaterprovedincorrectbymanyincludingMeietal.[ 66 ],Riveroetal.[ 87 ],andChangandMaxey[ 18 ].MeiandAdrian[ 68 ]obtainedanexpressionforthehistoryforcefornitebutsmallReynoldsnumbersbyconsideringoscillatoryowsaroundaparticle.Kimetal.[ 52 ]madefurtherimprovementstoMeiandAdrian'sexpressionbyconsideringlargeamplitudeoscillationsoftheparticle.Inspiteofsignicantprogressintheunderstandingofthehistoryforce,itisimportanttorecognizethatthehistoryforceisproblem-dependent,seeLovalentiandBrady[ 60 ],andauniversalpracticalexpressionhasnotyetbeenformulated. 1.2.5ForceSuperposition MagnaudetandEames[ 62 ]suggestedthatthedragforceonaparticleinanincompressibleowcanbeparameterizedas F(t)=Fqs(t)+Fiu(t)+Fvu(t),(1) wherethetermsonright-handsiderepresentthequasi-steady,inviscidunsteady(added-massandpressuregradient)andviscousunsteady(Bassethistory)forces.Lift,andbuoyancy/gravityforcesareignoredinthiswork.Alldevelopmentsdescribedabovewereconcernedwithincompressibleows.Nosystematicstudiesexistforcompressibleows.Inprinciple,Eq.( 1 )canbeusedforcompressibleows,providedsuitablemodicationsareincorporatedintothemodelingoftheforces. 22

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1.3ParticleEquationofMotioninCompressibleFlows TheimportanceofcompressibleeffectsisquantiedthroughtheMachnumber,denedas M=urel=c,(1) wherecisthespeedofsoundoftheambientow. Tothebestofourknowledge,therehasbeennosystematicworkonderivinganequationofmotionforparticlesincompressibleows,Therearesometheoreticalinvestigationsofunsteadyforcesonaparticleinaninviscidcompressibleuidintheacousticlimit,i.e.M!0,andinaviscouscompressibleuidinthelimitofzeroMachandReynoldsnumbers.ForniteMachnumbers,attentionhasbeenfocusedmainlyonthequasi-steadyforce. 1.3.1Quasi-SteadyDrag Therepresentationofthequasi-steadydragcoefcientintermsoftheparticleReynoldsnumber,e.g.Eq.( 1 ),isvalidforincompressibleows.Forcompressibleows,theincompressibledragcoefcientcanbeassumedtobeapproximatelyvaliduptothecriticalMachnumber.ThecriticalMachnumberisthelargestfreestreamMachnumberrelativetotheparticleforwhichtheowaroundparticleremainssubsonic.Forasphericalparticle,thecriticalMachnumberisabout0.6.Insupercriticalbutsubsonicfreestreamows,duetoowaccelerationaroundtheparticle,thelocaloweldbecomessupersonicandshockwavesappearontheparticlesurface.Insupersonicow,adetachedbowshockappearsupstreamoftheparticle.Thepresenceofshockwavesonorupstreamoftheparticleincreasesthequasi-steadydragsignicantly.Thus,forsupercriticalow,thedragcoefcientnowdependsalsoontheMachnumberinadditiontotheReynoldsnumber.Forexperimentalworkonthedragcoefcientinsupercriticalows,see,e.g.,BaileyandHiatt[ 4 ],BaileyandStarr[ 5 ],andMillerandBailey[ 72 ].Tomakesuchdatausefulforparticlesimulations,itneedstobeexpressedintheformofanempiricalrelation.Severalcorrelationshavebeenproposedtorelate 23

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theparticleReynoldsandMachnumbertothedragcoefcient,see,e.g.,Crowe[ 24 ],Walsh[ 116 ],Henderson[ 44 ],andLoth[ 59 ].Despitetheavailabilityofcorrelationsthatincludecompressibilityeffects,researchersarestillusingthecorrelationderivedforincompressibleowstorepresenttheforceonparticlesinhigh-speedows,seeElperinetal.[ 30 ]andSaito[ 93 ].Acloseexaminationofexistingcorrelationsindicatesthattheydonotrepresentexperimentaldatafaithfully.Thecorrelationsneedtobeimprovedtorepresentexperimentaldatamorefaithfully. 1.3.2Inviscid-UnsteadyForce Therearemanyunsteadymultiphaseowswherecompressibilityissignicant.TheearliestworkontheunsteadyinviscidforceinacompressibleowisduetoLove[ 61 ]andTaylor[ 106 ].LaterMiles[ 71 ]carriedoutananalyticalinvestigationoftheinviscidcompressibleunsteadyowduetotheimpulsivemotionofacylinderintheacousticlimit.Longhorn[ 58 ]obtainedthecorrespondingexpressionforasphere.Incompressibleows,thedependenceoftheunsteadyforceontheinstantaneousaccelerationisbrokenduetothenitewave-propagationspeed.Therefore,theunsteadyforcedependsontheaccelerationhistoryontheacoustictimescale.Longhorngavefollowingexpressionfortheinviscidunsteadyforceonarectilinearlymovingsphere Fiu=mfZt0dv dtt=e)]TJ /F9 7.97 Tf 6.58 0 Td[(c(t)]TJ /F16 7.97 Tf 6.58 0 Td[()=acos(c(t)]TJ /F8 11.955 Tf 11.96 0 Td[()=a)d.(1) Longhorn[ 58 ]showedthattheamountofworkdonetomoveasphereimpulsivelyisdoubledcomparedtothatingradualmotiontoreachthesamenalvelocity.Tracey[ 113 ]extendedtheanalyticalworkofMiles[ 71 ]tonitebutsmallMachnumbers.Longhorn'sanalyticalworkhasnotbeenextendedtoniteMachnumbers.Brentner[ 13 ]carriedoutnumericalsimulationsofcompressibleowaboutanacceleratingcylinderatM=0.4.ThefocusofBrentner'sworkwasonthepropagationofacousticenergyfromtheacceleratingcylinder.ThereisaneedtoextendtheworkofMilesand 24

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LonghorntoniteMachnumberstounderstandthenatureoftheunsteadyforceininviscidcompressibleows. 1.3.3Viscous-UnsteadyForce ZwanzigandBixon[ 119 ]obtainedinthefrequencydomainanexpressionfortheforceonasphericalparticleundergoingoscillatorymotioninaquiescentviscouscompressibleow.AminorerrorintheresultwascorrectedbyMetiuetal.[ 70 ].SimilarresultswereobtainedbyGuz[ 39 ]inLaplacespace.BedeauxandMazur[ 9 ]derivedFaxen'scorrectionfortheforceonasphericalparticleundergoingoscillatorymotioninviscouscompressibleows.Theseresultshavenotbeenderivedinaformusefulforparticletrackinginthetimedomain. Severalinvestigationsoftheviscousunsteadyforceinacompressibleowwerecarriedoutinthecontextofshock-particleinteraction.Thebehaviorandtheformofthehistoryforceneedsfurtherinvestigationincompressibleows.TherearefewstudiesontheinuenceoftheBassethistoryforceontheparticlemotionintheshock-particleinteraction,see,e.g.,Forneyetal.[ 35 ],Tedeschietal.[ 108 ],Thomas[ 111 ],andTedeschietal.[ 109 ].Theknowledgeoftheinuenceoftheunsteadyforcesontheparticlemotionisveryimportantforthesmalltracerparticlesusedintheparticle-imagingvelocimetry(PIV)studiesofsupersonicows. ThestudiesconcludedthattheBasset-historyforcecanbemanytimeslargerthanthequasi-steadydragforceinthepresenceoflargevelocitygradients.However,theinuenceoftheBassethistoryforceontheparticlemotionmaystillbelimitedbecauseitexceedsthequasi-steadyforceonlyforabriefperiod.Theincompressibleformofthehistoryforcewasusedintheabovecitedinvestigationsbecausenocorrespondingexpressionexistsforcompressibleow.Thus,thereisaneedtoinvestigatethenatureandtheformoftheviscousunsteadyforceincompressibleow. 25

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Figure1-2. State-of-the-artfortheforcesonaparticle. 1.4GoalofthePresentWork Anequationofmotionsuitableforparticlesincompressibleowshasnotbeenproposedsofar.Inparticular,theabovediscussionshowsthattheunsteadyforcesonparticlesincompressibleowshavenotbeenstudiedextensively.Figure 1-2 summarizesschematicallythestate-of-the-artforthemodelingofforcesandassociatedequationofmotionforparticlesinincompressibleandcompressiblemultiphaseows. Duetothelackofwell-developedtheoriesfortheinviscidunsteadyandviscousunsteadyforcesincompressibleows,manycomputationalinvestigationsusesonlythequasi-steadyforce,seeCarrier[ 17 ],Kriebel[ 53 ],Rudinger[ 89 ],RudingerandChang[ 92 ],PelantiandLeVeque[ 83 ],MiuraandGlass[ 73 ],Elperinetal.[ 30 ],IgraandTakayama[ 46 ],Saito[ 93 ]andJourdanetal.[ 50 ].Theoverarchinggoalofthisworkistostudythebehavioroftheunsteadyforcesincompressibleowsanddevelopmodelsforuseincomputationalstudiesofcompressiblemultiphaseows.Inthisthesis,using 26

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theoryandsimulations,wecarryoutasequenceofinvestigationswiththefollowingobjectives: 1. Tostudyandmodelthequasi-steady,inviscidunsteady,andviscousunsteadyforcesonasphericalparticleinuniformcompressibleStokesows. 2. Tostudyandmodelthequasi-steady,inviscidunsteady,andviscousunsteadyforcesonasphericalparticleinnon-uniformcompressibleStokesows. 3. Tomodeltheeffectofcompressibilityonthequasi-steadydraginsupercriticalows. 4. TostudyandmodelthebehaviorofinviscidunsteadyforceincompressibleowsatniteMachnumbers. 5. Todevelopamodelforshock-particleinteraction. 6. TostudyandmodeltheviscousunsteadyforceincompressibleowsatniteMachandReynoldsnumber. Inthecurrentwork,systematicstepsaretakentostudyunsteadyforcesinparticular.Itisclear,however,thatthetopiccannotbestudiedinitsentiretyinonedissertation.Similartoincompressibleows,multipledissertationsarerequiredtodofulljusticetothisvasttopic. 1.5DissertationLayout Restofthedissertationisorganizedintosevenchaptersandanappendix. Chapter2:GeneralizedBasset-Boussinesq-Oseenequationforunsteadyforcesonasphereinacompressibleow.AtheoreticalinvestigationiscarriedoutusingthelinearizedcompressibleNavier-Stokesequationstocomputetheforceonasphericalparticleundergoingrectilinearunsteadymotioninaquiescenthomogeneousuid.Anexplicitformulationispresentedforquasi-steady,inviscidunsteady,andviscousunsteadyforcesthatincorporatescompressibilityeffects.ThisworkisunderreviewforpublicationinthePhysicalReviewLetters,seeParmaretal.[ 81 ]. Chapter3:Equationofmotionforasphereinnon-uniformcompressibleows.UsingthelinearizedcompressibleNavier-Stokesequations,atheoretical 27

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investigationiscarriedoutofunsteadymotionofasphericalparticleinaninhomogeneousow.Lorentz'sreciprocaltheoremisemployedtoobtainFaxen'scorrectiontothequasi-steady,inviscidunsteady,andviscousunsteadyforces.TheresultsareusedtogeneralizetheMaxey-Riley-Gatignolequationtocompressibleows.ThisworkisabouttobesubmittedtotheJournalofFluidMechanics. Chapter4:Theinviscidunsteadyforceoncylindersandspheresincompress-ibleows.Theunsteadyforcesonacylinderandasphereinsubcriticalcompressibleowareinvestigated.Itisshownthattheunsteadyinviscidforcecanbemorethanfourtimeslargerthanthatpredictedfromincompressibletheory.ThisworkhasbeenpublishedintheProceedingsofRoyalSociety,seeParmaretal.[ 79 ]. Chapter5:Modelingoftheunsteadyforceforshock-particleinteraction.BasedontheworkdescribedintheChapter4,asimplemodelfortheunsteadyforcesforshock-particleinteractionispresented.Theresultsarecomparedwithexperimentalandcomputationaldataforbothstationaryandnon-stationaryspheres.ThisworkhasbeenpublishedintheShockWaves,seeParmaretal.[ 80 ]. Chapter6:Animproveddragcorrelationforspheresandapplicationtoshock-tubeexperiments.Anewcorrelationispresentedforthedragforceonsphericalparticlesincompressiblecontinuumowsthatrepresentsexperimentaldatamorefaithfullythanpriorcorrelations.Withthiscorrelation,recentexperimentaldataobtainedinshock-tubesforparticletrajectoryarereproducedaccurately.ThisworkhasbeenpublishedintheAIAAJournal,seeParmaretal.[ 82 ]. Chapter7:Viscousunsteadyforcesonasphereincompressibleow.AcomputationalinvestigationiscarriedoutforviscousunsteadyforcesonasphericalparticleincompressibleowsatvanishingMachnumbersbutniteReynoldsnumbers.AnewformulationoftheviscousunsteadyforceispresentedbasedontheoreticalworkofChapter2andtheformulationofMeiandAdrian[ 68 ]. 28

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Chapter8:Summaryandconclusionsanddirectionsforfuturework.Conclusionsofthepresentworkarepresentedandthedirectionsforfutureworkarediscussed. AppendixAdescribesnumericalmethodsusedinthiswork. 29

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CHAPTER2GENERALIZEDBASSET-BOUSSINESQ-OSEENEQUATIONFORUNSTEADYFORCESONASPHEREINACOMPRESSIBLEFLOW ViscouscompressibleowaroundasphereisconsideredinthelimitofvanishingReynoldsandMachnumbers.Usingtheanalyticalsolutionderivedinearlierworks,anexactexpressionforthetransientforceonasphereundergoingarbitrarymotionwiththeinclusionofcompressibilityeffectsispresented.Thetransientforceisdecomposedintoquasi-steady,inviscidunsteady,andviscousunsteadycomponents.Theinuenceofcompressibilityoneachofthesecomponentsisexamined.Numericalresultsforthetransientforceareinexcellentagreementwiththeory.ThepresentformulationthusoffersanexplicitexpressionfortheunsteadyforceinthetimedomainandcanbeconsideredasageneralizationoftheBasset-Boussinesq-Oseenequationtothecompressibleowregimethatcanbeusedinnumericalsimulationsofcompressiblemultiphaseows. 2.1Introduction TheunsteadyforceonaparticleinacceleratedmotionwasrstanalyzedbyStokes[ 101 ],whopresentedanexpressionforthefrequency-dependentforceonanoscillatingsphericalparticle.LaterBasset[ 8 ],Boussinesq[ 10 ],andOseen[ 77 ]independentlyexaminedthetime-dependentforceonasphereduetorectilinearmotioninaquiescentviscousincompressibleuid.TheybasedtheiranalysesonthelinearizedunsteadyincompressibleNavier-Stokesequationsvalidforcreepingmotion,i.e.,inthelimitofvanishingReynoldsnumber.Theresultingequationofmotionforasphericalparticle,theso-calledBBOequation,canbewrittenas mpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2mfdv dt)]TJ /F4 11.955 Tf 11.96 0 Td[(6a2p Zt1 p t)]TJ /F8 11.955 Tf 11.95 0 Td[(dv dd,(2) wherempistheparticlemass,v(t)isparticlevelocity,aistheparticleradius,isthedynamicviscosity,mfisthemassofuiddisplacedbytheparticle,istheuiddensity,and==isthekinematicviscosity.Thethreetermsontheright-hand 30

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sidearethequasi-steady(Stokes)drag,inviscidunsteady(added-mass),andviscousunsteady(Bassethistory)forces,respectively.TheBBOequationhasbeenextendedtonon-uniformcreepingowsbyTchen[ 107 ],MaxeyandRiley[ 63 ],andGatignol[ 36 ]andtoniteReynoldsnumbersbyMeiandAdrian[ 68 ],Kimetal.[ 52 ],andMagnaudetandEames[ 62 ]. OurprimarygoalistoextendtheBBOequationtocompressibleows.TherstworkrelevanttoourgoalappearstobethatofZwanzigandBixon[ 119 ],whoinvestigatedthevelocity-correlationfunctionofanatomimmersedinacompressiblevisco-elasticliquid.AminorerrorintheirworkwascorrectedbyMetiuetal.[ 70 ].TemkinandLeung[ 110 ]solvedthelinearizedcompressibleNavier-Stokesequationsinthefrequencydomaintostudytheinteractionbetweenaplanemonochromaticacousticwaveandasphericalparticle.AmoregeneralapproachwasadoptedbyGuz[ 39 ],whosolvedthelinearizedcompressibleNavier-StokesequationsintheLaplacedomainandpresentedasolutionvalidforarbitraryparticlemotion.Theresultsofthesestudiesareessentiallyidenticalexceptfordifferencesduetosimplifyingassumptionsandsometypographicalmistakes.Felderhof[ 32 33 ]usedthefrequency-spacesolutionofZwanzigandBixon[ 119 ]toinvestigatethemotionofaparticleinresponsetoaforceimpulse. Thepurposeofthisworkistopresentanexplicitexpressionforthetime-dependentforceonasphericalparticleundergoingarbitraryunsteadymotionontheacoustictimescalesuchthatcompressibilityeffectsareimportant.AttentionisrestrictedtothezeroReynolds-andMach-numberlimitssothatnon-lineareffectscanbeignored.WeusepreviouslyderivedsolutionsofthelinearizedcompressibleNavier-StokesequationsintheFourier/Laplacedomainstodeterminetheforceonaparticleinresponsetoadelta-functionaccelerationinthetimedomain.Thisforceresponseisthenusedtoconstructanexpressionforthetime-dependentforceonaparticleundergoingarbitrarymotion.Theresultingexpressioncanbeinterpretedasthegeneralizationof 31

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theBBOequationtocompressibleows.ThegeneralizedBBOequationallowsclearinterpretationoftheeffectofcompressibilityonthequasi-steady,inviscidunsteady,andviscousunsteadydragforces.WeshowthatcompressibilitycausestheinviscidunsteadyforcetoassumeanintegralrepresentationrstderivedbyLonghorn[ 58 ].Weobtaintheeffectofcompressibilityontheviscousunsteadyforce.Thetheoreticalresultsarecomparedwithdirectnumericalsimulationsofthecompressibleowaroundanacceleratingparticle.Finally,wepresentthegeneralizedBBOequationthatcanbeusedtotrackparticlesincompressibleowandthatreducestoEq.( 3 )inthelimitofincompressibleow. 2.2ProblemFormulation WeconsidertheunsteadymotionofaparticleinaquiescentcompressibleNewtonianuid.WeconsiderthelimitofRe!0andM!0suchthattheperturbationeldgeneratedbytheparticlemotionisgovernedbythelinearizedcompressibleNavier-Stokesequations.Here,MandRearesuitablydenedMachandReynoldsnumbers.Furthermore,iftheKnudsennumber(whichisproportionaltoM=Re)issmall,viscousheatingandadiabaticcoolingeffectsintheenergyequationcanbeneglected.Asaresult,thetemperatureelddecouplesandthecontinuityandmomentumequationsreducetotheformgivenbyZwanzigandBixon[ 119 ],@0 @t+0ru0=0, (2)0@u0 @t+rp0)]TJ /F8 11.955 Tf 11.96 0 Td[(r2u0)]TJ /F4 11.955 Tf 11.96 0 Td[((b+1 3)rru0=0. (2) InEqs.( 2 )and( 2 ),propertiesassociatedwiththequiescentuidaredenotedbythesubscript0,perturbationquantitiesaredenotedbythesuperscript0,uisthevelocity,pisthepressure,andbisthebulkviscosity.AllothersymbolsaredenedasinEq.( 3 ).Becausetemperatureuctuationsareneglected,theviscositiesareconstantandthespeedofsound c0=p (@p=@)s=p p0=0(2) 32

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canbeusedasaclosurerelation.TheselinearizedequationshavebeensolvedanalyticallybyZwanzigandBixon[ 119 ]andMetiuetal.[ 70 ],whoobtainedanexplicitexpressionfortheforceontheparticleinthefrequencydomain.Givenageneralparticlemotionwithvelocityv(t)thesolutionofEqs.( 2 )( 2 )inLaplacespacecanbewrittenas F(s)=)]TJ /F3 11.955 Tf 9.3 0 Td[(mfsG(r1,r2)L(v)(2) whereF(s)=L(F(t))andL(v)aretheLaplacetransformsofthetimedependentforceF(t)andrectilinearparticlevelocityv(t),respectively,andmf=40a3=3isthemassofuiddisplacedbytheparticle.ThetransferfunctionG(r1,r2)isgivenby G(r1,r2)=(9+9r1+2r21)(1+r2)+(1+r1)r22 r21(1+r2)+(2+2r1+r21)r22,(2) wherer1(s)=as=c0 p 1+(b=+4=3)s=c20andr2(s)=ar s (2) AsimilaranalysiswasindependentlyperformedbyTemkinandLeung[ 110 ]inthecontextofacousticscatteringbyasphericalparticle.TheirresultsareslightlydifferentfromthosebyZwanzigandBixon[ 119 ].ThedifferencescanbetracedtoanapproximationmadebyTemkinandLeung[ 110 ]inthedenitionofstressdistributionaroundtheparticle.TheaboveresultshavealsobeenderivedintherecentbookbyGuz[ 39 ].However,duetotypographicalerrorstheresultsmightappeardifferent. 2.3SolutionforImpulsiveMotion Sincetheproblemislinear,theforceonaparticleundergoingarbitraryrectilinearmotionv(t)canbeexpressedasaconvolutionintegral F(t)=Ztdv dF(t)]TJ /F8 11.955 Tf 11.95 0 Td[()d,(2) 33

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whereF(t)istheforceresponsetoadelta-functionacceleration(i.e.,correspondingtoaunitstepchangeinparticlevelocity).UsingEq.( 2 ),F(t)canbeexpressedas F(s)=)]TJ /F3 11.955 Tf 9.3 0 Td[(mfG(r1,r2).(2) AnexplicitLaplaceinversetransformofEq.( 2 )and,therefore,aclosed-formexpressionforF(t)isnotreadilyavailable.Beforeconstructingthetime-domainsolution,werstanalyzethelimitingcaseofincompressibleow. Theincompressiblelimitisobtainedbylettingc0!1inEq.( 2 )toobtain F,inc(s)=)]TJ /F3 11.955 Tf 9.3 0 Td[(mf9+9r2+r22 2r22,(2) wherer22canbeinterpretedastheLaplacevariablecorrespondingtotimenon-dimensionalizedbytheviscoustimescalea2=.Thecorrespondingexpressioninthetimedomainis F,inc(t)=)]TJ /F4 11.955 Tf 9.3 0 Td[(6aH(t))]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2mf(t))]TJ /F4 11.955 Tf 11.96 0 Td[(6a20r tH(t),(2) whereH(t)istheHeavisidestepfunction.TheaboveexpressionisidenticaltotheBBOequationforadelta-functionacceleration,cf.Eq.( 3 ).Therefore,thegeneralcompressibleforcegiveninEq.( 2 )reducestothecorrectlimitforanincompressibleow. 2.4CompressibilityEffectonInviscidUnsteadyForce Weisolatethethreeterms(quasi-steady,inviscidunsteady,andviscousunsteadyforces)ontheright-handsideofEq.( 3 )andinvestigatetheeffectofcompressibility.First,weconsiderthecompressibilityeffectontheinviscidunsteadyforce.Theinviscidlimitisobtainedbysubstituting=0inEq.( 2 )toget F,iu=)]TJ /F3 11.955 Tf 9.3 0 Td[(mf1+r1 2+2r1+r21,(2) 34

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wherer1canbeinterpretedastheLaplacevariablecorrespondingtotimenon-dimensionalizedbytheacoustictimescalea=c0.Thecorrespondingexpressioninthetimedomainis F,iu()=)]TJ /F3 11.955 Tf 9.3 0 Td[(mfc0 ae)]TJ /F16 7.97 Tf 6.59 0 Td[(cosH(),(2) where=c0t=a.ThisresultfortheinviscidunsteadyforceonasphereimpulsivelysetinmotioninacompressiblequiescentuidwasrstpresentedbyLonghorn[ 58 ].TheeffectofcompressibilityontheinviscidunsteadyforcecanbeestablishedbycomparingEq.( 2 )withthesecondtermontheright-handsideofEq.( 2 ).Thenitespeedofsounddestroystheinstantaneousrelationshipbetweenaccelerationandforceandthusinacompressibleowtheinviscidunsteadyforcecannot,strictlyspeaking,beconsideredasanadded-massforce.Furthermore,compressibilityregularizesthesingulardelta-functionkerneltoasmoothoscillatoryexponentialdecay.Fromaphysicalperspective,thiscanbeexplainedbythecompressionandrarefactionwavesthatemanatefromtheacceleratedparticlewhichpropagateoutwardatnitespeed.Therefore,inacompressibleowtheinviscidunsteadyforceisdependentnotonlyontheinstantaneousacceleration,butonthepastaccelerationhistory.However,duetotheexponential-decayterminEq.( 2 ),thecompressibilityeffectissignicantonlyfor.10. TheinviscidunsteadyforcewasobtainedbyLonghorn[ 58 ]usingtheacousticapproximationofthevelocitypotentialequationandisthusvalidonlyinthezero-Machnumberlimit.Theright-handsideofEq.( 2 )canbeconsideredtobetheresponsekernelforadelta-functionaccelerationforM!0.NotethatR10e)]TJ /F16 7.97 Tf 6.59 0 Td[(cosd=1=2,andthusovertimesmuchlongerthantheacoustictimescale,thenetimpulseontheparticlereducestothecorrectlimitasthatgivenbytheincompressibleadded-massforce.ThecorrespondingkernelsforniteMachnumberscanbeobtainedthroughnumericalsimulations,seeParmaretal.[ 79 ]. 35

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2.5AsymptoticBehaviorsofCompressibleViscousUnsteadyForce Wenowexaminetheeffectofcompressibilityontheviscousunsteadyforce.Tostudytheforceatarbitrarytimes,weresorttonumericalinversionofEq.( 2 )becausewehavenotfoundanexplicitanalyticalLaplaceinverseofthecompletetransferfunctionG(r1,r2).WithVdenotingthescaleofthevelocityvariation,wedenetheReynoldsandMachnumbersasRe=0Va=andM=V=c0.Thuswecanwrite F() mfc0=a=L)]TJ /F10 7.97 Tf 6.58 0 Td[(1(G(R1,R2)),(2) whereL)]TJ /F10 7.97 Tf 6.59 0 Td[(1denotestheLaplaceinversewithrespecttothenon-dimensionaltime=c0t=aandR1=S p 1+(b=+4=3)Kn0SandR2=r S Kn0, (2) whereS=as=c0isthenon-dimensionalLaplacevariableandKn0=M=Re==0c0adenotesamodiedKnudsennumber.ItisinterestingtonotethattheforceresponsedependsonlyontheratiooftheMachnumbertotheReynoldsnumberandnotontheirindividualvalues.TheexpressionmodiedKnudsennumberismotivatedbythestandarddenitionoftheKnudsennumberasKn=p =2M=Re,whereistheratioofspecicheatsoftheuid.Forairatstandardconditions,=1.4andhenceKn00.67Kn.ThecontinuumassumptioniscommonlytakentoimplyKn<0.01andthusweareinterestedintheforceresponseforKn0.O(10)]TJ /F10 7.97 Tf 6.59 0 Td[(2). AlthoughanexplicitexpressionforF()validforarbitraryisnotavailable,fourdifferentasymptoticregimescanbeidentied: RegimeI:Veryshorttime,denedasKn01, RegimeII:Intermediateshorttime,denedasKn01, RegimeIII:Intermediatelongtime,denedas11=(ReM), RegimeIV:Verylongtime,denedas11=(ReM). 36

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Explicitexpressionsfortheforceinthetimedomaincanbeobtainedfortherstthreeregimes.Aswewillseenbelow,nonlinearitybecomesimportantfor1=(ReM).ThusthelinearizedEqs.( 2 )and( 2 )andhencethesolutionpresentedheredonotdescribeaccuratelytheforceevolutioninRegimeIV.Asimilarsituationarisesintheincompressibleformulationalso.ItiswellestablishedthatnomatterhowsmalltheReynoldsnumber,thedecayoftheviscousunsteadyforceatsufcientlylongtimewillbefasterthanthet)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2decaygivenbytheBassethistorykernel(seeMeiandAdrian[ 68 ]). Inwhatfollows,weexaminethebehaviorofF()intherstthreeasymptoticregimes.TheveryshorttimebehaviorofF()canbeobtainedbyconsideringthefollowinglimitintheLaplacespace:1Kn0jSjjSjjSj=Kn0.Correspondingly,thetransferfunctioncanbesimpliedas G(R1,R2) 2+s b +4 3!r Kn0 S.(2) TheLaplaceinversegivesthetimedomainforceresponseinRegimeIas RegimeI:F())]TJ /F11 11.955 Tf 23.91 20.44 Td[( 4 9+2 9s b +4 3!6a20c0r Kn0 H()forKn01.(2) Comparingwiththethirdtermontheright-handsideofEq.( 2 ),whichcanbewrittenas)]TJ /F4 11.955 Tf 9.3 0 Td[(6a20c0p Kn0=,itcanbeenseenthatcompressibilitymodiestheviscousunsteadyforceatveryshorttimesbyafactorthatdependsonb=.Forb=0,compressibilityreducestheunsteadyforceby4(1+1=p 3)=90.70.Thecorrectionfactortotheviscousunsteadyforceincreaseswithincreasingbulkviscosity.Interestingly,forb==59=124.92thecorrectionfactorisequaltounityandEq.( 2 )becomesidenticaltothatintheincompressiblecase. Fromthedenitionoftheintermediateshorttime(Kn01),thefollowingconditioncanbeplacedontheLaplacevariable:Kn0jSj1jSj1=Kn0jSj=Kn0. 37

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ThenG(R1,R2)canbesimpliedas G(R1,R2)1+R1 1+(1+R1)2+2 R21+2(1+R1) 1+(1+R1)2+1 1+(1+R1)2)]TJ /F4 11.955 Tf 50.71 8.09 Td[(1 [1+(1+R1)2]2.(2) ThedominantcontributionforintermediateshorttimecomesfromtheLaplaceinverseofthersttwotermsinthetransferfunctionG(R1,R2),resultingin RegimeII:F())]TJ /F3 11.955 Tf 21.91 0 Td[(mfc0 ae)]TJ /F16 7.97 Tf 6.59 0 Td[(cos)]TJ /F4 11.955 Tf 13.16 8.08 Td[(8 3a20c0r Kn0 H()forKn01.(2) ThersttermissameasF,iu()givenbyEq.( 2 ).Comparingthesecondtermwiththethirdtermontheright-handsideofEq.( 2 ),itcanbeenseenthattheviscousunsteadyforceatintermediateshorttimesisreducedbyafactorof4=90.44becauseofcompressibilityeffects.Notethatthisreductionisindependentofb=.AswillbeseenbelowinFig. 2-1 ,wheretheresultsofthenumericalLaplaceinversionareshown,RegimeIIcanbeobservedonlyifKn01.WithincreasingKn0,thedurationoftheRegimeIIreducesandvanishesentirelyforKn010)]TJ /F10 7.97 Tf 6.58 0 Td[(2.Thustheoveralleffectofcompressibilityontheshort-timebehavioroftheviscousunsteadyforceisnotaspronouncedasfortheinviscidunsteadyforce.The)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2decayobservedintheincompressiblecasepersistsandonlythemagnitudeoftheviscousunsteadyforceismodied. Theintermediatelong-timebehaviorcanbeobtainedinasimilarmannerbyconsideringanasymptoticexpansionforjSj!0andcarryingouttheLaplaceinverseweobtain RegimeIII:F())]TJ /F4 11.955 Tf 21.92 0 Td[(6aH())]TJ /F4 11.955 Tf 11.95 0 Td[(6a20c0r Kn0 for11 ReM.(2) ComparingwithEq.( 2 ),boththequasi-steadyandtheviscousunsteadyforcesarerecoveredandfoundtobeunaffectedbycompressibilityeffects.Strictlyspeaking,theabovesolutionforthelinearizedperturbationNavier-Stokesequationsisvalidfor1 38

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andtheadditionallimitof1=(ReM)arisesonlyfromtheneglectofthenonlinearterms.Thetimescaleonwhichnonlineareffectsbecomesignicantcanbeestimatedasfollows.InderivingthelinearizedformofthecompressibleNavier-Stokesequations,theassumptionthattheinertialtermsarenegligiblecomparedtotheviscoustermsimpliesthatthelengthscaleL=V.Ifwetakethelengthscaletogrowbydiffusionasp t,theassumptionoflinearizedcompressibleNavier-Stokesequationscanbejustiedonlyfort=V2.Expressedintermsoftheacoustictimescale,thisrestrictionbecomes1=(ReM).Notethattheaboveargumentappliesinanincompressibleowalso,andthenonlineareffectscanbeshowntobecomeimportantforc1=Re,wherecistimenon-dimensionalizedbytheconvectivetimescalea=V.ThisisconsistentwithpastresultsforincompressibleowthattheBasset-historykernelisvalidonlyforc1=ReevenatlowReynoldsnumbers(seeMeiandAdrian[ 68 ]).Thus,owingtononlinearity,theverylongtimeforcebehaviorinRegimeIVwillbebothReynolds-andMach-numberdependentinacomplexmannerandwillnotbeaddressedhere. 2.6NumericalEvaluationofViscousUnsteadyForce Inthefollowing,weextractthecompressibleformoftheviscousunsteadyforceatarbitrarytimesusingnumericalLaplaceinversionandcompareitwithitsincompressibleform.WeisolatetheviscousunsteadyforcefromtheoverallforceexpressiongiveninEq.( 2 )bysubtractingthequasi-steadycontributionandtheinviscidunsteadyforce,giveninEqs.( 2 )and( 2 )intheLaplaceandtimedomains,respectively.Theresultingviscousunsteadyforceinthetimedomainisthen F,vu() mfc0=a=)]TJ /F11 11.955 Tf 11.29 16.86 Td[(L)]TJ /F10 7.97 Tf 6.59 0 Td[(1(G(R1,R2)))]TJ /F4 11.955 Tf 13.16 8.09 Td[(9 2Kn0)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F16 7.97 Tf 6.59 0 Td[(cos.(2) Werecastthisviscousunsteadyresponsetodeltafunctionaccelerationinthefollowingform F,vu() mfc0=a=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(9 2r Kn0 C(),(2) 39

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whereC()isacompressiblecorrectionfunction,denedastheratioofF,vu()relativetotheincompressibleformoftheviscousunsteadyforce. Figure 2-1 showsC()plottedasafunctionofnon-dimensionaltimeforvariousvaluesofKn0andb=0.InRegimeI(Kn01)thecorrectionfunctionapproaches0.70atveryshorttimes.Also,weobservethatC()!1as!1asexpected.Atintermediateshorttimes(whichexistonlyforKn010)]TJ /F10 7.97 Tf 6.58 0 Td[(2)thecorrectionfunctiontakesaconstantvalueof0.44.Amorecomplexcompressibilityeffectcanbeobservedattransitionaltimesbetweenthedifferentasymptoticregimes.Thedecreasefromtheconstantvalueatveryshorttimetothenewconstantvalueatintermediateshorttimeoccursinamonotonicfashion.Atabout=O(10)]TJ /F10 7.97 Tf 6.59 0 Td[(2Kn0),C()startstodeviatefromitslimitingvalueof0.70anddecreasesto0.44atabout=O(Kn0).ThetransitionfromRegimeIItoRegimeIIIthatoccursatO(1)ismorecomplex.At=O(10)]TJ /F10 7.97 Tf 6.58 0 Td[(2),C()increasesrapidlyirrespectiveofKn0towardapeakvalueofabout1.45beforedecreasinginastronglydampedoscillatorymannertowardunity.Thus,thecompressibilitycorrectiontotheviscousunsteadyforceisboundedbetween0.44and1.45. ThesensitivityofC()tothebulkviscosityisassessedinFig. 2-2 .Asdiscussedabove,theasymptoticbehaviorisdependentonb=atveryshorttimes.ProvidedKn0issmallenough,atintermediatesmalltimesC()isindependentofb=.ForallvaluesofKn0,thebehaviorofC()for&O(10)]TJ /F10 7.97 Tf 6.59 0 Td[(2)isonlyweaklydependentonb=,asexpected. FromtheanalysisofLonghornitcanbeinferredthatwhenasphericalparticleisimpulsivelysetintomotion,aninvisciddisturbancefrontofradiusa+c0t=a(1+)propagatesawayfromtheparticle.Initiallythedisturbanceeldiscompressional/expansionalupstream/downstreamoftheparticle,andthuscontributingtoastronginviscidunsteadyforcedirectedoppositetoparticlemotion.Astheinvisciddisturbancefrontpropagatesoutanalternatingsequenceofcompressionalandexpansionwavesradiateouton 40

