Fluid mechanics: with problems and solutions, and an aerodynamic laboratory

363 Pages·2005·7.43 MB·English

by  Egon Krause

Egon Krause
Fluid Mechanics
Egon Krause
Fluid Mechanics
With Problems and Solutions,
and an Aerodynamic Laboratory
With 607 Figures
Prof. Dr. EgonKrause
RWTHAachen
AerodynamischesInstitut
Wu¨llnerstr.5-7
52062Aachen
Germany
ISBN3-540-22981-7 SpringerBerlinHeidelbergNewYork
LibraryofCongressControlNumber: 2004117071
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Preface
During the past 40 years numerical and experimental methods of fluid mechanics were sub-
stantially improved. Nowadays time-dependent three-dimensional flows can be simulated on
high-performance computers, and velocity and pressure distributions and aerodynamic forces
and moments can be measured in modern wind tunnels for flight regimes, until recently not
accessibleforresearchinvestigations.Despiteofthisimpressivedevelopmentduringtherecent
pastandeven100yearsafterPrandtlintroducedtheboundary-layertheory,thefundamentals
arestillthestartingpointforthesolutionofflowproblems.Inthepresentbooktheimportant
branches of fluid mechanics of incompressible and compressible media and the basic laws de-
scribingtheircharacteristicflowbehaviorwillbeintroduced.Applicationsoftheselawswillbe
discussedinawaysuitableforengineeringrequirements.
Thebookisdividedintothesixchapters:FluidmechanicsIandII,exercisesinfluidmechan-
ics, gas dynamics, exercises in gasdynamics, and aerodynamics laboratory. This arrangement
followsthestructureoftheteachingmaterialinthefield,generallyacceptedandapprovedfor
a long time at German and foreign universities. In fluid mechanics I, after some introductory
statements, incompressible fluid flow is described essentially with the aid of the momentum
andthemomentofmomentumtheorem.InfluidmechanicsIItheequationsofmotionoffluid
mechanics,theNavier-Stokesequations,withsomeoftheirimportantasymptoticsolutionsare
introduced. It is demonstrated, how flows can be classified with the aid of similarity param-
eters, and how specific problems can be identified, formulated and solved. In the chapter on
gasdynamicstheinfluenceofvariabledensityonthebehaviorofsubsonicandsupersonicflows
isdescribed.
IntheexercisesonfluidmechanicsIandIIandongasdynamicsthematerialdescribedinthe
previous chapters is elaborated in over 200 problems, with the solutions presented separately.
Itisdemonstratedhowthefundamentalequationsoffluidmechanicsandgasdynamicscanbe
simplifiedforthevariousproblemformulationsandhowsolutionscanbeconstructed.Numer-
ical methods are not employed. It is intended here, to describe the fundamental relationships
in closed form as far as possible, in order to elucidate the intimate connection between the
engineering formulation of fluid-mechanical problems and their solution with the methods of
appliedmathematics.Intheselectionoftheproblemsitwasalsointended,toexhibitthemany
differentformsofflows,observedinnatureandtechnicalapplications.
Becauseofthespecialimportanceofexperimentsinfluidmechanics,inthelastchapter,aero-
dynamicslaboratory,experimentaltechniquesareintroduced.Itisnotintendedtogiveacom-
prehensive and complete description of experimental methods, but rather to explain with the
descriptionofexperiments,howinwindtunnelsandothertestfacilitiesexperimentaldatacan
beobtained.
A course under the same title has been taught for a long time at the Aerodymisches Institut
of the RWTH Aachen. In the various lectures and exercises the functioning of low-speed and
supersonic wind tunnels and the measuring techniques are explained in experiments, carried
out in the facilities of the laboratory. The experiments comprise measurements of pressure
distributionsonahalfbodyandawingsection,ofthedragofasphereinincompressibleand
compressible flow, of the aerodynmic forces and their moments acting on a wing section, of
velocityprofilesinaflat-plateboundarylayer,andoflossesincompressiblepipeflow.Another
important aspect of the laboratory course is to explain flow analogies, as for example the
VI Preface
so-called water analogy, according to which a pressure disturbance in a pipe, filled with a
compressible gas, propagates analogously to the pressure disturbance in supercritical shallow
waterflow.
