Table Of ContentCOUNTERFLOWS
Counterflowsplayimportantrolesinnatureandtechnology.Anatural
exampleistheGulfStreamandtheoppositeflowintheoceandepths.
Technological applications include hydrocyclones, vortex tubes, and
vortexcombustors.Theseelongatedcounterflowsarewildlyturbulent
but survive intense mixing, a seeming paradox. Local counterflows,
whosespatialextentissmallcomparedwiththatofsurroundingflows,
occurbehindbluffbodiesandinswirlingstreams.Thelatterareoften
referredtoasvortexbreakdownbubbles,whichoccurintornadoesand
abovedeltawings.Mostscalecounterflowsarecosmicbipolarjets.Most
miniaturecounterflowsoccurincapillarymenisciofelectrospraysand
fuelatomizers.Thisbookdiscussesthephysicalmechanismsthatdrive
counterflows, examining how they emerge, develop, become double
and multiple counterflows, and comprise both global and local
circulations.
Dr. Vladimir Shtern received his Doctor of Sciences in 1978, and was
made a Professor in 1990, at the Institute of Thermal Physics in
Novosibirsk.Sincethenhehasheldnumerousconsulting,faculty,and
visiting faculty positions at institutions including the University of
Houston, the University of Seville, the DLR Institute of Fluid
Mechanics in Gottingen, and the University of Bristol, and as the
Senior Scientist at General Vortex Energy, Inc., in Missouri City,
Texas. He served as a consultant for Shell US and BP-Amoco
Exploration companies. Dr. Shtern is a specialist in fluid mechanics
and heat transfer with a focus on vortex flows, thermal convection,
combustion, hydrodynamic instability, and bifurcation theory. His
results include analytical solutions of the Navier-Stokes, Boussinesq,
magneto-hydrodynamic(MHD),andcompressiblegasequations,and
explaining mechanisms of intriguing and practically important flow
effects, such as multiple solutions, hysteretic transitions, and vortex
breakdown. Dr. Shtern is the author of three books and more than
onehundredpapersinrefereedjournals.
Counterflows
PARADOXICAL FLUID MECHANICS PHENOMENA
Vladimir Shtern
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
32AvenueoftheAmericas,NewYork,NY10013-2473,USA
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107027596
©VladimirShtern2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedStatesofAmerica
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloginginPublicationdata
Shtern,V.N.(VladimirNikolaevich),1940–
Counterflows:paradoxicalfluidmechanicsphenomena/VladimirShtern.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-107-02759-6(hardback)
1. Counterflows(Fluiddynamics) 2. Fluidmechanics. I. Title.
TA357.5.C68S54 2012
5320.052–dc23 2012006878
ISBN978-1-107-02759-6
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
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Contents
Acknowledgments pagexiii
1 Introduction 1
1.1. NaturalandTechnologicalCounterflows 1
1.2. PhysicalMechanismsofCounterflows 2
1.2.1. Accumulation 2
1.2.2. SwirlEffect 4
1.2.3. Separation 7
1.2.4. ThermalConvection 8
1.3. CounterflowApplications,Control,andStability 9
1.4. Approach 10
2 AccumulationCounterflows 11
2.1. ConicalSimilarityFlows 11
2.2. ConicalJets 13
2.2.1. EquationReduction 13
2.2.2. JetintheFreeSpace 14
2.2.3. JetAboveaPlane 15
2.3. Super-CollimatedJet 16
2.4. CapillaryJet 20
2.4.1. FeaturesofCapillaryJets 20
2.4.2. ConicalSimilarityModeloftheMeniscusFlow 22
2.4.3. NumericalSimulationsoftheCone-JetFlow 24
3 BifurcationofSwirlinConicalCounterflows 28
3.1. ObservationsofSpontaneousSwirlAppearance 28
3.2. BifurcationofSwirlasSymmetryBreaking 30
3.2.1. ReductiontoaBoundary-ValueODEProblem 30
3.2.2. NecessaryConditionsforSwirlBifurcation 32
3.3. SwirlAppearanceinCapillaryFlows 33
3.3.1. Two-MediumFlows 33
3.3.2. SwirlOrigination 34
v
vi Contents
3.3.3. SwirlDevelopment 35
3.3.4. Two-CellCirculation 37
3.3.5. CollimatedAnnularJets 38
3.3.6. SwirlBifurcationintheMeniscusFlow 40
3.4. SwirlAppearanceinElectro-VortexFlows 43
3.4.1. ProblemFormulation 43
3.4.2. ForcedSwirl 44
3.4.3. Multi-CellCounterflows 46
3.4.4. Self-Swirling 51
3.5. MechanismofSwirlAppearanceinConicalFlows 56
3.5.1. ComparisonofSelf-SwirlingCapillaryand
Electro-VortexFlows 56
3.5.2. MechanismofSwirlAccumulation 56
3.5.3. DestroyedBifurcation 58
4 BifurcationofCounter-Swirl 60
4.1. OutlineofStabilityandBifurcationFeatures 60
4.2. ParallelJetlikeFlows 62