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Figure2-1. ThebehaviorofthecorrectionfunctionC()thataccountsforthecompressibilityeffectontheviscousunsteadyforce(seeEq.( 2 )).Resultsareplottedforb=0andKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,10)]TJ /F10 7.97 Tf 6.59 0 Td[(5,10)]TJ /F10 7.97 Tf 6.59 0 Td[(8,10)]TJ /F10 7.97 Tf 6.59 0 Td[(11g.Theopencirclesrepresentthecurve-tgivenbyEq.( 2 )evaluatedforKn0=10)]TJ /F10 7.97 Tf 6.59 0 Td[(5. acoustictimescale,whosestrengthrapidlydecays,thuscontributingtoboththeoscillatorybehavior(costerm)andtheexponentialdecayoftheinviscidkernel(Kiu,seeEq.( 2 )).Incomparison,theviscousboundarylayerthicknesscanbeestimatedtogrowasp t=ap Kn0.For&O(1)theviscousboundarylayerissubmergedwithintheinvisciddisturbanceeldandasaresulttheeffectofcompressibilityontheviscousunsteadyforceisoscillatoryandstronglydecays.Onlyforverysmalltime(RegimeI),whenKn0,theviscousboundarylayeristhickerthantheinvisciddisturbancefront.Itisonlyinthisregimethebulkviscosityhasastronginuenceontheboundarylayergrowthandconsequentlyontheviscousunsteadykernel. 41

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Figure2-2. ThedependenceofthecorrectionfunctionC()onthebulkviscosity.Resultsareplottedforb==f0,1,59=12,10gandKn0=f10)]TJ /F10 7.97 Tf 6.58 0 Td[(2,10)]TJ /F10 7.97 Tf 6.58 0 Td[(8g. 2.7InviscidandViscousUnsteadyForceKernelsandNumericalConrmation Basedonresultspresentedintheprevioussections,wewrite F mfc0=a=1 mfc0=a)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(F,qs+F,iu+F,vu=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(9 2 ac0)]TJ /F3 11.955 Tf 11.96 0 Td[(e)]TJ /F16 7.97 Tf 6.58 0 Td[(cos)]TJ /F4 11.955 Tf 13.15 8.09 Td[(9 2r Kn0 C() p .(2) Whilethenormalizedinviscidunsteadyforcedependsonlyon,thenormalizedviscousunsteadyforcedependsonbothKn0andb=throughC().Theaboveforceresponsetoadeltafunctionaccelerationcanbeusedtodeneinviscidandviscousunsteadyforcekernelsas Kiu()=e)]TJ /F16 7.97 Tf 6.58 0 Td[(cosandKvu()=C() p .(2) InthedenitionofKvuwehavefollowedtheconventionalnotationfortheBassethistorykernelKB()=1=p andC()isthecorrectionfunctiondenedinEq.( 2 ). 42

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Tocheckthetheoreticalresultsandestablishlimitsofvalidity,wehavecarriedoutnumericalsimulationsoftheaxisymmetriccompressibleowaboutasphericalparticleforb=0.Inthesimulations,thesphericalparticleisinitiallystationaryinaquiescentuidandimpulsivelyacceleratedtoanalsteadystate.Toextracttheunsteadyforceinthesimulations,wesubtractthequasi-steadyforcebasedonthecorrelationCD,qs=(24=Re)(1+0.15Re0.687),duetoSchillerandNaumann[ 94 ],fromthecomputedinstantaneousforce.Theresultsoffoursimulationswillbereportedhere.TherstthreesimulationsarecharacterizedbyM=10)]TJ /F10 7.97 Tf 6.58 0 Td[(3andRe=f0.1,1.0,10.0g.ThecorrespondingmodiedKnudsennumbersareKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,10)]TJ /F10 7.97 Tf 6.59 0 Td[(3,10)]TJ /F10 7.97 Tf 6.59 0 Td[(4g.TheresultsofthesimulationsareshowninFig. 2-3 ,wherethenon-dimensionalforceisplottedasafunctionoftheacoustictime=c0t=a.Theagreementbetweenthetheoryandthesimulationsisexcellent.Thesimulationscaptureaccuratelythe)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2decayforboth1and&O(10).Atintermediatetimes,theinuenceofinviscidunsteadyforcebecomessignicantandthesimulationscapturethiswell.Infact,forKn0=10)]TJ /F10 7.97 Tf 6.59 0 Td[(4and10)]TJ /F10 7.97 Tf 6.58 0 Td[(3,overashorttimeintervalaround3.5,theunsteadyforceontheparticlebecomesnegativeandisorientedalongthedirectionofparticlemotion(asopposedtotheforcebeingorientedagainstthedirectionofparticlemotion).ThisinterestingcounterintuitivebehaviorhasbeenobservedandcommenteduponbyParmaretal.[ 79 ]. Intherstthreesimulations,wehave1=(ReM)=f104,103,102g,respectively,andthustheagreementbetweenthenonlinearsimulationsandthelineartheoryisgoodovertheentirerangeofthecomputedtimeinterval,asexpected.WehavealsosimulatedacaseofM=10)]TJ /F10 7.97 Tf 6.59 0 Td[(1andRe=10,correspondingtoKn0=10)]TJ /F10 7.97 Tf 6.59 0 Td[(2.TheresultsforthiscasearealsoplottedinFig. 2-3 .Againtheagreementisexcellentatsmalltimes.However,since1=(ReM)=1forthiscase,theeffectofnonlinearitybecomesimportantforO(1)andtheresultsofthesimulationshowafasterdecaythanthe)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2behaviorpredictedbythelineartheory. 43

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Figure2-3. Timeevolutionofthenormalizedunsteadyforce.Theoreticalpredictions(lasttwotermsofEq.( 2 ))areplottedassolidlinesforKn0=f10)]TJ /F10 7.97 Tf 6.58 0 Td[(2,10)]TJ /F10 7.97 Tf 6.58 0 Td[(3,10)]TJ /F10 7.97 Tf 6.58 0 Td[(4gandb=0.Inviscidunsteadykernel(secondlastterminEq.( 2 ))isshownasdashesline.Correspondingsimulationresultsforfourdifferentcasesareshownassymbols. As!0,whiletheinviscidkernelisequaltounity,theviscouskerneldivergesas1=p .Atlargetimes,theinviscidkerneldecaysexponentially,whiletheviscouskerneldecaysalgebraically.Thus,bothatshortandlongtimes,theviscousunsteadyforcedominatestheinviscidunsteadyforce.Atintermediatetimes,theinviscidunsteadyforcebecomesimportantanditcanbeshownthatonlyforKn0>5.9610)]TJ /F10 7.97 Tf 6.59 0 Td[(2willtheviscousunsteadyforcedominatetheinviscidunsteadyforceatalltimes.ThislimitingvalueofKn0mustbeviewedwithcaution,however,becausethecontinuumassumptionbreaksdownforKn0>10)]TJ /F10 7.97 Tf 6.59 0 Td[(2.AtsmallervaluesofKn0,thereexistsanintermediaterangeoftime,lowupp,wheretheinviscidunsteadyforcewillexceedtheviscousunsteadyforce. 44

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Figure2-4. lowanduppforb==f0,1,59=12,10g. Figure 2-4 presentslowanduppforarangeofKn0andb=.ForsmallvaluesofKn0,uppreachesaconstantvalueof=2independentofb=. 2.8GeneralizationoftheBBOEquationtoCompressibleFlows Theabove-presentedresultscanbeusedtowriteageneralexpressionfortheforceonaparticleundergoingarbitrarytime-dependentmotionv(t)inaviscouscompressibleuid.ThegeneralizationoftheBBOequationtocompressibleowcanbeexpressedas mpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av)]TJ /F3 11.955 Tf 11.89 0 Td[(mfZtKiu(t)]TJ /F8 11.955 Tf 11.96 0 Td[()c0 adv ddc0 a)]TJ /F4 11.955 Tf 11.88 0 Td[(6a20p ZtKvu(t)]TJ /F8 11.955 Tf 11.89 0 Td[()dv dd,(2) wheretheinviscidandviscousunsteadyforcekernelsaregiveninEq.( 2 ).TheaboveequationisvalidinthelimitofzeroReynoldsandMachnumbers.Thesignicanceofthisextensionistwo-fold.First,itincludesexplicitexpressionsfor 45

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theinviscidunsteadyandviscousunsteadycomponentsoftheforcethatreducetotheirwell-knowncounterpartsintheincompressiblelimit.Second,theextensioncanbecombinedwithotherforcessuchasbuoyancy,gravity,liftandelectrostaticforcestogiveacompleteequationofmotionfortheparticle.ThecompressibleBBOequationcanbeusedtotrackaccuratelylargenumbersofparticlesinEulerian-EulerianorEulerian-Lagrangiansimulationsofcompressiblemultiphaseows.Theaboveexpressionalsoallowsforback-couplingoftheforceontothecarrieruidphase. Boththeinviscidandviscouscompressibilitycorrectionsdecayrapidlyandcanbeignoredfor&10.Therefore,ifthetimescaleofunsteadinessismuchlargerthant&10a=c0,theincompressibleformulationissufcient.Notethatitwasmentionedearlierthatthelong-timeintegraloftheinviscidunsteadykernelKiu()reducestotheadded-masscoefcientof1=2.AsimilarresultcanbeestablishedfortheviscousunsteadykernelKvu()givenbyEq.( 2 ).Thelong-timeintegrationofKvu()canbeshowntoapproachtheincompressiblelimit, Zt0Kvu()d)]TJ /F11 11.955 Tf 11.96 16.27 Td[(Zt0KB()d!0fort1.(2) Whentheproposedequationofmotion( 2 )isusedinpractice,anexpressionforC()isrequired.Thefollowingcurve-tcanbeemployedforthispurpose,assumingthatb=0:C()=4 9+4 9p 31 1+2.38)]TJ /F16 7.97 Tf 10.85 -4.98 Td[( Kn00.57e1.02=Kn0+2C1e)]TJ /F9 7.97 Tf 6.59 0 Td[(C2fcos[C3()]TJ /F3 11.955 Tf 11.95 0 Td[(C4)]+sin[C3()]TJ /F3 11.955 Tf 11.95 0 Td[(C4)]g+5 9C5 C5+C6, (2) 46

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whereC1=0.96+1.71exp0.51(logKn0+1.25),C2=1.14+0.22exp0.57(logKn0+1.45), (2)C3=0.87)]TJ /F4 11.955 Tf 11.96 0 Td[(0.26exp0.50(logKn0+3.55),C4=0.25)]TJ /F4 11.955 Tf 11.95 0 Td[(1.38exp0.60(logKn0+1.98), (2)C5=3.38+7.82exp0.72(logKn0)]TJ /F4 11.955 Tf 11.95 0 Td[(0.16),C6=5.09+5.71exp0.89(logKn0+2.42). (2) ThelimitingbehaviorofthiscurvetisconsistentwithEq.( 2 ).TheaccuracyofthecurvetcanbejudgedfromFig. 2-1 forKn0=10)]TJ /F10 7.97 Tf 6.59 0 Td[(5.TheagreementisequallygoodforothervaluesofKn0.Themaximumerrorofthecurve-tislessthan1percent. Finally,itshouldalsobepointedoutthekernelspresentedinEq.( 2 )combinedwiththeabovecorrectionfunctionareappropriateinthelimitofzeroReynoldsandMachnumbers.TheniteMach-numberinuenceontheinviscidkernelhasbeenaddressedbyParmaretal.[ 79 ].Similarly,thecorrectionfunctionC()canbeexpectedtodependonbothReynoldsandMachnumbers.MeiandAdrian[ 68 ]andKimetal.[ 52 ]haveestablishedthatnon-linearitiesduetoniteReynoldsnumberhaveacomplexeffectontheviscousunsteadykernel,whichalsobecomesdependentontheprecisenatureoftheaccelerationanddeceleration.DependenciesontheReynoldsandMachnumbersofsimilarcomplexitycanbeexpectedtoariseinthecompressibleviscouskernelalso. 2.9Conclusions Wehaveobtainedanexplicitequationforthetime-dependentforceonasphericalparticleundergoingarbitraryunsteadymotioninacompressibleow.TheresultingequationofmotionisthegeneralizationoftheBasset-Boussinesq-Oseenequationtothecompressibleregime.Thesignicanceofthisextensionisthatitincludesexplicitexpressionsforthequasi-steady,theinviscidunsteady,andtheviscousunsteadycomponentsoftheforce,whichreducetotheirwell-knowncounterpartsin 47

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theincompressiblelimit.Theeffectofcompressibilityontheinviscidunsteadyforceissignicant.Thenitespeedofsoundinacompressibleowdestroystheinstantaneousrelationshipbetweenaccelerationandtheinviscidunsteadyforceandregularizesthesingulardelta-functionforcetoasmoothoscillatoryexponentialdecay.Theeffectofcompressibilityontheviscousunsteadyforceismodest.Theoverall1=p tdecayrateoftheviscousunsteadyforceismaintainedthesameasthatoftheBassethistoryforce.ThemodicationduetocompressibilityappearsasamultiplicativecorrectionfactorC(c0t=a)totheBassethistoryforce,whosevalueisbounded,4=9C(c0t=a)<1.5(forzerobulkviscosity).Theeffectofbulkviscosityisnotstrongandislimitedtoveryshorttimes,c0t=aKn0.Theeffectofcompressibilityontheinviscidunsteadyandtheviscousunsteadyissignicantonlyuptofewacoustictimes,sayc0t=a<10. 48

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CHAPTER3EQUATIONOFMOTIONFORASPHEREINNON-UNIFORMCOMPRESSIBLEFLOWS ViscouscompressibleNavier-Stokesequationsareconsideredfortransientforceonasphericalparticleundergoingunsteadymotioninanon-homogeneousow.CompressibleextensiontoMaxey-Riley-GatignolequationisproposedforparticledynamicsinaviscouscompressibleowatlowMachandReynolds 3.1Introduction Stokes[ 101 ]studiedthesteadyandoscillatorymotionofasphericalparticleinanincompressibleuidandobtainedexpressionsforthesteadyandfrequency-dependentdraginthelimitofcreepingow.ThecorrespondingexpressioninthetimedomainforthedragonaparticleundergoingarbitrarymotionwasobtainedindependentlybyBasset[ 8 ],Boussinesq[ 10 ],andOseen[ 77 ].Theresultingso-calledBBOequationdescribesthetime-dependentmotionofaparticleinanincompressiblequiescentuidandcanbewrittenas, mpdup dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6aup)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2mfdup dt)]TJ /F4 11.955 Tf 11.95 0 Td[(6a2p Zt1 p t)]TJ /F8 11.955 Tf 11.96 0 Td[(dup dtt=d,(3) wheremp,up(t),andaaretheparticlemass,velocityandradius,respectively,mfisthemassofuiddisplacedbytheparticleand,,and==arethedensityanddynamicandkinematicviscositiesoftheuid,respectively.Thethreetermsontheright-handsidearethequasi-steady(Stokes)drag,inviscidunsteady(added-mass),andviscousunsteady(Basset-history)forces,respectively. Theeffectofambient-owinhomogeneitywasrststudiedbyFaxen[ 31 ],whoobtainedtheresultthatthesteadydragonasphericalparticleisgivenby6a uS,where uSistheundisturbedambientuidvelocityaveragedovertheparticlesurface.Theinuenceofinhomogeneityforunsteadyparticlemotionwasrstconsideredinthefrequency-domainsolutionofMazurandBedeaux[ 65 ].MaxeyandRiley[ 63 ]andGatignol[ 36 ]independentlyobtainedanexplicitexpressionfortheFaxencorrection 49

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tothetime-dependentforceonaparticleinthecreeping-owlimit.TheresultingMaxey-Riley-Gatignol(MRG)equationdescribesthetime-dependentmotionofaparticleinanunsteadyinhomogeneousincompressibleow. TheMRGequationanditsextensionshavebeenusedextensivelyinthetrackingofparticles,bubbles,anddropletsthroughcomplexambientows.TheMRGequationiscommonlyheldtobeapplicabletoincompressibleows.However,itisvalidforcompressibleambientowsalso,providedthatthecompressibilityoftheperturbationowcausedbytheparticlecanbeignored.Therearesituations,suchasshock-particleinteraction,wherethecompressibilityoftheperturbationowbecomesimportant.Furthermore,evenforMachnumbersapproachingtheincompressiblelimit,acompressibleformulationofequationofparticlemotionisrequirediftheforceevolutionontheacoustictimescale,i.e.,ta=c,wherecisthespeedofsound,isofinterest.Inthiswork,wepresentageneralizedMRGequationthatfullyaccountsforthecompressiblenatureoftheperturbationow. TheearliestefforttoincludetheeffectofcompressibilityontheviscousmotionofaparticleappearstobethatofZwanzigandBixon[ 119 ].ThisandsubsequentworksbyTemkinandLeung[ 110 ],Metiuetal.[ 70 ],Felderhof[ 32 33 ],andGuz[ 39 ]presentedexactsolutionsofthelinearizedcompressibleNavier-StokesequationsfortheunsteadyhomogeneousowaroundaparticleinthefrequencyorLaplacedomains.RecentworkbyParmaretal.[ 81 ]exploitedthefrequency-domainsolutiontoobtainanexplicitexpressionforthetime-dependentforceonaparticlethatisvalidontheacoustictimescale.TheresultsofParmaretal.[ 81 ]canbeconsideredasageneralizedBBOequationthataccountsforthecompressibilityoftheperturbationow. InthesamemannerinwhichtheMRGequationextendstheBBOequationtoinhomogeneousincompressibleows,hereweextendthecompressibleformulationofParmaretal.[ 81 ]toincludetheeffectsoftheinhomogeneityoftheambientow.Thesimultaneousinclusionofinhomogeneity,unsteadiness,andcompressibilityposes 50

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signicantchallengeseveninthelimitofzeroReynoldsandMachnumbers.Arigorousderivationhasbeenmadepossiblebythefollowingsteps:(i)FollowingMaxeyandRiley[ 63 ],weformulatethegoverningequationinthemovingreferenceframeattachedtotheparticleandseparatetheowintoundisturbedambientandperturbationcomponents.(ii)Thegoverningequationsarelinearizedandsimpliedthroughrigorousscalinganalysis.(iii)Adensity-weightedvelocitytransformationisthekeystepthattransformstheproblemtoanequivalentconstant-densityproblem.(iv)TheresultinglinearizedcompressibleNavier-StokesequationsaresolvedbymakinguseofthegeneralizedLorentzreciprocaltheoremtoobtainanexpressionfortheforceontheparticleinthefrequencydomain.(Similarresultscanalsobeobtainedbyfollowingtheinduced-forceapproachofBedeauxandMazur[ 9 ].)(v)FollowingParmaretal.[ 81 ],wetransformthesolutiontothetimedomainand(vi)nallybyinvokingGalileaninvarianceweobtainanexplicitexpressionforthetime-dependentforceonaparticle. TheabovestepsaredescribedindetailinSections2to7.Theresultingcompressibleequationofmotioninvolvesinviscidandviscousforcekernels,whichareobtainedinSection8.ThenalequationofmotionispresentedanditslimitationsarediscussedinSection9.HerewealsoshowthatthepresentcompressibleequationofmotioncorrectlyreducestoBBOandMRGequationsintheappropriatelimits. 3.2GoverningEquationsforFlowAroundaMovingParticle TheequationsgoverningtheundisturbedambientowintheabsenceoftheparticlearethecompressibleNavier-Stokesequations,@0 @~t+~r0u0=0, (3)@0u0 @~t+~r0u0u0=)]TJ /F8 11.955 Tf 9.29 0 Td[(0g+~r0, (3)0=)]TJ /F3 11.955 Tf 9.3 0 Td[(p0I+)]TJ /F4 11.955 Tf 7.33 -7.16 Td[(~ru0+(~ru0)T+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(b)]TJ /F10 7.97 Tf 13.15 4.71 Td[(2 3~ru0I, (3) wherethesymbolsaredenedasabove,thesubscript()0indicatestheundisturbedambientow,gisthegravitationalacceleration,Iistheunittensor,andbisthebulk 51

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viscosity.Thetimederivative@=@~tandthegradientoperator~rareexpressedinthexedlaboratorycoordinate~x.TheKnudsennumberisassumedtobesmall,sothattheenergyequationdecouplesfromthecontinuityandmomentumequations.Closureisprovidedbythedenitionofthesoundspeedasc20=@p0=@0. Considerasphericalparticleofradiusawithcenterat~xp(t),translatingandrotatingwithvelocitiesup(t)andp(t).Theambientowismodiedbytheparticleandisgovernedbythefollowingsetofequations,@ @~t+~ru=0, (3)@u @~t+~ruu=)]TJ /F8 11.955 Tf 9.3 0 Td[(g+~r, (3)=)]TJ /F3 11.955 Tf 9.3 0 Td[(pI+)]TJ /F4 11.955 Tf 7.32 -7.16 Td[(~ru+(~ru)T+)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(b)]TJ /F10 7.97 Tf 13.15 4.71 Td[(2 3~ruI, (3) wherec2=@p=@providesclosure.Theboundaryconditionsare u(~x,~t)=up(~t)+p(~x)]TJ /F4 11.955 Tf 11.8 0 Td[(~xp)forj~x)]TJ /F4 11.955 Tf 11.81 0 Td[(~xp(~t)j=a,(3) and (~x,~t)=0(~x,~t),p(~x,~t)=p0(~x,~t),u(~x,~t)=u0(~x,~t)forj~x)]TJ /F4 11.955 Tf 11.81 0 Td[(~xp(~t)j!1.(3) Theequationofmotionoftheparticleis mpdup d~t=mpg+ISndS,(3) wheretheintegrationisoverthesurfaceoftheparticle,nisoutwardunitnormalvector,andghasbeenassumedconstant. 3.3MovingReferenceFrameandSeparationofDisturbanceFlow Forfurtheranalysis,itisconvenienttoattachthereferenceframetothecenterofthemovingparticle,henceforthtermedthemovingreferenceframeanddenotedbyx=~x)]TJ /F4 11.955 Tf 12.16 0 Td[(~xp.Theuidvelocityinthemovingreferenceframeisv(x,t)=u(x,t))]TJ /F15 11.955 Tf 12.32 0 Td[(up(t) 52

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andthegoverningequationsbecome@ @t+rv=0, (3)@v @t+rvv=)]TJ /F8 11.955 Tf 9.3 0 Td[(dup dt)]TJ /F8 11.955 Tf 11.95 0 Td[(g+r, (3)=)]TJ /F3 11.955 Tf 9.3 0 Td[(pI+)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rv+(rv)T+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(b)]TJ /F10 7.97 Tf 13.15 4.71 Td[(2 3rvI, (3) wherethetimederivative@=@tandthegradientoperatorrareexpressedinthemovingreferenceframe.WhenthegoverningequationsarelinearizedasoutlinedinSection 3.4 ,theinuenceofparticletranslationandrotationdecouples.Thus,ignoringparticlerotation,theboundaryconditionsbecome v(x,t)=0forjxj=a,(3a) and v(x,t)=u0(x,t))]TJ /F15 11.955 Tf 11.96 0 Td[(up(t)forjxj!1.(3b) Theoweld(,v,p)canbeseparatedintotheundisturbedoweld(0,v0,p0)andthedisturbanceoweld(0,v0,p0).Theundisturbedvelocityinthemovingreferenceframeisv0=u0)]TJ /F15 11.955 Tf 12.44 0 Td[(up,whereu0anduparegivenbyEqs.( 3 )and( 3 ),respectively. Thegoverningequationsforthedisturbanceoware@0 @t+r(0v0+0v0+0v0)=0, (3)@ @t(0v0+0v0+0v0)+r(20v0v0+0v0v0+0v0v0+20v0v0+0v0v0)=)]TJ /F8 11.955 Tf 9.3 0 Td[(0dup dt)]TJ /F8 11.955 Tf 11.95 0 Td[(0g+r0, (3)0=)]TJ /F3 11.955 Tf 9.3 0 Td[(p0I+)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rv0+(rv0)T+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(b)]TJ /F10 7.97 Tf 13.15 4.71 Td[(2 3rv0I, (3) withboundaryconditions v0(x,t)=up(t))]TJ /F15 11.955 Tf 11.95 0 Td[(u0(x,t)forjxj=a,(3) 53

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and v0(x,t)=0forjxj!1.(3) 3.4ScalingAnalysis Wenowsimplifythegoverningequationsbyconsideringtherelativeimportanceofthedifferentterms.Intheprocess,wederiveasetofconditionsforthesimpliedgoverningequationstobevalid.Inthefollowing,[f]indicatestheappropriatescaleoff.Theundisturbeddensity,pressure,andrelativevelocityattheinitiallocationoftheparticlewillbechosenasthereferencedensity,pressure,andvelocityscales,i.e.,r=0(0,0),pr=p0(0,0),andvr=ju0(0,0))]TJ /F15 11.955 Tf 11.98 0 Td[(up(0)j.Acoustic,viscous,andconvectivetimescalescanbedenedas ta=a c0,tv=a2 ,tc=a vr.(3) Takingtheradiusoftheparticletobethelengthscale,wedeneMachandReynoldsnumbersas M=vr c0=ta tcandRe=vra =tv tc.(3) ThepresentanalysisislimitedtoverysmallReynoldsandMachnumbers.Inaddition,werestrictattentiontothecontinuumregime.Therefore,theKnudsennumber,whichisproportionaltoM=Re,isrequiredtobesmallalso.Asaresult,weexpresstherstofourconditionsas Condition1:MRe1ortatvtc.(3) Next,weestablishthescalingoftheperturbationdensityfromabalanceofthersttwotermsinEq.( 3 ).Weinferthat [0]8>>>>>><>>>>>>:rfortO(tc),RerfortO(tv),MrfortO(ta),(3) 54

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andthusinordertorestrictthedensityperturbationstoremainsmallwelimitattentiontoonlyacousticandviscoustimescales.Inotherwords,weimposetherestrictionthat Condition2:ttc.(3) Thescalefortheperturbationvelocityisequaltotheambientvelocityscaleinthemovingreferenceframe,i.e.,[v0]vr. Wenowwishtocontainthescalesofspatialandtemporalvariationsofundisturbedambientdensityandvelocityelds.LetLvandLbethelengthscalesoftheundisturbedambientvelocityanddensityvariations,approximatedasLvvr=[rv0]andLr=[r0],respectively.LetthecorrespondingtimescalesbeTvvr=[@v0=@t]andTr=[@0=@t],respectively.Werequirethespatialvariationsoftheambientvelocitytobecontainedovertheparticleradius.Similarly,werequiretemporalvariationsoftheambientvelocitytobecontainedontheconvectivetimescale.Theserequirementsleadto Conditions3:8>><>>:a.Lv)a vr[rv0].1,a vr.Tv)a v2r@v0 @t.1.(3) Alongthesamelines,werequirevariationsoftheambientdensityovertheparticleradiusandontheconvectivetimescaletobecontained,leadingto Conditions40:8>><>>:a.L)a r[r0].1,a vr.T)a rvr@0 @t.1.(3) Conditions3and40areconsistentwiththecompressibleNavier-Stokesequations(Eqs.( 3 )to( 3 ))fortheundisturbedambientow.ItshouldbenotedthatthelengthandvelocityscalesoftheundisturbedambientowatthesystemlevelmaybesuchthatthemacroscaleReynoldsnumberislarge.Asaresult,thenonlineareffectsareimportantonthatscale.Herewewilllinearizetheequationsbyignoringnonlinearityonthescaleoftheparticle.Givenanambientow,conditions3and40canbeinterpreted 55

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asimposingrestrictionsontheallowableparticlesizeforthelinearizationtobevalid.Althoughconditions40aresufcientforlinearization,aswewillseebelow,theywillbereplacedbystrongerconditionstofurthercontaintheperturbationdensity.Linearizationfurtherrequiresthat Conditions5:8>><>>:a2 vrdup dt.1,a2 vr[g].1.(3) TheaboveveconditionsallowEqs.( 3 )and( 3 )tobesimpliedto@0 @t+r(0v0)=0, (3)@0v0 @t=r)]TJ /F3 11.955 Tf 9.3 0 Td[(p0I+)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(rv0+(rv0)T+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(b)]TJ /F10 7.97 Tf 13.16 4.71 Td[(2 3rv0I, (3) withc20=p0=0. 3.5Density-WeightedVelocityTransformation Thekeytosolvingthelinearizedperturbationequationsistodeneadensity-weightedvelocityas V=0v0 r,(3) andreducetheproblemtooneofconstantuniformambientdensity.Tocompletethisreductionweneedtofurtherconstraintheambient-densityvariation.Theambient-densityvariationneartheparticleisapproximatedwithatruncatedTaylorseriesas 0(x,t)=r+t@0 @t(0,0)+x(r0)(0,0),(3) where()(0,0)indicatesevaluationatx=0andt=0.WerequiretheapproximationtobevalidfortO(tv).Overthistimescale,thedistancestraveledbytheconvective,shear,andacousticwavesareO(Rea),O(a),andO(Rea=M)respectively.Sincewewanttosimplifythedensityintheviscousterms,werequirethetruncatedTaylorseriestobeadequateforjxjO(a).Conditions40for[@0=@t]issufcientforthesecondtermontheright-handsideofEq.( 3 )tobeofsmallerorder.However,werequirea 56

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strongerrestrictionon[r0]thanthatexpressedbyconditions40fortheTaylorseriestobeconvergent.Thus,wearriveatthefollowingconditionsfortheambient-densityvariationthatreplaceconditions40, Conditions4:8>><>>:aL)a r[r0]1,a vr.T)a rvr@0 @t.1.(3) Withthedensity-weightedvelocitytransformationandconditions4,Eqs.( 3 )and( 3 )canbesimpliedto@0 @t+rrV=0, (3)r@V @t=rh)]TJ /F3 11.955 Tf 9.3 0 Td[(p0I+(rV)+(rV)T+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(b)]TJ /F10 7.97 Tf 13.15 4.71 Td[(2 3rVIi. (3) Thecorrespondingboundaryconditionsforthedensity-weightedvelocityontheparticlesurfaceandfareldare,V(x,t)=0(x,t) r(up(t))]TJ /F15 11.955 Tf 11.96 0 Td[(u0(x,t))=Vp(x,t)forjxj=a, (3)V(x,t)=0forjxj!1. (3) Theadvantageofthetransformationisthatthevariabledensityisnowabsentfromtheequationsandthattheundisturbedambientdensityappearsonlyintheboundaryconditionontheparticlesurface. 3.6HydrodynamicForceduetoUndisturbedFlow ThetotalhydrodynamicforceistakentobethesumoftheforceF0duetoundisturbedambientowandtheforceF0duetothedisturbanceow.Theformerisdenedas F0=IS0ndS,(3) where0isthestressduetotheundisturbedow. 57

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Wesimplifythehydrodynamicforceduetotheundisturbedowthroughthedivergencetheoremandemployingthemomentumequationoftheambientowinthemovingreferenceframetoobtain F0=ZV0Dv0 Dt+dup dt)]TJ /F15 11.955 Tf 11.95 0 Td[(gdV.(3) Here,Visthevolumeoftheparticleandwedenethemassofundisturbeduiddisplacedbytheparticletobemf=RV0dV.Assumingthebody-forcedensitygandparticleaccelerationdup=dttobeconstant,thehydrodynamicforceduetotheundisturbedowcanbewrittenintheoriginallaboratoryframeofreferenceas F0=ZV0Du0 D~tdV)]TJ /F3 11.955 Tf 11.96 0 Td[(mfg.(3) 3.7ReciprocalTheoremforCompressiblePerturbationFlow Equations( 3 )and( 3 )areidenticaltothosesolvedbyZwanzigandBixon[ 119 ].However,theboundaryconditionsgivenbyEqs.( 3 )and( 3 )aremorecomplexduetotheinhomogeneousambientow.If0andu0weretobehomogeneousthenthesolutionofZwanzigandBixon[ 119 ]readilyappliesinLaplacespaceandthecorrespondingtime-domainsolutionisgivenbyParmaretal.[ 81 ].However,theinhomogeneousnatureoftheboundaryconditiongivenbyEq.( 3 )complicatesthesolution. WeemploytheLorentzreciprocaltheorem,seeHappelandBrenner[ 40 ],whichallowsdeterminationofforceontheparticleinacomplexStokesowwithoutactuallysolvingforthedetailsoftheperturbation.Inthepresentapplication,onejustneedsthedetailedsolutionofthehomogeneousuniformambientowproblem,anditcanbeusedelegantlyinthereciprocaltheoremtodevelopanexpressionforthedesiredintegralquantitiesinamorecomplexinhomogeneousow.Peres[ 84 ]usedthereciprocaltheoremtoobtainasimpleandelegantproofofFaxen'stheoremforsteady 58

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incompressibleows.Gatignol[ 36 ]andMaxeyandRiley[ 63 ]alsomadeuseofthereciprocaltheoremtoextendFaxen'stheoremtounsteadyincompressibleows. Here,ageneralizationofthereciprocaltheoremtocompressibleunsteadyowsisrequired.Thereciprocaltheoremforalinearizedcompressibleowhasbeenobtainedandexploitedbyseveralresearchers,seeChowandHermans[ 20 ],Kaneda[ 51 ],andCunhaetal.[ 26 ].ConsidertwozeroReynoldsnumbercompressibleoweldsaroundasphericalparticle,with(i,vi,pi),satisfyingEqs.( 3 )and( 3 )andtheclosurerelationc20=p0i=0i.Here,thesubscripti=1,2indicatethetwodifferentowelds.Lettheambientvelocityeldsbesuchthattheyvanishfarawayfromthesphereandbezerofort0.Thenthereciprocalrelationbetweenthetwoowscanbecastasanintegralrelation, ISL(v2)L(1)ndS=ISL(v1)L(2)ndS,(3) whereL()istheLaplacetransform,theintegrationisoverthesurfaceofthesphere,andiarethestresseldscreatedbytherespectiveowsthatsatisfyEq.( 3 ). Wechoosetheunknowncomplexoweldarisingfrominhomogeneousboundaryconditiononthesurfaceofthespheretobeoweld1(1,v1,p1).Fornowtheinhomogeneousboundaryconditionischosentobe v1(x,t)=u1,p(x,t)forjxj=a,(3) whichwilllaterbesettoEq.( 3 ),correspondingtotheproblemtobesolved.Thedesiredquantityofinterestisthenettime-dependenthydrodynamicforceontheparticle,whichcanbeexpressedintheLaplacespaceas L(F1)=ISL(1)ndS.(3) Welettheoweld2(2,v2,p2)tobethatduetotheunsteadymotionofasphereinanotherwisequiescentuid,whosesolutionisknown,seeZwanzigandBixon[ 119 ].The 59

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boundaryconditionforv2is v2(x,t)=u2,p(t)forjxj=a.(3) UsingEq.( 3 )andtheboundaryconditionsgivenbyEqs.( 3 )and( 3 )leadsto L(u2,p)L(F1)=ISL(u1,p(x))L(2)ndS.(3) Tosimplifythisrelation,werecognizethatinthepresentcaseofaxisymmetricowthetractionvectorL(2)nonthesurfaceofthespherecanbewrittenasalinearcombinationofcomponentsintheradialdirectionandalongthedirectionofparticlemotionas L(2)n=H1(s)(L(u2,p)n)n+H2(s)L(u2,p),(3) wheresistheLaplacevariable.TheimportantpropertythatwewillexploitisthatH1(s)andH2(s)areindependentofthedetailsofthemotionandaswillbeshownlatertheyareinvariantonthesurfaceofthesphere(i.e.,independentofthesphericalcoordinatesand).SubstitutingEq.( 3 )inEq.( 3 )andrearranging,L(F1)L(u2,p)=H1IS(L(u2,p)n)(L(u1,p(x))n)dS+H2ISL(u1,p(x))L(u2,p)dS,=H1 aIS[frL(u2,p)gL(u1,p(x))]ndS+H2ISL(u1,p(x))dSL(u2,p),=H1 aZVr[frL(u2,p)gL(u1,p(x))]dV+H2ISL(u1,p(x))dSL(u2,p),=H1 aZV[L(u1,p(x))+rfrL(u1,p(x))g]dV+H2ISL(u1,p(x))dSL(u2,p). (3) Deningvolumeandsurfaceaveragesas fV(x,t)=3 4a3ZVf(x)]TJ /F15 11.955 Tf 11.95 0 Td[(x0,t)dx0and fS(x,t)=1 4a2ISf(x)]TJ /F15 11.955 Tf 11.96 0 Td[(x0,t)dx0,(3) 60

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werewritetheaboveasL(F1)=4 3a2H1n L(u1,p(x,t))V+ r[rL(u1,p(x,t))]Vo+4a2H2h L(u1,p(x,t))Si. (3) Thusweobtainanexpressionfortheforceonasphericalparticleduetotheinhomogeneousvelocityboundaryconditiononitssurfaceintermsofsurfaceandvolumeaveragesoftheinhomogeneousboundarycondition. 3.8HydrodynamicForceduetotheDisturbanceow Thedetailedsolutionoftheoweld(2,v2,p2)duetoasphereundergoingarbitrarymotionu2,p(t)inaquiescentuidhasbeenobtainedbymanybeginningwithZwanzigandBixon[ 119 ].InLaplacespace,thesurfacetractionvectorcanbecastasL(2)n=)]TJ /F3 11.955 Tf 11.95 0 Td[(ars(1+r1)r22)]TJ /F4 11.955 Tf 11.96 0 Td[((1+r2)r21 r21(1+r2)+(2+2r1+r21)r22(L(u2,p)n)n)]TJ /F3 11.955 Tf 11.95 0 Td[(ars(3+3r1+r21)(1+r2) r21(1+r2)+(2+2r1+r21)r22L(u2,p), (3) where r1(s)=as=c0 p 1+(b=+4=3)s=c20andr2(s)=ar s .(3) DeningtransferfunctionsGvandGs,Gs(r1,r2)=3(3+3r1+r21)(1+r2) r21(1+r2)+(2+2r1+r21)r22,Gv(r1,r2)=(1+r1)r22)]TJ /F4 11.955 Tf 11.96 0 Td[((1+r2)r21 r21(1+r2)+(2+2r1+r21)r22, (3) thetractiononthespherecanbewrittenas L(2)n=)]TJ /F3 11.955 Tf 9.3 0 Td[(arsGv(L(u2,p)n)n+Gs 3L(u2,p).(3) Itisinterestingtonotethatthenormalcomponentofthetractionvectorontheparticlesurfaceisproportionaltotheparticlevelocityresolvedalongthenormaldirection.Bycontrast,thetractioncomponentalongthedirectionofparticlemotionisinvariantalong 61