This book was stimulated by the friendly encouragement of Dr. M. Feuchte of B.G. Teubner-
Verlag. My thanks go also to Dr. D. Merkle of the Springer-Verlag, who agreed to publish
theEnglishtranslationoftheGermantext.Gratefulacknowledgementisduetomysuccessor
Professor Dr.-Ing. W. Schro¨der, who provided personal and material support by the Aerody-
namischesInstitutinthepreparationofthemanuscript.IamindebtedtoDr.-Ing.O.Thomer
who was responsible for the preparatory work during the initial phase of the project until he
left the institute. The final manuscript was prepared by cand.-Ing. O. Yilmaz, whom I grate-
fullyacknowledge.Dr.-Ing.M.Meinkeofferedvaluableadviceinthepreparationofsomeofthe
diagrams.
Aachen,July2004 E.Krause
Table of Contents
1. Fluid Mechanics I 1
1.1 Introduction.......................................................... 1
1.2 Hydrostatics.......................................................... 2
1.2.1 SurfaceandVolumeForces........................................ 2
1.2.2 ApplicationsoftheHydrostaticEquation ........................... 3
1.2.3 HydrostaticLift ................................................. 5
1.3 Hydrodynamics ....................................................... 5
1.3.1 KinematicsofFluidFlows ........................................ 5
1.3.2 StreamTubeandFilament........................................ 7
1.3.3 ApplicationsofBernoulli’sEquation................................ 8
1.4 One-DimensionalUnsteadyFlow ........................................ 10
1.5 MomentumandMomentofMomentumTheorem .......................... 12
1.5.1 MomentumTheorem............................................. 12
1.5.2 ApplicationsoftheMomentumTheorem............................ 13
1.5.3 FlowsinOpenChannels.......................................... 17
1.5.4 MomentofMomentumTheorem................................... 18
1.5.5 ApplicationsoftheMomentofMomentumTheorem.................. 19
1.6 ParallelFlowofViscousFluids ......................................... 20
1.6.1 ViscosityLaws .................................................. 21
1.6.2 PlaneShearFlowwithPressureGradient ........................... 22
1.6.3 LaminarPipeFlow .............................................. 24
1.7 TurbulentPipeFlows .................................................. 25
1.7.1 MomentumTransportinTurbulentFlows........................... 26
1.7.2 VelocityDistributionandResistanceLaw ........................... 27
1.7.3 PipeswithNon-circularCrossSection ............................. 29
2. Fluid Mechanics II 31
2.1 Introduction.......................................................... 31
2.2 FundamentalEquationsofFluidMechanics ............................... 31
2.2.1 TheContinuityEquation ......................................... 31
2.2.2 TheNavier-StokesEquations...................................... 32
2.2.3 TheEnergyEquation ............................................ 35
2.2.4 DifferentFormsoftheEnergyEquation ............................ 36
2.3 SimilarFlows ......................................................... 38
2.3.1 DerivationoftheSimilarityParameterswiththeMethod
ofDimensionalAnalysis .......................................... 38
2.3.2 TheMethodofDifferentialEquations .............................. 40
2.3.3 PhysicalMeaningoftheSimilarityParameters....................... 41
2.4 CreepingMotion ...................................................... 42
VIII TableofContents
2.5 VortexTheorems...................................................... 44
2.5.1 RotationandCirculation ......................................... 44
2.5.2 VorticityTransportEquation...................................... 45
2.6 PotentialFlowsofIncompressibleFluids.................................. 46
2.6.1 PotentialandStreamFunction .................................... 46
2.6.2 DeterminationofthePressure..................................... 48
2.6.3 TheComplexStreamFunction .................................... 48
2.6.4 ExamplesforPlaneIncompressiblePotentialFlows................... 49
2.6.5 Kutta-JoukowskiTheorem ........................................ 53
2.6.6 PlaneGravitationalWaves........................................ 54
2.7 LaminarBoundaryLayers .............................................. 55
2.7.1 Boundary-LayerThicknessandFrictionCoefficient................... 56
2.7.2 Boundary-LayerEquations........................................ 56
2.7.3 ThevonKa´rma´nIntegralRelation................................. 58
2.7.4 SimilarSolutionfortheFlatPlateatZeroIncidence ................. 59
2.8 TurbulentBoundaryLayers............................................. 61
2.8.1 Boundary-LayerEquationsforTurbulentFlow....................... 61
2.8.2 TurbulentBoundaryLayerontheFlatPlateatZeroIncidence......... 62
2.9 SeparationoftheBoundaryLayer ....................................... 64
2.10 SelectedReferences .................................................... 66
2.11 Appendix ............................................................ 66
3. Exercises in Fluid Mechanics 69
3.1 Problems............................................................. 69
3.1.1 Hydrostatics .................................................... 69
3.1.2 Hydrodynamics.................................................. 71
3.1.3 MomentumandMomentofMomentumTheorem .................... 76