4.3. SecondaryFlows 62
4.4. TheLyapunov-SchmidtMethod 63
4.5. BifurcationsintheJetlikeFlows 66
4.6. MHDFlowinanAnnularPipe 68
4.7. SolvingStabilityProblemsforLargeRe 68
4.8. BifurcationsintheAnnular-PipeFlows 70
5 ConicalCounterflowsDrivenbySwirl 73
5.1. SwirlingJetAboveaPlane 73
5.1.1. ReductiontoaBoundary-ValueODEProblem 73
5.1.2. AsymptoticAnalysisofTwo-CellFlow 75
5.1.3. Hysteresis 79
5.1.4. VortexBreakdown 83
5.1.5. VortexConsolidation 85
5.1.6. CuspCatastrophe 87
5.1.7. Near-PlaneOutflow 90
5.2. AHalf-LineVortexinaFreeSpace 92
5.2.1. TornadoandDelta-WingVortices 92
5.2.2. MultipleSolutions 94
5.2.3. ModelingTurbulentVortexBreakdown 98
5.3. SwirlingJetsinConicalRegions 98
5.3.1. SuctionDevicesandTheirModeling 98
5.3.2. AsymptoticAnalysis 100
5.3.3. DecompositionoftheFlowForce 102
5.3.4. DescendingOne-CellFlow 103
5.3.5. AscendingOne-CellFlow 106
5.3.6. FlowInsidetheθ =45°Cone 107
c
5.3.7. FlowOutsidetheθ =45°Cone 109
c
5.3.8. PressurePeakinSwirlingAnnularJets 112
Contents vii
5.4. Super-CollimationinSwirlingCounterflows 114
5.4.1. BipolarJetInducedbyVortex-SinkAccretion 114
5.4.2. AnalysisofSuper-Collimation 115
5.4.3. Vortex-WallInteractionasaModelTornado 118
6 JetlikeSwirlingCounterflows 122
6.1. Power-LawJets 122
6.1.1. Introduction 122
6.1.2. ProblemFormulation 123
6.1.3. FeaturesofPower-LawJets 125
6.2. AnalyticalModelingofMultipleCounterflows 132
6.2.1. Motivation 132
6.2.2. GeneralizedVortexSink 134
6.2.3. ShapeoftheSurfaceofRevolution 136
6.2.4. InnerSolutions 139
6.2.5. CompositeVortexSink 142
6.2.6. ApplicationsoftheGeneralized
VortexSink 148
6.2.7. ApplicationsoftheCompositeSolutions 152
6.3. SwirlingCounterflowsinaCapillaryMeniscus 160
6.3.1. EffectsofSwirlingGasJet 160
6.3.2. AnalysisofChangingFlowTopology 164
7 SwirlingCounterflowsinCylindricalDevices 168
7.1. Swirl-DecayMechanism 168
7.1.1. ElongatedCounterflows 168
7.1.2. ProblemFormulation 168
7.1.3. ModelingSwirlDecay 170
7.1.4. VelocityProfiles 172
7.1.5. PressureDistribution 173
7.1.6. End-WallEffects 175
7.2. ModelingCounterflowsinVortexSeparators 177
7.2.1. Introduction 177
7.2.2. TwoFlowComponents 178
7.2.3. CoreFlowFeatures 181
7.2.4. FlowApproximationNearEndWalls 184
7.2.5. ParticleTrajectories 185
7.2.6. PressureDistribution 187
7.2.7. CentrifugalStratification 188
7.2.8. SummaryoftheAsymptoticAnalysis 189
7.3. NumericalStudyofVortexBreakdownandDouble
Counterflow 190
7.3.1. TechnologicalImportanceofLocalandGlobal
Circulations 190
7.3.2. FormulationoftheNumericalProblem 190
7.3.3. DevelopmentofGlobalCounterflowasSwirlNumber
Increases 192
viii Contents
7.3.4. DevelopmentofGlobalCounterflowasReIncreases 193
7.3.5. ComparisonwiththeAsymptoticTheory 194
7.3.6. VortexBreakdownDevelopment 196
7.3.7. DoubleCounterflowDevelopment 199
7.3.8. SummaryofDoubleCounterflowFeatures 205
7.4. DoubleCounterflowinaVortexTrap 206
7.4.1. TechnologicalImportanceofVortexTraps 206
7.4.2. DevelopmentofaGlobalCounterflow 207
7.4.3. AnalyticalApproximationoftheGlobalCounterflow
intheVortexTrap 209
7.4.4. SolidParticleTrajectoryintheSingleCounterflow 210
7.4.5. DoubleCounterflowintheVortexTrap 212
7.4.6. SolidParticleTrajectoriesintheDoubleCounterflow 213
7.4.7. DevelopmentofKarmanVortexStreet 216
7.4.8. SummaryoftheVortexTrapFeatures 217
8 SeparationCounterflows 219
8.1. CounterflowsinaPlaneDivergingChannel 219
8.1.1. BriefLiteratureReview 219
8.1.2. ProblemFormulation 220
8.1.3. PatternsofJeffery-HamelCounterflows 221
8.1.4. Scaling 223
8.1.5. Counting 223
8.2. CounterflowsDuetoBifurcationsofVortexSourceFlow 225
8.2.1. EquationsforDisturbances 225
8.2.2. BifurcationCharacter 227
8.2.3. PhasePatternandAsymptoticFeatures 228
8.2.4. SpiralVortices 229
8.3. StabilityofPlaneCounterflows 233
8.3.1. Approach 233
8.3.2. StabilityofVortex-SourceFlow 235
8.3.3. SpatialStabilityoftheJeffery-HamelFlow 236
8.4. TransitionFlows 238
8.4.1. JetintheSinkFlow 238
8.4.2. TripolarJet 240
8.4.3. AttachmentFlowintheDivergingChannel 242
8.4.4. JetEmergingfromaSlitinaWall 243
8.4.5. JetEmergingfromaThinPlaneChannel 245
8.5. SummaryofPlaneCounterflowFeatures 247
8.5.1. SpatialInstability 247
8.5.2. FurtherApplications 248
8.5.3. Limitations 250
8.6. CounterflowsDuetoInternalSeparationinSpatialConical
Flows 250
8.6.1. Introduction 250
8.6.2. GoverningEquations 252
8.6.3. BasicFlows 253