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theentiresurfaceofthesphere.ComparingtheaboveexpressionwithEq.( 3 ),itcanbeseenthatH1=)]TJ /F3 11.955 Tf 9.3 0 Td[(arsGvandH2=)]TJ /F3 11.955 Tf 9.3 0 Td[(arsGs=3andtherebyitcanbeveriedthatH1andH2aredependentonlyontheLaplacevariableasassertedintheapplicationofthereciprocaltheoreminEq.( 3 ). DeningthehydrodynamicforceF0duetotheperturbationowas F0=IS0ndS,(3) wherer0isgivenbytheright-handsideofEq.( 3 ).FromEq.( 3 )weobtainanexpressionfortheforceduetotheperturbationowintheLaplacespaceasL(F0)=4 3a3rsnGsLh Vp(x,t)Si+GvLh Vp(x,t)V+ rrVp(x,t)Vio. (3) whereVp(x,t)istheinhomogeneousvelocitydistributionontheparticlesurface,giveninEq.( 3 ).ThetransferfunctionsGsandGvapplyseparatelytosurfaceandvolumeaveragesoftheundisturbedambientowquantitiesandcanbeinterpretedasfollows.ThetransferfunctionGscorrespondstotheforceresponseinLaplacespaceduetoastepchangeinthesurfaceaverageofthedensity-weightedvelocityVp.ThetransferfunctionGvcorrespondstotheforceresponseduetoastepchangeinthevolume-averagedquantitypresentedwithinthesecondsquarebracketsontheright-handsideofEq.( 3 ). Theaboveexpressionsimpliesforahomogeneousundisturbedambientow(0=constant,u0=constant).Then,rVp=0,andsinceVpisaconstant,itssurfaceandvolumeaveragesarethesameastheconstantvalue.Thus,werecovertheresultofZwanzigandBixon[ 119 ]forauniformcompressibleow,L(F0)=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(4 3a3rs[Gv+Gs]L(Vp)=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(4 3a3rsGL(Vp), (3) whereGisthetotalresponsefunction.Iftheundisturbedambientowweretobeofconstantdensityandincompressible,thenrVp=0.However,iftheambientowis 62

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inhomogeneous,ingeneral VVp6= VSpandasaresulttheforceduetotheperturbationowreducestoL(F0)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(4 3a3rshGsL( VSp)+GvL( VVp)i. (3) Ifwefurtherassumethecompressibilityeffectsoftheperturbationeldtobeunimportant,GsandGvcanbesimpliedandtheaboveexpressionreducestothatgiveninthefrequency-domainsolutionofMazurandBedeaux[ 65 ]foraninhomogeneousincompressibleow.ThecorrespondingLaplacetransformtothetimedomainwillresultintheMRGequation. FollowingParmaretal.[ 81 ],Eq.( 3 )canbetransformedintothetimedomainandseparatedintoquasi-steadyandvariousunsteadycontributions.TheexpressionfortheforceduetothedisturbanceowcannowbewrittenasF0(t)=6a( u0S)]TJ /F15 11.955 Tf 11.95 0 Td[(up)+6a2p ZtKsvu(t)]TJ /F8 11.955 Tf 11.95 0 Td[() d 0u0S dtt=)]TJ /F3 11.955 Tf 14.34 8.09 Td[(d S0up dtt=!d+4 3a3ZtKiu(t)]TJ /F8 11.955 Tf 11.96 0 Td[()c0 a d 0u0V dtt=)]TJ /F3 11.955 Tf 14.34 8.09 Td[(d V0up dtt=!c0 ad+6a2p ZtKvvu(t)]TJ /F8 11.955 Tf 11.95 0 Td[() d 0u0V dtt=)]TJ /F3 11.955 Tf 14.35 8.09 Td[(d V0up dtt=!d+4 3a3ZtKiu(t)]TJ /F8 11.955 Tf 11.96 0 Td[()c0 ad2 dt2 r0Vt=c0 ad+6a2p ZtKvvu(t)]TJ /F8 11.955 Tf 11.95 0 Td[()d2 dt2 r0Vt=d, (3) whereKiu()isthekerneloftheinviscidunsteadyforceandKvvu()andKsvu()aretheviscouskernelsthatoperateonvolumeandsurfaceintegralsrespectively.Inobtainingtheaboveexpressions,wehaveusedtheresultthatthetransferfunctionsGsandGvcanbesplit, Gs=Gsqs+GsvuandGv=Gviu+Gvvu,(3) 63

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wherethequasi-steadypartofthetransferfunctionassociatedwiththesurfaceaverageandtheinviscidunsteadypartofthetransferfunctionassociatedwiththevolumeaverage,denedbelow,havebeenseparated: Gsqs=9 2r22andGviu=1+r1,inv 1+(1+r1,inv)2,(3) where r1,inv=lim!0r1=as=c0.(3) TheLaplaceinverseoftheterminvolvingGsqsgivesrisetothequasi-steadycontributiondenedintermsofthesurface-averagedvelocity.TheLaplaceinverseoftheotherthreetransferfunctionscanbeusedtodeneinviscidandviscouskernelsasshownbelow:c0 aKiu=L)]TJ /F10 7.97 Tf 6.59 0 Td[(1[Gviu], (3)9 2ar Ksvu=L)]TJ /F10 7.97 Tf 6.59 0 Td[(1[Gsvu], (3)9 2ar Kvvu=L)]TJ /F10 7.97 Tf 6.59 0 Td[(1[Gvvu]. (3) ThersttwotermsinEq.( 3 )arisefromthersttermontheright-handsideofEq.( 3 ).ThethirdandfourthtermsinEq.( 3 )arisefromthesecondtermontheright-handsideofEq.( 3 ).ThelasttwotermsinEq.( 3 )arisefromthelasttermontheright-handsideofEq.( 3 ).ThusthelastfourtermsofEq.( 3 )arerelatedtovolumeaverages.Aswillbeshownbelow,thelasttwoadditionaltermsaremuchsmallerthantheothercontributionsandcanthusbeignoredundertheconditionsconsideredinthiswork. WecanconsiderdifferentlimitingbehaviorofthetransferfunctionsGsvuandGvvuandthecorrespondingkernelsKsvuandKvvu.Considerrsttheinviscidlimit,wherelim!0Gsvu!0)Ksvu!0, (3)lim!0Gvvu!0)Kvvu!0. (3) 64

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Inthislimitthequasi-steadyandtheviscousunsteadyforcesvanishasexpectedonerecoverstheinviscidunsteadycomponentastheonlyforceontheparticle.Nextconsidertheincompressiblelimit,wherelimc0!1Gsvu!9 2r2)Ksvu!1 p t, (3)limc0!1Gvvu!0)Kvvu!0. (3) ThekernelKsvureducestotheBasset-historykernelintheincompressiblelimit.ThekernelKvvuiszerointheknownlimitsofinviscidandincompressibleowsandthusitappearsasanewunsteadyviscouskernelonlyinviscouscompressibleows. InEq.( 3 ),d=dtmustbecarefullyinterpretedasthepartialtimederivativeintheframeattachedtotheparticleorasthetimederivativefollowingtheparticle.Inacompressibleow,boththeinviscidandviscousunsteadyforcesdependonthehistoryofrelativeaccelerationweightedbyappropriatekernels. 3.8.1ImportanceoftheDifferentTerms Inthissection,estimatesofthemagnitudeofthedifferenttermsinEq.( 3 )arepresentedusingasimplescalingargument.Weexpectthequasi-steadycontribution(thersttermontheright-handsideofEq.( 3 ))tobethedominanttermanditscalesasavr.Themagnitudeoftheinviscidunsteadycontribution(thirdtermontheright-handsideofEq.( 3 ))canbeobtainedasfollows.Theinviscidkernel(tobediscussedbelowinSection 3.8.2 )isO(1)onlyoveranacoustictimescaleofa=c0.Thetimederivativeofthedensityweightedvelocity(d 0u0V=dt)canbeestimatedtoscaleasrv2r=a.Thustheinviscidunsteadytermscalesasrv2ra2,andwhencomparedtothequasi-steadyforceisO(Re). Intheviscousunsteadyterms(secondandfourthtermsontheright-handsideofEq.( 3 )),theviscouskernelsgoas1=p tandthisdecayisrelevantonlyovertheviscoustimescaletv,beyondwhichtheviscouskernelsdecayfasterduetonon-lineareffect.Thus,RKsvud,RKvvudp tv=a=p .Withd 0u0V=dtscalingasrv2r=athe 65

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secondandfourthtermsontheright-handsideofEq.( 3 )canbeestimatedtoscaleasrv2ra2,andthustheyareO(Re)comparedtothequasi-steadyforce. NowweconsiderthelasttwotermsofEq.( 3 ).UsingtruncatedTaylorseriesfor0asafunctionofr,wehave r0V rVr+ rrVr0jr=0= rrVr0jr=0.(3) UsingEq.( 3 )intheaboveweobtain r0Var,(3) andusingEq.( 3 )againgives O@2 @t2 r0Vrv2r a.(3) Thus,thefthandsixthtermsontheright-handsideofEq.( 3 )areanorderofmagnitudesmallerthanO[Re]comparedtorstterm.Sotermsinvolving r0Vcanbeneglectedcomparedtotheotherterms.Itmustbestressedthattheseadditionaltermsinvolving r0VwerealsoobtainedbyBedeauxandMazur[ 9 ],butherewehavearguedthatthesetermsareasymptoticallysmallandcanbeignored. 3.8.2InviscidandViscousKernels Thekernelfortheinviscidunsteadyforcecanbeexplicitlyexpressedas Kiu()=L)]TJ /F10 7.97 Tf 6.59 0 Td[(1[Gviu]=e)]TJ /F16 7.97 Tf 6.58 0 Td[(cos,(3) where=c0t=a=t=taisthedimensionlesstimenondimensionalizedbytheacoustictimescale.TheaboveexpressionfortheinviscidkernelisapplicableforasphericalparticleinthelimitofzeroMachnumberandwasoriginallyderivedbyLonghorn[ 58 ].Inanincompressibleow,theeffectofthedensity-weighteduidaccelerationd 0u0V=dtortheparticleaccelerationd V0up=dtwillresultinaninstantaneousinviscidforce,whichistermedtheadded-massforce.Inacompressibleow,duetothenite 66

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Figure3-1. ThebehaviorofthecorrectionfunctionCv()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==0andKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,10)]TJ /F10 7.97 Tf 6.59 0 Td[(5,10)]TJ /F10 7.97 Tf 6.59 0 Td[(8,10)]TJ /F10 7.97 Tf 6.59 0 Td[(11g. propagationspeedoftheacousticwaves,theinstantaneousinviscidforceonaparticleisdueofparthistoryofrelativeacceleration(@ 0u0V=@t)]TJ /F8 11.955 Tf 12.93 0 Td[(@ V0up=@t),weightedbytheinviscidunsteadykernel.DuetotheexponentialdecayofKiu,theinviscidforceduetoinstantaneousrelativeaccelerationpersistsonlyforafewacoustictimescalesandfor>10theinviscideffectofaccelerationisnegligible.Furthermore,duetothecosterm,theinviscidforcecanbenegativeatintermediatetimesandleadtothecounterintuitivebehavioroftheinviscidforcepointinginthedirectionoppositetotherelativeacceleration. InthelimitofzeroReynoldsandMachnumbers,thekernelsfortheviscousunsteadyforcethatoperateonvolumeandsurfaceaveragedquantitiescanbewritten 67

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Figure3-2. ThebehaviorofthecorrectionfunctionCs()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==0andKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,10)]TJ /F10 7.97 Tf 6.59 0 Td[(5,10)]TJ /F10 7.97 Tf 6.59 0 Td[(8,10)]TJ /F10 7.97 Tf 6.59 0 Td[(11g. asKvvu(t)=Cvtc0 a1 p tandKsvu(t)=Cstc0 a1 p t. (3) where1=p tisthestandardincompressibleBassethistorykernel.Thus,Cv()andCs()arecorrectionfunctionstotheincompressibleviscousunsteadykernel.ExpressionsforthesecorrectionfunctionscanbewrittenasCv()=L)]TJ /F10 7.97 Tf 6.59 0 Td[(1(Gvvu) 9 2r Kn0 !)]TJ /F10 7.97 Tf 6.59 0 Td[(1, (3)Cs()=L)]TJ /F10 7.97 Tf 6.59 0 Td[(1(Gsvu) 9 2r Kn0 !)]TJ /F10 7.97 Tf 6.59 0 Td[(1. (3) Forthecaseofhomogeneousundisturbedambientow,thevolumeandsurfaceaveragesoftheambientowquantitiesareidenticalandasaresulttheabovetwo 68

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Figure3-3. ThebehaviorofthecorrectionfunctionCv()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==f0,1,59=12,10gandKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,10)]TJ /F10 7.97 Tf 6.59 0 Td[(5g. correctionsthatapplytovolumeandsurfaceaveragescanbecombinedintoasinglecorrectionfunction,C()=Cv()+Cs().ThiscompressibilitycorrectionfunctionC()totheBassetkernelwasrecentlyobtainedbyParmaretal.[ 81 ]intheirderivationofthecompressibleextensionoftheBBOequation. InthepresentcaseofaninhomogeneousambientowthecorrectionfunctionneedstobeseparatedintoCv()andCs(),whichareshowninFigs. 3-1 and 3-2 ,respectively.TheLaplacetransformsoftheresponsefunctionsGvvuandGsvutothetimedomainhavenotbeenobtainedanalytically.Asaresult,theLaplaceinverse-transformswerecomputednumericallyandthenumerically-evaluatedcorrectionfunctionsareshowninFigs. 3-1 and 3-2 .NotethatthecorrectionstotheviscouskernelarealsofunctionsofKn0=M=Re==(ac0)andb=. 69

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Figure3-4. ThebehaviorofthecorrectionfunctionCs()thataccountsforthevolumeintegralcontributionofthecompressibilityeffectontheviscousunsteadyforce,seeEq.( 3 ),forb==f0,59=12,10gandKn0=f10)]TJ /F10 7.97 Tf 6.59 0 Td[(2g. AsdiscussedbyParmaretal.[ 81 ],fourasymptoticregimescanbeidentiedforCv()andCs(): RegimeI:Veryshorttime,Kn01, RegimeII:Intermediateshorttime,Kn01, RegimeIII:Intermediatelongtime,11=(ReM), RegimeIV:Verylongtime,11=(ReM). Explicitexpressionsforthecorrectionfunctionsinthetimedomaincanbeobtainedfortherstthreeregimes.Nonlinearitybecomesimportantfor1=(ReM).ThusthesolutionofthelinearizedgoverningequationspresentedheredonotdescribeaccuratelytheforceevolutioninRegimeIV. NextweexaminetheasymptoticbehaviorofCv()andCs()intherstthreeregimes.Theveryshort-timebehaviorofthecorrectionfunctionscanbeobtainedby 70

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consideringthefollowinglimitintheLaplacespace:1Kn0jSjjSjjSj=Kn0,whereS=as=c0.ThisresultsinthefollowingsimpliedexpressionsforthetransferfunctionsGvvuandGsvu,Gvvu )]TJ /F4 11.955 Tf 9.3 0 Td[(1+s b +4 3!r Kn0 S, (3)Gsvu3r Kn0 S. (3) TheirLaplaceinversegivesthetime-domainbehaviorofthecorrectionfunctionsinRegimeIas RegimeI(Kn01):8>>><>>>:Cv()2 9 )]TJ /F4 11.955 Tf 9.3 0 Td[(1+s b +4 3!,Cs()2 3.(3) Itcanbeseenthatveryshort-timebehaviorofCv()dependsonb=.Forb=0,Cv(Kn01)=2(2)]TJ 11.96 9.96 Td[(p 3)=(9p 3)0.034(seeFig. 3-1 ). Fromthedenitionoftheintermediateshorttime,thefollowingconditioncanbeplacedontheLaplacevariable:Kn0jSj1jSj1=Kn0jSj=Kn0.ThenGvvuandGsvucanbesimpliedasGvvu1 r2)]TJ /F4 11.955 Tf 9.3 0 Td[(1+(1+r1) 1+(1+r1)2+2 1+(1+r1)2)]TJ /F4 11.955 Tf 48.77 8.09 Td[(2 [1+(1+r1)2]2, (3)Gsvu1 r23+3(1+r1) 1+(1+r1)2. (3) ThedominantcontributionforintermediateshorttimeforCv()andCs()are RegimeII(Kn01):8>><>>:Cv())]TJ /F4 11.955 Tf 23.11 8.09 Td[(2 9,Cs()2 3.(3) ItcanbeenseenthatCv()atintermediateshorttimesreducesto)]TJ /F4 11.955 Tf 9.3 0 Td[(2=90.22becauseofcompressibilityeffects.Notethatthisreductionisindependentofb=.AswillbeseeninFig. 3-1 ,wheretheresultsofthenumericalLaplaceinversionareshown, 71

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RegimeIIcanbeobservedonlyifKn01.WithincreasingKn0,thedurationofRegimeIIreducesandvanishesentirelyforKn010)]TJ /F10 7.97 Tf 6.59 0 Td[(2.ForCsthereisnodistinctionbetweenRegimeIandRegimeII. Theintermediatelong-timebehaviorcanbeobtainedinasimilarmannerbyconsideringanasymptoticexpansionforjSj!0andcarryingouttheLaplaceinverseweobtain RegimeIII(11=(ReM)):8>><>>:Cv()!0,Cs()!1.(3) Strictlyspeaking,theabovesolutionforthelinearizedperturbationNavier-Stokesequationsisvalidaslongas1.Theadditionallimitof1=(ReM)arisesonlyfromneglectingthenonlinearterms. InFig. 3-2 ,thecorrectiontotheviscouskernelthatappliestothesurfaceaveragebecomesaconstantequalto2=3for<10)]TJ /F10 7.97 Tf 6.58 0 Td[(3andfor>10werecoverCs=1.Intheintermediatetimeweobserveadampedoscillatorybehavior.ItcanbeobservedthatforKn0<10)]TJ /F10 7.97 Tf 6.59 0 Td[(5,thecorrectionfunctionCsbecomesnearlyindependentofthemodiedKnudsennumberaswellasbulkviscosity(alsoseeFig. 3-4 ).ForKn0>10)]TJ /F10 7.97 Tf 6.59 0 Td[(5,weobserveaweakquantitativedependenceonKn0.HerewelimitattentiontoKnudsennumberssmallerthan10)]TJ /F10 7.97 Tf 6.59 0 Td[(2,sincethecontinuumassumptionstarttobreakdownforlargervalues.ThedependenceofCvandCsonthebulkviscosityisshowninFigs. 3-3 and 3-4 3.9Discussion Bycombiningtheforcesduetotheundisturbed(Eq.( 3 ))anddisturbanceows(Eq.( 3 ),withoutthesmallerlasttwoterms),wecanderiveanexplicitequationofmotionforasphereundergoingunsteadymotioninacompressibletime-dependentinhomogeneousambientow.TheresultingcompressibleextensionoftheMRG 72

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equationcanbeexpressedas mpdup d~t=Fqs+Fiu+Fvu+Fbg,(3) wherethetermsontheright-handsidearethequasi-steady,inviscidunsteady,viscousunsteadyandbuoyancy/gravityforces,Fqs=6a( uS0)]TJ /F15 11.955 Tf 11.96 0 Td[(up), (3)Fiu=ZV0Du0 D~tdV+4 3a3Z~tKiu(~t)]TJ /F8 11.955 Tf 11.96 0 Td[()c0 a D 0u0V D~t~t=)]TJ /F3 11.955 Tf 14.34 8.09 Td[(D V0up D~t~t=!c0 ad, (3)Fvu=6a2p Z~tKvvu(~t)]TJ /F8 11.955 Tf 11.95 0 Td[() D 0u0V D~t~t=)]TJ /F3 11.955 Tf 14.35 8.09 Td[(D V0up D~t~t=!d+6a2p Z~tKsvu(~t)]TJ /F8 11.955 Tf 11.95 0 Td[() D 0u0S D~t~t=)]TJ /F3 11.955 Tf 14.35 8.09 Td[(D S0up D~t~t=!d, (3)Fbg=(mp)]TJ /F3 11.955 Tf 11.96 0 Td[(mf)g. (3) Ininterpretingandusingtheaboveequationofmotionthefollowingshouldbenoted.First,Eq.( 3 )iscastintermsofthexedlaboratorycoordinateandthusthasbeenreplacedwith~t.Second,termssuchasd 0u0V=d~tinEq.( 3 )donotpreserveGalileaninvariance.Thelackofinvarianceiscausedbyneglectingtheadvectiontermsduringlinearization.Accordingly,inEqs.( 3 )and( 3 ),termsinvolvingd=d~thavebeenreplacedbytotalderivativefollowingtheuidelementD=D~t.AspointedoutbyMaxeyandRiley[ 63 ]anddiscussedbyAutonetal.[ 1 ],thedifferencebetweend=d~tandD=D~tisasymptoticallysmallandforstrictGalileaninvariancewesuggestusingD=D~t. Notethatthesurface-andvolume-averages( ()S, ()V)areusedinsidethetotalderivativefollowingGatignol[ 36 ].Inthesolutionofthelinearizedequationstermssuchd 0u0V=d~tcanbereplacedby d0u0=d~tV,sincetimeandspaceoperationscommute.Asaresultintheaboveequationwecouldwritetheunsteadyforcecontributionsinsteadinvolvingtermslike D0u0=D~tV,wheretheaverageisofthetotalderivative. 73

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AlthoughthesetermsalsopreserveGalileaninvariance,thetotalderivativeandthesurfaceorvolumeaveragesdonotcommute.OnthebasisofthepresentlinearizedderviationwecannotrationallychoosetermssuchasD 0u0V=D~tover D0u0=D~tV,sinceasymptoticallytheybothareofthesameorder. Theaboveequationisvalidforunsteadyinhomogeneousambientowswhere0andu0arefunctionsofbothspaceandtime.Theformoftheaboveequationofmotioncouldhavebeenanticipated,sinceitappearstocombinethefeaturesofMRGequationofmotionandthecompressibleBBOequationofParmaretal.[ 81 ].Forexample,thevolumeandsurfaceaveragesaresimilartothosearisingintheincompressibleMRGequationofmotionforaninhomogeneousambientow,whiletheinviscidandviscouscompressiblecorrectionsarethesameasthoseobtainedinthecompressibleBBOequation(althoughtheviscouscorrectionseparatesintotwocomponentsthatapplyindependentlyforthevolumeandsurfaceaverages).Nevertheless,theaboveequationisnewandforthersttimeincorporatesthecombinedeffectsofunsteadinessandinhomogeneityintheambientvelocityanddensityontheinviscidandviscousunsteadyforcesonasphericalparticleinacompressibleowinaconsistentmanner.Therigorousderivationiscruciallydependentonthedensity-weightedvelocitytransformation. Mostimportantlyequation( 3 )iscastinthetimedomainandthuscanbereadilyusedtotrackparticle.Itisparticularlyusefulwhenthecompressibilityoftheperturbationowandtheforceevolutionontheacoustictimescaleareimportant. Theequationofmotionreducestothecorrectsimpliedformswhenappropriateapproximationsaremade.(i)Ifthespatialvariationoftheambientowasseenbytheparticleisnegligiblysmall,thevolumeandsurfaceaveragescanbereplacedbythecorrespondingundisturbedambientvaluesatthecenteroftheparticleandEq.( 3 )reducestotheequationforahomogeneousambientow.(ii)Ifwefurtherassumetheambientowtobesteady,theequationofmotionreducestothegeneralizedBBO 74

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equationforthecompressibleowgivenbyParmaretal.[ 81 ].(iii)Instead,ifambientowvariationsandparticleaccelerationoccuronlyontimescalesmuchlargerthantheacoustictimescale,thentheexpressionwithintheparenthesesinthesecondtermontheright-handsideofEq.( 3 )canbemovedoutoftheintegralandfort=ta&10theinviscidunsteadykernelKiu()canbeintegratedtoobtaintheincompressibleadded-massforce 2 3R3D 0u0V D~t)]TJ /F3 11.955 Tf 13.15 8.09 Td[(D V0up D~t.(3) TheresultinginviscidunsteadyforceisinagreementwiththeexpressionderivedbyEamesandHunt[ 29 ]fortheforceonaspheremovingunsteadilyinacompressedow.Inasimilarfashion,fort=ta&10theviscousunsteadykernelKsvu(~t)inEq.( 3 )reducestotheincompressibleBasset-historykernel,whileKvvu(~t)becomeszero.(iv)Ifwefurtherassume0tobeconstant,aswouldbeappropriateinanuniformdensityincompressibleow,theresultingsimpliedequationofmotioncanbeshowntobeidenticaltotheMRGequation.(v)If,inaddition,weassumetheambientowu0tobeconstantweobtaintheBBOequation. ThefollowinglimitationsofEq.( 3 )mustbehighlighted.First,thisequationofmotionisstrictlyvalidonlyinthelimitofRe!0andM!0.AtniteReynoldsandMachnumberslinearizationoftheperturbationequationswillbeinvalid,sinceCondition1(Eq.( 3 ))willbeviolated.AsaresultrigorousanalyticsolutionsareunavailableatniteReynoldsandMachnumbers.However,empiricalrelationshavebeendevelopedinthecontextofincompressibleperturbationows,andtheirdevelopmenthasbeenfunctionallybasedonthecorrespondingzeroReynoldsnumberMRGequation.Inparticular,thesuperpositionofthetotalforceontheparticleintoquasi-steady,inviscidunsteadyandviscousunsteadycontributionshasbeenadoptedundernonlinearconditionsaswell.Forexample,atniteReynoldsnumberstheStokesdragexpressionisreplacedbythestandarddragcorrelation,whichisanempiricalfunctionofRe.ForacompressibleowniteReynoldsandMachnumberempirical 75

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correctionstothequasi-steadydraghavebeendiscussedbyParmaretal.[ 82 ].Theinviscidunsteadyforce,bydenition,isnotinuencedbytheReynoldsnumberandtheMachnumberdependenceofKiu(t)hasbeendiscussedbyParmaretal.[ 79 ].Thenonlineardependenceoftheviscouskernelscanbeexpectedtobestrongandcomplex.Forexample,intheincompressiblelimititisknowthattheviscouskerneldecaysfasterthan1=p tandtheactualbehaviorofthekerneldependsnotonlyonRe,butalsoonthenatureofaccelerationordeceleration.Whencompressibilityeffectsaretakenintoaccountitisexpectedthattheviscouskernels,Ksvu(t)andKvvu(t)willdependonbothReandMinacomplexmanner,whosebehaviorneedsfurtherinvestigation. LimitationsposedbytheotherconditionspresentedinSections 3.4 and 3.5 cansimilarlybeanalyzed.Fromtheinviscidkernelandthecompressibilitycorrectiontotheviscouskernelsitisclearthatthecompressibilityeffectsaresignicantonlyoverabout10acoustictimescales.Sinceinthecontinuumregimewerequiretatv,therestrictionC2(Eq.4.5)totimesoftheorderoftheviscoustimescaleisnotaseriouslimitationintheinvestigationofcompressibilityeffects. RestrictionsC3andC4onthespatio-temporalvariationoftheambientdensityandvelocitywilllimitthestrictvalidityoftheequationofmotiontosufcientlysmallparticles.Itisclearthatinapplicationssuchasshock-particleinteraction,thescaleoftheambientowvariationislikelytobesmallerthantheparticle.Yet,itisencouragingthatasimpliedversionoftheabovecompressibleinviscidunsteadyforcehasbeenquitesuccessfulinreproducingtheforceevolutionduringthepassageofashockwaveoverstationaryparticles,seeParmaretal.[ 80 ]. Itthusappearsthattheabovedenedvolumeandsurfaceaveragesprovideagoodapproximationfortheeffectiveambientowquantitiesseenbytheparticleevenwhentheyvarystronglyonthescaleofaparticle,asitcanhappenfrequentlywhencompressibleowfeaturesinteractwiththeparticle. 76

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Theabovelinearformulationdoesnotincludecontributionsfromvorticity-inducedliftforces.Inacompressibleow,contributionsfrombothambientshearanddensitygradientscanexist.Forexample,asshownbyEamesandHunt[ 28 ],additionallift-likeinviscidforcesariseduetobaroclinicvorticityproduction,evenundersteadymotionofaparticlethroughaeldofnon-uniformdensity.Suchforcescanmakeadditionalcontributionsincaseofshock-particleinteractionandhavenotbeenincludedinEq.( 3 ). Finally,wewanttoplacethepresentresultsinthecontextofthepriorworksofBedeauxandMazur[ 9 ]andMaxeyandRiley[ 63 ],whichhavegreatlymotivatedourefforts.First,thegoverninglinearizedperturbationequations( 3 )and( 3 )thatwesolvearesomewhatdifferentfromthosesolvedinBedeauxandMazur[ 9 ].Thedifferences,althoughasymptoticallysmallandnegligibleunderthepresentscalinganalysis,haveallowedustoapplythedensity-weightedvelocitytransformationandobtaintheaboveresults.Here,followingMaxeyandRiley[ 63 ],wesolvethegoverningequationsinthemovingreferenceframeattachedtotheparticleandseparatetheforcecontributionsfortheundisturbedandthedisturbanceows.Thisallowsexacttreatmentofparticleboundaryconditionandpreciseextractionofbuoyancy/gravityforcesaswellaspressure-gradientforces.Finally,throughtheLaplaceinversewepresentourresultsinthetimedomainandhencereadilyusableforparticletracking. NotethatMaxeyandRiley[ 63 ]approximatedthesurfaceandvolumeaveragesintermsoftheLaplacianofambientvelocityattheparticlecenter.Thisapproximationrequiresarestrictiononthespatialvariationofambientvelocitythatismorestringentthanconditions3.ThismorestringentrestrictionisnecessarytoemploythefollowingtruncatedTaylor-seriesexpansionoftheambientvelocityinapproximatingthevolumeandsurfaceintegrals, v0(x,t)=v0(x,t)(0,0)+x(rv0)(0,0)+1 2xx:(rrv0)(0,0).(3) 77

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Inthisconditions3mustbereplacedbythestrongercondition aLv)a vr[rv0]1.(3) ThisstrongerrestrictionstatedinMaxeyandRiley[ 63 ]canberelaxedifweleavethevolumeandsurfaceaveragesinEq.( 3 )unmodied. 78

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CHAPTER4ONTHEUNSTEADYINVISCIDFORCEONCYLINDERSANDSPHERESINSUBCRITICALCOMPRESSIBLEFLOWS Theunsteadyinviscidforceoncylindersandspheresinsubcriticalcompressibleowisinvestigated.Inthelimitofincompressibleow,theunsteadyinviscidforceonacylinderorsphereistheso-calledadded-massforcewhichisproportionaltotheproductofthemassdisplacedbythebodyandtheinstantaneousacceleration.Incompressibleow,theniteacousticpropagationspeedmeansthattheunsteadyinviscidforcearisingfromaninstantaneouslyappliedconstantaccelerationdevelopsgraduallyandreachessteadyvaluesonlyfornon-dimensionaltimesc1t=R&10,wherec1isthefreestreamspeedofsoundandRistheradiusofthecylinderorsphere.Inthislimit,aneffectiveadded-masscoefcientmaybedened.ThemainconclusionofourstudyisthatthefreestreamMachnumberhasapronouncedeffectonboththepeakvalueoftheunsteadyforceandtheeffectiveadded-masscoefcient.AtafreestreamMachnumberof0.5,theeffectiveadded-masscoefcientisabouttwiceaslargeastheincompressiblevalueforthesphere.Coupledwithanimpulsiveacceleration,theunsteadyinviscidforceincompressibleowcanbemorethanfourtimeslargerthanthatpredictedfromincompressibletheory.Furthermore,theeffectoftheratioofspecicheatsontheunsteadyforcebecomesmorepronouncedastheMachnumberincreases. 4.1Introduction Inanincompressibleow,well-establishedanalyticalexpressionsexistforthesteadyandunsteady(added-massandhistory)forcesoncylindersandspheresintheStokesandinviscidlimits(see,e.g.,Landau&Lifshitz[ 54 ]).Analyticalandempiricalextensionsofthequasi-steady,added-mass,andhistoryforcestoniteReynoldsnumbershavebeenstudiedextensively(see,e.g.,Croweetal.[ 25 ]andMagnaudet&Eames[ 62 ]).Inthecompressibleregime,attentionhasgenerallybeenfocusedonthesteadydragforce.DetailedparameterizationsofthesteadydragforceintermsofMachandReynoldsnumbershavebeenconsidered(see,e.g.,Bailey&Hiatt[ 4 ]and 79

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thereferencescitedtherein).However,ourunderstandingofunsteadyforcesinthecompressibleregime,arisingeitherfromtheaccelerationofthecylinderorsphereorfromtheaccelerationofthesurroundinguid,islimited. TheearliestfundamentalcontributionstothestudyofunsteadyforcesinthecompressibleregimeappeartobeduetoLove[ 61 ]andTaylor[ 106 ].Miles[ 71 ]investigatedthemotionofacylinderimpulsivelystartedfromrestbasedontheacousticapproximationofthevelocitypotentialequation.Hetreatedthecasesoftransientmotiongeneratedbyaconstantforceappliedoveranitetimeintervalaswellasanimpulsivelyappliedvelocity.Independently,Longhorn[ 58 ]consideredtheunsteadymotionofaspherebasedonthesameapproximation.TheworkofLonghornwasconsideredbyFfowcsWilliams&Lovely[ 34 ],whodeterminedanalyticallytheacousticeldproducedbyasphereacceleratedimpulsivelyfromrest.BothMilesandLonghornpointedoutthelimitationsoftheconventionaladded-massconceptindescribingtheinviscidforceincompressibleows. Theselimitationsarerootedintherelationshipbetweentheadded-massforceandtheinstantaneousacceleration.Inanincompressibleow,theadded-massforcedependsonlyontheinstantaneousacceleration.Inacompressibleow,ontheotherhand,theinviscidforcedevelopsonanacoustictimescaleR=c1,whereRistheradiusofthecylinderorsphereandc1isthespeedofsoundintheambientuidinthefareld.TheresultsofMilesandLonghornshowthatunderconstantacceleration,theinviscidforcereachesaconstantvalueforc1t=R&10.Ifanadded-masscoefcientiscomputedbasedonthisconstantlong-timeforce,valuesof1.0and0.5arerecoveredforthecylinderandsphere,respectively.Thesevaluesareconsistentwiththelow-Mach-numberlimitimplicitintheacousticapproximationemployedbyMilesandLonghorn.Itshouldalsobenotedthatinacompressibleowtheforceevolutioninresponsetoconstantaccelerationisnon-monotonic.Asaresult,atintermediatetimes 80

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theinstantaneousforceonthecylinderorspherecanbesubstantiallylargerthantheconstantnalforce. OtherrelevantworkwasperformedbyTracey[ 113 ],whoextendedMiles'sanalyticalworkonthemotionofanimpulsivelystartedcylindertoniteMachnumber.NumericalsimulationsofcompressibleowaboutanacceleratingcylinderatniteMachnumberswereperformedbyBrentner[ 13 ].However,thefocusofBrentner'sstudywasonthepropagationofacousticenergyasthecylinderacceleratedimpulsivelyfromresttoaMachnumberof0.4. TheobjectiveofthisworkistoextendtheresultsofMilesandLonghorntonitefreestreamMachnumbers.WewillinvestigatetheeffectofMachnumberonthenon-monotonicevolutionandasymptoticlong-timeconstantvalueoftheunsteadyforceinresponsetoasuddenconstantacceleration.Tothisend,wesolvenumericallytheEulerequationsinaframeofreferenceattachedtothecylinderandsphere,prescribetheirmotion,andcomputethedragcoefcient.WethustakeanapproachsimilartoBrentner,butourgoalisthedeterminationofforcesandnotthepropagationoftheacousticeld.WerestrictourattentiontothesubcriticalMachnumberregimeinthisarticle. 4.2NumericalMethod ThenumericalmethodsolvestheEulerequationsinintegralformcastinaframeofreferenceattachedtothecylinderorthesphere.Thespatialdiscretizationisbasedontheux-differencesplittingmethodofRoe(1981)(see[ 41 ]forreference)andtheweightedessentiallynon-oscillatoryreconstructiondescribedbyHaselbacher[ 41 ].Thediscreteequationsareintegratedintimeusingthefour-stageRunge-Kuttamethod.Thebasicmethodologyemployedinthisworkhasbeenappliedtoseveralunsteadycompressibleowsanddemonstratedgoodagreementwiththeoryandexperimentaldata,see,e.g.,Haselbacheretal.[ 43 ]. 81