3.1.4 LaminarFlowofViscousFluids ................................... 80
3.1.5 PipeFlows ..................................................... 83
3.1.6 SimilarFlows ................................................... 86
3.1.7 PotentialFlowsofIncompressibleFluids............................ 88
3.1.8 BoundaryLayers ................................................ 91
3.1.9 Drag........................................................... 92
3.2 Solutions............................................................. 96
3.2.1 Hydrostatics .................................................... 96
3.2.2 Hydrodynamics.................................................. 99
3.2.3 MomentumandMomentofMomentumTheorem .................... 106
3.2.4 LaminarFlowofViscousFluids ................................... 113
3.2.5 PipeFlows ..................................................... 119
3.2.6 SimilarFlows ................................................... 123
3.2.7 PotentialFlowsofIncompressibleFluids............................ 126
3.2.8 BoundaryLayers ................................................ 132
3.2.9 Drag........................................................... 134
4. Gasdynamics 139
4.1 Introduction.......................................................... 139
4.2 ThermodynamicRelations.............................................. 139
TableofContents IX
4.3 One-DimensionalSteadyGasFlow....................................... 141
4.3.1 ConservationEquations .......................................... 141
4.3.2 TheSpeedofSound.............................................. 142
4.3.3 IntegraloftheEnergyEquation ................................... 143
4.3.4 SonicConditions ................................................ 144
4.3.5 TheLimitingVelocity............................................ 145
4.3.6 StreamTubewithVariableCross-Section ........................... 145
4.4 NormalCompressionShock............................................. 147
4.4.1 TheJumpConditions ............................................ 147
4.4.2 IncreaseofEntropyAcrosstheNormalCompressionShock............ 149
4.4.3 NormalShockinTransonicFlow................................... 150
4.5 ObliqueCompressionShock............................................. 151
4.5.1 JumpConditionsandTurningoftheFlow .......................... 151
4.5.2 WeakandStrongSolution ........................................ 153
4.5.3 Heart-CurveDiagramandHodographPlane......................... 154
4.5.4 WeakCompressionShocks ........................................ 155
4.6 ThePrandtl-MeyerFlow ............................................... 156
4.6.1 IsentropicChangeofVelocity...................................... 157
4.6.2 CornerFlow .................................................... 158
4.6.3 InteractionBetweenShockWavesandExpansions.................... 159
4.7 LiftandWaveDraginSupersonicFlow .................................. 160
4.7.1 TheWaveDrag ................................................. 161
4.7.2 LiftofaFlatPlateatAngleofAttack.............................. 161
4.7.3 ThinProfilesatAngleofAttack................................... 161
4.8 TheoryofCharacteristics............................................... 163
4.8.1 TheCroccoVorticityTheorem .................................... 163
4.8.2 TheFundamentalEquationofGasdynamics......................... 164
4.8.3 CompatibilityConditionsforTwo-DimensionalFlows................. 166
4.8.4 ComputationofSupersonicFlows.................................. 167
4.9 CompressiblePotentialFlows ........................................... 170
4.9.1 SimplificationofthePotentialEquation ............................ 170
4.9.2 DeterminationofthePressureCoefficient ........................... 171
4.9.3 PlaneSupersonicFlowsAboutSlenderBodies....................... 172
4.9.4 PlaneSubsonicFlowAboutSlenderBodies ......................... 174
4.9.5 FlowsaboutSlenderBodiesofRevolution........................... 175
4.10 SimilarityRules....................................................... 178
4.10.1SimilarityRulesforPlaneFlowsAftertheLinearizedTheory.......... 178
4.10.2ApplicationoftheSimilarityRulestoPlaneFlows ................... 180
4.10.3SimilarityRulesforAxiallySymmetricFlows........................ 182
4.10.4SimilarityRulesforPlaneTransonicFlows.......................... 183
4.11 SelectedReferences .................................................... 184
5. Exercises in Gasdynamics 185
5.1 Problems............................................................. 185
5.1.1 One-DimensionalSteadyFlowsofGases ............................ 185
5.1.2 NormalCompressionShock ...................................... 188
5.1.3 ObliqueCompressionShock ...................................... 191
5.1.4 ExpansionsandCompressionShocks ............................... 193
5.1.5 LiftandWaveDrag–Small-PerturbationTheory .................... 196
X TableofContents
5.1.6 TheoryofCharacteristics ......................................... 198
5.1.7 CompressiblePotentialFlowsandSimilarityRules ................... 199
5.2 Solutions............................................................. 203
5.2.1 One-DimensionalSteadyFlowsofGases ........................... 203
5.2.2 NormalCompressionShock ....................................... 208