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Forthecylinder,atwo-dimensionalhexahedralgridofO-typetopologyisusedwith386cellsaroundthecircumference.RelativetothecylinderradiusR,theradialgridspacingadjacenttothecylindersurfaceisr=R=1.62410)]TJ /F10 7.97 Tf 6.58 0 Td[(2,thusproducingcellswithaspectratiosofnearlyunity.Theradialstretchingofgridcellsisadjustedsuchthateachlayerofcellsconsistsofapproximatelysquarecellstominimizeinternalwavereections.Wehaveemployedgridsconsistingofupto1,760,160cellstoassessgrid-independenceofoursolutions.Theresultsshownbelowwereobtainedonagridof110,010cells. Forthesphere,ahexahedralgridconsistingofsixblocksisused.Eachblockcontains100100cellsonthespheresurfaceand320cellsintheradialdirection.RelativetothesphereradiusR,theradialgridspacingonthesurfaceisr=R2.910)]TJ /F10 7.97 Tf 6.59 0 Td[(2.Asforthecylindergrid,theradialstretchingofcellsisadjustedsuchthateachlayerconsistsofapproximatelycuboidcells.Theresultsshownbelowwereobtainedwithaverynegridof19,200,000cells.Forbothcylinderandspherecomputations,thecharacteristicboundaryconditionsofPoinsot&Lele[ 85 ]areappliedattheouterboundary,locatedat200R. 4.3Results Ourobjectiveistoextractthetime-dependentinviscidforceonacylinderorsphereinresponsetosuddenlyimposedaccelerationatniteMachnumbers.Toaccomplishthis,thecylinderorsphereisrstheldxedandasteady-statesolutionisobtainedatthechosenfareldMachnumberM1,0.Atsometimet0,weimposeonthecylinderorthesphereaconstantaccelerationainthedirectionoppositetotheambientow.Wemaintaintheconstantaccelerationforanitetimeintervaltf)]TJ /F3 11.955 Tf 12.77 0 Td[(t0andremoveitthereafter.Thedurationofacceleration,non-dimensionalizedintermsoftheacoustictimescale,ischosentobec1(tf)]TJ /F3 11.955 Tf 12.29 0 Td[(t0)=R=20,whichissufcientfortheinviscidforcetoreachaconstantvalue.TheinstantaneousrelativeMachnumberincreaseslinearly 82

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duringtheperiodofaccelerationandreachesavalueM1,0+bytheendoftheinterval,where=a(tf)]TJ /F3 11.955 Tf 11.95 0 Td[(t0)=c1. Inallcasesconsidered,thenon-dimensionalacceleration=aR=c21ischosencarefullytosatisfythefollowingtwocompetingrequirements.First,welimitthevalueofsuchthat,thechangeinMachnumber,iskeptsmall.Thisallowsinterpretationoftheresultingtime-dependentforceonthecylinderorspheretobeataxedorfrozenMachnumberofM1,0.Accordingly,wedropthesubscript0andsimplywriteM1inthefollowing.Figure 4-2 showsthetimeevolutionofthenon-dimensionalforceonacylinderforseveralvaluesofthenon-dimensionalaccelerationstartingfromM1=0.3.Here,theforce(perunitwidth)isnon-dimensionalizedaseF=F=(mfa),wheremfisthemassoftheuiddisplacedperunitwidthofthecylinder.Itcanbeseenthatprovidedissufcientlysmall,thenon-dimensionalforceisobservedtobeindependentoftheactualvalueof.Withincreasingmagnitudeofacceleration,e.g.,for=1.210)]TJ /F10 7.97 Tf 6.59 0 Td[(3,theincreaseinrelativeMachnumberoverthedurationofaccelerationhasasignicantinuenceonthenetforceandthereforetheresultcannolongerbeconsideredtocorrespondtoafrozenMachnumberof0.3.Atevenhigheraccelerations,theinstantaneousMachnumberexceedsthecriticalvalueofabout0.398andtheeffectoflocallysupersonicowaroundthecylinderresultsinasteadyincreaseintheforce.Clearly,shouldbemaintainedsufcientlysmalltoextractthetime-dependentforceatafrozenMachnumber.Thesecondrequirementisthattherateofaccelerationmustnotbetoosmall,forotherwisetheresultingforcewillbeveryweakwithlowsignal-to-noiseratio.Inallthecasesconsideredhere,arangeofaccelerationsatisesthetwocompetingrequirements.Providedischosentoliewithinthisrange,theresultingappropriatelynon-dimensionalizedforceisindependentof. NotethatthefareldMach-numberrangeinvestigatedhereislimitedtovaluesbelowthecriticalMachnumbersofabout0.398and0.6forthecylinderandsphere,respectively.BelowthecriticalfareldMachnumber,thesteadyowsaroundthe 83

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Figure4-1. SchematicdepictionofvariationoffreestreamMachnumberduringcomputations. cylinderandthesphereremainsubsoniceverywhere.Therefore,thesteady-stateinvisciddragforceisidenticallyzerobeforetheapplicationoftheaccelerationaswellasaftertheremovaloftheaccelerationfollowingthedecayoftransients. 4.3.1EffectofMachNumber Inanincompressibleow,acylinderorspherewithaconstantaccelerationaexperiencesaninviscidforceofmagnitudeCMmfaoppositetothedirectionofacceleration,whereCMistheadded-masscoefcient.ForacylinderCM=1andforasphereCM=0.5.Theadded-massforceisrealizedinstantaneouslyupontheapplicationofaccelerationduetotheinniteacousticpropagationspeedimplicitintheincompressibilityassumption.Theforceisproportionaltotheappliedinstantaneous 84

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Figure4-2. EffectofaccelerationparameterontheunsteadyforcecoefcientoncylinderforM1,0=0.3and=1.4 accelerationandceasestoexistoncetheaccelerationisremoved.Asdescribedinx1,oncecompressibilityeffectsbecomeimportant,theforceisnolongerdependentononlytheinstantaneousacceleration. 4.3.1.1Cylinder Ingure 4-3 weplotthetime-dependentforceonthecylinderasafunctionofnon-dimensionaltime=c1(t)]TJ /F3 11.955 Tf 13.18 0 Td[(t0)=RforfareldMachnumbersrangingfromzeroto0.39.Guidedbyincompressibleresults,thedimensionalforceFperunitcylinderwidthhasbeennon-dimensionalizedaseFcy=F=(mfa).AlsoplottedinthegureisthetheoreticalresultofMiles[ 71 ]correspondingtothelimitM1!0.Thenon-monotonictimeevolutionoftheinertialforceinresponsetoconstantaccelerationisclearlyvisible.Afterabout12acoustictimeunitsthenon-dimensionalforceeFcyreachesaconstantvalueof1.0,whichcorrespondstotheadded-masscoefcientofacylinder 85

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AForcecoefcient(forlegendsee(B)) BDerivativeofforcecoefcient Figure4-3. ComparisonofcomputedresultsforcylinderwiththeoreticalresultsofMiles(1951)for=1.4 86

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A=0.56 B=1.67 C=2.78 D=11.12 Figure4-4. Evolutionofnon-dimensionalperturbationpressure(scaledtorangebetweenminusandplusoneateachinstant)forcylinderatM1=0.2and=1.4 inincompressibleow.Inotherwords,for&12theacousticdisturbancearisingfromthesuddenonsetofaccelerationhasradiatedsufcientlyfarawaythattheneareldisapproximatedwellbyanincompressiblepotentialow.Atintermediatetimes,however,theinviscidforceisobservedtobesubstantiallylarger.Forinstance,Miles'solutionyieldsapeakvalueofeFcy1.17at3.1. 87

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Thenormalizedperturbationpressure(differencebetweentheinstantaneouspressureandtheinitialsteadypressuredistributions,normalizedtorangebetweenminusandplusone)isplottedingure 4-4 forM1=0.2atthefourdifferenttimesmarkedbyblackcirclesingure 4-3 (A).FromthecompressibleformofBernoulli'sequation,theperturbationpressurecanbeshowntohavetwocontributions.Therstcontributionarisesfromthetime-dependenceofthevelocitypotential.Thesecondcontributionisduetochangesinthesquareofthevelocity.Ingure 4-4 ,theinuenceoftherstcontributiondominatesatearlytimesandafore-aftasymmetrycanbeseenclearly.Atlatertimes,withthesecondcontributionbecomingincreasinglyimportant,theperturbationpressureincreasesandappearstobemorefore-aftsymmetric. Asindicatedbygure 4-3 (A),thequalitativebehavioroftheunsteadyforceremainsthesameatallsubcriticalMachnumbers.Theasymptoticlong-timeconstantvalueandthepeakvalueofthenon-dimensionalforceincreaseswithM1.ForboththepeakandasymptoticvaluesofeFtheeffectofMachnumberissubstantial,ascanbeseenfromgure 4-5 (A).ThepeakinstantaneousforceatM1=0.39isabout2.4timesaslargeasthatpredictedfromincompressibletheory.Theasymptoticlong-timeconstantvalueofthenon-dimensionalforcecanbeseentoincreasesteadilyfromavalueofunityatM1!0toabout2.04atM1=0.39.Thissteadyvaluecanbeconsideredasaneffectiveadded-masscoefcientforcompressibleow. Theratioofpeaktolong-timesteadyforceforthedifferentMachnumbersisalsoshowningure 4-5 (A).Itcanbeseenthatthisratioisnearlyconstantatabout1.17-1.19fortherangeofMachnumbersconsidered.WithincreasingMachnumber,thetimeatwhicheFreachesapeakincreases.Forexample,forM1=0.39,thepeakoccursat5.26.Correspondingly,theapproachtoaconstantvalueisalsoslightlydelayed.Interestingly,oncetheaccelerationisturnedoff,thereturntozeroforceoccursinamannersimilartowhentheaccelerationisrstapplied. 88

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Itisconvenienttodifferentiatethenon-dimensionalforcepresentedingure 4-3 (A)anddeneakernelasKcy()=deFcy=d.TheresultingresponsekernelsforthedifferentMachnumbersarepresentedingure 4-3 (B).Thekernelcanbeinterpretedreadilyintermsofthenon-dimensionalinviscidforceperunitwidthonacylindersubjectedtoanimpulsivejumpinrelativevelocityofu0givenasKcy(=tc1=R)=F=(mf(c1u0=R)).Infact,thisistheforminwhichMileshadpresentedhisresultforM1!0.AsshownbyMiles,inthislimitthekernelsatisestheintegralequation Z0()]TJ /F8 11.955 Tf 11.95 0 Td[(+1)2 p ()]TJ /F8 11.955 Tf 11.95 0 Td[()()]TJ /F8 11.955 Tf 11.96 0 Td[(+2)Kcy()d=p (+2).(4) TheasymptoticsolutionstotheintegralequationforsmallandlargewerealsoobtainedbyMilesas Kcy(,M1!0)=8>><>>:1)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F4 11.955 Tf 16.29 8.09 Td[(1 162+5 483if1,)]TJ /F4 11.955 Tf 13.03 8.09 Td[(2 3)]TJ /F4 11.955 Tf 13.15 8.09 Td[(62)]TJ /F4 11.955 Tf 11.96 0 Td[(24ln4 5+Oln2 7if1.(4) Thenon-monotonicityseeningure 4-3 (A)translatestothekernelbeingpositiveforashortduration,thenbecomingnegative,andslowlyapproachingzero.Thisbehaviorhasinterestingimplications.Theresponsetoanimpulsivejumpincylindervelocityisaninitialnon-dimensionalforceofunitmagnitude(Kcy(!0)!1),whichdecaysrapidlywithtime.Initiallytheforceisoppositetothedirectionofcylinderacceleration.ButaftersometimeasKcychangessign,theinviscidhydrodynamicforceonthecylinderisalongthedirectionofimpulsiveacceleration.ThetimeatwhicheFpeaksingure 4-3 (A)correspondstothezero-crossingtimeingure 4-3 (B).TheslowerapproachtosteadystatewithincreasingM1isclearlyvisible. 4.3.1.2Sphere Thetimeevolutionofthenon-dimensionalforceonthesphereforvaryingM1isplottedingure 4-6 (A).Thenon-dimensionalforceisdenedtobeeF=F=(mfa),wheremfisthemassoftheuiddisplacedbythesphere.TheanalyticalresultofLonghorn 89

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ACylinder BSphere Figure4-5. Computedbehaviorofpeakandsteady-statevaluesofeFfor=1.4 90

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[ 58 ]correspondingtothelimitM1!0isalsoshowninthegure.Inthislimit,aftertheinitialtransient,thelong-timevalueofnon-dimensionalforcesettlesat0.5consistentwiththeadded-masscoefcientofasphereinincompressibleow.Asforthecylinder,theapproachtotheconstantforceisnon-monotonic.Thepeakvalueofabout0.6isreachedatanon-dimensionaltimeof1.6. TheeffectofMachnumberonthepeakvalueoftheunsteadyforcecoefcientandtheeffectiveadded-masscoefcientareshowningure 4-5 (B).TheeffectofMachnumberisagainsubstantial.Thelong-timeasymptoticvalueforM1=0.5isabout0.97andapeakvalueofabout1.2isreachedat3.1.Asbefore,theapproachtosteadystateisdelayedwithincreasingMachnumber.However,comparedtothecylinder,thesteadystateisapproachedmorerapidly. Wedifferentiatethenon-dimensionalforcepresentedingure 4-6 (A)anddeneakernelasKsp()=deFsp=d.Theresultingresponsekernelsarepresentedingure 4-6 (B)asafunctionofMachnumber.Theinviscidforceonaspheresubjectedtoaninstantaneousjumpinrelativevelocityisthengivenbymf(c1u0=R)Ksp,whereu0isthejumpinrelativevelocity.InthelimitM1!0,Longhornobtained Ksp(,M1!0)=expcos.(4) Theapproachtosteadystateisoscillatory,buttherapidexponentialdecaymaskstheoscillatorybehavior.Theoscillatorynatureofthekernelcanbediscernedfromthecomputationalresultsingure 4-6 (B),atleastforthehighervaluesofM1.ItmaybeconjecturedthatKcyisalsooscillatory.(Suchbehaviorcanbeseeningure 4-3 forM1=0.39.) 4.3.2Mach-NumberExpansion Providedtheowaroundthecylinderorsphereisirrotational,thevelocityeldcanbeexpressedintermsofavelocitypotential.Inthecompressibleregime,the 91

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AForcecoefcient(forlegendsee(B)) BDerivativeofforcecoefcient Figure4-6. ComparisonofcomputedresultsforspherewiththeoreticalresultsofLonghorn(1952)for=1.4 92

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governingequationforthevelocitypotentialcanbeexpressedas c2r2=@2 @t2+@ @t+1 2(rr)(r)2,(4) wherethespeedofsoundcisgivenby c2=c20)]TJ /F4 11.955 Tf 11.96 0 Td[(()]TJ /F4 11.955 Tf 11.95 0 Td[(1)@ @t+1 2(r)2,(4) withc0denotingthestagnationspeedofsound.Itisunderstoodthat(r)2=(r)(r).Wenon-dimensionalizeEq.( 4 )withRasthelengthscale,M1c0asthevelocityscale,andTasanasofyetunspeciedtimescale.Denotingthenon-dimensionalquantitiesbyatilde,theresultingequationcanbeexpressedas er2e)]TJ /F3 11.955 Tf 19.57 8.09 Td[(R2 c20T2@2e @et2=M1R c0T@(ere)2 @et+()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M1R c0T@e @eter2e+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2M21(ereere)er2e+M21 2ereer(ere)2.(4) Immediatelyafterthesuddenonsetofaconstantaccelerationa,irrespectiveofitsmagnitude,theappropriatetimescalewillbesuchthatthepropagationofacousticsandtheassociatedtimederivativetermsarebothimportant.However,aftertheacousticshavepropagatedsufcientlyfarawayfromthecylinderorsphere,theappropriatetimescaleforthevariationoftheambientowisT=M1c0=a.Thenitcanbededucedthatcomparedtoer2ethenextthreetermsinEq.( 4 ),allofwhichinvolvetimederivatives,scaleas2=M21,,and,respectively.Providedthatthenon-dimensionalacceleration=aR=c20satisestheconditionM1,thethreetermscanbeignoredandtheresultingequationis er2e=)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2M21(ereere)er2e+M21 2ereer(ere)2.(4) Thus,afteraninitialtransient,thecompressiblepotentialowaroundthecylindercanbeconsideredquasi-steadyandexpressedintermsoftheJanzen-Rayleigh 93

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expansion(seeOswatitsch[ 78 ])toO(M41)as U1R=er+1 ercos+M2113 12er)]TJ /F4 11.955 Tf 18.37 8.09 Td[(1 2er3+1 12er5cos)]TJ /F11 11.955 Tf 11.95 16.86 Td[(1 4er)]TJ /F4 11.955 Tf 21.51 8.09 Td[(1 12er3cos3, wherethetimedependenceentersonlythroughtheinstantaneousfreestreamMachnumberM1.TheaboveexpansioncanbesubstitutedintothecompressibleformoftheBernoulliequationtoobtainaMach-numberexpansionforpressure,whichcanbeintegratedaroundthecylindertoobtaintheforce.Ananalyticexpansionfortheeffectiveadded-masscoefcientcanthenbeobtainedas CM,e=1+M21+O(M41),(4) whichisalsoplottedingure 4-5 (A).Unfortunately,withthepresentnumericalsimulationitisnotpossibletogotomuchsmallerMachnumbersandthedifferencebetweentheaboveexpansionandthenumericalresultsincreaseswithincreasingM1.Nevertheless,theabovesimpletheorysupportsthequalitativebehaviorofincreasingadded-masseffectatniteMachnumber. Theroleofspecic-heatratiocannowbeexamined.InthelimitM1!0,Eq.( 4 )reducesto er2e)]TJ /F3 11.955 Tf 19.57 8.08 Td[(R2 c20T2@2e @et2=0.(4) ThisisthefundamentalequationunderlyingtheworksofMilesandLonghorn.ItindicatesthatinthelimitM1!0theowandforceevolutioninresponsetoinnitesimalaccelerationwillbeindependentof.Ontheotherhand,ifweassumeM1tobenite,thenwendthatatshorttimesthevelocitypotentialisgivenbythecompleteEq.( 4 )andtheleading-ordertermincludingthespecic-heatratioisO(M1).Atlongtimes,however,Eq.( 4 )appliesandtheleading-ordertermdependingonthespecic-heatratioscalesasO(M21).Thus,theeffectofthespecic-heatratioontheeffectiveadded-masscoefcientcanbeexpectedtobeweakatsmallvaluesofM1. 94

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Figure4-7. Effectofonunsteadyforcecoefcientoncylinder Thesetheoreticalresultsarecorroboratedbyourcomputations.Ingure 4-7 ,wepresenttheeffectofvaryingforthreedifferentvaluesofM1ontheunsteadyforceforasuddenlyacceleratedcylinder.Aswiththepreviouslypresentedresults,theaccelerationissmallenoughthatquasi-steadyconditionsaremaintained.ItcanbeseenclearlythatforM1=0.2,theeffectofisindiscernible.ForM1=0.39,however,theeffectofisnoticeableforboththepeakandsteady-statevaluesoftheforcecoefcient. 4.4Discussion Inthefollowing,wediscusssomespecicquestionsrelatedtotheresultspresentedinx3.Wefocusinparticularontherelevanceoftheseresultstopredictingparticlemotionusingforcelaws. Therstquestionthatarisesis:Underwhatconditionswillcompressibilityeffectsonunsteadyforcesbeimportant?Therelativeimportanceofinviscid(added-massandpressuregradient)andviscous(history)unsteadyforcesarisingfromparticle 95

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accelerationcomparedtothedominantquasi-steadydragscalesas1=(+CM)and1=p +CM,respectively,whereistheratioofparticledensitytosurroundinguiddensityandCMistheadded-masscoefcient(Bagchi&Balachandar[ 2 ]).Thus,inthecontextofaparticlemovinginair,owingtothelargedensityratioencounteredinmostpracticalsituations,theunsteadyforcesarisingfromparticleaccelerationaretypicallyignored. Itisimportanttonotethattheunsteadyforcesarisingfromtheaccelerationofthesurroundinguiddonotfollowtheabovescalings.Forexample,inthecaseofaparticleinjectedintoanacceleratingow,theratioofinviscidandviscousunsteadyforcestoquasi-steadydragcanbeshowntoscaleasRed=Landp Red=L,whereReisReynoldsnumberbasedonparticlediameterdandrelativevelocityandListhetypicallengthscaleofambientowvariation(Bagchi&Balachandar[ 2 ]).Notethatthedensityratiodoesnotappearintheseestimates.Theabovescaling,althoughdevelopedforanincompressibleow,willbeappropriateevenifcompressibilityeffectsareimportant.Thus,situationscanexistwherethemotionofanite-sizeparticleinarapidlyacceleratingcompressibleowcanbeinuencedbyunsteadyforces,irrespectiveoftheparticletouiddensityratio.Forexample,Tedeschietal.[ 108 ]andThomas[ 111 ]assessedtheinuenceofthehistoryforceonthemotionofaparticlethroughashockwavetheoreticallyandnumericallyusingtheBasset-Boussinesq-Oseenequation(Croweetal.[ 25 ])andobservedtheinstantaneoushistoryforcetobemanytimeslargerthantheviscousdragforce. Thesecondquestionis:Howwouldtheaboveresultsbeusedinpractice?Asimplemodelofthecompressibleinviscidforcearisingfromtheunsteadymotionoftheparticleorthesurroundinguidcanbewrittenasfollows Fiu(t)=mfZtKc1(t)]TJ /F8 11.955 Tf 11.96 0 Td[() R;M1Du Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dv dtdc1 R+mfDu Dt,(4) 96

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whereDu=DtistheambientuidaccelerationevaluatedattheparticlepositionandKandmfarechosenappropriatelyforthecylinderandthesphere.AscautionedbyMiles[ 71 ],Longhorn[ 58 ],Yih[ 117 ],andseveralothers,werefrainfromcallingthersttermontheright-handsideoftheaboveequationthe`added-massforce,'sincethetime-dependentnatureoftheforcedoesnotalwaysreducetotheformofaconstantmassmultiplyingtheinstantaneousacceleration.Inthecaseofaconstantacceleration,forc1t=R1theaboveintegralreducestoaconstanttimesacceleration,thuspermittinginterpretationintermsofaneffectiveadded-masscoefcient,whichreducestotheincompressiblevalueasM1!0. Wenextaddressthequestionofhowmuchtheincompressibleresultsfortheadded-massforcearemodiedbycompressibility.Asalreadyseeningures 4-5 ,theeffectiveadded-masscoefcientmorethandoublesastheMachnumberincreasesinthesubcriticalrange.Furthermore,aspointedoutbyMilesandLonghorn,theeffectiveinuenceoftheunsteadyinviscidforcewillbestrongeriftheaccelerationislargeoverashortperiodoftimethanifitissmalloveralongperiodoftime.ToexplorethisbehavioratniteMachnumber,consideraproblemsimilartothatdescribedatthebeginningofx3:AcylinderorsphereisheldxedinasteadyambientowoffareldMachnumberM1.Attimet=0,thecylinderorsphereisgivenaconstantaccelerationainthedirectionoppositetotheambientowuntilt=t,whentheaccelerationisremoved.TheresultingforceonthecylinderorsphereisgivenbyEq.( 4 ).Fromthisforce,theworkdonebythecylinderorsphereatt=tcanbeexpressedas W(t,t)=Zt0mfatZt0aKc1(t)]TJ /F8 11.955 Tf 11.95 0 Td[() R;M1dc1 Rdt+ZttmfatZt0aKc1(t)]TJ /F8 11.955 Tf 11.95 0 Td[() R;M1dc1 Rdt.(4) 97

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RearrangingtheintegralsweobtainW(t,t)=mf(t,t)(at)2=2,where (t,t)=2 (t)2Zt0tZc1t=R0K(;M1)ddt+2 tZttZc1t=Rc1(t)]TJ /F10 7.97 Tf 6.59 0 Td[(t)=RK(;M1)ddt.(4) Theworkdoneisequaltothekineticenergyimpartedtotheuidduetotheacceleration.Thus,fort!1,mfcanbeinterpretedastheadded-massduetotheuidwhosevelocitychangedbyatbecauseoftheaccelerationofthecylinderorsphere. Thevariationof(t!1,t)forvaryingMachnumbersispresentedingure 4-8 .Forthesphere,Longhorn(1952)showedthatinthelimitofM1!0,theresultreducesto (t!1,t)=1+1 (t)2[1)]TJ /F4 11.955 Tf 11.95 0 Td[(exp()]TJ /F4 11.955 Tf 9.3 0 Td[(t)cos(t))]TJ /F4 11.955 Tf 11.95 0 Td[(exp()]TJ /F4 11.955 Tf 9.3 0 Td[(t)sin(t)].(4) AtniteMachnumbers,wasobtainedthroughnumericalintegrationofEq.( 4 )usingthekernelspresentedingures 4-3 (B)and 4-6 (B).Inthelimitofsustainedslowacceleration(i.e.,t!1,a!0),theforceonthecylinderorsphereremainsconstantexceptfortheinitialandnaltransients,andapproachestheeffectiveadded-masscoefcientCM,epresentedingures 4-5 (A)and(B).However,increasesasthedurationofaccelerationisreduced,andinthelimitoft!0weobserveadoublingoftheaddedmass.Infact,inthelimitoft!0,thecontributionfromtherstterminEq.( 4 )becomeszeroanditcanbeshownthat (t!1,t!0)=2(t!1,t!1)=2CM,e.(4) Thus,thecombinedeffectofniteMachnumberandimpulsiveaccelerationcanintensifytheadded-masseffecttomorethanfourtimesofwhatwouldbepredictedbasedonincompressibletheory. Wenowcommentontheuseoftheunsteadyinviscidforcegiveninequation(4.1).ConsiderthemotionofacylinderorsphereinresponsetoanexternalforceFextina 98

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ACylinder BCylinder,normalized(forlegendsee(A)) CSphere DSphere,normalized(forlegendsee(C)) Figure4-8. Behaviorof(t!1,t)denedbyEq.( 4 ) stagnantinviscidcompressibleuid.Themotionisgovernedby mpdv dt+mfZtKc1(t)]TJ /F8 11.955 Tf 11.95 0 Td[() R;M1dv dtdc1 R=Fext,(4) wherempisthemassofthecylinder(perunitwidth)orofthesphereandmfisthemassoftheuiddisplacedbythecylinder(perunitwidth)orthesphere.Iftheexternalforceissmallandmaintainedoveralongperiodoftime,theresultingaccelerationofthecylinderorsphereisgivenbyFext=(mp+mfCM,e),whichreducestotheincompressible 99

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resultinthelimitM1!0.Ontheotherhand,iftheexternalforceislargeandofverybriefdurationwithanetimpulseofI,theresultingvelocitychangeduetotheimpulseisgivenbyI=(mp+mfCM)inanincompressibleow.Inacompressibleow,however,anasymptoticconstantvelocitywillbereachedonanacoustictimescaleaftertheimpulse,andifanadded-masscoefcientweretobecomputedincomparisonwiththeaboveincompressibleresult,itwillbedependentonbothM1andthedensityratio.SimilarbehaviorcanbeobservedintheresultsofTracey(1988).Clearly,ascautionedbyotherauthors,theconceptofaddedmassisfraughtwithdifcultyincompressibleowanditisadvantageoustosimplyconsiderEq.( 4 )asanexpressionfortheunsteadyinviscidforce. Finally,wenotethatinanincompressibleow,theadded-massforcehasbeenshowntobeindependentoftheReynoldsnumberorviscouseffects(seeRiveroetal.[ 87 ],Chang&Maxey[ 19 ],Mougin&Magnaudet[ 74 ],Bagchi&Balachandar[ 2 ],Bagchi&Balachandar[ 3 ],andWakaba&Balachandar[ 114 ]).Theinstantaneousnatureoftheadded-massforceinincompressibleowprecludesanyinteractionwiththeviscousresponsetotheacceleration.Inacompressibleow,theeffectoftheReynoldsnumberReontheunsteadyinviscidforcewilldependonthetimescales.Theacoustic,inertial,andviscoustimescalesareR=c1,R=UandR2=,respectively,whereUisthecharacteristicrelativevelocityandisthekinematicviscosityoftheuid.TheratioofviscoustoacoustictimescaleswillbeproportionaltoRe=M1whereRe=UR=andthusatsufcientlyhighReynoldsnumbertheunsteadyinviscidforcecanbeexpectedtobeindependentofReynoldsnumber.However,atniteRe,quasi-steadyandunsteady(history)componentsofthehydrodynamicforcemustbetakenintoaccountalso. 100

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CHAPTER5MODELINGOFTHEUNSTEADYFORCEFORSHOCK-PARTICLEINTERACTION Theinteractionbetweenaparticleandashockwaveleadstounsteadyforcesthatcanbeanorderofmagnitudelargerthanthequasi-steadyforceintheoweldbehindtheshockwave.Simplemodelsfortheunsteadyforcehavesofarnotbeenproposedbecauseofthecomplicatedoweldduringtheinteraction.Here,asimplemodelispresentedbasedontheworkofParmaretal.(Phil.Trans.R.Soc.A,366,2161-2175,2008).Comparisonswithexperimentalandcomputationaldataforbothstationaryspheresandspheressetinmotionbyshockwavesshowgoodagreementintermsofthemagnitudeofthepeakandthedurationoftheunsteadyforce. 5.1Introduction Theinteractionbetweenaparticleandashockwavehasbeenstudiedextensivelyduetoitspracticalimportance,see,e.g.,[ 46 95 ],andmanyothers.Astheshockwavepropagatesintoagas-particlemixture,thegasvelocityincreasesinstantaneouslyacrosstheshock.Bycontrast,theparticlevelocityapproachesthepost-shockgasvelocityonlyslowlyduetotheniteinertiaoftheparticles.Thisleadstothesocalledfrozen,relaxation,andequilibriumregimesthathavebeendiscussedindetailbyCarrier[ 17 ],Soo[ 100 ],Kriebel[ 53 ],andRudinger[ 89 ].Severalinvestigations,see,e.g.,[ 27 46 47 50 57 88 90 103 ],havedocumentedcarefulmeasurementsofthetime-dependentparticlemotionbehindashockwave.Oneimportantobservationisthattheparticlesgenerallyapproachtheequilibriumstatefasterthanpredictedbythestandarddragrelation.Thus,theabovestudieshaveinferredthatthedragforceontheparticle,andhencethedragcoefcient,aresubstantiallyenhancedinthepost-shockow. Directtime-resolvedmeasurementsoftheforceexertedbyashockwavepropagatingoveraparticleareverychallenging.Themeasurementsneedtoresolveverysmallforcesoveradurationofmillisecondswitharesolutionofmicroseconds.Techniques 101

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withtherequiredaccuracyandresolutionhaveonlybeendevelopedrelativelyrecently.Tannoetal.[ 104 105 ]andSunetal.[ 102 ]usedanaccelerometerinstalledinsidethesphere.BredinandSkews[ 11 ]andSkewsetal.[ 97 ]usedastress-wavedragbalanceandreportedtime-dependentforcemeasurementonastationaryparticlewithanestimatederroroflessthan15%.Theseexperimentshaveshownthattheforceontheparticleincreasessignicantlyastheshockwavepassesoverit.Thepeakforceontheparticlecanbemorethananorderofmagnitudelargerthanthesteady-stateforceinthepost-shockow.Furthermore,itwasobservedthatthetransitionfromthepeakforcetothesteady-stateforcecanbenon-monotonic. Thepropagationofashockwaveoverasphericalobstaclesuchasaparticleisverycomplicated,consistingofregularandirregularshock-wavereection,diffraction,andfocusingphenomena,see[ 15 102 104 ].Withrecentadvancesinnumericalmethodsandcomputerperformance,highlyaccuratedirectnumericalsimulationsofshock-particleinteractionhavebeenaccomplished.OneexampleistheworkofSunetal.[ 102 ],whoobtainedgoodagreementwiththeirownmeasurements.Thesimulationscapturedtheincreaseoftheinstantaneousdragforcebymorethananorderofmagnitudeastheshockwavepropagatesovertheparticleaswellasthenon-monotonicapproachtothesteadystate.Sunetal.[ 102 ]observedthatthemaximumdragcoefcientoccursslightlyaftertheshock-wavereectionchangesfromaregulartoaMachreectionandthattheminimumdragcoefcientappearstobeduetothefocusingofthediffractedshockwaveattherearofthesphere. Inmanyapplications,shockwavesinteractwithmillionstobillionsofparticles.Insuchsituations,thedetailedresolutionoftheinteractionbetweentheshockwaveandeachparticleisnotfeasible.Instead,oneneedstoresorttotheEulerian-Lagrangianpoint-particleapproachandtrackthetrajectoryofasufcientlylargenumberofparticlesastheyinteractwiththeshockwave.Thekeycomponentofthisapproachisamodelthataccuratelypredictstheinstantaneousforceontheparticles.Thecommonlyused 102

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standard-dragmodel,see,e.g.,[ 21 ],parameterizesthedragcoefcientintermsoftheparticleReynoldsnumberbasedontheparticlediameterandthevelocityoftheparticlerelativetothatofthesurroundinggas.Evenwithcompressibilitycorrectionsthataccountfornite-Mach-numbereffects,see,e.g.,[ 59 ],thestandard-dragcorrelationisappropriateonlyforthequasi-steadystatethatisestablishedlongafterthepassageoftheshockwave.Thelargeincreaseininstantaneousdragandthesubsequentnon-monotonicapproachtosteadystateisclearlyduetotheunsteadyeffectresultingfromthepropagationoftheshockwaveovertheparticle.Thisstrongtime-dependentcomponentoftheforcecannotbeaccountedforwithquasi-steadyforcemodels. Inincompressibleows,ithasbeenwellestablishedthatunsteadiness,eitherduetoaccelerationoftheparticleorofthesurroundinguid,givesrisetoadditionalinviscid(pressure-gradientandadded-mass)andviscous(Bassethistory)forces,see,e.g.,[ 54 ].Recently,ourunderstandingofandmodelsfortheseforceshavebeenextendedtoniteparticleReynoldsnumbers(see,e.g.,[ 19 62 74 114 ]).However,theexistingunsteady-forceparameterizationsaredevelopedfortheincompressiblelimit,andarethereforeincapableofaccuratelypredictingthestrongvariationindragforceastheshockwavepropagatesovertheparticle. Theearliestinvestigationsofunsteadyforcesinthecompressibleregime,duetoLove[ 61 ]andTaylor[ 106 ],focusedontheinviscidcontributiononly.Subsequently,Miles[ 71 ]andLonghorn[ 58 ]consideredtheeffectofcompressibilityonunsteadyinviscidforcesonacylinderandasphere,respectively,inthezero-Mach-numberlimitusingtheacousticapproximationofthevelocitypotentialequation.Morerecently,Parmaretal.[ 79 ]extendedtheresultsofMilesandLonghorntoniteMachnumber. Itisimportanttonotethattheconventionalconceptofaddedmassisoflimiteduseincompressibleows,becausecompressibilitydestroysthestraightforwarddependenceoftheinviscidunsteadyforceontheinstantaneousacceleration.Instead,inacompressibleuid,aparticleacceleratedimpulsivelyasa(t)=(t),where(t) 103

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istheDiracdeltafunction,issubjectedtoatime-dependentinviscidforcethatdecaysontheacoustictimescalec1=R,wherec1isthespeedofsoundintheambientuidandRistheparticleradius.Theinviscidunsteadyforcecanberepresentedintermsofahistoryintegralthatusesakerneltoweightthehistoryoftheparticleaccelerationrelativetotheuidacceleration.Thedecayofthekernelintermsofthenon-dimensionaltime=c1t=RdependsonboththegeometryoftheparticleandtheMachnumberMformedfromthevelocityoftheparticlerelativetothatoftheuidandc1.ThekernelsforacylinderandasphereinthelimitofzeroMachnumberwereobtainedbyMiles[ 71 ]andLonghorn[ 58 ],respectively.ThecorrespondingkernelsatnitebutsubcriticalMachnumbers(M.0.4foracylinderandM.0.6forasphere)wereobtainedbyParmaretal.[ 79 ].TheyobservedthatastheMachnumberincreases,thekernelchangessuchthattheeffectiveinviscidunsteadyforcecanbemorethandoubleofitsvalueinthezero-Mach-numberlimit. Theobjectiveofthispaperistopresentaphysics-basedforcemodelforunsteadycompressibleowsthatcombinestheinviscidunsteadyforceexpressedintermsofahistoryintegralwiththestandardquasi-steadydragforce.Wewilldemonstratethattheresultingsimplemodelprovidestheabilitytopredictthetime-dependentforceonasphericalparticleasashockwavepropagatesoverit.Recenttime-resolvedmeasurementsoftheforceexertedbyashockwavepropagatingoverastationaryspherebyTannoetal.[ 104 105 ],Sunetal.[ 102 ],BredinandSkews[ 11 ],andSkewsetal.[ 97 ]andtheircompanionnumericalsimulationresultswillbeusedtoevaluatetheaccuracyoftheforcemodel.Intheseexperimentsandcomputations,theshock-waveMachnumberMsissufcientlylowthattheMachnumberoftheowbehindtheshockwaveissubcritical,thuspermittingtheuseofthenite-Mach-numberkernelsobtainedbyParmaretal.[ 79 ].ResultsfromtheforcemodelwillalsobepresentedforthecaseofaspheresetinmotionbytheimpactofashockwaveandcomparedtoexperimentalandcomputationaldataofBritanetal.[ 14 ].Despiteitssimplicity,themodelappearsto 104

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capturetheessentialfeaturesoftheunsteadyforceduringtheshock-particleinteractionremarkablywellinallcases.TheEulerian-Lagrangianpoint-particleapproachwiththeproposedforcemodelcanthusbeusedasanefcientapproachtocomputecompressiblemultiphaseowsinvolvingshockwavespropagatingthroughsuspensionscontaininglargenumbersofparticles. 5.2ForceModel 5.2.1ForceParameterization MagnaudetandEames[ 62 ]suggestedthattheforceF(t)onaparticleinanincompressibleowcanbeparameterizedforarangeofparticleReynoldsnumbersas F(t)=Fqs(t)+Fiu(t)+Fvu(t)+Fl(t)+Fbg(t), (5) wherethetermsontheright-handsiderepresentquasi-steady,inviscidunsteady(added-massandpressure-gradient),viscousunsteady(Bassethistory),lift,andbuoyancy/gravityforces,respectively.Equation( 5 )isbasedontheassumptionthattheparticleismuchsmallerthanacharacteristiclengthscaleofthesurroundingow;thiswillbediscussedfurtherbelow.WeemphasizeherethattheinviscidunsteadyforceFiuconsistsoftwocontributions,namelytheadded-massandpressure-gradientforces.Theformeristheforcearisingfromtheno-penetrationconditionontheparticlesurfacewhenevertherelativeaccelerationbetweentheparticleandtheambientuidisnon-zero.Thelattermaybeinterpretedastheforcethatexistedduetoanambientpressuregradientintheabsenceoftheparticle,i.e.,iftheparticlewerereplacedbytheuid. Theaboveparameterizationisapplicableinthecompressibleregimealsoprovidedappropriatemodicationsaremadetothemodelingofthedifferentterms.Inacompressibleow,thequasi-steadydragwilldependonboththeReynoldsandMachnumbers.Anempiricalrelationforthequasi-steadydragcoefcientonaspherical 105

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particleinincompressibleowcanbeexpressedas CD,qs(Re)=kFqsk 1 2fkurk2R2=24 Re(Re)+0.421+42500 Re1.16)]TJ /F10 7.97 Tf 6.59 0 Td[(1,(5) whereRe=2fkurkR=istheparticleReynoldsnumberbasedontheuiddensityf,therelativevelocityur=u)]TJ /F15 11.955 Tf 11.98 0 Td[(vbetweentheparticleandthesurroundinggas,theparticlediameter2R,andthedynamicuidviscosity,see[ 21 ].Intheabove,fandarethedensityanddynamicviscosityofthesurroundinggasand(Re)=1+0.15Re0.687,see[ 94 ].ProvidedthattheparticleMachnumberM=kurk=c1remainsbelowthecriticalvalue(approximately0.6forasphere),theowaroundtheparticleisentirelysubsonic.Therefore,theeffectofcompressibilityisrelativelysmallandtheuseofanincompressiblerelationforthedragcoefcientisjustied.AbovethecriticalMachnumber,theeffectofcompressibilitycannotbeneglectedandmodiedformsofEq.( 5 )mustbeconsidered,see,e.g.,[ 59 ].WerestrictattentiontosubcriticalMachnumbersinthisarticle.(Undersupercriticalconditions,preliminarysimulationsindicatethattheinviscidunsteadyforceisnegligiblecomparedtochangesinthequasi-steadydragastherelativevelocityisvaried.Apossible,butperhapsonlypartial,explanationforthisobservationisthatthequasi-steadydragforceFqs,whichnowincludesasubstantialcontributionfromshockwaves,increasesfasterthantheunsteadyinviscidforceFiuwiththeMachnumber.) Asdiscussedintheintroduction,thedependenceoftheinviscidunsteadyforceontheinstantaneousaccelerationislostwiththeintroductionofcompressibility.Instead,theinviscidunsteadyforceonaparticledependsonaweightedintegraloftheaccelerationhistory.FollowingthesuggestionofParmaretal.[ 79 ],theinviscidunsteadyforceonaparticleinacompressibleowcanbemodeledas Fiu(t)=ZtKDmfu Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dmfv dtdc1 R+ZfDu DtdV,(5) 106

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whereK=K(c1(t)]TJ /F8 11.955 Tf 12.79 0 Td[()=R;M)isthecompressibleinviscidunsteadyforcekernel,MistheparticleMachnumber,mfisthemassoftheuiddisplacedbytheparticle,D()=Dtistherateofchangefollowingtheundisturbedambientuid,andd()=dtisrateofchangefollowingtheparticle.NotethattheaboveequationincorporatestheeffectofdensitychangesintheambientuidassuggestedbyEames&Hunt[ 29 ]andthatthesecondtermontheright-handsideisthepressure-gradientforce.Thedecayofthekernelintermsofthenon-dimensionaltime=c1t=RdependsonboththegeometryoftheparticleandtheMachnumberM.Forasphericalparticle,anexplicitexpressionfortheinviscidunsteadyforcekernelinthezero-Mach-numberlimitwasobtainedbyLonghorn[ 58 ]as Ksp()=exp()]TJ /F8 11.955 Tf 9.3 0 Td[()cos.(5) ThecorrespondingkernelsatnitebutsubcriticalMachnumberswerepresentedbyParmaretal.[ 79 ].ThekernelswereobtainedfromcomputedtimehistoriesoftheforceonaparticleinresponsetosuddenlyimposedconstantaccelerationsonceasteadystateisestablishedatagivenMachnumber.Providedtheaccelerationissufcientlysmall,sothatthechangeinMachnumberduringtheperiodofconstantaccelerationisnegligible,theextractedkernelscanbeinterpretedtobeassociatedwiththegivenMachnumber.ThekernelsobtainedbyParmaretal.[ 79 ]forcylindricalandsphericalbodiesoverarangeofMachnumbersareshowninFig. 5-1 TheviscousunsteadyforceinthecompressibleregimewilldependonboththeparticleReynoldsandMachnumbersandalsotakestheformofahistoryintegral.Thehistorykernelfortheviscousunsteadyforcehasbeenextensivelystudiedinincompressibleow.Itsformhasbeenshowntobequitecomplicatedandproblem-dependent,see[ 67 68 ].Itcanbeexpectedthatcompressibilityfurthercomplicatestheviscoushistoryforce.Wearenotawareofanyexistingmodelsfortheviscousunsteadyforcethattakeintoaccountcompressibilityeffects. 107

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Inthepresentcaseofaparticlesubjectedtoaplanarshockwave,theliftforceiszeroandweignoretheviscousunsteadyandbuoyancy/gravityforcesinthefollowing.Theviscousunsteadyforceisneglectedbecauseoftheapparentlackofsuitablemodelsforcompressibleows.Futurestudieswillfocusonthedevelopmentofmodelsfortheviscousunsteadyforceforcompressibleows. ThusthepresentmodelisbasedonthefollowingsimpliedformofEq.( 5 ), F(t)=Fqs(t)+Fiu(t), (5) whereFqs(t)andFiu(t)aredeterminedfromEqs.( 5 )and( 5 ),respectively.Tomakethismodelusefulfromapracticalperspective,theambientowasexperiencedbytheparticlemustbecharacterized.BeforeweaddressthecharacterizationoftheambientowinSection 5.2.4 ,wediscusstheimportanceoftheinviscidunsteadyforceaswellastheeffectofniteMachnumbers. 5.2.2ImportanceofInviscidUnsteadyContribution Itisgenerallyassumedthattheratioofinviscidandviscousunsteadyforcestothequasi-steadydragforcetoscaleastheuid-to-particledensityratioandthereforecanbeignoredforsolidparticlesintypicalgasows,see,e.g.,[ 91 99 ].Hereitisimportanttoseparateunsteadyforcesduetoaccelerationofaparticlefromthoseduetoaccelerationoftheambientow.Toseewhy,considerthequasi-steadyforcegivenbyEq.( 5 )tobethedominantcontribution.Thenitcanbeshownthattheresultingparticleaccelerationinresponsetothisforcewillbeinverselyproportionaltotheparticledensity.Therefore,theunsteadyforcearisingfromparticleaccelerationwillscaleasFqs=,where=p=fistheparticle-to-uiddensityratio. However,iftheambientowaccelerationisimposedexternally,suchasduringashock-particleinteraction,theunsteadyforcearisingfromtheambientowaccelerationneednotfollowtheabovescaling,see[ 3 ].Ifthevelocity,length,andtimescalesoftheambientowvariationaredenotedbyU,L,andT=L=U,therelativeimportanceofthe 108

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ACylinder BSphere Figure5-1. ResponsekernelsofParmaretal.[ 79 ] 109

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unsteadyforcearisingfromtheambientowaccelerationcomparedtothequasi-steadydragcanbeexpressedas 2 9R LRe (Re),(5) whereitisassumedthatkurkO(U)andthequasi-steadydragforceincludedonlythersttermontheright-handsideofEq.( 5 ).Therefore,atniteReynoldsnumbers,unsteadyforcesonaparticlecanbeimportantprovidedthatthelengthscaleoftheambientowvariationisnotmuchlargerthantheparticlediameter.Theaboveargumentisconsistentwithrecentexperiments,discussedbelow,wherethepeakforceonaparticleastheshockwavepropagatesoveritisobservedtobe10-20timeslargerthanthecorrespondingquasi-steadyforce.Thepeakdragforceoccursslightlyaftertheshock-wavereectionchangesfromregulartoMachreectionandbeforetheshockwavereachestheequator.Subsequently,thedragdecaysnon-monotonicallytothequasi-steadyvalueontheacoustictimescale.Furthermore,thisresultisindependentoftheparticle-to-uiddensityratio. Havingestablishedtherelativemagnitudeoftheunsteadyinviscidandquasi-steadyforces,wenowturnattentiontotheimportanceoftheinviscidunsteadyforceonthemotionofaparticle.Intheincompressiblelimit,theequationofmotionoftheparticleconsideringonlytheinviscidforcescanbeexpressedas pdv dt=1 2fDu Dt)]TJ /F3 11.955 Tf 13.15 8.09 Td[(dv dt+fDu Dt,(5) whereconsistencywithEq.( 5 )isobtainedifwetake Ksp()=H()(),(5) whereH(t)istheHeavisidestepfunctionand(t)istheDiracdeltafunction.Inthislimit,theparticleaccelerationisdirectlyproportionaltotheinstantaneousaccelerationoftheambientow.Thus,ifweconsidertheambientowtojumpfromu1=0toauniformvelocityofu2>0,thecorrespondingjumpintheparticlevelocityduetoinviscidforces 110

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canbeestimatedtobe v2 u2=3 2+1,(5) whereistheparticle-to-uiddensityratio. Withtheintroductionofcompressibility,thesimplelineardependenceoftheinviscidunsteadyforceoninstantaneousaccelerationislostandtheforcemustbeexpressedintermsofthehistoryintegral(seeEq.( 5 )).Theequationofmotionfortheparticle,accountingforonlytheinviscidforces,cannowbewrittenas dv dt+Kdv dt=Du Dt+KDu Dt,(5) wheremfisassumedtobeaconstantandthefollowingnotationfortheconvolutionintegralhasbeenadopted: f(t)g(t)=Ztf(t)]TJ /F8 11.955 Tf 11.96 0 Td[()g()d.(5) Inthezero-Mach-numberlimittheanalytickernelobtainedbyLonghorn[ 58 ]givenbyEq.( 5 )applies.TheequationofmotioncanbeintegratedwiththeabovekerneltoagainobtainEq.( 5 )forthevelocityjumpoftheparticleduetoinviscidforces.Theabovesimpleanalysisclearlyillustratesthefollowingtwoimportantpoints.First,theinviscidunsteadycontributionandthecompressibilityoftheowareofcriticalimportanceinaccuratelyaccountingfortherapidincreaseinthedragforceontheparticlethatoccursontheacoustictimescale.Second,whenconsideringtheintegratedeffectonparticlemotion,thecontributionfrominviscidforcesisproportionaltotheuid-to-particledensityratio)]TJ /F10 7.97 Tf 6.58 0 Td[(1.Ofcourse,theparticlevelocitywillcontinuetochangeandapproachthepost-shockgasvelocityduetotheactionofviscousforces.Whiletheinviscidunsteadyforceactsonaveryshortacoustictimescale,theviscousforceactsonamuchlongertimescale. 111

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5.2.3EffectofFiniteMachNumber AsdiscussedinSection 5.2.1 ,theeffectoftheMachnumberonquasi-steadydragisquiteweakinthesubcriticalregime.Inthecontextofashockwavepropagatingoverastationaryparticle,theowaroundtheparticlewillremainsubcriticalprovidedthatMs.1.5.TheeffectoftheMachnumberontheinviscidunsteadyforceappearsthroughthedependenceofthekernelontheMachnumber.FromFig. 5-1 itcanbeseenthatwithincreasingMachnumberinthesubcriticalregimetherateofdecayofthekernelbecomessmaller,suggestingamoreextendedinuenceofunsteadiness.Asaresult,thepeakforceatM=0.5wasobservedtobeabouttwiceaslargeasthatatM!0.Furthermore,thetimeatwhichthepeakforceoccursisalsonearlydouble(Parmaretal.[ 79 ]).ItcanthereforebeexpectedthattheMachnumberofthepost-shockowplaysanimportantroleindeterminingthemagnitudeandtimingofthepeakdragforceastheshockwavepropagatesovertheparticle. TheimpactoftheMachnumberontheintegratedeffectoftheinviscidunsteadyforcecannowbeinvestigated.AsinParmaretal.,werstdeneaneffectiveadded-masscoefcientas CM,e(M)=Z10Kc1t R;Mdc1t R.(5) Inthezero-Mach-numberlimit,werecovertheadded-masscoefcientforincompressibleowasCM,e(M!0)!0.5.Usingthenite-Mach-numberkernels,suchasthosepresentedinFig. 5-1 ,theeffectiveadded-masscoefcientcanbecalculatedforM>0.Inparticular,forasphericalparticleweobserveCM,e(M!0.5)1.0,i.e.,doubleitsvalueinthezero-Mach-numberlimit. Toobtainthejumpinparticlevelocity,weintegratetheequationofmotionintimefromto1.BecausethekernelKhascompactsupport,equation( 5 )canbeintegratedusingtheconvolutiontheoremtoobtainthefollowingexactrelation, v2 u2=1+CM,e(M) +CM,e(M).(5) 112

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ThisexpressionisvalidforniteMachnumbersandisindependentofthemannerinwhichtheuidvelocityvariesfromthequiescentstateaheadoftheshockwavetou2behindtheshockwave.TheonlylimitationisthatthechangeinMachnumberincurredduringthechangeinuidvelocityisassumedtobesmallenoughsothatmfcanbeassumedtobeconstantandthattheeffectiveadded-masscoefcientcanbedenedforthegivenMachnumber.Intheincompressiblelimit,Eq.( 5 )reducestoEq.( 5 ).Iftheparticledensityismuchlargerthanthatofthesurroundinggas,i.e.,if!1,weobtainfromEqs.( 5 )and( 5 ), v2 u2M=2 3(1+CM,e(M))v2 u2M=0,(5) andhencethejumpinparticlevelocityforapost-shockMachnumberof0.5willbeapproximately4=3timeslargerthanthejumpforincompressibleow. 5.2.4ApproximationofAmbientFlow TheforceparameterizationgivenbyEq.( 5 )assumestheparticletobesufcientlysmallcomparedtothelengthscalesoftheambientow.Thisallowsunambiguousdenitionsofc1,M,andmfasthespeedofsound,particleMachnumber,anddisplacedmassoftheambientuidasexperiencedbytheparticle.Ontheotherhand,iftheambientowvariationsoccuronascalecomparabletothesizeoftheparticle,unambiguousdenitionsofthesequantitiesarenotpossible.Thisiscertainlythecaseforashock-particleinteraction,sincetheshockwavecanbeinterpretedasadiscontinuitywithathicknessoftheorderofthemeanfreepath,see,e.g.,[ 96 ].Althoughtheowaheadofandbehindtheshockwavecanbeconsideredhomogeneous,theambientowisstronglyinhomogeneousastheshockwavepassesovertheparticle.Therefore,unambiguousvaluesofthequantitieslistedaboveaswellasofquantitiessuchastheMachnumber,density,andviscosityoftheambientowasseenbytheparticlearenoteasilydened.Oneneedstoresorttomodelingandidentifymodelsthatbestcapturetheunderlyingphysics. 113

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Figure5-2. Schematicofshockpositioninthesymmetryplaneofasphericalparticleanddenitionofvariables Thepresentmodelisbasedonasimpliedone-dimensionaldescriptionoftheinteractionoftheshockwavewiththeparticle.Morespecically,weassumetheshockwavetobeplanar,innitelythin,andtopropagateoverthevolumeoccupiedbytheparticleinitsundisturbedform,asillustratedschematicallyinFig. 5-2 .TheimplicationsofthesesimplicationsarediscussedinSection 5.4 .Atanytimet,theparticlecanthereforebethoughttobedividedintotwopartscorrespondingtothehomogeneousstatesaheadofandbehindtheshockwave.Inthefollowing,thesestatesaredenotedbythesubscripts1and2,respectively.Thecross-sectionalareaoftheparticlecutbytheshockwaveisdenotedbyA(t)andthevolumesoftheparticleaheadofandbehindtheshockaredenotedbyV1(t)andV2(t),respectively. 114

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Todescribetheambientowexperiencedbytheparticle,wedenethereferencetimet=0tocorrespondtotheinstantwhentheshockwavemakescontactwiththeparticle.Beforethisreferencetime,theparticleisentirelyimmersedinthequiescentambientuidupstreamoftheshockwave.Astheshockwavemovesovertheparticle,theparticleexperiencesbothregions,andhencetheeffectiveambientowseenbytheparticleisnecessarilyacombinationoftheupstreamanddownstreamconditions.Oncetheshockwavehasmovedpasttheparticletheambientowseenbytheparticleisgivenbythedownstreamstate.Thepositionoftheshockwavewithrespecttothefrontoftheparticleisgivenby s(x0s(t))=x0s(t) R=1+1 R(xs(t))]TJ /F3 11.955 Tf 11.96 0 Td[(xp(t))=1+1 Rust)]TJ /F11 11.955 Tf 11.95 16.28 Td[(Ztv()d,(5) wherexs(t)andxp(t)arethepositionsoftheshockwaveand(thecentroidof)theparticle,respectively,usistheambientshockvelocity(assumedtobeconstant),andv(t)isthevelocityoftheparticleobtainedfromNewton'ssecondlaw, v(t)=1 mpZtF()d,(5) wherempisthemassoftheparticleandF(t)istheunsteadydragforcedeterminedfromEq.( 5 ).(WeusethescalarforceF(t)inEq.( 5 )forthex-componentofthevectorforceF(t)inEq.( 5 )becauseweassumetheinteractiontobeone-dimensional.) Weproposeasimplemodelfortheambientowvelocityanduidpropertiesseenbytheparticletobeanaverageoftheupstreamanddownstreamowconditions,weightedbythesweptvolume.ElementarygeometryallowsthesweptvolumeV2(x0s(t)) 115

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tobeexpressedintermsofx0s(t)as V2(x0s(t)) V=8>>>>>><>>>>>>:0ifs60,1 4(3)]TJ /F8 11.955 Tf 11.95 0 Td[(s)2sif0>>>>><>>>>>>:0ifs60,V2(x0s) Vif0>>>>><>>>>>>:0ifs60,A(x0s) Vdx0s dtif0
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appropriatekernelintheintegrationofEq.( 5 ).ThekernelsaretabulatedasafunctionofMandforthevaluesgiveninthelegendofFig. 5-1 (B);valuesinbetweenthetabulateddatapointsareinterpolatedlinearly. TheuseofthevolumeratioinEq.( 5 )tomodelthevelocityanddensityseenbytheparticleissomewhatarbitrary.Othermodelsarepossible,ofcourse.Forexample,thesurface-arearatioS2=Smaybemoreappropriateforthecomputationofthequasi-steadydragaccordingtotheFaxencorrection,see[ 36 63 ].Wehavefoundtheresultsobtainedwiththesurface-arearatiotobesimilartothoseobtainedwiththevolumeratio. Finally,wenotethatthepressure-gradientforceinEq.( 5 )canbecomputedas ZfDu DtdV=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Z@p @xdV=(p2)]TJ /F3 11.955 Tf 11.95 0 Td[(p1)A(x0s(t))=2 +1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(M2s)]TJ /F4 11.955 Tf 11.95 0 Td[(1p1A(x0s(t)),(5) foranite-sizedparticlewhereistheratioofspecicheatsandMs=us=c1istheshock-waveMachnumber. 5.3Results ThemodeldescribedinSection 5.2 isassessedbycomparisonwithcomputationalandexperimentalresultsfortheinteractionofshockwaveswithspheres.First,weconsiderthecaseofastationarysphericalparticlesubjectedtoanormalshockwave.Thepredictedforceontheparticleiscomparedwithexperimentalandcomputationalresults.Second,weconsiderthecaseofasphericalparticlesetinmotionbytheimpactofashockwave.Thepredictedparticlepositionandvelocityarecomparedagainstexperimentalandnumericalresults.Allcasesarebasedonaratioofspecicheatsof=1.4andaspecicheatofthegasofCp=1004.64J=(kgK).TheresultsarepresentedintermsofthedragcoefcientCD(t)=F(t)=(1 22u22R2)andthenon-dimensionaltimes=ust=R.Withthisdenitionofthenon-dimensionaltime,theundisturbedshockwavehaspropagatedpastthesphereats=2. 117

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5.3.1StationarySphere Inthefollowing,wecomparethepresentmodeltotheexperimentsandsimulationsofSunetal.[ 102 ]andtotheexperimentsofSkewsetal.[ 97 ].Theformerallowustostudytheeffectofspherediameterontheunsteadyforce,thelattertheeffectofshock-waveMachnumber. 5.3.1.1ExperimentsofSunetal. IntheexperimentsofTannoetal.[ 104 ]andSunetal.[ 102 ],theunsteadyforceexertedonaspherewithdiameter8cmwasmeasuredforashockwavewithMs=1.22,p1=101325Pa,andT1=293.15K.SimulationsbasedontheaxisymmetriccompressibleNavier-Stokesequationsforsphereswithdiametersrangingfrom8mto8mmwerealsocarriedoutbySunetal.[ 102 ].TheReynoldsnumberbasedonparticlediameterandowconditionsbehindtheshockis49forthesmallestsphereandincreasesto4.9105forthelargestsphere.Correspondingly,theKnudsennumbervariesfrom9.410)]TJ /F10 7.97 Tf 6.58 0 Td[(3to9.410)]TJ /F10 7.97 Tf 6.59 0 Td[(7,justifyingthecontinuumassumptioninthesimulations.Sunetal.foundthatthecomputationsagreedverywellwiththeexperiments.Forthelargerspheres,theexperimentsandcomputationsshowedabriefperiodofnegativedragduetoshock-wavefocusingattherearofthesphere.Forsmallerspheres,negativedragwasnotobservedbecausetheviscouscontributionbecomesmorepronounced. TheunsteadydragcoefcientscomputedbySunetal.arepresentedinFigs. 5-3 (A)5-5 (A)forthediameters8m,80m,and8mm.Forthelargestdiameter,theexperimentalresultfor80mmisalsoshown.Therapidincreaseofthedragcoefcientfollowingtheimpactoftheshockwaveisimmediatelyapparent.Thedragcoefcientisobservedtoexhibitamaximumats1,followedbyadecreasetoaminimumats5)]TJ /F4 11.955 Tf 12.25 0 Td[(6,andaslowapproachtotheconstantdragcoefcientduetotheowbehindtheshockwave.Forthelargerspheres,thepeakdragcoefcientismorethananorderofmagnitudelargerthanthequasi-steady-statevalue.Forthesmallestsphere,thepeak 118

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dragcoefcientisaboutvetimeslargerthanthequasi-steady-statevalue.Thisisaclearindicationthatunsteadyeffectsareverystrongastheshockwavepropagatesoverthesphereandthatthepredictionofthedragforcebasedonquasi-steadydragformulaestronglyunderpredictstheactualtime-dependentforceonthesphere.ThemaximumdragcoefcientwasshownbySunetal.[ 102 ]tooccurslightlyafterthetimewhentheshock-wavereectionchangesfromaregulartoaMachreectionandbeforetheshockwavereachestheequator.TheminimumdragcoefcientwasexplainedbyTannoetal.[ 105 ]andSunetal.[ 102 ]asbeingduetothefocusingoftheshockwaveattherearofthesphere. AlsoplottedinFigs. 5-3 (A)5-5 (A)arethepredictionsoftheunsteadydragcoefcientbasedonEqs.( 5 ),( 5 ),and( 5 )withthenite-Mach-numberkernelsofParmaretal.[ 79 ](seeFig. 5-1 (B)),thezero-Mach-numberkernelofLonghorn[ 58 ](Eq.( 5 )),andtheincompressiblekernel(Eq.( 5 )).Inthegurelegends,theseresultsarelabeledbyModel,ModelM!0,andModelM=0,respectively.Theoverallagreementbetweenthecomputations,experimentsofSunetal.[ 102 ],andthepresentmodelintermsoftheorderofmagnitudeofthemaximumandminimumdragcoefcients,thetimesatwhichtheyoccur,andtheapproachtosteadystateisgood.Infact,theagreementisbetterthanexpectedgiventhatthepresentmodelhasbeenconstructedbasedondatagatheredfromhomogeneousows.Thedrag-coefcientminimapredictedbythepresentmodelareduetothenegativevaluesofthekernelshowninFig. 5-1 CloserinvestigationoftheresultspresentedinFigs. 5-3 (A)5-5 (A),revealssomediscrepancies.First,wenotethatthemodeloverpredictsthepeakdragcoefcientforthesphereswithdiameters80mmand80m,butunderpredictsthepeakdragcoefcientforthespherewithdiameter8m.Thesetrendsleadtoslightover-andunderpredictionofthedragcoefcientfor1.s.2.Inparticular,itisnoteworthythatwhiletheagreementbetweenthemodelandthecomputationsisquitegoodfor2. 119

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ATotaldrag BBreakdownofmodelterms Figure5-3. Comparisonofmodelwithcomputationsforspherewithdiameter8mofSunetal.[ 102 ] 120

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ATotaldrag BBreakdownofmodelterms Figure5-4. Comparisonofmodelwithcomputationsforspherewithdiameter80mofSunetal.[ 102 ] 121

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ATotaldrag BBreakdownofmodelterms Figure5-5. Comparisonofmodelwithexperimentforspherewithdiameter80mmandcomputationsfor8mmofSunetal.[ 102 ] 122

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s.5forthediameters80mmand80m,themodelunderpredictsthecomputationalresultsforthediameter8m.Thisunderpredictionisrelatedtothenarrowerpeakandfasterdecaypredictedbythemodelcomparedtothatobservedinthecomputations.Itappearsthattherearealsoothersourcesforthediscrepancybecausethecomputationdoesnotexhibitapronouncedminimuminthedragcoefcient. Theguresshowclearlythatwiththepresentsimplemodeloftheambient-owaccelerationseenbytheparticle,eventheresultsobtainedwiththeincompressiblekernelpredictreasonablywellthelargeincreaseinthedragforceastheshockwavepassesoverthesphere.However,theinviscidunsteadyforcepredictedbytheincompressiblekerneliszerofors>2becausetheambientowisthentakentobethesteadypost-shockstate,andhencethedragforceisgivenbythequasi-steadycontributiononly.Therapidriseofthedragforceanditsnon-monotonicdecayoverseveralacoustictimescalestothequasi-steadystateobservedinthemeasurementsandarecapturedwellbythecompressiblekernels.IntheexperimentsofSunetal.[ 102 ],theMachnumberbehindtheshockwaveisquitelowatM2=0.31andthereforethedifferencebetweentheresultspredictedbythezero-Mach-numberkernelofLonghorn[ 58 ]andthenite-Mach-numberkernelofParmaretal.[ 79 ]issmall.Thepeakdragcoefcientscomputedwiththesekernelsarebasicallyidenticalandlowerthanthosepredictedbytheincompressiblekernel.ThetimeatwhichthepeakdragcoefcientoccursispredictedtobelargerbythekernelsofLonghorn[ 58 ]andParmaretal.[ 79 ]comparedtothatpredictedbytheincompressiblekernel.Theprimarydifferencebetweenthedrag-coefcientvariationobtainedfromthekernelsofLonghorn[ 58 ]andParmaretal.[ 79 ]isthatthelattershowaslowerdecayofthedragcoefcientandareclosertothecomputationsandexperimentsduringthatphase. AbreakdownofthetermsinthemodelofParmaretal.isshowninFigs. 5-3 (B)5-5 (B).Inthelegendsofthesegures,thelabelHistorydenotesthatpartoftheinviscidunsteadyforcegivenbythersttermontheright-handsideofEq.( 5 ).Inthepresent 123

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model,thequasi-steadydragincreasesmonotonicallyandisfullyestablishedats=2.Becausethepressure-gradientdragiszerofors2,theexistenceofanegativedragforceforalimitedtimeisdeterminedbytherelativemagnitudeofthequasi-steadydragandtheminimumvalueoftheinviscidunsteadydragforce.Theviscousunsteadycontributionhasbeenignoredinthepresentmodel.Ascanbeexpected,itscontributionislikelytobehigherforthesmallerspheresandthusexplains,atleastpartially,thelargerdiscrepancyseenforthesmallestsphereofdiameter8mandalsotheveryslowdecayinthecomputeddragcoefcientfors>10. 5.3.1.2ExperimentsofSkewsetal. Skewsetal.[ 97 ]measuredtheunsteadydragonasphereofdiameter5cmforshockwaveswithMs=1.08and1.31andp1=83000PaandT1=293.15K.ThecorrespondingsphereReynoldsnumbersvariedfromabout105to6105andtheMachnumbersM2ofthepost-shockowvariedfrom0.13to0.40. ThebehaviorofthedragforcepredictedbythepresentmodeliscomparedtothemeasurementsinFig. 5-6 .Thelargeoscillationsinthemeasuredforcemaybeduetochaoticvortexsheddingbehindthesphere,whichisrelevantatthehigherReynoldsnumbersconsideredhere.Thetime-averagedmeanvalueofthequasi-steadyforceonthespherelongtimeafterthepassageoftheshockisreasonablywellcapturedatbothMachnumbers.Themodelcapturestherapidincreaseanddecreaseofthedragcoefcientreasonablywell.ItisinterestingthatthepeakisunderpredictedforMs=1.08andoverpredictedforMs=1.31.Thedifference,thus,cannotbesatisfactorilyexplainedbytheneglectedviscousunsteadycontribution.Thedifcultyofmeasuringtheunsteadyforce,associateduncertainties,andtheshortcomingsoftheforcemodelperhapscontributetothedifference. AswiththeresultsfortheexperimentsofSunetal.[ 102 ],theincompressiblekerneloverpredictsthepeakdragcoefcientcomparedtothekernelsofLonghorn[ 58 ]andParmaretal.[ 79 ].Thedifferencesbetweentheresultsobtainedwiththelatter 124

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AMs=1.08 BMs=1.31 Figure5-6. ComparisonofmodelwithexperimentsofSkewsetal.[ 97 ] 125

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twokernelsarerestrictedtothedecayofthedragcoefcientandapproachtothequasi-steadystate,withthepredictionsbythekernelofParmaretal.[ 79 ]beingslightlybetter. 5.3.2MovingSphere Nextweconsiderthemotionofaspherefollowingtheimpactofashockwave.Themajorityofexperimentaldataforspheressetinmotionbyshockwavesarenotsuitableforassessingthepresentmodel.Thereasonisthatmostexperimentsconsiderlargeparticle-to-uiddensityratiosthatresultinnegligiblemotionoftheparticleintheearlystagesoftheinteractionwiththeshockwave.HerewemakeuseoftheexperimentaldataofBritanetal.[ 14 ]tovalidatethepresentmodel.Britanetal.[ 14 ]measuredthemotionofasphereofdiameter38mmandadensityofp=89.4kg=m3inresponsetoashockwavewithMs=1.5inashocktubewithacross-sectionalareaof64cm2.Therelativelylargeblockageratioleadstobriefchokingoftheowandreducesthetesttimeduetoreectionoftheshockwavefromthesidewalls.Furtherdisturbancesareintroducedbythesupportthatprotrudesfromtheshock-tubeoortoholdthespherepriortothearrivaloftheshockwave.Despitethesedeciencies,theexperimentofBritanetal.[ 14 ]isinterestingbecauseitshowsclearlytheeffectoftheshockwaveonthemotionofthesphere. Britanetal.[ 14 ]alsocomputedthespheremotionusingasimpleone-dimensionalmodelthatincludedonlythequasi-steadydragforceandthedrag-coefcientcorrelationofGilbertetal.[ 37 ],CD,qs=0.48+28Re)]TJ /F10 7.97 Tf 6.59 0 Td[(0.85,forparticlemotioninthepost-shockow.TheresultsofBritanetal.[ 14 ]werepresentedintermsofanon-dimensionaltime0sthatwaszerowhentheundisturbedshockwavemovedpastthesphere,i.e.,0s=s)]TJ /F4 11.955 Tf 12.56 0 Td[(2.Theynotedthattheircomputationswithaninitialconditionofzeroinitialvelocityat0s=0gavepooragreementwiththeexperimentaldata,indicatingthattheuseofthequasi-steadydragforceonlyisinsufcient.Britanetal.observedthataninitialvelocityof11m/shadtobeappliedat0s=0toobtaingoodagreementwiththemeasured 126

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evolutionofthespherepositionwithtime.Thissuggeststhatthespheregained,nearlyinstantaneously,avelocityofabout11m/sduringtheinterval)]TJ /F4 11.955 Tf 9.3 0 Td[(20s0.Thepresentmodelcanbeusedtopredictthemotionofthesphereevenduringtheinitialinteractionwiththeshockwave. ThemotionofthespherepredictedbythepresentmodelispresentedinFig. 5-7 alongwiththeexperimentaldataandcomputationsofBritanetal.[ 14 ].Ifwetakethedensityratiop=2tobe32,thespheretrajectoryevaluatedwithEq.( 5 )compareswellwiththeexperimentaldata.Modelresultsobtainedusingdensityratiosofp=2=26and38arealsopresentedtoexaminesensitivitytovariationsinthegasdensity.(TheconditionsaheadoftheshockwerenotspeciedbyBritanetal.[ 14 ])Itisclearthattheinitialvelocityof11m/ssuggestedbyBritanetal.[ 14 ]basedontheirmeasurementsisconsistentwiththeinitialvelocitiespredictedwiththepresentinviscidunsteadyforcemodel.Therefore,thepresentmodelprovidesarationalwayofestimatingthevelocityimpartedtothesphereduetotheactionoftheinviscidunsteadyforce. Thebreakdownofthedragforceasafunctionofthenon-dimensionaltimesandthenon-dimensionalpositionxp=(2R)areshowninFig. 5-8 .Itcanbeseenveryclearlythattheinviscidunsteadyforceisnegligiblecomparedtothequasi-steadyforceforxp=(2R)&0.08,indicatingthatquasi-steadyconditionsareestablishedbeforethespherehasmovedaboutatenthofadiameter.Therapiditywithwhichquasi-steadyconditionsareattainedexplainswhyBritanetal.[ 14 ]wereabletocomputeaccurateresultsforthespherepositionandvelocitywiththequasi-steadydragforceoncetheinitialvelocityjumpwastakenintoaccount. Itisnotedthattheexperimentaldataexhibittime-dependentoscillationsintheparticlepositionandvelocity.Britanetal.[ 14 ]attributedtheoscillationstoshockwavesreectedfromtheshock-tubesidewalls.Vortexsheddingbehindthespheremayalso 127

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contributetotheoscillatorymotion.Neitheroftheseeffectsarecapturedbythepresentmodelandhencenooscillationsarepresentinthemodelpredictions. 5.4Discussion TheresultspresentedinSection 5.3 clearlydemonstratethatincludingtheinviscidunsteadyforceiscrucialtocapturingthepeakintheunsteadyforceonparticlesduetotheinteractionwithashockwave.Infact,asalreadymentioned,theresultsobtainedwiththepresentmodeldemonstratemuchbetteragreementthanmightbeexpectedgiventhatthemodeldoesnotexplicitlyincorporatethedetailsoftheshock-wavediffraction,reection,andfocusingprocesses. Thebetter-than-expectedagreementcanbeexplainedasfollows.Weconsideronlythecaseofastationarysphericalparticle,sincetheargumentissimilarforaparticlesetinmotionbyashockwave.Followingtheimpactoftheshockwaveonthefrontofthesphere,thepressurenearthefrontstagnationpointisthatbehindareectedshockwave.Assumingthatthespheredoesnotmove,thepressureattherearstagnationpointremainsunchangeduntiltheshockwavereachestherearend.Thusattheveryearlystagesoftheinteractionprocess,theforceonthesphereisdeterminedonlybyanincreaseinpressurenearthefrontofthesphere.TheinviscidunsteadyforcemodelofParmaretal.[ 79 ],however,isbasedonthecompressiblepotentialowresultingfromanunsteadyhomogeneousambientowandhenceapproximatestheactualinteractioninatleasttwoways.First,sincethemodeldoesnotincorporatethedetailsoftheshock-interactionprocess,itunderpredictsthepressureloadingonthefrontofthesphere.Second,themodelisbasedonasimultaneousdecreaseinpressureattherearofthesphereasgivenbythepotentialowduetothehomogeneousambientowacceleration.Theunderpredictionofthepressureincreaseatthefrontcombinedwiththeprematurereductionofthepressureattherearresultinareasonablyaccuratenetforceprediction. 128