5.2.3 ObliqueCompressionShock....................................... 211
5.2.4 ExpansionsandCompressionShocks ............................... 214
5.2.5 LiftandWaveDrag–Small-PerturbationTheory .................... 217
5.2.6 TheoryofCharacteristics ......................................... 219
5.2.7 CompressiblePotentialFlowsandSimilarityRules .................. 221
5.3 Appendix ............................................................ 225
6. Aerodynamics Laboratory 233
6.1 WindTunnelforLowSpeeds(G¨ottingen-TypeWindTunnel) ............... 233
6.1.1 PreliminaryRemarks............................................. 233
6.1.2 WindTunnelsforLowSpeeds..................................... 234
6.1.3 CharakteristicDataofaWindTunnel.............................. 235
6.1.4 MethodofTestandMeasuringTechnique........................... 237
6.1.5 Evaluation...................................................... 243
6.2 PressureDistributiononaHalfBody .................................... 247
6.2.1 DeterminationoftheContourandthePressureDistribution........... 247
6.2.2 MeasurementofthePressure...................................... 248
6.2.3 TheHele-ShawFlow ............................................. 249
6.2.4 Evaluation...................................................... 251
6.3 SphereinIncompressibleFlow .......................................... 253
6.3.1 Fundamentals................................................... 253
6.3.2 ShiftoftheCriticalReynoldsNumberbyVariousFactors
ofInfluence ..................................................... 257
6.3.3 MethodofTest.................................................. 258
6.3.4 Evaluation...................................................... 259
6.4 Flat-PlateBoundaryLayer ............................................. 262
6.4.1 IntroductoryRemarks............................................ 262
6.4.2 MethodofTest.................................................. 263
6.4.3 PredictionMethods.............................................. 265
6.4.4 Evaluation...................................................... 267
6.4.5 Questions....................................................... 268
6.5 PressureDistributiononaWing......................................... 271
6.5.1 WingofInfiniteSpan ............................................ 271
6.5.2 WingofFiniteSpan ............................................. 273
6.5.3 MethodofTest.................................................. 278
6.5.4 Evaluation...................................................... 280
6.6 AerodynamicForcesActingonaWing ................................... 283
6.6.1 NomenclatureofProfiles.......................................... 283
6.6.2 MeasurementofAerodynamicForces............................... 283
6.6.3 ApplicationofMeasuredDatatoFull-ScaleConfigurations ............ 287
6.6.4 Evaluation...................................................... 295
6.7 WaterAnalogy–PropagationofSurfaceWavesinShallowWater
andofPressureWavesinGases ......................................... 299
6.7.1 Introduction .................................................... 299
TableofContents XI
6.7.2 TheWaterAnalogyofCompressibleFlow........................... 299
6.7.3 TheExperiment................................................. 304
6.7.4 Evaluation...................................................... 305
6.8 ResistanceandLossesinCompressiblePipeFlow.......................... 307
6.8.1 FlowResistanceofaPipewithInsertedThrottle
(Orificee,Nozzle,Valveetc.) ...................................... 307
6.8.2 FrictionResistanceofaPipeWithoutaThrottle..................... 307
6.8.3 ResistanceofanOrifice........................................... 311
6.8.4 Evaluation...................................................... 313
6.8.5 Problems ....................................................... 316
6.9 MeasuringMethodsforCompressibleFlows............................... 320
6.9.1 TabularSummaryofMeasuringMethods ........................... 320
6.9.2 OpticalMethodsforDensityMeasurements ......................... 320
6.9.3 OpticalSetup ................................................... 325
6.9.4 MeasurementsofVelocitiesandTurbulentFluctuationVelocities....... 326
6.9.5 Evaluation...................................................... 327
6.10 SupersonicWindTunnelandCompressionShockattheWedge.............. 329
6.10.1Introduction .................................................... 329
6.10.2ClassificationofWindTunnels .................................... 329
6.10.3ElementsofaSupersonicTunnel................................... 332
6.10.4TheObliqueCompressionShock................................... 333
6.10.5DescriptionoftheExperiment..................................... 337
6.10.6Evaluation...................................................... 339
6.11 SphereinCompressibleFlow............................................ 342
6.11.1Introduction .................................................... 342
6.11.2TheExperiment................................................. 342
6.11.3FundamentalsoftheCompressibleFlowAboutaSphere .............. 344
6.11.4Evaluation...................................................... 348
6.11.5Questions....................................................... 348
Index 351