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ASpheretrajectory BSpherevelocity Figure5-7. ComparisonofmodelwithexperimentsandcomputationsofBritanetal.[ 14 ] 129

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AAsafunctionofnon-dimensionaltime BAsafunctionofnon-dimensionaldistancemoved Figure5-8. BreakdownofdragforceforexperimentalconditionsconsideredbyBritanetal.[ 14 ] 130

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Atpresent,itisnotclearwhetherthesamelevelofagreementwillholdforhighershock-waveMachnumbers.Detailednumericalsimulationsindicatethatforshock-waveMachnumbersthatleadtosupercriticalbutsubsonicpost-shockMachnumbers,vortexsheddingoccursthatmakesaclearidenticationofinviscidunsteadyforcesdifcult.Furthermore,simulationsnotreportedonhereshowthattheinviscidunsteadyforceisnegligiblecomparedtothechangeinthequasi-steadydragforceatsupersonicpost-shockMachnumbers. Thelocalminimumofthedragcoefcientbeforetheslowapproachtothequasi-steady-statevalue,wasattributedbyTannoetal.[ 104 ]toshock-wavefocusingattherearofthesphere.Thelocalforceminimumoccursaftertheundisturbedshockwavehasleftthesphere,i.e.,fors>2.Therefore,atthetimeofminimumforce,thecontributionofthepressure-gradienttermisidenticallyzeroandtheminimumisonlydeterminedbytherelativemagnitudeoftheinviscidunsteadyandquasi-steadydragterms.Itseemsreasonabletoassumethatthequasi-steadydragmodelisaccurate,giventhatthelong-timebehaviorofthemeasureddragforceisaccuratelycapturedbythepresentmodel.Therefore,itcanbeconcludedthatthepresentmodelfortheinviscidunsteadydragforceisreasonablyeffectiveinpredictingthelocalminimumofthedragcoefcient,althoughtheshock-focusingprocessisnottakenintoaccount.Thisinturnsuggeststhatthekernelandthepresentmodelfortheambient-owaccelerationseenbytheparticleareeffective. Thepresentmodelcanalsobeappliedtopredicttheinviscidunsteadyforceexertedbyshockwavesonotherparticlegeometriesprovidedtheappropriateinviscidforcekernelsandamodelforambientowaccelerationseenbytheparticlearedeveloped.Forthecaseofacylindersubjectedtoashockwavetheresultsobtainedwiththepresentmodelarecomparedwithinviscidcomputationsusingthenumericalmethoddescribedin[ 41 43 ]inFig. 5-9 forMs=1.22.Aswiththepreviouslypresentedresultsforthesphere,theoverallagreementintermsofthepeakvalueofthedrag 131

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coefcient,thetimeatwhichthepeakvalueisattained,andthelong-timebehaviorisgood. Weclosethediscussionbyoutlininghowthemodelpresentedinthisarticlecouldbeappliedtothenumericalsimulationofshockwavesinteractingwithmillionstobillionsofparticles.Insuchsimulations,shockwavesareusuallycapturedwithshock-capturingschemesandhencesmearedover2-5cells.Thusparticlesinteractwithregionsinwhichthesolutionvariesrapidly,butoverlengthscalesmuchlargerthantheparticlediameter.WithintheframeworkoftheEulerian-Lagrangianpoint-particleapproach,theforcedecompositiongivenbyEq.( 5 )canbeusedwiththeambientuidpropertiesevaluatedattheparticlecenter,wherethequasi-steadydragisevaluatedfromEq.( 5 ),andtheinviscidunsteadyforceisevaluatedfromEq.( 5 )usingthekernelpresentedinFig. 5-1 (B).Simplieddescriptionsoftheunsteadyforce,suchasthosebasedonthekernelbyLonghorn[ 58 ],seeEq.( 5 ),ortheincompressiblekernelarepossible.TheimpactofthesimplieddescriptionsontheunsteadydragcoefcientwasdiscussedinSections3.1and2.3,respectively.Iftheparticlediameteriscomparabletoorlargerthanthewidthoftheshockwave,theapproachoutlinedinSection 5.2.4 canbeused. 132

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AComparisonofmodelandcomputation BBreakdownofmodelterms Figure5-9. ResultsforshockinteractionwithcylinderatMs=1.22 133

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CHAPTER6ANIMPROVEDDRAGCORRELATIONFORSPHERESANDAPPLICATIONTOSHOCK-TUBEEXPERIMENTS Anewcorrelationispresentedforthedragforceonsphericalparticlesincompressiblecontinuumows.Thenewcorrelationrepresentsexperimentaldatamorefaithfullythanpriorcorrelations.Withthiscorrelation,experimentaldataobtainedinshocktubesbyJourdanetal.(Proc.R.Soc.A,463:3323-3345,2007)fortheparticlevelocitybehindshockwavescanbereproducedaccurately.Thissuggeststhatthehigherdragcoefcientsobservedinseveralexperimentsinvolvingshock-particleinteractioncouldsimplybeaconsequenceofcompressibility. 6.1Introduction Thedragcoefcientonasphericalparticlesubjectedtoincompressiblesteadyuniformowhasbeenstudiedextensively.EmpiricalcorrelationsexistthatcanveryaccuratelycapturethedragcoefcientasafunctionofReynoldsnumber,see,e.g.,CliftandGauvin[ 21 ].ThesecorrelationsareaccurateprovidedthattheparticleissmallerthanthesmallestlengthscaleoftheambientowandthattherelativeMachnumber,basedonthevelocityoftheparticlerelativetothatoftheambientow,isquitesmall.AstherelativeMachnumberincreases,compressibilityeffectsbecomeimportantandthedragcoefcientdependsonboththeReynoldsandMachnumbers.Severalauthorshavepresenteddrag-coefcientcorrelationsforniteMachnumbers,see,e.g.,Henderson[ 44 ]andLoth[ 59 ].Inthisnote,werstassesstheaccuracyofthecorrelationsofHendersonandLothusingthedatacollectedbyBaileyandStarr[ 5 ]anddevelopanimproveddragcorrelationthatisquiteaccurateoverarangeofReynoldsandMachnumbers. Wethenvalidatetheimprovedcorrelationforshock-particleinteractionusingtherecentshock-tubeexperimentsofJourdanetal.[ 50 ].Shock-tubeexperimentsareoftenusedtodeterminethedragcoefcientsofasinglesphericalparticleoracloudofsphericalparticles,see,e.g.,Ingebo[ 47 ],SelbergandNicholls[ 95 ],Rudinger[ 90 ], 134

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Sommerfeld[ 99 ],IgraandTakayama[ 46 ],Rodriguezetal.[ 88 ],Devalsetal.[ 27 ],andJourdanetal.[ 50 ].Thedragcoefcientsforsingleparticlesobtainedfromsuchexperimentsareoftenobservedtobeconsiderablyhigherthanthosegivenbythestandard-dragcurvethatdescribesthedragcoefcientinthelimitofquasi-steadyincompressibleconditions.Discrepancieshavebeenascribedtoseveralsources:accelerationeffects[ 46 47 ],interferencebetweenparticles[ 90 ],trajectoryperturbations[ 90 ],boundarylayersonshock-tubewalls[ 88 ],turbulence[ 112 ],surfaceroughness[ 95 ],unsteadyshearwaves[ 50 ],andsupportmechanismsforparticlespriortothearrivaloftheshockwave[ 50 ].HereweusetherecentdataofJourdanetal.[ 50 ]asanexampleanddemonstratethattheexperimentallyobservedincreaseinthedragcoefcientisprimarilyduetocompressibilityeffects. 6.2ImprovedDrag-CoefcientCorrelation Severalmodelsforthequasi-steadydragcoefcientthatincorporateadependenceontherelativeMachnumberareavailableintheliterature.ThemodelofHenderson[ 44 ]includesthreecorrelationsdependingonwhethertherelativeMachnumberisbelowunityorabove1.75,withlinearinterpolationin-between.HendersondemonstratedthesuperioraccuracyofhiscorrelationcomparedtothoseofCarlsonandHoglund[ 16 ]andCrowe[ 24 ].Recently,Loth[ 59 ]presentedamodelthatincludestwocorrelationsdependingonwhethertherelativeReynoldsnumberisaboveorbelow45,whichistakentobethelimitbetweenrarefaction-andcompressibility-dominatedregimes. ThecorrelationsofHendersonandLotharecomparedwiththedataofBaileyandStarr[ 5 ]inFigs. 6-1A and 6-1B .ForagivenMachnumber,Henderson'scorrelationdecreasesmonotonicallywithincreasingReandthusfailstocapturetheriseinthedragcoefcientasthecriticalReynoldsnumberisapproached.Loth'scorrelationshowsaconsistentearlyriseastheReynoldsnumberincreasesforagivenMachnumber.1 1ItshouldbenotedthatEq.(25b)inLoth'sarticleismissingtheterm2p =3s. 135

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Furthermore,thereisanoverlapinLoth'scorrelationfor0.89M1.0thatdoesnotexistintheexperimentaldata.ForHenderson'scorrelation,thelargestdeviationfromthedataofBaileyandStarrisabout16%forReynoldsnumbersbelow104.ForLoth'scorrelation,thelargesterrorisabout55%,concentratedaroundaMachnumberofabout0.9.Discrepanciesofthismagnitudecallforthedevelopmentofanimproveddrag-coefcientcorrelation. Hereweproposeanimprovedcorrelationbasedonthefollowingassumptions:(i)Welimitourattentiontocontinuumows,i.e.,weassumethattheKnudsennumber Kn=M Rer 2<0.01,(6) whereMistheMachnumberbasedonrelativevelocity,ReistheReynoldsnumberbasedontherelativevelocityandparticlediameter,andistheratioofspecicheats.(ii)Theparticletemperatureisassumedconstantandequaltothesurroundinggastemperature.(iii)ThedragcorrelationshouldapproachthestandarddragrelationgivenbyCliftandGauvin[ 21 ]inthelimitofzeroMachnumber, CD,std(Re)=24 Re)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1+0.15Re0.687+0.421+42500 Re1.16)]TJ /F10 7.97 Tf 6.59 0 Td[(1.(6) (iv)AttentionisrestrictedtosubcriticalReynoldsnumbers,i.e.,Re/2105,abovewhichtheattachedboundarylayerbecomesturbulent.(v)Wefocusontherange06M61.75andmakeuseoftheextensivedatacompiledbyBaileyandStarr[ 5 ]. Theresultingimproveddrag-coefcientcorrelationconsistsofthreeseparatecorrelationsforsubcritical(06M6Mcr0.6),supersonic(1>>>>><>>>>>>:CD,std(Re)+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(CD,Mcr(Re))]TJ /F3 11.955 Tf 11.95 0 Td[(CD,std(Re)M McrifM6Mcr,CD,sub(Re,M)ifMcr
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ACorrelationofHenderson[ 44 ] BCorrelationofLoth[ 59 ] Figure6-1. ComparisonofdragcorrelationswithdataofBaileyandStarr[ 5 ]assumingthat=1.4andthattheparticletemperatureisequaltothesurroundinggastemperature. 137

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Thisseparationismotivatedbythefollowingobservations.(i)ForsubcriticalMachnumbers,theowaroundasphericalparticleisshock-freeandthusthedragcoefcientisonlyweaklyaffectedbycompressibilityeffects.(ii)ForsupercriticalbutsubsonicMachnumbers,anannularshockwaveoflimitedradialextentexistsonthesphereandthedragcoefcientbecomesmorestronglydependentontheMachnumber.(iii)ForsupersonicMachnumbers,abowshockexiststhatleadstoalargeincreaseindragcoefcient.(Itshouldbenotedthatthebowshockdoesnotappearatpreciselysonicconditions,buttheabovesimpleseparationintoregimesissufcientforourmodelingpurposes.)(iv)TheupperlimitofM=1.75isusedbecausethatisthemaximumMachnumberforwhichBaileyandStarrpresenteddata.Thislimitissufcientforourpurposesbecausewearemainlyinterestedinunsteadyshockwavesacceleratinginitiallystationaryparticles.Forsuchinteractions,thelargestpossibleMachnumberisM1.89for=1.4. Inthesubcriticalregime,thedragcoefcientisexpressedasalinearinterpolationbetweenthedragcoefcientsatM=0andM=Mcr.(ItshouldbenotedthatwedonotmakeuseofthedataofBaileyandStarrforM<0.6becauseitexhibitsdragcoefcientsthatliebelowthestandarddragcurve.)BasedonthefunctionalformofEq.( 6 ),CD,Mcr(Re)isexpressedas CD,Mcr(Re)=24 Re)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1+0.15Re0.684+0.5131+483 Re0.669)]TJ /F10 7.97 Tf 6.59 0 Td[(1.(6) NotethesimilarityofthecoefcientscomparedtoEq.( 6 )duetotheweakinuenceofthecompressibilityinthisregime,asstatedabove. Inthesupersonicregime,thedragcoefcientisexpressedasanonlinearinterpolationbetweenthedragcoefcientsatM=1andM=1.75, CD,sup(Re,M)=CD,M=1(Re)+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(CD,M=1.75(Re))]TJ /F3 11.955 Tf 11.96 0 Td[(CD,M=1(Re)sup(M,Re),(6) 138

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whereCD,M=1(Re)=24 Re)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1+0.118Re0.813+0.691+3550 Re0.793)]TJ /F10 7.97 Tf 6.59 0 Td[(1, (6)CD,M=1.75(Re)=24 Re)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(1+0.107Re0.867+0.6461+861 Re0.634)]TJ /F10 7.97 Tf 6.59 0 Td[(1, (6) and sup(M,Re)=3Xi=1fi,sup(M))]TJ /F3 11.955 Tf 11.96 0 Td[(fi,sup(1) fi,sup(1.75))]TJ /F3 11.955 Tf 11.96 0 Td[(fi,sup(1)3Yj6=ij=1logRe)]TJ /F3 11.955 Tf 11.96 0 Td[(Cj,sup Ci,sup)]TJ /F3 11.955 Tf 11.96 0 Td[(Cj,sup,(6) withf1,sup(M)=0.126+1.15M)]TJ /F4 11.955 Tf 11.95 0 Td[(0.306M2)]TJ /F4 11.955 Tf 11.95 0 Td[(0.007M3)]TJ /F4 11.955 Tf 11.95 0 Td[(0.061exp1)]TJ /F3 11.955 Tf 11.96 0 Td[(M 0.011, (6)f2,sup(M)=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.901+2.93M)]TJ /F4 11.955 Tf 11.96 0 Td[(1.573M2+0.286M3)]TJ /F4 11.955 Tf 11.95 0 Td[(0.042exp1)]TJ /F3 11.955 Tf 11.96 0 Td[(M 0.01, (6)f3,sup(M)=0.13+1.42M)]TJ /F4 11.955 Tf 11.96 0 Td[(0.818M2+0.161M3)]TJ /F4 11.955 Tf 11.96 0 Td[(0.043exp1)]TJ /F3 11.955 Tf 11.95 0 Td[(M 0.012, (6) and C1,sup=6.48,C2,sup=8.93,C3,sup=12.21.(6) Intheintermediateregime(supercriticalandsubsonic),thedragcoefcientisexpressedasanon-linearinterpolationbetweenthedragcoefcientsatMcrandM=1, CD,sub(Re,M)=CD,Mcr(Re)+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(CD,M=1(Re))]TJ /F3 11.955 Tf 11.96 0 Td[(CD,Mcr(Re)sub(M,Re),(6) where sub(M,Re)=3Xi=1fi,sub(M))]TJ /F3 11.955 Tf 11.95 0 Td[(fi,sub(Mcr) fi,sub(1))]TJ /F3 11.955 Tf 11.95 0 Td[(fi,sub(Mcr)3Yj6=ij=1logRe)]TJ /F3 11.955 Tf 11.95 0 Td[(Cj,sub Ci,sub)]TJ /F3 11.955 Tf 11.95 0 Td[(Cj,sub,(6) withf1,sub(M)=)]TJ /F4 11.955 Tf 9.29 0 Td[(0.087+2.92M)]TJ /F4 11.955 Tf 11.95 0 Td[(4.75M2+2.83M3, (6)f2,sub(M)=)]TJ /F4 11.955 Tf 9.29 0 Td[(0.12+2.66M)]TJ /F4 11.955 Tf 11.95 0 Td[(4.36M2+2.53M3, (6)f3,sub(M)=1.84)]TJ /F4 11.955 Tf 11.95 0 Td[(5.13M+6.05M2)]TJ /F4 11.955 Tf 11.95 0 Td[(1.91M3, (6) 139

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Figure6-2. ComparisonofnewdragcorrelationwithdataofBaileyandStarrBaileyandStarr[ 5 ]. and C1,sub=6.48,C2,sub=9.28,C3,sub=12.21.(6) TherangeofapplicabilityoftheimprovedcorrelationislimitedtoRe62105,M61.75,andKn<0.01.TheimprovedcorrelationiscomparedtothedataofBaileyandStarrinFig. 6-2 .Withintherangeofapplicability,thelargestdeviationbetweentheimprovedcorrelationandthedataofBaileyandStarrisonly2.5%forM>0.6.ThisdeviationissubstantiallysmallerthanthoseforthecorrelationsofHendersonandLoth.InFig. 6-3 ,acomparisonwiththeearlierdataofGoinandLawrence[ 38 ]andMayandWitt[ 64 ]isshown.ItcanbeseenthatevenatlowerReynoldsnumbers,thenewdrag-coefcientcorrelationagreesquitewellwithexperimentaldata. 140

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Figure6-3. Comparisonofnewdrag-coefcientcorrelationwithdataofGoinandLawrence[ 38 ]andMayandWitt[ 64 ]. 6.3Validation Wevalidatethenewdragcorrelationusingtheshock-particleinteractionexperimentsofJourdanetal.[ 50 ].Comparedtoothershock-tubeexperiments,themeasurementsofJourdanetal.[ 50 ]aremoresuitablebecause(i)boundary-layereffectsareeliminatedbyhangingthesphericalparticlesfromaspider-webthread,(ii)alargepartoftheparticletrajectoryisrecordedusingmultipleshadowgraphsinasinglerun,and(iii)interferencebetweenparticlesisminimizedbytestingnomorethanthreeparticlessimultaneously.Table 6-1 liststhecasesfromtheexperimentsofJourdanetal.[ 50 ]thatweusetovalidateourmodel.(SeeTable2inRef.[ 50 ]foracompletelistofcasesandthereportbyJourdanandHouas[ 49 ]formoredetails.)Thecasesincludeexperimental 141

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Table6-1. SummaryofselectedtestcasestakenfromJourdanetal.[ 50 ]usedtocomparewithmodel.Thecolumnlabeled`gases'liststhegasesinthedriveranddrivensectionsoftheshocktube. Mdpp CaseGases)]TJ /F4 11.955 Tf 47.73 0 Td[(mmkg=m3 148air/air0.47)]TJ /F4 11.955 Tf 11.96 0 Td[(0.661.921130166aair/air0.47)]TJ /F4 11.955 Tf 11.96 0 Td[(0.611.961204181ahelium/air1.28)]TJ /F4 11.955 Tf 11.96 0 Td[(1.600.621096184ahelium/air0.70)]TJ /F4 11.955 Tf 11.96 0 Td[(1.531.6025 conditionsleadingtosubsonicandsupersonicparticleMachnumbersbehindtheshockwave. ThetimehistoryoftheparticlevelocitymeasuredbyJourdanetal.iscomparedagainstthevelocitycomputedfromtheparticleequationofmotion mpdup dt=F(t),(6) wherempisthemassoftheparticle,upistheparticlevelocity,andF(t)isthedragforceontheparticle.Weassumethatthedragforceontheparticleisgivenbythequasi-steadydragthatdependsontheinstantaneousrelativevelocitybetweentheparticleandambientuidas F(t)= 82(ug2)]TJ /F3 11.955 Tf 11.95 0 Td[(up(t))jug2)]TJ /F3 11.955 Tf 11.95 0 Td[(up(t)j(dp)2CD(Re(t),M(t))(6) where2andug2arethedensityandvelocityofthegasbehindtheshock(assumedtobeconstants),dpisthediameteroftheparticle,CD(Re(t),M(t))isgivenbyEq.( 6 ),andtherelativeReynoldsandMachnumbersare Re(t)=2jug2)]TJ /F3 11.955 Tf 11.96 0 Td[(up(t)jdp 2andM(t)=jug2)]TJ /F3 11.955 Tf 11.96 0 Td[(up(t)j a2,(6) where2anda2arethedynamicviscosityandspeedofsoundofthegasbehindtheshock.Otherforces,suchasthosearisingfromunsteadyeffectsandgravity,are 142

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negligibleforthetimescalesofinterestinthisstudy.SeeParmaretal.[ 79 80 ]formoreinformationonwhytheinviscidunsteadyforcecanbeneglected. TheresultsarepresentedinFig. 6-4 ,wheretheparticlevelocityupnormalizedbythepost-shockgasvelocityu2isplottedagainstthenondimensionaltimes=ust=dp.(Notethats=1correspondstothetimerequiredfortheshockwavetopropagateovertheparticle.)AscanbeseenfromFigs. 6-4A and 6-4B ,theparticlevelocityispredictedquiteaccuratelywiththenewdrag-coefcientcorrelationforcases148and166a.BecausetherelativeMachnumbersaresubsonicandmostlysubcritical,thedifferencebetweentheresultsobtainedwiththestandardandthenewdrag-coefcientcorrelationsarerelativelysmall.Thestandarddrag-coefcientcorrelationleadstobetterresultsthanHenderson'scorrelation.TheexplanationforthisresultisthatHenderson'scorrelationunder-predictsthestandard-dragcoefcientcurvefortherangeofMachandReynoldsnumbersencounteredincases148and166a,seeFig. 6-1A .Figures 6-4C and 6-4D showresultsforcases181aand184a,wheretherelativeMachnumberafterthepassageoftheshockwaveovertheparticleissupercritical.TheresultsobtainedwithboththenewcorrelationandHenderson'scorrelationagreewellwiththeexperimentaldata. Figures 6-4C and 6-4D showresultsforthecaseswheretherelativeMachnumberafterthepassageoftheshockwaveovertheparticleissupercritical.Astheresult,thecompressibilityeffectonthequasi-steadydragissignicant.Thesecasesshowthatthedifferencesbetweentheresultsobtainedwiththestandardandthenewdrag-coefcientcorrelationsaresubstantial.Inbothcases,thenewdrag-coefcientcorrelationaccuratelycapturestheevolutionofthenormalizedparticlevelocity. TheresultsforthefourcasessuggestthatthediscrepanciesbetweentheexperimentsofJourdanetal.[ 50 ]andthestandarddragcorrelationcanbetracedtocompressibilityeffects. 143

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ACase148 BCase166a CCase181a DCase184a Figure6-4. ComparisonofmodelwithexperimentaldataofJourdanetal.[ 50 ]Notetheabscissascaling. 144

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6.4Discussion Asmentionedintheintroduction,ithasbeenarguedthatunsteadinesscouldcontributetotheincreaseddragontheparticleobservedinshock-particleinteractionexperiments[ 46 47 ].Twosourcesofunsteadinesscanbeidentiedinshock-particleinteractions.First,astheshockwavemovesovertheparticle,theambientowaroundtheparticlerapidlychangesfromthequiescentconditionaheadoftheshockwavetotheusuallyuniformpost-shockow.RecentexperimentsbySunetal.[ 102 ]andSkewsetal.[ 97 ]andthetheoreticalmodelofParmaretal.[ 80 ]haveshownthattheunsteadycontributiontotheforceontheparticleimmediatelyfollowingtheshock-particleinteractioncanbeanorderofmagnitudelargerthanthequasi-steadyforce.Furthermore,theunsteadyforcedecaysrapidlyonanacoustictimescaleanditseffectcanbeignoredfors'O(5)]TJ /F4 11.955 Tf 12.53 0 Td[(10).ReferringtoFig. 6-4 ,itcanbeseenthattheunsteadycontributiontotheforceisinsignicantonthethetimescaleonwhichtheparticleattainsasignicantfractionofthepost-shockgasvelocity.Thevelocitygainedbytheparticleasafractionofthepast-shockgasvelocityduetotheunsteadyinviscidforcecanbeshowntobeproportionaltothegas-to-particledensityratio[ 80 ].Thesecondsourceofunsteadinessarisesfromtheaccelerationoftheparticlethattakesplaceontheparticletimescale(dp)2=18g.Theratiooftheunsteadyforceduetoparticleaccelerationtothequasi-steadyforce,however,againscalesasthegas-to-particledensityratio.Thusitisclearthatunsteadyeffectsareinsignicantforthevelocityevolutionoftheparticlefollowingitsinteractionwiththeshockwaveiftheparticledensityissubstantiallylargerthanthegasdensity.ThisisindeedthecaseintheexperimentsofJourdanetal.[ 50 ]andinmostshock-tubestudies. 6.5Conclusions Animprovedcorrelationforthedragcoefcientofsphericalparticleswaspresented.Thenewcorrelationrepresentstheavailableexperimentaldatamoreaccuratelythanpreviouscorrelations.Themodeliscapableofreproducingparticlevelocitiesfollowing 145

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theimpactofshockwavesmeasuredintheexperimentsofJourdanetal.[ 50 ]Thissuggeststhattheincreaseddragcoefcientsobservedintheshock-tubeexperimentsofJourdanetal.aresimplyduetocompressibilityeffectsandnotduetounsteadiness.Itishypothesizedthatthediscrepanciesseeninatleastsomeoftheearliershock-tubeexperimentsarealsoduetocompressibilityeffects.Additionalcarefullydesignedandexecutedexperimentsarerequiredtoverifythishypothesis. 146

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CHAPTER7UNSTEADYFORCESONAPARTICLEINVISCOUSCOMPRESSIBLEFLOWSATFINITEMACHANDREYNOLDSNUMBERS Effectsofnonlinearityareexploredforunsteadyforcesonaparticleinviscouscompressibleows.TheoreticalinvestigationsareusedtostudylinearizedowsinChapters 2 and 3 .SuchtheoriesarenotpossibleforniteMandReows.Chapter 4 presentedinviscidunsteadyforcesinnitebutsubcriticalMach-numberowsusingcarefullydesignednumericalsimulations.ThischapterpresentsresultsfromhighdelitynumericalsimulationsusedtostudytheinviscidunsteadyandviscousunsteadyforcesonasphereinacompressibleowsatniteMandRe. 7.1Introduction TheunsteadyforceonaparticleinacceleratedmotionwasrstanalyzedbyStokes[ 101 ],whopresentedanexpressionforthefrequency-dependentforceonanoscillatingsphericalparticle.LaterBasset[ 8 ],Boussinesq[ 10 ],andOseen[ 77 ]independentlyexaminedthetime-dependentforceonasphereduetorectilinearmotioninaquiescentviscousincompressibleuid.TheybasedtheiranalysesonthelinearizedunsteadyincompressibleNavier-Stokesequationsvalidforcreepingmotion,i.e.,inthelimitofvanishingReynoldsnumber.Theresultingequationofmotionforasphericalparticle,theso-calledBBOequation,canbewrittenas mpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2mfdv dt)]TJ /F4 11.955 Tf 11.96 0 Td[(6a2p ZtKBH(t)]TJ /F8 11.955 Tf 11.95 0 Td[()dv dd,(7) where KBH(t)=1 p t,(7) andmpistheparticlemass,v(t)isparticlevelocity,aistheparticleradius,isthedynamicviscosity,mfisthemassofuiddisplacedbytheparticle,istheuiddensity,and==isthekinematicviscosity.Thethreetermsontheright-handsidearethequasi-steady(Stokes)drag,inviscidunsteady(added-mass),andviscousunsteady(Bassethistory)forces,respectively. 147

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ExtensionofBBOequationtocompressiblecreepingowsisrecentlyproposedbyParmaretal.[ 81 ].Thegeneralizedparticleequationofmotionincompressibleowsbecomes mpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av)]TJ /F3 11.955 Tf 11.95 0 Td[(mfZtKLH(t)]TJ /F8 11.955 Tf 11.96 0 Td[()c0 adv dtdc0 a)]TJ /F4 11.955 Tf 11.96 0 Td[(6a2p ZtKPHB(t)]TJ /F8 11.955 Tf 11.96 0 Td[()dv dd,(7) wherec0isthespeedofsound,KLH(t)istheinviscidunsteadykernelrstfoundbyLonghorn[ 58 ],andKPHB(t)istheviscousunsteadykernelincompressibleowproposedbyParmaretal.[ 81 ].KLH(t)andKPHB(t)aregivenby KLH()=e)]TJ /F16 7.97 Tf 6.59 0 Td[(cos(7) and KPHB(t)=C(c0t=a) p t,(7) where=c0t=aandC(c0t=a)isthecompressiblecorrection-functiontoBasset-historyforcewhoseexpressioncanbefoundinParmaretal.[ 81 ]. Non-linearityeffectsforunsteadyforceswerestudiedbyMeiandAdrian[ 68 ],LovalentiandBrady[ 60 ],andKimetal.[ 52 ]inincompressibleows.TheBBOequationhasbeenextendedtoniteReynoldsnumbersbyMeiandAdrian[ 68 ],Kimetal.[ 52 ],andMagnaudetandEames[ 62 ].Firstimportantndingwasthatadded-massforceisindependentofRe,seeMeietal.[ 66 ],Riveroetal.[ 87 ],ChangandMaxey[ 19 ],andWakabaandBalachandar[ 114 ].Second,itwasfoundthattheBassethistoryforceisnotuniformlyvalidforallReynoldsnumbers.Theviscous-unsteadyforcewasfoundtobedependentonRe.Evenforcreepingows(Re!0),thelongtimedecayrateisaffectedbynon-linearity.MeiandAdrian[ 68 ]proposedamodiedexpressionforviscous-unsteadyforcewhichbehaveslikeBassethistoryforceforshorttimes(1=t1=2)andforlongtimesitdecaysatafasterrate(1=t2).ForniteReowalso,shorttimebehaviorcanbepredictedbyBassethistoryforce.Thetimescaleonwhichnonlinear 148

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effectsbecomesignicantcanbeestimatedasfollows.InderivingthelinearizedformofthecompressibleNavier-Stokesequations,theassumptionthattheinertialtermsarenegligiblecomparedtotheviscoustermsimpliesthatthelengthscaleL=V,whereiskinematicviscosityandVisscaleofowvelocity.Ifwetakethelengthscaletogrowbydiffusionasp t,wheretistime,theassumptionoflinearizedNavier-Stokesequationscanbejustiedonlyfort=V2.Expressedintermsoftheconvectivetimescale,thisrestrictionbecomestc=tV=L1=Re.Particleequationofmotionincorporatingmodiedhistory-kernelproposedbyMeiandAdrian[ 68 ]canbewrittenas, mpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av(Re))]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 2mfdv dt)]TJ /F4 11.955 Tf 11.96 0 Td[(6a2p ZtKMA(Re;t)]TJ /F8 11.955 Tf 11.95 0 Td[()dv dd,(7) where KMA(Re;t)=K)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2BH(t)+K)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NL(Re;t))]TJ /F10 7.97 Tf 6.59 0 Td[(2,(7) where KNL(Re;t)= r 323f6H v6!1 t2,(7) wherefH=0.75+0.105Re.ForniteReynolds-numberows(Re)=1+0.15Re0.687,seeSchillerandNaumann[ 94 ].Theshortandlongtimelimitsareasfollows,limta=vKMA(t)!KBH(t), (7)limta=vKMA(t)!KNL(t). (7) InthelimitofshorttimethemodiedhistorykernelofMei&AdrianreducestoBassethistory.Equation 7 canbeconsideredanextensionofBBOequationtonon-linearniteReynolds-numberows. Aforcekernelcanbeinterpretedasforceduetodelta-functionacceleration.Figure 7-1 showsnormalizedformsofabovementionedinviscid-andviscous-unsteadyforcesvsnon-dimensionaltimeforM=0.01andRe=1.Theforcesarenormalizedbymfc0=a. 149

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Figure7-1. TimeevolutionofnormalizedunsteadyforceforM=0.01,Re=1.Bassethistoryforce(Eq.( 7 )),modiedhistoryforceduetoMeiandAdrian[ 68 ](Eq.( 7 )),inviscidunsteadyforceincompressibleowsduetoLonghorn[ 58 ](Eq.( 7 )),andinviscidandviscousunsteadyforceincompressibleowsduetoParmaretal.[ 81 ](Eq.( 7 ))areplotted. Correspondingextensionofparticleequationofmotiontonon-linearcompressibleowshasnotbeenstudiedsofar.HerewecarryoutcarefullydesignednumericalsimulationstostudyunsteadyforcesonaparticleinniteMach-andReynolds-numberows.Asdiscussedearlier,inacompressibleow,bothinviscid-andviscous-unsteadyforceshaveanintegralrepresentationdependingonhistoryofrelativeaccelerationoftheuidwithrespecttotheparticle.Astraight-forwardseparationofinviscid-andviscous-unsteadyforcesisnotpossible.NumericalmethodologyusedtostudytheunsteadyforcesisdescribedinSection 7.2 .Next,resultsfromnumericalsimulationforM=0.01andRe=1,10aredescribedinSection 7.3 .Motivatedbythendinginincompressibleowsthatadded-massforceisindependentofReynoldsnumber,wemodelinviscidunsteadyforceinacompressibleowtobeindependentofReynolds 150

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number.AnewkernelforviscousunsteadyforceisproposedbasedonworkofParmaretal.[ 81 ]andMeiandAdrian[ 68 ].ResultsforniteMach-andReynolds-numbersaregiveinSection 7.4 .Finally,conclusionsarepresentedinSection 7.5 7.2NumericalMethodology Twodifferentnumericalmethodsareusedinthisinvestigation,namelythedissipativesolverandthenon-dissipativesolver.ThenumericalmethodssolvesthecompressibleNavier-Stokesequationsinintegralformcastinaframeofreferenceattachedtothesphere. Inthedissipativesolverthespatialdiscretizationisbasedontheux-differencesplittingmethodofRoe(1981)(see[ 41 ]forreference)andtheweightedessentiallynon-oscillatoryreconstructiondescribedbyHaselbacher[ 41 ].Thediscreteequationsareintegratedintimeusingthefour-stageRunge-Kuttamethod.Themethodologyofdissipativesolveremployedinthisworkhasbeenappliedtoseveralunsteadycompressibleowsanddemonstratedgoodagreementwiththeoryandexperimentaldata,see,e.g.,Haselbacheretal.[ 43 ]. Thenon-dissipativesolverisbasedontheformulationgivenbyHouandMahesh[ 45 ].DetailsofthismethodisgiveninAppendix. Axisymmetricformulationisusedtosimulateowaroundasphere.Atwo-dimensionalhexahedralgridofO-typetopologyisusedwith200cellsaroundthesemi-circumference.Relativetothecylinderradiusa,theradialgridspacingadjacenttothecylindersurfaceisr=a=1.510)]TJ /F10 7.97 Tf 6.59 0 Td[(4.andisgraduallyincreasedtoreachaspectratioofnearlyunityfarfromcylinder.Theradialstretchingofgridcellsisadjustedsuchthatneartheparticlesurfacehighaspectratiocellsarecreatedtocaptureboundarylayergrowthandawayfromparticleeachlayerofcellsconsistsofapproximatelysquarecellstominimizeinternalwavereections.Wehaveemployedgridsconsistingofupto124,800cellstoassessgrid-independenceofoursolutions.Theresultsshownbelowwereobtainedon 151

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AHighaspectratiocellsnearcylindersurface. BNearlysquarecellsawayfromcylinder. Figure7-2. Meshquality. 152

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Figure7-3. SchematicdepictionofvariationoffreestreamMachandReynoldsnumberduringcomputations. gridsof62,400cells(coarsemesh)and124,800cells(nemesh).Fig. 7-2 showsthenear-eldmeshandfar-eldmeshforcoarsemesh. Forthecomputations,thecharacteristicboundaryconditionsofPoinsot&Lele[ 85 ]areappliedattheouterboundary,locatedat100a. Wedividethecomputationsintotwostages.Instageone,thesphereisheldxedandasteady-statesolutionisobtainedforM1=M1,0andRe1=Re1,0.Instagetwo,weimposeadelta-functionnegativeaccelerationinthex-directiononthesphereattimeti.ThespherewillthusattainarelativeMachandReynoldsnumbersM1,0(1+)andRe1,0(1+)respectively,where>0,asdepictedschematicallyinFig. 7-3 .Wechoose1toensurethatthechangeinMachandReynoldsnumberissmallandthatwecanthereforeisolatetheeffectoftheinitialfreestreamMachandReynoldsnumbersonthedragforce. 7.3UnsteadyForcesOveraSphereatSmallMachNumbersandFiniteReynoldsNumbers InChapter 2 resultsofnumericalsimulationforM=10)]TJ /F10 7.97 Tf 6.59 0 Td[(3andRe=0.1,1.0,10wereshown.ForsuchlowMach-andReynolds-numberrangethenumericalsimulation 153

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Figure7-4. TimeevolutionofnormalizedunsteadyforceforM=0.01,Re=1.Bassethistoryforce(Eq.( 7 )),modiedhistoryforceduetoMeiandAdrian[ 68 ](Eq.( 7 )),andinviscidandviscousunsteadyforceincompressibleowsduetoParmaretal.[ 81 ](Eq.( 7 ))areplotted.Correspondingsimulationresultsareshownasopencirclesymbols. resultswereinexcellentagreementwithresultsoftheoreticalinvestigationoflinearizedcompressibleNavier-Stokesequationsforthetimerangeshownthere.Forsufcientlylongtime,thenon-lineareffectsresultsindeviationfromthetheorybasedonlinearizedow.Togetnon-lineareffectsearlierintime,herewechooseM=10)]TJ /F10 7.97 Tf 6.59 0 Td[(2andRe=1,10,100. Figure 7-4 showsnumericallyobtainednormalizedunsteadyforce,existingtheories,andempiricalcorrelationsforM=0.01andRe=1.Forshort-time(1),numericaldatafollowscompressibility-modiedhistory-force(Parmaretal.[ 81 ])closely.Forlong-time(1),numericaldatafollowsmodiedhistory-kernelofMeiandAdrian[ 68 ].Modelinginviscid-unsteadyforceasindependentofReynoldsnumberwecanseparateviscous-unsteadyforce.FortheM=0.01owconsideredhere,inviscid-unsteady 154

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AM=0.01,Re=1. BM=0.01,Re=10. Figure7-5. ComparisonofnormalizedunsteadyforceobtainedbynumericalsimulationandthatgivenbythelasttwotermsofEq.( 7 ).Simulationresultsareshownasopencirclesymbols. 155

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kernelcanbetakensameasLonghorn'skernelgivenbyEq.( 7 ).Anewkernelcanbeproposedforviscous-unsteadyforceincompressibleowsatniteReynolds-numbertomimicobservedbehavior.InthenewkernelwereplaceBasset-historyforceintheformulaofKMA(t)(Eq.( 7 ))byrecentlyfoundviscous-unsteadykernelincompressibleowsKPHB(t)(Eq.( 7 ))byParmaretal.[ 81 ].Wecanwritenewviscous-unsteadykernelas Knewvu(Re;t)=K)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2PHB(t)+K)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2NL(Re;t))]TJ /F10 7.97 Tf 6.59 0 Td[(2(7) AnewparticleequationofmotionvalidforverysmallMach-numbersandnitebutsmallReynolds-numberscanbewrittenas,mpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av(Re))]TJ /F3 11.955 Tf 11.96 0 Td[(mfZtKLH(t)]TJ /F8 11.955 Tf 11.95 0 Td[()c0 adv dtdc0 a)]TJ /F4 11.955 Tf 11.96 0 Td[(6a2p ZtKnewvu(Re;t)]TJ /F8 11.955 Tf 11.95 0 Td[()dv dd, (7) Figures 7-5A and 7-5B showcomparisonofnumericallyobtainedunsteadyforceandthatpredictedbyEq.( 7 )forM=0.01,Re=1andM=0.01,Re=10respectively.Numericaldataandmodelpredictionsareinexcellentagreement. 7.4UnsteadyForcesOveraSphereatSub-CriticalMachNumbersandFiniteReynoldsNumbers WecarryoutnumericalsimulationsfornitebutsubcriticalMach-numberandniteReynolds-numberows.Followingfourcasesareconsidered: 1. M=0.1andRe=10 2. M=0.2andRe=20 3. M=0.3andRe=30 4. M=0.4andRe=40 Figures 7-6A 7-6B 7-7A ,and 7-7B shownumericallyobtainedunsteadyforceduetoadelta-functionacceleration.Asmentionedearlier,wemodelinviscid-unsteadyforcetobeindependentofReynoldsnumberanddependentonlyonMachnumberas 156

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AM=0.1,Re=10. BM=0.2,Re=20. Figure7-6. Comparisonofnormalizedunsteadyforceobtainedbynumericalsimulationandthatgivenbynewmodel(thelasttwotermsofEq.( 7 )). 157

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AM=0.3,Re=30. BM=0.4,Re=40. Figure7-7. Comparisonofnormalizedunsteadyforceobtainedbynumericalsimulationandthatgivenbynewmodel(thelasttwotermsofEq.( 7 )). 158

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aparameter.EffectofMachnumberontheinviscid-unsteadykernelwasstudiedbyParmaretal.[ 79 ].Hereweusethesamemodeltorepresentinviscid-unsteadykerneldenotedbyKiu(M;)andseparateviscous-unsteadyforce.Theviscous-unsteadyforceforaviscous-compressibleowatniteMach-andReynolds-numberdependsonbothMachandReynolds-numbers.Asarstapproximation,wemodelviscous-unsteadyforceasonlyafunctionofReynoldsnumberusingthenewkernelKnewvu(Re;t)usedinSection 7.3 inEq.( 7 ).AnapproximateparticleequationvalidforniteMach-andReynolds-numberowscanbewrittenasmpdv dt=)]TJ /F4 11.955 Tf 9.3 0 Td[(6av(Re))]TJ /F3 11.955 Tf 11.96 0 Td[(mfZtKiuM;(t)]TJ /F8 11.955 Tf 11.95 0 Td[()c0 adv dtdc0 a)]TJ /F4 11.955 Tf 11.96 0 Td[(6a2p ZtKnewvu(Re;t)]TJ /F8 11.955 Tf 11.95 0 Td[()dv dd, (7) Figures 7-6A 7-6B 7-7A ,and 7-7B alsoshowpredictionusinglasttwotermsofEq.( 7 ).Forverysmalltime,numericallyobtainedunsteadyforcefollowcompressibility-modiedhistoryforce(lasttermofEq.( 7 )).UnlikethecaseofweakMachnumberowsconsideredinSection 7.3 ,thenumericallyobtainedunsteady-forceinniteMach-numberowsconsideredheredoesnotfollowpredictionbynewkernelKnewvu(Re;t)forlongtime.ThisdeviationisresultofniteMach-numbereffectonviscous-unsteadykernelwhichhasbeenneglectedinrstapproximationconsideredhere.Forlongtime,however,thedecayrateof1=t2ismaintained.ThemagnitudeofoffsetincreaseswithMachnumber. 7.5Conclusions Unsteadyforcesinviscous-compressibleowsatniteMach-andReynolds-numberarestudied.AsdiscussedinChapter 2 ,solutionoflinearizedcompressibleNavier-Stokesequationsisvalidforearlytime<1=(ReM)evenforniteMach-andReynolds-numberows.WecanidentifythreeregimesforMachandReynoldsnumbers. 1. RegimeI:VerysmallMachandReynoldsnumbers,i.e.,M1andRe1suchthatM=Re1toensurecontinuumassumption. 2. RegimeII:SmallMach-numbersandniteReynolds-numbers. 159

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3. RegimeIII:LargeMach-andReynolds-numbers. ForregimeI,<1=(ReM)issatisedforuptoverylargetimeandthuslinearizedsolutionarevalidtilllongtime.ForregimeII,<1=(ReM)isviolatedearlyintimeandnon-lineareffectscausessolutiontodeviatefromlinearsolution.SinceMachnumberisstillsmall,itseffectonnon-linearregimeatlargetimeisweakandnegligible.InthelimitofsmallMachnumber(M1)andniteReynolds-number,anewparticleequationofmotionisproposed(Eq.( 7 )).StandarddragcorrelationofSchillerandNaumann[ 94 ]isusedforquasi-steadydrag.Inviscid-unsteadyforceismodeledindependentofReynoldsnumberasgivenbyLonghorn[ 58 ].Anewkernelisproposedforviscous-unsteadyforceincorporatingnon-lineareffectsdependingonlyonReynoldsnumberasparameter.ThepredictionsforunsteadyforcedusingEq.( 7 )areinexcellentagreementwithnumericallyobtainedresults.ForniteMach-andReynolds-numberowinRegimeIII,asarstapproximation,aparticleequationofmotionisproposedconsistingofstandarddragcorrelationofSchillerandNaumann[ 94 ],Mach-numberdependentinviscid-unsteadykernelsofParmaretal.[ 79 ],andReynolds-numberdependentviscous-unsteadykernelproposedinSection 7.3 (Eq.( 7 )).FiniteMach-numbereffectsonnon-linearbehavioratintermediateandlargetimeforviscous-unsteadykernelisnotmodeledinthiswork.WorkisinprogresstomodeltheeffectsofniteMach-numberonviscous-unsteadykernel. 160

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CHAPTER8SUMMARY,CONCLUSIONS,ANDFUTUREWORK 8.1SummaryandConclusions Themainresultsandconclusionsarrivedatinthisworkcanbesummarizedasfollows: BBOequationforcompressiblecreepingows.AtheoreticalanalysisiscarriedoutforthelinearizedcompressibleNavier-Stokesequations.Anexactsolutionisobtainedfortheforceactingonasphereundergoingunsteadyrectilinearmotioninahomogeneousuid.Thetotalforceisexpressedasthesumofthequasi-steady,inviscidunsteady,andviscousunsteadyforces.Thequasi-steadyforceisidenticaltothewellknownStokesdragintheincompressiblelimit.Theinviscidunsteadyforceisverydifferentfromthatencounteredinincompressibleows.Thesingularityoftheinviscidunsteadyforceintheincompressiblelimitisregularizedforthecaseofcompressibleowsbecauseofthenitespeedofsound.TheexpressionfortheinviscidunsteadyforceinviscouscompressibleowisidenticaltothatobtainedbyLonghorn[ 58 ]forasphereacceleratinginaninviscidcompressibleuidintheacousticlimit(M!0).Theinviscidunsteadyforceisrepresentedbyanintegralthatdependsontheaccelerationhistory.Theeffectofcompressibilityontheviscousunsteadyforceisfoundtobemoderate.Theviscousunsteadyforceismodiedforshorttimesonly(c0t=a10),afterwhichitrevertstotheincompressiblebehavior,withadecayrateoft)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2.Anexactanalyticalexpressionfortheviscousunsteadyforceinacompressibleowcannotbeobtained.Acurvettonumericallyinverteddataisproposedthatisaccuratetowithin1%.Withtheexpressionfortheforces,anequationofmotionisproposedasthecompressibleextensionoftheBasset-Boussinesq-Oseen(BBO)equation.TheproposedparticleequationofmotioncanbeusedinLagrangiantrackingofpointparticlesincompressibleowsinthelimitRe1andM1. 161

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MRGequationforcompressiblecreepingows.ThetheoreticalanalysisunderlyingthederivationoftheBBOequationforcompressiblecreepingowsisextendedtoinvestigatetheforceonasphereundergoingunsteadymotioninanon-uniformcompressibleow.Asystematicapproachisadoptedtosimplifythegoverningequations.Adensityweightingtechniqueisusedtoincorporatedensitychanges.Faxen'stheoremisderivedfortheforceonthesphereusingthereciprocaltheoremforviscouscompressibleows.Faxencorrectionsarederivedforthequasi-steady,inviscidunsteady,andviscousunsteadyforces.TheFaxencorrectionsappearassurfaceandvolumeaveragesofquantitiesoftheowrelativetotheparticle.Incontrasttoincompressibleows,thevolume-averagequantitiesappearintheinviscidunsteadyaswellastheviscousunsteadyforces.Withtheexpressionfortheforces,anequationofmotionisproposedthatcanbeconsideredanextensionoftheMaxey-Riley-Gatignol(MRG)equationtocompressibleows. InviscidunsteadyforceatniteMachnumbers.CarefullydesignednumericalsimulationsarecarriedouttoinvestigatetheinviscidunsteadyforceforcylindersandspheresatnitebutsubcriticalfreestreamMachnumbers.ThisworkcanbeconsideredanextensionoftheworkofMiles[ 71 ]andLonghorn[ 58 ]AsnotedbyMilesandLonghorn,theconceptofadded-massisnotapplicabletocompressibleowsbecausethedependenceoftheforceontheinstantaneousaccelerationisbroken.AresponsekernelKiucanbedenedthatrepresentstheforceresponsetoanimpulsiveaccelerationofunitmagnitude.Thetotalforceforageneralaccelerationcasecanbeobtainedbyaconvolutionintegraloftheresponsekernelandtheacceleration.ForniteMachnumbers,thebehavioroftheinviscidunsteadyresponsekernelsissimilartothatintheacousticlimit.Forconstantaccelerationandlongtimes,aneffectiveadded-masscoefcientcanbederived.Theeffectiveadded-masscoefcientfornitebutsubcriticalMachnumbersincreasestoavaluethatisnearlydoubletheadded-masscoefcientinanincompressibleow.Inasubcriticalcompressibleow,theinviscidunsteadyforce 162

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canbeuptofourtimeslargerthanthatinincompressibleows.TheeffectoftheratioofspecicheatsisfoundtoincreasewithincreasingMachnumbers. Shock-particleinteractionmodel.Shock-particleinteractionusuallyleadstolargeunsteadyforces.Asimplemodeltoaccountfortheunsteadyforceinshock-particleinteractionwasconstructed.Resultsobtainedwiththismodelclearlyshowthatincludingtheinviscidunsteadyforceiscrucialtocapturethepeakforceactingonaparticle.Thesimplemodelpredictstheunsteadyforcewithgoodaccuracywhencomparedwithrecentexperimentaldataonstationaryparticles.Theoverallagreementintermsofthepeakvalueofthedragcoefcient,thetimeatwhichthepeakvalueisattained,andthelong-timebehaviorisgood.Forthemovingparticlesitisobservedthattheprimaryeffectoftheinviscidunsteadyforceistogiveaninitialimpulsiveincreaseinparticlevelocities. Drag-coefcientcorrelationforniteMachnumbers.Animprovedcorrelationforthedragcoefcientofsphericalparticleswasconstructed.Thenewcorrelationrepresentsavailableexperimentaldatamoreaccuratelythanpreviouslypublishedcorrelations.Themodeliscapableofaccuratelyreproducingparticlevelocitiesmeasuredintheexperimentsfollowingtheimpactofashockwave.Thissuggeststhattheincreaseddragcoefcientobservedinmanyshock-tubeexperimentsissimplyaconsequenceofcompressibilityeffectsandnotduetounsteadinessasissometimesclaimed. UnsteadyforcesonaparticleinaniteReynoldsandniteMachow.HighdelitynumericalsimulationshavebeencarriedouttomeasuretheforcesonanacceleratingparticleinniteMach-andReynolds-numberows.Machnumbersupto0.4andReynoldsnumbersupto40areconsideredinthiswork.SolutionofthelinearizedcompressibleNavier-Stokesequationsisvalidforearlytimec0t=a<1=(ReM)evenforniteMach-andReynolds-numberows.WecanidentifythreeregimesforMachandReynoldsnumbers.(i)RegimeIischaracterizedbyverysmallMachandReynoldsnumbers,i.e.,M1andRe1suchthatM=Re1toensurecontinuumassumption. 163

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Inthisregime,lineartheoryisvalidtilllongtime.(ii)RegimeIIischaracterizedbysmallMach-numbersandniteReynolds-numbers.Inthisregime,c0t=a<1=(ReM)isviolatedearlyintimeandnon-lineareffectscausessolutiontodeviatefromlinearsolution.SinceMachnumberisstillsmall,itseffectonnon-linearregimeatlargetimeisweakandnegligible.Anewkernelforviscous-unsteadyforceisproposedwhichaccountsforcompressibility-correctionforshorttimesandincorporatesReynolds-numbereffectsforlargetime.Thepredictionsforunsteadyforcesusingproposedunsteadykernelsareinexcellentagreementwithnumericallyobtainedresults.(iii)RegimeIIIischaracterizedbylargeMach-andReynolds-numbers.ForniteMach-andReynolds-numberowinthisregime,asarstapproximation,inviscid-unsteadyforceismodeledasdependentonlyonMachnumberandviscous-unsteadyforceismodeledasdependentonlyonReynoldsnumber.AparticleequationofmotionisproposedconsistingofstandarddragcorrelationofSchillerandNaumann[ 94 ],Mach-numberdependentinviscid-unsteadykernelsofParmaretal.[ 79 ],andReynolds-numberdependentnewviscous-unsteadykernelproposedhere.Agoodagreementisfoundinoverallbehaviorofnumericallyobtainedforceandthatpredictedbythemodel.WorkisinprogresstomodeltheeffectsofniteMach-numberonviscous-unsteadykernel. 8.2FutureWork Asystematicstudyhasbeenundertakeninthisworktostudyparticleequationofmotionincompressibleows.AsdescribedinSection 8.1 ,signicantprogresshasbeenmadeinourunderstandingofforces,particularlyunsteadyforces,onaparticleincompressibleows.However,therearestillmanyopenquestionsleftforfuturestudies.Suggesteddirectionsforfutureworkare: Inviscidunsteadyforceinsupercriticalandsupersonicows.Inasupercriticalbutsubsonicow,ashockwaveappearsontheparticlesignicantlyincreasingthedrag.Moreover,thesymmetricallypositionedshockwavesonthesurfaceoftheparticlesoonbecomeunstableandresultinhighlyunsteadyowwithvortexsheddingeven 164

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ininviscidows.Supersonicowaroundtheparticleischaracterizedbyadetachedbowshockcausingastrongwaveorformdrag.Investigationsneedtobecarriedouttodeterminetheformandmagnitudeoftheinviscidunsteadyforceinsuper-criticalandsupersonicowscomparedtoinviscidquasi-steadydrag.Ourpreliminaryinvestigationsshowthattheinviscidunsteadyforceisverysmallcomparedtothequasi-steadydraginsupercriticalandsupersonicows. Investigationofhistoryforceincompressibleows.FurtherworkisrequiredtostudythehistoryforceincompressibleviscousowsatniteMachandReynoldsnumbers.Theintegralnatureofboththeinviscidandtheviscousunsteadyforcesincompressibleowmakestheirseparateextractionverydifcult.Thusacompositekernelcanbeproposedthatrepresentsthetotalunsteadyforceinacompressibleow.Ourpreliminaryresultsshowthattheshort-timebehavioroftheviscousunsteadyforcecanbeobtainedfromlineartheory.Atintermediatetimes,non-lineareffectsappearthatcausetheviscousunsteadykernelforniteMachandReynoldsnumberstodeviatefromthatobtainedusinglineartheory.Thelongtimedecayrateis1=t2,thesameasthatobservedinincompressibleows.Butthecompressibleviscousunsteadykernelisfoundtobelargerthantheincompressiblekernelforlongtime. Shock-particleinteractionincludingviscousunsteadyforce.Usingtheunderstandingdevelopedinthisdissertationontheviscousunsteadyforce,shock-particleinteractionmodelneedstobeimproved.Ourpreliminaryinvestigationshowsthattheinviscid-unsteadyforceisthedominantforce,anorderofmagnitudelargerthantheviscous-unsteadyforce,duringinitialperiodofshock-particleinteraction. Effectofniteaccelerations.Forincompressibleows,Kimetal.[ 52 ]modiedthekernelsfortheviscousunsteadyforcepresentedbyMeiandAdrian[ 68 ]toaccountforlargeaccelerations.Asimilarstudyisneededforcompressibleviscousows. Sphericalbubbleswithvariableslip.Thepresentworkisfocusedonthestudyofparticlemotionwithno-slipconditionsontheparticlesurfaceinviscousowsand 165

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perfectslipconditionsininviscidows.Abubblewithnocontaminationonthesurface(perfectslip)inaviscouscompressibleowhasnotbeenstudied.However,suchstudieshavebeenundertakeninincompressibleowsandfoundtoleadtodifferentviscousunsteadyforces.Also,variableslipconditionsneedtobeexplored.Ourpreliminaryworkshowsthatnewviscous-unsteadyforcesariseincompressibleows,similartothatinincompressibleows,duetovariablesliponparticlesurface. Rigidnon-sphericalparticles.Particleshapesarerarelyperfectlyspherical.Thisworkfocusedonsphericalparticlesasanidealization.Furtherworkisneededtoextendthepresentstudytonon-sphericalparticleshapes. 166

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NUMERICALMETHODOLOGY Themainfocusofthepresentworkistostudytheunsteadyforceonparticlesinviscouscompressibleows.ForowswithniteReynoldsandMachnumberstheoreticalanalysisisnotpossible.Hence,anaccuratenumericalmethodologyisneeded.Thenumericalmethodshouldhavethefollowingproperties: 1. Capableofsolvingviscouscompressibleowswithsecondorderaccuracy, 2. AbletosimulateM!0aswellasniteMows, 3. Captureshocks,and 4. Accurateboundaryconditions. Anotherfactoraffectingtheaccuracyofnumericalresultsisthequalityofthemeshbeingused.Signicanttimehasbeeninvestedtodevelopanaccuratenumericalmethodandhighqualitygrids. Twosolutionmethodshavebeenusedforthenumericalinvestigations.ThesolutionmethodswillbecalledDissipativesolverandNon-dissipativesolverinthisdissertation.TheDissipativesolverwasdevelopedbyProf.Haselbacher(seeHaselbacher[ 42 ])andextendedinthiswork.TheNon-dissipativesolverwasdevelopedfromscratchinthiswork.Speciccapabilitiesthathavebeenaddedtoboththesesolversinclude(i)solvinggoverningequationsinmovingreferenceframe,(ii)axisymmetriccomputationtoreducecomputationalcost,and(iii)characteristicboundaryconditionsandspongelayers. Thisappendixdescribesthesolutionmethods. 167

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A.1GoverningEquations A.1.1DimensionalForm Thegoverningequationsarethethree-dimensionaltime-dependentcompressibleNavier-Stokesequations,expressedinCartesiantensornotation,@ @t+@ui @xi=0 (A)@ui @t+@uiuj @xj=)]TJ /F8 11.955 Tf 11.54 8.09 Td[(@p @xi+@ij @xj)]TJ /F8 11.955 Tf 11.96 0 Td[(ai (A)@E @t+@Hui @xi=@ijuj @xi)]TJ /F8 11.955 Tf 13.15 8.09 Td[(@qi @xi)]TJ /F8 11.955 Tf 11.96 0 Td[(uiai (A) whereisthedensity,uiistheithcomponentofthevelocityvector,pisthepressure,ijistheijthcomponentofthestresstensor,aiistheithcomponentoftheaccelerationvectorforthemovingreferenceframe,Eisthetotalenergy,Histhetotalenthalpy,andqiistheithcomponentoftheheat-uxvector.Thesummationconventionisusedthroughoutthisdocument. Thetotalenergyisgivenby E=CvT+1 2uiui(A) whereCvisthespecicheatatconstantvolumeandTisthestatictemperature.Thetotalenthalpyisgivenby H=CpT+1 2uiui(A) whereCpisthespecicheatatconstantpressure.Theequationofstateis p=RT(A) whereR=Cp)]TJ /F3 11.955 Tf 11.96 0 Td[(Cvisthegasconstant. Thestresstensorisgivenby ij=@ui @xj+@uj @xi)]TJ /F4 11.955 Tf 13.15 8.08 Td[(2 3ij@uk @xk(A) 168

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whereisthedynamicviscosityandijistheKroneckerdelta.Theheat-uxvectorisgivenby qi=)]TJ /F8 11.955 Tf 9.3 0 Td[(@T @xi(A) whereisthethermalconductivity. Thespeedofsoundisdenedas c=p RT=p p=(A) where=Cp=Cv. A.1.2Non-DimensionalForm Theindependentanddependentvariablesarenon-dimensionalizedthrough xi=xi Lref (A) t=ureft Lref (A) = ref (A) ui=ui uref (A) p=p)]TJ /F3 11.955 Tf 11.96 0 Td[(pref refu2ref (A) T=T Tref (A) = ref (A) ai=ai u2ref=Lref (A) wherethenotation ()denotesanon-dimensionalvariableandthesubscriptrefindicatesareferencevalue.Itisassumedthat pref=refRTref(A) and c2ref=RTref=pref=ref(A) 169

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Substitutingthesedenitionsintothedimensionalformofgoverningequationsgives@ @ t+@ ui @ xi=0 (A)@ ui @ t+@ ui uj @ xj=)]TJ /F8 11.955 Tf 12.13 8.09 Td[(@ p @ xi+1 Reref@ ij @ xj)]TJ ET q .478 w 326.61 -71.21 m 341.7 -71.21 l S Q BT /F8 11.955 Tf 326.61 -78.53 Td[(ai (A) andM2ref@ @ t p+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2 ui ui+@ @ xj p+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2 ui ui uj+@ ui @ xi=()]TJ /F4 11.955 Tf 11.95 0 Td[(1)M2ref Reref@ ij uj @ xi+1 RerefPr@ @ xi @ T @ xi)]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M2ref uiai (A) whereReref=refurefLref ref (A)Mref=uref cref (A)Pr=Cp (A) Theequationofstatebecomes T=M2ref p+1(A) Thespeedofsoundbecomes c2=c cref2=1 )]TJ /F8 11.955 Tf 5.48 -9.68 Td[(M2ref p+1= T(A) A.2SolutionMethods A.2.1DissipativeSolver Thedissipativesolversolvesthedimensionalformofthegoverningequationsinintegralformonarbitraryunstructuredgridsconsistingoftetrahedra,hexahedra,prisms,andpyramids.Thespatialdiscretizationisbasedontheux-differencesplittingmethodofRoe(1981)andasimpliedweightedessentiallynon-oscillatoryreconstruction 170

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describedbyHaselbacher[ 41 ].Thediscreteequationsareintegratedintimeusingthefour-stageRunge-Kuttamethod.Thebasicmethodologyemployedinthisworkhasbeenappliedtoseveralunsteadycompressibleowsanddemonstratedgoodagreementwiththeoryandexperimentaldata,see,e.g.,Haselbacheretal.[ 43 ]. Thesolverisfullyparallelandhasbeenshowntogiveexcellentscalabilityonthousandsofprocessors.DetailsfortheDissipativesolvercanbefoundinHaselbacher[ 42 ]. A.2.2Non-DissipativeSolver Anon-dissipativeowsolverbasedonthemethodologypresentedbyHou&Mahesh(2005)hasalsobeendeveloped.Italsooperatesonarbitraryunstructuredgrids.Thenon-dissipativesolversolvesthenon-dimensionalformofthegoverningequationsinintegralform.Fig. A-1 showsthestorageofvariables.Thevelocitycomponents,pressure,anddensityarecolocatedinspaceatcellcentroid.Theface-normalvelocityisstoredatfacecenters.NotethatDensity,pressure,andtemperaturearestaggeredintimecomparedtothevelocity.Thismakesthediscretizationsymmetricinspaceandtime,resultinginanon-dissipativesolutionmethod.Apressure-correctionmethodisusedtosolvethegoverningequations.Apredictor-correctortypeiterativealgorithmisimplemented.Thenumericalschemeisimplicitandsecond-orderaccurateinspaceandtime.TheHyprelibrary(LawrenceLivermoreNationalLaboratory2003)isusedtosolvethelinearsysteminparallel. Afewadvantagesofthissolverarethatthenumericalschemeis(i)non-dissipative,(ii)kinetic-energyconserving,and(iii)thesolvercantacklenitebutsubcriticalMachaswellasM!0ows. Alimitationofthesolveristhatthesupercriticalowscannotbesimulatedduetolackoftheshock-capturingcapability.Addingsuchacapabilityisworkinprogress. 171

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A.3DiscretizationofNon-DissipativeSolver TheNon-dissipativesolverisbasedonthemethodofHouandMahesh[ 45 ](hereafterreferredtoasHM))withsomemodications.InHMthermaluxinenergyequationwastreatedexplicitly.Itwasfoundthatthismakesnumericalmethodunstable.Weuseimplicitformulationforthethermalux.AdetailedderivationfordiscretizationandsolutionalgorithmispresentedincurrentsectionandSection A.4 ,respectively. A.3.1NotationandVariableArrangement Thenotationtobeusedbelowfollowstheseguidelines: Superscriptsdenotetimelevels.Thusn+1denotesthevalueofattimeleveln+1.Superscriptscanbecombinedtoindicateiterates.Thusn+1,qdenotestheqthiterateofattimeleveln+1.Similarly,thesuperscriptdenotesapredictedvalue.Thusn+1,denotesthepredictedvalueofattimeleveln+1. Subscriptsdenotespatiallocationsorcomponentsofavectorortensor.Theindicesiandjdenotecomponentsofavectorortensor.Greeksubscriptsindicatevariablesassignedtoaspeciccontrolvolume.Thusdenotesthevariableincontrolvolume.Thesubscriptkdenotestheindexofthefacesharedbycontrolvolumesand. Thesummationconventionisnotappliedtospatiallocations,i.e.,k,,and. ThevariablearrangementisshowninFig. A-1 .Notethatthevelocitycomponentsandthethermodynamicvariablesarestoredatthesamelocationinspace,butarestaggeredintime.Inaddition,thevelocitynormaltoacontrol-volumefaceisstoredatthesametime-levelasthecontrol-volumevelocity. Inthefollowing,the ()notationisdroppedforconvenience;allvariablesareunderstoodtobenon-dimensional. A.3.2ContinuityEquation Thecontinuityequationisintegratedover[tn+1=2,tn+3=2]togive Zn+3=2n+1=2Z@ @tdVdt+Zn+3=2n+1=2Z@ui @xidVdt=0.(A) 172

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FigureA-1. Variablearrangement. FigureA-2. Timediscretization. 173

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Expressedthesurfaceintegralasasumofintegralsoverthefacesofthecontrolvolume,andapproximatingthesurfaceintegraloneachfacetogive )]TJ /F8 11.955 Tf 5.48 -9.69 Td[(n+3=2)]TJ /F8 11.955 Tf 11.95 0 Td[(n+1=2V+Zn+3=2n+1=2 Xkkvksk!dt=0,(A) wherewiththeface-normalvelocityisdenedasv=uini,Thetimeintegralisapproximatedbythemid-pointrule, n+3=2)]TJ /F8 11.955 Tf 11.96 0 Td[(n+1=2 t+1 VXkn+1kvn+1ksk=0,(A) whereskistheareaoffacekandt=tn+3=2)]TJ /F3 11.955 Tf 11.96 0 Td[(tn+1=2. Ateachface,n+1kisobtainedfrom n+1k=n+1+n+1 2(A) andthevaluen+1isobtainedbyinterpolationintimeas n+1=n+1=2+n+3=2 2(A) andsimilaryforn+1,soweobtain V t+1 4Xkvn+1ksk!n+3=2+1 4Xkn+3=2vn+1ksk= V t)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4Xkvn+1ksk!n+1=2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4Xkn+1=2vn+1ksk (A) A.3.3MomentumEquation Themomentumequationisintegratedover[tn,tn+1]togiveZn+1nZ@ui @tdVdt+Zn+1nZ@uiuj @xjdVdt=)]TJ /F11 11.955 Tf 11.29 16.27 Td[(Zn+1nZ@p @xidVdt+1 RerefZn+1nZ@ij @xjdVdt)]TJ /F11 11.955 Tf 11.95 16.27 Td[(Zn+1nZaidVdt (A) 174

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Usingtheapproximationsintroducedinthediscretizationofthecontinuityequationforthetime-derivative,convective,anddiffusiveterms,weobtain(ui)n+1)]TJ /F4 11.955 Tf 11.96 -.17 Td[((ui)nV+tXk(ui)n+1=2kvn+1=2ksk=)]TJ /F11 11.955 Tf 11.29 16.28 Td[(Zn+1nZ@p @xidVdt+t RerefXkn+1=2ij,knj,ksk)]TJ /F4 11.955 Tf 11.96 -.16 Td[((ai)n+1=2Vt (A) whereitisassumedthatttn+3=2)]TJ /F3 11.955 Tf 11.95 0 Td[(tn+1=2=tn+1)]TJ /F3 11.955 Tf 11.96 0 Td[(tn. FollowingHM,thepressuretermisapproximatedas Zn+1nZ@p @xidVdt=Vt@p @xin+1=2(A) whereitshouldbenotedthatthedivergencetheoremisnotused.Following[ 115 ],thepressuregradientappearinginEq.( A )isapproximatedas@p @xin+1=2=1 2"@p @xin+@p @xin+1# (A)=1 4"@p @xin)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2+2@p @xin+1=2+@p @xin+3=2# (A) Equation( A )canbeinterpretedasalteredpressuregradientorasthegradientofalteredpressure~pn+1=2denedby ~pn+1=2=1 4)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(pn)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2+2pn+1=2+pn+3=2(A) WithEqs.( A )and( A )andbydividingbyt(andnotbyVtlikeHMtosimplifytheimplementation)weobtainthediscretemomentumequation,V t(ui)n+1)]TJ /F4 11.955 Tf 11.95 -.17 Td[((ui)n+Xk(ui)n+1=2kvn+1=2ksk=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(V 4"@p @xin)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2+2@p @xin+1=2+@p @xin+3=2#+1 RerefXkn+1=2ij,knj,ksk)]TJ /F4 11.955 Tf 11.95 -.16 Td[((ai)n+1=2V (A) 175

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HMdonotdescribehowtheviscoustermsarecomputed.Thefollowingoutlinesaplausiblewaytocomputetheviscousterms.Tondaformoftheviscousstresssuitablefordiscretization,itismoreconvenienttoconsiderthetractionvector ti=ijnj(A) wheretheindicesassociatedwithtimediscretizationandfacelocationareignored.ItisasimplemattertoshowfromEq.( A )thatt1=@u1 @n+1 3@uk @xkn1+@u2 @x1n2)]TJ /F8 11.955 Tf 13.15 8.09 Td[(@u2 @x2n1+@u3 @x1n3)]TJ /F8 11.955 Tf 13.15 8.09 Td[(@u3 @x3n1 (A)t2=@u2 @n+1 3@uk @xkn2+@u1 @x2n1)]TJ /F8 11.955 Tf 13.15 8.09 Td[(@u1 @x1n2+@u3 @x2n3)]TJ /F8 11.955 Tf 13.15 8.09 Td[(@u3 @x3n2 (A)t3=@u3 @n+1 3@uk @xkn3+@u1 @x3n1)]TJ /F8 11.955 Tf 13.15 8.08 Td[(@u1 @x1n3+@u2 @x3n2)]TJ /F8 11.955 Tf 13.15 8.08 Td[(@u2 @x2n3 (A) where @ui @n=@ui @xjnj(A) Itisconvenienttowrite ti=ti,N+ti,D+ti,T(A) where,referringtoEqs.( A ),( A ),and( A ),ti,Nrepresentstherstterm,ti,Drepresentsthesecondterm,andti,Trepresentsthethirdandfourthterms.Thesignicanceofthesetermsisasfollows:Thersttermwillbelargebecauseitcapturesthewall-normalvariationinboundarylayers(providedcontrol-volumefacesarealignedwiththewall),thesecondtermwillbesmallunlesslargedensitygradientsexist,andthecontributionsofthethirdandfourthtermstotheuximbalanceofacontrolvolumewillbezeroiftheviscosityisconstant. Toensurestrongcouplingandtodamposcillationsonthegridscale,ti,Natafacekisapproximatedas @ui @nk=ui,)]TJ /F3 11.955 Tf 11.95 0 Td[(ui, nk(A) 176

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wherenk=kr)]TJ /F15 11.955 Tf 12.01 0 Td[(rk.Thederivativesappearinginti,Dandti,Tcanbecomputedusingaveragedcellgradients,i.e., @ui @xjk=1 2"@ui @xj+@ui @xj#(A) Notethatusingthisapproachtocomputethenormalderivativewouldleadtodecouplingonsomeuniformgrids.CellgradientsareapproximatedbyapplicationoftheGreen-Gausstheoremas @ui @xj=1 2VXkui,nj,ksk(A) A.3.4EnergyEquation Theenergyequationisintegratedover[tn,tn+1]togiveM2refZn+1nZ@ @tp+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2uiuidVdt+M2refZn+1nZ@ @xjp+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2uiuiujdVdt+Zn+1nZ@ui @xidVdt=()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M2ref RerefZn+1nZ@ijuj @xidVdt+1 RerefPrZn+1nZ@ @xi@T @xidVdt)]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M2refZn+1nZuiaidVdt (A) Usingtheapproximationsintroducedintheapproximationofthecontinuityequationforthetime-derivative,convective,anddiffusiveterms,andEq.( A )appliedtotheface 177

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pressure,weobtainM2refpn+1+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2n+1(uiui)n+1)]TJ /F11 11.955 Tf 11.96 16.86 Td[(pn+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2n(uiui)nV+M2reftXk~pn+1=2k+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2n+1=2k(uiui)n+1=2kvn+1=2ksk+tXkvn+1=2ksk=()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M2ref RereftXk(ijuj)n+1=2kni,ksk+1 RerefPrtXk@T @xin+1=2kni,ksk)]TJ /F4 11.955 Tf 11.96 0 Td[(()]TJ /F4 11.955 Tf 11.95 0 Td[(1)M2ref(uiai)n+1=2Vt (A) Aswiththeviscoustermsinthemomentumequation,HMdidnotdescribethediscretizationoftheviscousandconductiontermsintheenergyequation.TheapproximationoftheviscoustermsinEq.( A )buildsonthosedevelopedforthemomentumequation.Becausethestresstensorijissymmetric,wecanwrite ijujni=jiujni=tjuj(A) Theconductiontermisdiscretizedas @T @xin+1=2kni,k=@T @nn+1=2k=Tn+1=2)]TJ /F3 11.955 Tf 11.95 0 Td[(Tn+1=2 nk(A) HenceweobtainM2refpn+1+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2n+1(uiui)n+1)]TJ /F11 11.955 Tf 11.96 16.85 Td[(pn+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2n(uiui)nV+M2reftXk~pn+1=2k+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2n+1=2k(uiui)n+1=2kvn+1=2ksk+tXkvn+1=2ksk=()]TJ /F4 11.955 Tf 11.95 0 Td[(1)M2ref RereftXk(tjuj)n+1=2ksk+1 RerefPrtXk@T @nn+1=2ksk)]TJ /F4 11.955 Tf 11.95 0 Td[(()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M2ref(uiai)n+1=2Vt (A) A.4SolutionAlgorithmforNon-DissipativeSolver ReferringtoFig. A-2 ,weseektoadvancethevelocitiesfromnton+1andthedensity,pressure,andtemperaturefromn+1=2ton+3=2bysolvingthediscrete 178

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equationsinaniterativefashion.Accordingly,thesuperscriptn+m,qdenotestheqthiterateofagivenvariableatthetimen+m.Theequationsaresolvedusingapredictor-correctorapproachbasedonapressurecorrection.Wedene pn+3=2,q+1=pn+3=2,q+pn+3=2(A) A.4.1ContinuityEquation Withthisnotation,Eq.( A )becomes V t+1 4Xkvn+1,qksk!n+3=2,q+1+1 4Xkn+3=2,q+1vn+1,qksk= V t)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4Xkvn+1,qksk!n+1=2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4Xkn+1=2vn+1,qksk (A) whichcanbewrittenas an+3=2,q+1+Xkan+3=2,q+1=b(A) wherea=V t+1 4Xkvn+1,qksk (A)a=1 4vn+1,qksk (A)b= V t)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 4Xkvn+1,qksk!n+1=2)]TJ /F4 11.955 Tf 13.15 8.08 Td[(1 4Xkn+1=2vn+1,qksk (A) 179

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A.4.2MomentumEquation PredictorStep.UsingEqs.( A )and( A ),Eq.( A )canbewrittenasV t(ui)n+1,q+1)]TJ /F4 11.955 Tf 11.95 -.16 Td[((ui)n+Xk(ui)n+1=2,q+1kvn+1=2,q+1ksk=)]TJ /F3 11.955 Tf 10.5 8.08 Td[(V 4"@p @xin)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2+2@p @xin+1=2+@p @xin+3=2,q+1#+1 RerefXktn+1=2,q+1i,N,k+tn+1=2,q+1i,D,k+tn+1=2,q+1i,T,ksk)]TJ /F4 11.955 Tf 11.95 -.16 Td[((ai)n+1=2V (A) where tn+1=2,q+1i,N,k=kun+1=2,q+1i,)]TJ /F3 11.955 Tf 11.95 0 Td[(un+1=2,q+1i, nk(A) Thevelocityun+1,q+1iispredictedfromV t(ui)n+1,)]TJ /F4 11.955 Tf 11.95 -.17 Td[((ui)n+Xk(ui)n+1=2,kvn+1=2,qksk=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(V 4"@p @xin)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2+2@p @xin+1=2+@p @xin+3=2,q#+1 RerefXktn+1=2,i,N,k+tn+1=2,qi,D,k+tn+1=2,qi,T,ksk)]TJ /F4 11.955 Tf 11.95 -.17 Td[((ai)n+1=2V (A) wherethesuperscriptdenotesthepredictedvalueatq+1andonlythersttractiontermistreatedimplicitly.Thisisbecausetheothertermswouldleadtoalargerbandwidthofthelinearsystembelow.InEq.( A ),wedene (ui)n+1=2,k=1 2(ui)nk+(ui)n+1,k(A) and vn+1=2,qk=1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(vnk+vn+1,qk(A) 180

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HMdonotspecifyhow(ui)nkand(ui)n+1,kappearinginEq.( A )arecomputed.Aplausibleapproachis(ui)nk=1 2(ui)n+(ui)n (A)(ui)n+1,k=1 2(ui)n+1,+(ui)n+1, (A) wherenisinterpolatedfromn=1 2)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(n)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2+n+1=2 (A) InthealgorithmofHM,itappearstobeimplicitlyassumedthat n+1,q+1=n+1,(A) sothatn+1,=1 2)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(n+1=2+n+3=2,q+1 (A) Withtheobviouschangesinsubscripts,n+1,q+1andn+1,arecomputedsimilarly. Theimplicitlytreatedtractiontermisexpressedas tn+1=2,i,N,k=kun+1=2,i,)]TJ /F3 11.955 Tf 11.95 0 Td[(un+1=2,i, nk(A) andwiththedenition un+1=2,i,=uni,+un+1,i, 2(A) weget tn+1=2,i,N,k=k 2nk)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(un+1,i,)]TJ /F3 11.955 Tf 11.96 0 Td[(un+1,i,+uni,)]TJ /F3 11.955 Tf 11.96 0 Td[(uni,(A) Withthesedenitions,Eq.( A )canbewrittenas aui(ui)n+1,+Xkaui(ui)n+1,=bui(A) 181

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whereaui=V t+1 4Xkvn+1=2,qksk+1 RerefXkk 2nk1 n+1,sk (A)aui=1 4vn+1=2,qksk)]TJ /F4 11.955 Tf 22.15 8.09 Td[(1 RerefXkk 2nk1 n+1,sk (A) and bui=bui,1+bui,2+bui,3+bui,4(A) wherebui,1=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(V 4"@p @xin)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2+2@p @xin+1=2+@p @xin+3=2,q# (A)bui,2=1 RerefXktn+1=2,qi,D,k+tn+1=2,qi,T,k+k 2nk)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(uni,)]TJ /F3 11.955 Tf 11.95 0 Td[(uni,sk (A)bui,3= V t)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4Xkvn+1=2,qksk!(ui)n)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 4Xk(ui)nvn+1=2,qksk (A)bui,4=)]TJ /F3 11.955 Tf 9.3 0 Td[(V(ai)n+1=2 (A) From(ui)n+1,,weobtain un+1,i,=(ui)n+1, n+1,=(ui)n+1, n+1,q+1(A) byEq.( A )andn+1,q+1isgivenbyEq.( A ). CorrectorStep.SubtractingEq.( A )fromEq.( A )gives (ui)n+1,q+1=(ui)n+1,)]TJ /F4 11.955 Tf 13.15 8.08 Td[(t 4@p @xin+3=2(A) wherethecontributionsfromtheinviscidandviscoustermswereneglected.HMpresenttheequivalentofEq.( A )asbeingexactanddonotdiscusswhythecontributionsfromtheconvectiveanddiffusivetermscanbeneglected.FromEq.( A )onecan 182

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obtain un+1,q+1i,=un+1,i,)]TJ /F4 11.955 Tf 29.56 8.08 Td[(t 4n+1,q+1@p @xin+3=2(A) where,onaccountofEq.( A ),theappearanceofadensityratiomultiplyingun+1,i,isavoided.n+1,q+1iscomputedfromEqs.( A )and( A ).Thekineticenergybecomes,toO)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p2, (uiui)n+1,q+1=(uiui)n+1,)]TJ /F3 11.955 Tf 11.95 0 Td[(un+1,i,t 2n+1,q+1@p @xin+3=2(A) NowHMclaimthattheface-normalvelocitycanbecomputedfromEq.( A )bydottingitwiththefacenormalni,k.Thisapproachisnotrigorousbecauseeachfacehastwoadjacentcells.Instead,onemaydene vn+1,q+1k=1 2un+1,q+1i,+un+1,q+1i,ni,k=vn+1,k)]TJ /F4 11.955 Tf 13.15 8.09 Td[(t 8"1 n+1,q+1@p @xin+3=2+1 n+1,q+1@p @xin+3=2#ni,k(A) where vn+1,k=1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(un+1,i,+un+1,i,ni,k(A) Byassumingthatn+1,q+1=n+1,q+1=n+1,q+1k,oneobtains vn+1,q+1k=vn+1,k)]TJ /F4 11.955 Tf 29.56 8.08 Td[(t 8n+1,q+1k"@p @xin+3=2+@p @xin+3=2#ni,k(A) butthisisnotdesirablebecauseitwillverylikelyleadtodecouplingoftheface-normalvelocityandpressuregradient.SotheonlywaytoderiveHM'sexpressionforvn+1,q+1kistoforegotherigorousderivationandsubstituteapressuregradientnormaltothefacewhichisnotcomputedfromasimpleaverage, vn+1,q+1k=vn+1,k)]TJ /F4 11.955 Tf 29.56 8.08 Td[(t 4n+1,q+1k@p @nn+3=2k(A) where n+1,q+1k=1 2)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(n+1,q+1+n+1,q+1(A) 183

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whereEq.( A )isusedtocomputen+1,q+1.Theface-normalpressuregradientiscomputedas @p @nn+3=2k=pn+3=2)]TJ /F8 11.955 Tf 11.96 0 Td[(pn+3=2 nk(A) wherenk=kr)]TJ /F15 11.955 Tf 12.26 0 Td[(rkandristhepositionvectorofthecentroidofcontrolvolume.Thisguaranteesstrongcouplingbutisingeneralhighlyinaccurate(eveninconsistent)ondistortedgrids. A.4.3EnergyEquation TheapproximationsinEq.( A )willbedescribedterm-by-term. Time-DerivativeTerm.Thetime-derivativetermbecomes(excludingthecommonfactorM2refV),byusingtheappropriateaveragesandEq.( A ),1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(pn+3=2,q+pn+3=2+pn+1=2+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2n+1,q+1(uiui)n+1,q+1)]TJ /F11 11.955 Tf 11.96 16.85 Td[(1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(pn+1=2+pn)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2n(uiui)n (A) wherenandn+1,q+1arecomputedfromEqs.( A )and( A ).NowusingEq.( A ),weobtain(cf.Eq.(23)inHM)1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(pn+3=2,q+pn+3=2+pn+1=2+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2)]TJ /F8 11.955 Tf 5.47 -9.69 Td[(n+1,q+1(uiui)n+1,)]TJ /F8 11.955 Tf 11.96 0 Td[(n(uiui)n)]TJ /F8 11.955 Tf 13.15 8.09 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[(1 4un+1,i,t@p @xin+3=2)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(pn+1=2+pn)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2 (A) andwithEq.( A )wearriveatpn+3=2 2)]TJ /F8 11.955 Tf 13.15 8.09 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[(1 8Vun+1,i,tXkpn+3=2ni,ksk+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(n+1,q+1(uiui)n+1,)]TJ /F8 11.955 Tf 11.95 0 Td[(n(uiui)n+1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(pn+3=2,q)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2 (A) 184

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ConvectiveTerm.Theconvectivetermbecomes(excludingthecommonfactorM2reft),usingEq.( A ),Xk 4pn)]TJ /F10 7.97 Tf 6.59 0 Td[(1=2k+2pn+1=2k+pn+3=2,q+1k+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 8n+1=2k)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(un+1,q+1i,k+uni,k)]TJ /F3 11.955 Tf 12.95 -9.68 Td[(un+1,q+1i,k+uni,kvn+1,q+1k+vnk 2sk (A) andusingEqs.( A )and( A ),Xk" ~pn+1=2k+pn+3=2k 4!+)]TJ /F4 11.955 Tf 11.96 0 Td[(1 8n+1=2k)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(un+1,q+1i,k+uni,k)]TJ /F3 11.955 Tf 12.95 -9.69 Td[(un+1,q+1i,k+uni,k#"1 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(vn+1,k+vnk)]TJ /F4 11.955 Tf 29.56 8.09 Td[(t 8n+1,q+1k@p @nn+3=2k#sk (A) NowitistacitlyassumedthatEq.( A )appliesnotonlytovelocitiesatcellcenters,butalsotovelocitiesatfaces.Hencewewrite un+1,q+1i,k=un+1,i,k)]TJ /F4 11.955 Tf 29.56 8.09 Td[(t 4n+1,q+1k@p @xin+3=2k(A) wheren+1,q+1kisgivenbyEq.( A )andtherefore un+1,q+1i,k+uni,k=un+1,i,k+uni,k)]TJ /F4 11.955 Tf 29.56 8.09 Td[(t 4n+1,q+1k@p @xin+3=2k(A) Withthedenition un+1=2,i,k=uni,k+un+1,i,k 2(A) weobtain,toO)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(p2, )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(un+1,q+1i,k+uni,k)]TJ /F3 11.955 Tf 12.95 -9.68 Td[(un+1,q+1i,k+uni,k=4(uiui)n+1=2,k)]TJ /F4 11.955 Tf 25.16 8.09 Td[(t n+1,q+1kun+1=2,i,k@p @xin+3=2k(A) Withthedenitionsvn+1=2,k=1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(vn+1,k+vnk (A)k=~pn+1=2k+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 2n+1=2k(uiui)n+1=2,k (A) 185

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theconvectivetermcanbeexpressedasXk"k+ 4pn+3=2k)]TJ /F8 11.955 Tf 13.15 8.09 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[(1 8tn+1=2k n+1,q+1kun+1=2,i,k@p @xin+3=2k#"vn+1=2,k)]TJ /F4 11.955 Tf 29.56 8.09 Td[(t 8n+1,q+1k@p @nn+3=2k#sk (A) Equation( A )cannotbeusedtoapproximate@p @xin+3=2kbecausethelatterisafacegradient.HMappeartoemploytherelation @p @xin+3=2k=@p @nn+3=2kni,k(A) whichneglectscontributionsoftangentialcomponents.MultiplyingoutandusingEq.( A )toapproximatetheface-normalderivative,weobtaintoO)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p2,Xkkvn+1=2,ksk)]TJ /F4 11.955 Tf 11.95 0 Td[(kt 8n+1,q+1kpn+3=2)]TJ /F8 11.955 Tf 11.96 0 Td[(pn+3=2sk nk+ 4vn+1=2,kpn+3=2ksk)]TJ /F8 11.955 Tf 14.34 8.09 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[(1 8tn+1=2k n+1,q+1kvn+1=2,k2pn+3=2)]TJ /F8 11.955 Tf 11.95 0 Td[(pn+3=2sk nk# (A) Approximatingpn+3=2kas pn+3=2k=1 2pn+3=2+pn+3=2(A) andcollectingtermsgivesXkkvn+1=2,ksk+pn+3=2Xk 8vn+1=2,k+kt 8n+1,q+1k1 nk+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 8tn+1=2k n+1,q+1kvn+1=2,k21 nk!sk+Xk 8vn+1=2,k)]TJ /F4 11.955 Tf 11.95 0 Td[(kt 8n+1,q+1k1 nk)]TJ /F8 11.955 Tf 13.15 8.09 Td[()]TJ /F4 11.955 Tf 11.96 0 Td[(1 8tn+1=2k n+1,q+1kvn+1=2,k21 nk!pn+3=2sk (A) 186

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DivergenceTerm.Thedivergencetermcanbeexpressedas(excludingthecommonfactort) Xkvn+1=2ksk=Xkvn+1,q+1k+vnk 2sk(A) andusingEq.( A )gives Xkvn+1=2ksk=Xk"vn+1=2,k)]TJ /F4 11.955 Tf 29.56 8.09 Td[(t 8n+1,q+1k@p @nn+3=2k#sk(A) UsingEq.( A )andexpandingleadsto Xkvn+1=2ksk=Xkvn+1=2,ksk+pn+3=2Xkt 8n+1,q+1ksk nk)]TJ /F11 11.955 Tf 11.95 11.35 Td[(Xkt 8n+1,q+1kpn+3=2sk nk(A) ViscousandHeat-FluxTerms.Theviscoustermsdonotrequirefurthersimplication.ThermaltermsaretreateddifferentlythaninHM.Animplicitformulationisdevelopedforthermalux.Thermaluxcanbeexpressedas(excludingthecommonfactort RerefPr) Xk@T @nsk=XkTn+1=2)]TJ /F3 11.955 Tf 11.96 0 Td[(Tn+1=2 nsk(A) Likepressure,temperaturecanbeexpressedasTn+1=2=1 4)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Tn)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2+2Tn+1=2+Tn+3=2,q+1Tn+1=2=1 4)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Tn)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2+2Tn+1=2+Tn+3=2,q+T, (A) whereTisrelatedtopthroughequationofstateasfollows T=M2ref p.(A) 187

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MovingReferenceFrameSourceTerms.Sourcetermscanbeexpressedas(excludingthecommonfactor)]TJ /F4 11.955 Tf 9.3 0 Td[(()]TJ /F4 11.955 Tf 11.95 0 Td[(1)M2refVt)) (uiai)n+1=2=(ai)n+1=2un+1=2i,=(ai)n+1=2)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(un+1,q+1i,+uni, 2(A) UsingEq.( A ) (uiai)n+1=2=(ai)n+1=2un+1=2,i,)]TJ /F4 11.955 Tf 13.15 8.09 Td[(t 8(ai)n+1=2 n+1,q+1@p @xin+3=2(A) UsingEq.( A ) (uiai)n+1=2=(ai)n+1=2un+1=2,i,)]TJ /F4 11.955 Tf 19.52 8.09 Td[(t 16V(ai)n+1=2 n+1,q+1Xkpn+3=2ni,ksk(A) FinalForm.Collectingtermsanddividingthroughbyt(againwedonotdividebyVtosimplifytheimplementation)gives,nally, appn+3=2+Xkappn+3=2=bp(A) where ap=ap,1+ap,2+ap,3(A) andap,1=M2ref 2V t (A)ap,2=M2refXk 8vn+1=2,k+kt 8n+1,q+1k1 nk+)]TJ /F4 11.955 Tf 11.95 0 Td[(1 8tn+1=2k n+1,q+1kvn+1=2,k21 nk!sk (A)ap,3=Xkt 8n+1,q+1ksk nk (A) wherevn+1=2,kiscomputedfromEq.( A )andn+1,q+1kisgivenbyEq.( A ).Similarly, ap=ap,1+ap,2+ap,3+ap,4(A) 188

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whereap,1=)]TJ /F8 11.955 Tf 10.49 8.09 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[(1 8M2refun+1,i,ni,ksk (A)ap,2=M2ref 8vn+1=2,k)]TJ /F4 11.955 Tf 11.96 0 Td[(kt 8n+1,q+1k1 nk)]TJ /F8 11.955 Tf 13.15 8.09 Td[()]TJ /F4 11.955 Tf 11.95 0 Td[(1 8tn+1=2k n+1,q+1kvn+1=2,k21 nk!sk (A)ap,3=)]TJ /F4 11.955 Tf 26.9 8.08 Td[(t 8n+1,q+1ksk nk (A)ap,4=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(()]TJ /F4 11.955 Tf 11.96 0 Td[(1) 16M2reft(ai)n+1=2 n+1,q+1ni,ksk (A) and bp=bp,1+bp,2+bp,3+bp,4+bp,5(A) wherebp,1=)]TJ /F4 11.955 Tf 9.3 0 Td[(M2refV t)]TJ /F4 11.955 Tf 11.96 0 Td[(1 2)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(n+1,q+1(uiui)n+1,)]TJ /F8 11.955 Tf 11.95 0 Td[(n(uiui)n+1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(pn+3=2,q)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)]TJ /F10 7.97 Tf 6.58 0 Td[(1=2 (A)bp,2=)]TJ /F4 11.955 Tf 9.3 0 Td[(M2refXkkvn+1=2,ksk (A)bp,3=)]TJ /F11 11.955 Tf 11.3 11.36 Td[(Xkvn+1=2,ksk (A)bp,4=()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M2ref RerefXk(tjuj)n+1=2ksk+1 RerefPrXk@T @nn+1=2ksk (A)bp,5=)]TJ /F4 11.955 Tf 9.3 0 Td[(()]TJ /F4 11.955 Tf 11.95 0 Td[(1)M2refV(ai)n+1=2un+1=2,i, (A) A.4.4Summary Thefollowingisasummaryofthenecessarystepsinthesolutionapproachforeachtimestepn: 1. Initializeiterates:un+1,0i,=uni,,n+3=2,0=n+1=2,pn+3=2,0=pn+1=2,Tn+3=2,0=Tn+1=2,vn+1,0k=vnk. 2. SolveEq.( A )forn+3=2,q+1. 189

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3. SolveEq.( A )togetthepredictedmomenta(ui)n+1,. 4. Computepredictedcell-centeredvelocitiesun+1,i,fromEq.( A ). 5. Computepredictedface-normalvelocitiesvn+1,kfromEq.( A ). 6. SolveEq.( A )forpn+3=2. 7. Updatepn+3=2,q+1fromEq.( A ). 8. ComputegradientofpressurecorrectionfromEq.( A )with=pn+3=2. 9. Update(ui)n+1,q+1fromEq.( A ). 10. Updateun+1,q+1fromEq.( A ). 11. Checkforconvergenceofdensity,pressurecorrection,andvelocities.Ifconverged,proceedtonextiteration(n n+1,andgotoStep1),otherwisecarryoutadditionalpredictor-correctorstep(q q+1,andgotoStep2). A.5BoundaryConditionImplementationforNon-DissipativeSolver A.5.1SolidWalls Inthiswork,wallboundaryconditionariseattheparticlesurface.Forinviscidow,particlesurfaceactsasaslipwallandforviscousowitactsasano-slipwall.Onaslipwall,theuidmusthaveazerovelocitynormaltothewall,i.e.,vb=0.Onano-slipwall,(ui)b=0,andthusboththenormalandtangentialcomponentsofuidvelocityarezero.Thepressureanddensityatthewallareextrapolatedfromtheinteriorcell.Velocitygradientsattheboundaryfacecanbecomputedfrom @ui @xjnb=(ui)n nnj,(A) wherenbedistanceofboundaryfacefrominteriorcellcentroid.StressesattheboundaryfacecanbecomputedusingEqs.( A ),( A ),( A ),( A )andvelocitygradientsfromEq.( A ).Thecontributionsfromaboundaryfacetotheglobalsystemmatrixforcontinuityandmomentumequationareshownintable A-1 ,whereti,Dandti,TaregivenbyEq.( A )andsbisareaofboundaryface.Aslipwallcanbedistinguishedfromano-slipwallbymaking=0. 190

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Atasolidwallboundary,aghostcellisassumedinwhichthepressureanddensityareassignedthesamevaluesasthatininteriorcellandthevelocityisassignedasthemirrorimageofvelocityininteriorcell.ThusinEq.( A )p=pcanbeassumed,andsoatermsgotoa.Notconsideringsourcetermsduetomovingreferenceframe,thecontributionsfromwallboundaryfacetotheglobalsystemmatrixforenergyequationareshownintable A-1 TableA-1. Wallboundarycondition. ab Continuityeq.00Momentumeqs.1 Reref nn+1,sb1 Reref(uni, n+ti,D+ti,T)sbEnergyeq.)]TJ /F16 7.97 Tf 10.49 5.25 Td[()]TJ /F10 7.97 Tf 6.59 0 Td[(1 8M2refun+1,i,ni,ksk1 RerefPr)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(@T @nn+1=2ksk Foranisothermalboundary@T=@niscomputedas(T)]TJ /F3 11.955 Tf 12.34 0 Td[(Tb)=n.Boundarywithanimposedheatuxarehandledeasilybyreplacingtemperaturederivativeatthewall. A.5.2Fareld Fareldboundaryconditionariseattheouterboundarywhereuniformowconditionsareassumedtoapply.Thisassumptionisvalidiftheouterboundaryisplacedatalargedistancefromtheparticle.Letb,pb,(ui)bdenotetheuniformowconditionsimposedatthefareldboundary.Theface-normalowvelocityattheboundaryisvb=(ui)b(ni)b,Thecontributionsfromafareldboundaryfacestoglobalsystemmatrixareshownintable A-2 ,wheresbisareaofboundaryface. TableA-2. Fareldboundarycondition. ab Continuityeq.0)]TJ /F8 11.955 Tf 9.3 0 Td[(bvbsbMomentumeqs.0)]TJ /F8 11.955 Tf 9.3 0 Td[(b(ui)bvbsbEnergyeq.0)]TJ /F11 11.955 Tf 11.29 9.69 Td[()]TJ /F8 11.955 Tf 5.48 -9.69 Td[(pb+)]TJ /F10 7.97 Tf 6.58 0 Td[(1 2b(uiui)b+1vbsb A.6AbsorbingBoundaryConditions Thepresentworkstudiestheowaroundparticleinauidofinniteexpanse.Ideally,theouterboundariesshouldbeplaceatinnity.Toreducecomputationalcost, 191

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articialboundariesareusedtotruncatetheinnitedomain.Invariably,therearespuriousreectionsfromtheboundaries.Suchreectionscancorruptthesolutionneartheparticle.Absorbingboundaryconditionsareneededtoavoidoratleastreducespuriouswavereectionsfromarticialboundaries. A.6.1CharacteristicBoundaryConditions CharacteristicboundaryconditionsfortheEulerandtheNavier-StokesequationsbasedontheNSCBCmethodofPoinsotandLele[ 85 ]areused.TraditionallyNSCBCboundaryconditionswereimplementedinnite-differenceformulation.Inthecurrentwork,NSCBCisimplementedintheDissipativesolverinanite-volumeformulation.GradientsofsolutionvariablesattheboundaryarerequiredtoestimatethewavestrengthsintheNSCBCmethod.Thesegradientsarecomputedusingahigher-ordernite-differenceapproximation. A.6.2SpongeLayer Spongelayersarespecictypeofabsorbinglayersusedtodampthedisturbancespriortointeractionwithanboundary.Onesimplewaytoachievethisisbyaddingalinearfrictiontermtothegoverningequationstodampthedifferenceofsolutionwithrespecttosomereferencesolution, @q @t+L(q)=(q)]TJ /F15 11.955 Tf 11.95 0 Td[(qref).(A) whereqissolutionvector,Lisanon-linearoperator,isfrictioncoefcient,andqrefisreferencesolutionstate. Thefrictioncoefcientcanbeafunctionofspace.Typically,variessmoothlyfromzerotoalargevalueneartheboundarytoavoidspuriousreectionfromtheinterfacebetweenthespongelayerandtheinteriordomain.SeeIsraeliandOrszag[ 48 ]andColonius[ 23 ]formoredetailsonspongelayers. 192

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A.7MovingReferenceFrame Thissectioncontainsderivationofsourcetermsingoverningequationsinmovingreferenceframe.Eulerequationsaretakenasexampletoshowderivation,thoughsourcetermsduetomovingreferenceframeremainssameinNavier-Stokesequations. A.7.1EulerEquations Eulerequationscanbewrittenas@ @t+r(u)=0 (A)@u @t+r(uu)=r(p) (A)@E @t+r(Eu)=r(pu) (A) whereuisuidvelocity,risdivergenceoperatorandothersymbolshaveusualmeaningasexplainedinSection A.1.1 A.7.2CoordinateTransformation Considerastationaryreferenceframe(x,y,z,t)andamovingreferenceframe(x0,y0,z0,t0)movingwithvelocityup=(up,x,up,y,up,z).Variablesinmovingandstationaryreferenceframesarerelatedasfollows, t0=tr0=r)]TJ /F15 11.955 Tf 11.96 0 Td[(rp(A) whererp=(xp,yp,zp).Derivativesarerelatedasfollows, r=r0@ @t=@ @t0)]TJ /F15 11.955 Tf 11.95 0 Td[(upr0(A) A.7.3TransformationoftheEulerEquationstoMovingReferenceFrame Takecontinuityequations @ @t+r(u)=0(A) 193

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Transformingtomovingreferenceframe@ @t0)]TJ /F15 11.955 Tf 11.96 0 Td[(upr0()+r0((u0+up))=0,@ @t0)]TJ /F15 11.955 Tf 11.96 0 Td[(upr0()+r0(u0)+r0(up)=0,@ @t0+r0(u0)=0. (A) Re-writingmomentumequation@u @t+r(uu)=r(p). (A) Transformingmomentumequationtomovingreferenceframe@(u0+up) @t0)]TJ /F15 11.955 Tf 11.96 0 Td[(upr0((u0+up))+r0((u0+up)(u0+up))=r0(p),@u0 @t0+@up @t0)]TJ /F15 11.955 Tf 11.95 0 Td[(upr0((u0+up))+r0((u0u0+u0up))+upr0((u0+up))=r0(p). (A) Furtherrearranging@u0 @t0+r0(u0u0)+@up @t0+up@ @t0+r0(u0u0)=r0(p),@u0 @t0+r0(u0u0)=r0(p))]TJ /F8 11.955 Tf 11.95 0 Td[(@up @t0 (A) wheretheexpressionwithinsquarebracketisidenticallyzerobeingcontinuityequation. Considerenergyequation @E @t+r(Eu)=r(pu)(A) Transformingtomovingreferenceframe@E @t0)]TJ /F15 11.955 Tf 11.95 0 Td[(upr0(E)+r0(Eu0)+r0(Eup)=r0(pu0))-221(r0(pup),@E @t0+r0.(Eu0)=r0(pu0))]TJ /F15 11.955 Tf 11.95 0 Td[(upr0(p) (A) 194

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EnergyEcanbeexpandedas E=1 2u2+p )]TJ /F4 11.955 Tf 11.95 0 Td[(1=1 2u02+1 2up2+u0.up+p )]TJ /F4 11.955 Tf 11.96 0 Td[(1=E0+1 2up2+u0.up(A) where E0=1 2u02+p )]TJ /F4 11.955 Tf 11.95 0 Td[(1.(A) Substitutingaboveexpressioninenergyequation@E0 @t0+1 2@up2 @t0+@u0up @t0+r0(E0u0)+r01 2up2u0+r0((u0.up)u0)=r0(pu0))]TJ /F15 11.955 Tf 11.95 0 Td[(upr0(p) (A) Rearranging@E0 @t0+r0(E0u0)+1 2up2@ @t0+r0(u0)+up@u0 @t0+r0(u0u0)+r0(p)+@up @t0=r0(pu0))]TJ /F8 11.955 Tf 11.96 0 Td[(u0@up @t0 (A) Theexpressionsinsiderstandsecondsquarebracketsareidenticallybeingcontinuityandmomentumequationsrespectively. @E0 @t0+r0(E0u0)=r0(pu0))]TJ /F8 11.955 Tf 11.95 0 Td[(u0@up @t0(A) Insummary,theEulerequationsinmovingreferenceframeare@ @t0+r0(u0)=0,@u0 @t0+r0(u0u0)=r0(p))]TJ /F8 11.955 Tf 11.96 0 Td[(@up @t0,@E0 @t0+r0(E0u0)=r0(pu0))]TJ /F8 11.955 Tf 11.95 0 Td[(u0@up @t0 (A) CorrespondingNavier-StokesequationsinmovingreferenceframearegiveninEq.( A ). 195

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A.8AxisymmetricComputations ThisSectioncontainsderivationofaxisymmetricowequationsanddetailsoftheirimplementationin2Dcodes.Eulerequationsareusedasanexampleinthefollowingdescription,thoughsameargumentsapplyforNavier-Stokesequationsalso. A.8.1EulerEquationsforAxisymmetricFlows Eulerequationsinthecylindricalcoordinatescanbewrittenas, @ @t+1 r@ @r(rur)+1 r@ @(u)+@ @z(uz)=0(A) @ur @t+1 r@ @r)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(rur2+1 r@ @(uru)+@ @z(uruz)=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(@p @r(A) @u @t+1 r@ @r(ruru)+1 r@ @)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(u2+@ @z(uuz)=)]TJ /F4 11.955 Tf 10.5 8.09 Td[(1 r@p @(A) @uz @t+1 r@ @r(ruruz)+1 r@ @(uuz)+@ @z)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(uz2=)]TJ /F8 11.955 Tf 10.49 8.09 Td[(@p @z(A) @e @t+1 r@ @r(reur)+1 r@ @(eu)+@ @z(euz)=)]TJ /F11 11.955 Tf 11.29 16.85 Td[(1 r@rpur @r+1 r@pu @+@puz @z(A) Simplifyingaboveequationsfortheaxisymmetriccase,i.e.u=0and@ @=0, @ @t+1 r@ @r(rur)+@ @z(uz)=0(A) @ur @t+1 r@ @r)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(rur2+@ @z(uruz)=)]TJ /F8 11.955 Tf 10.5 8.09 Td[(@p @r(A) @uz @t+1 r@ @r(ruruz)+@ @z)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(uz2=)]TJ /F8 11.955 Tf 10.49 8.08 Td[(@p @z(A) @e @t+1 r@ @r(reur)+@ @z(euz)=)]TJ /F11 11.955 Tf 11.29 16.86 Td[(1 r@rpur @r+@puz @z(A) Rearrangingterms, @ @t+1 r@ @r(r(ur))+@ @z(uz)=0(A) @ur @t+1 r@ @r)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(r(ur2+p)+@ @z(uruz)=p r(A) @uz @t+1 r@ @r(r(uruz))+@ @z)]TJ /F4 11.955 Tf 5.48 -9.68 Td[((uz2+p)=0(A) @e @t+1 r@ @r(r(e+p)ur)+@ @z((e+p)uz)=0(A) 196

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Mappingz-axisandr-axisofthecylindricalcoordinatestox-axisandy-axisoftheCartesiancoordinatesrespectively,wecanexpressEulerequationsas @ @t+@ @x(ux)+1 y@ @y(y(uy))=0(A) @ux @t+@ @x)]TJ /F4 11.955 Tf 5.48 -9.68 Td[((ux2+p)+1 y@ @y(y(uxuy))=0(A) @uy @t+@ @x(uxuy)+1 y@ @y)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(y(uy2+p)=p y(A) @e @t+@ @x((e+p)ux)+1 y@ @y(y(e+p)uy)=0(A) DeningQ,Fx,FyandHas Q=0BBBBBBB@uxuye1CCCCCCCA;Fx=0BBBBBBB@uxux2+puxuy(e+p)ux1CCCCCCCA;Fy=0BBBBBBB@uyuxuyuy2+p(e+p)uy1CCCCCCCA;H=0BBBBBBB@00p y01CCCCCCCA,(A) whereFxrepresentstheuxinthex-directionandFyrepresentstheuxinthey-direction.Fluxinthedirectioniszero.Eulerequationscannowbeexpressedas, @ @t(Q)+@ @x(Fx)+1 y@ @y(yFy)=(H)(A) A.8.2VolumetricandSurfaceIntegration Atypicalnitevolumecellinthecylindricalcoordinatescanbechoseninsuchawaythatfacesofcontrolvolumearealongcurvilinearcoordinatedirections.FaceareasareAx=ydyd,Ay=ydxdandA=dxdyandcellvolumeisV=ydxdyd.IntegratingtheEulerequationsoverthecontrolvolumewouldyield, @ @t(Q)V+(Fx)Ax+(Fy)Ay=(H)V(A) 197

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SubstitutingAX,AyandVintheaboveequationyields, @ @t(Q)ydxdy+(Fx)ydy+(Fy)ydx=(H)ydxdyd(A) @ @t(Q)ydxdy+(Fx)ydy+(Fy)ydx=(H)ydxdy(A) @ @t(yQ)dxdy+(yFx)dy+(yFy)dx=(yH)dxdy(A) Solvingaxi-symmetricEulerequationEq.( A )inthecylindricalcoordinatesisequivalenttosolvingfollowingequationin2Dcartesiancoordinatesystem. @ @t(yQ)+@ @x(yFx)+@ @y(yFy)=(yH)(A) TherearethreepossiblewaystosolveaxisymmetricEulerequations. 1. FirstistomodifyQ,Fx,FyandHquantitiesbymultiplyingeachbyyandintegratingover2Dcartesianmesh.Integrationover2Dmeshcanbeequivalentlydonebyusingsinglelayered3Dmeshwithsinglelayerofunitlengthinz-directionprovidednouxcomputationisdoneonz-facesofeachcell. 2. Secondistomodifythemeshinsuchawaytogetafactorofyinvolumeandsurfaceareas.Thiscanbedonebyscalingz-coordinatesoftoplayerof3Dmeshdescribedabovebyvalueofy-coordinate.IntheoriginalmeshacontrolvolumemadebyfacesalongcurvilinearcoordinatelineswouldhaveV=dx.dy.1(aswidthalongz-directionis1),Ax=dy.1,Ay=dx.1andAz=dxdy.Azisnotneededasz-facesarevirtualfaces.Nowconsidermodiedmesh.ThesegeometricquantitiesbecomeV=ydxdy,Ax=ydyandAy=ydx.SowhileusingmodiedmeshowquantitiesQ,Fx,FyandHneednotbemodied.IntegrationcanbedoneasinEq.( A ).Substitutingforgeometricquantitiesweget. @ @t(Q)ydxdy+(Fx)ydy+(Fy)ydx=(H)ydxdy(A) whichissameasintegratingEq.( A )intheCartesiancoordinatesystemontheoriginalmesh. 3. Thirdistosimplyscalevolumesandsurfaceareasw/oactuallydistortingthemesh. 198

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BIOGRAPHICALSKETCH ManojParmarwasbornin1981inBaseri,Rajasthan,India.Hisearlyschoolingtill7thgradewasdoneinBaseri.ThereafterhisfamilymovedtoSriGanganagarandlaterpartofschoolingwascontinuedatCentralschoolatSriGangananagar.HereceivedBachelorofTechnologyandMasterofTechnologydegreesfromtheIndianInstituteofTechnology(IIT),Mumbaiin2003.HejoinedtheUniversityofIllinoisatUrbana-Champaign(UIUC)in2004forfurtherstudies.HeearnedM.S.degreeinAerospaceEngineeringin2006andcontinuedtostudytowardsPh.Ddegree.LaterhemovedtoUniversityofFlorida,Gainesvillein2007tocontinuehisPh.D.study.ManojParmarreceivedhisPh.D.degreefromtheUniversityofFloridainthefallof2010.HecontinuedaspostdocintheUniversityofFlorida.ManojParmarhasbeenmarriedtoSonaParmarsinceDecember2008. 207