FluidMechanicsforCivil andEnvironmental Engineers
AhlamI.Shalaby
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LibraryofCongressCataloging-in-PublicationData
Names:Shalaby,A.I.(AhlamIbrahim),1957-author.
Title:Fluidmechanicsforcivilandenvironmentalengineers / AhlamI.Shalaby.
Description:BocaRaton:Taylor&FrancisaCRCtitle,partoftheTaylor&Francisimprint, amemberoftheTaylor&FrancisGroup,theacademicdivisionofT&FInforma,plc,[2018]| Includesbibliographicalreferencesandindex.
Identifiers:LCCN2017038418|ISBN9780849337376(hardback:acid-freepaper)| ISBN9781315156637(ebook)
Subjects:LCSH:Fluidmechanics.|Hydrodynamics.
Classification:LCCTA357.S4252018|DDC620.1/06--dc23
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IdedicatethistoGodAlmighty,Whomadewritingthistextbookevenapossibility, anditscompletionareality.
Inlovingmemoryofmypreciousanddevotedmother,Mrs.ElhamAliOsman Shalaby,whogavemeinspirationtokeepgoingevenwhenitseemedimpossible,and mydearanddedicatedfather,Dr.IbrahimMahmoudShalaby,whogavememotivationtopursuemydreams.
Idedicatethistomylovingandveryspecialboys,OmarMohsenBagniedandBasem MohsenBagnied,whogivemeunconditionallove,andgivemeareasontobethebest ineverythingthatIdo.
Toallofmystudents:youarethereasonwhyIwrotethistextbook.
DreamBig,TakeaLeapofFaith,andThrowCautiontotheWind.
1.10.4.6EmpiricallyDerivedExpressionforShearStress forTurbulentFlow........................................
1.10.5.1DefinitionofSurfaceTension..............................
1.10.5.2TheFormationofaDropletintheAtmosphere...........
1.10.5.3TheFormationofaDropletonaSolidSurfaceand Capillarity.................................................
1.10.5.4TheFormationofaDropletonaSolidSurface............
1.10.6.1PartialPressure,AtmosphericPressure,andVapor Pressure....................................................
1.10.6.2Evaporation,Condensation,BoilingPoint,andVapor PressureofaLiquid.......................................
1.10.6.3Cavitation..................................................
1.10.7Elasticity,Compressibility,orBulkModulusofElasticity............
1.10.7.1DefinitionofBulkModulusofElasticity..................
1.10.7.2IdealGasLaw:DefinitionofGasConstantand
1.10.7.4DeterminationofBulkModulusofElasticityforGases...
1.10.7.5DefinitionoftheSonic(Acoustic)VelocityforFluids.....
1.10.7.6DefinitionoftheMachNumberforFluids:AMeasure
2.2.1Pascal’sLaw:TheHydrostaticPressureataPoint..................
2.2.2TheHydrostaticPressureEquation:VariationofPressure fromPointtoPoint..................................................
2.2.3TheHydrostaticPressureHead......................................
2.2.4TheHydrostaticPressureDistribution..............................
2.2.5ApplicationofthePrinciplesofHydrostatics........................
2.3.1PressureScales.......................................................
2.3.1.1StandardAtmosphericPressure..........................
2.3.5PiezometerColumns/Tubes:SimplestManometers.................
2.3.6OpenU-TubeManometers:SimpleManometers....................
2.3.6.1Single-FluidSimpleManometer..........................
2.3.6.2Multi fl
2.3.7DifferentialManometers.............................................
2.3.7.1Single-FluidDifferentialManometerwithin aSinglePipe.............................................. 152
2.3.7.2Single-FluidDifferentialManometerbetween TwoPipes................................................. 154
2.3.7.3Multi fluidDifferentialManometerbetween TwoPipes................................................. 159
2.4HydrostaticForcesonSubmergedSurfaces..................................
2.4.1TheVariationoftheHydrostaticPressurealongaSurface.......... 162
2.4.1.1VariationofthePressurePrismforaSurface
SubmergedinaGas...................................... 162
2.4.1.2VariationofthePressurePrismforaSurface
SubmergedinaLiquidOpentotheAtmosphere........ 162
2.4.1.3VariationofthePressurePrismforaSurface
SubmergedinaLiquid,Enclosed,andPressurized...... 164
2.4.2MagnitudeandLocationoftheHydrostaticForcefor PlaneSurfaces....................................................... 165
2.4.2.1MagnitudeoftheResultantHydrostaticForceon PlaneSurfaces............................................ 166
2.4.2.2LocationoftheResultantHydrostaticForceonPlane Surfaces................................................... 169
2.4.3PlanesSubmergedinaGas.......................................... 173
2.4.4PlanesSubmergedinaLiquidOpentotheAtmosphere............
2.4.4.1Submerged(h = H)HorizontalPlane(Open)............. 174
2.4.4.2Submerged(h = yo)VerticalPlane(Open)................ 176
2.4.4.3Submerged(h = yo sin α)SlopingPlane(Open)........... 180
2.4.5PlanesSubmergedinanEnclosedandPressurizedLiquid.......... 184
2.4.5.1Submerged(h = H)HorizontalPlane(Enclosed)......... 184
2.4.5.2Submerged(h = 0)VerticalPlane(Enclosed)............. 186
2.4.5.3Submerged(h = yo)VerticalPlane(Enclosed)............ 189
2.4.5.4Submerged(h = yo sin α)SlopingPlane(Enclosed)....... 193
2.4.6SubmergedNonrectangularPlanes.................................. 196
2.4.6.1Submerged(h = H)HorizontalNonrectangularPlane.... 197
2.4.6.2Submerged(h = H)HorizontalCircularPlane............ 197
2.4.6.3Submerged(h = H)HorizontalTriangularPlane......... 199
2.4.6.4Submerged(h = yo)VerticalNonrectangularPlane....... 200
2.4.6.5Submerged(h = yo)VerticalCircularPlane............... 201
2.4.6.6Submerged(h = yo)VerticalTriangularPlane............ 204
2.4.6.7Submerged(h = yo sin α)SlopingNonrectangular Plane...................................................... 208
2.4.6.8Submerged(h = yo sin α)SlopingCircularPlane.......... 208
2.4.6.9Submerged(h = yo sin α)SlopingTriangularPlane....... 212
2.4.7MagnitudeandLocationoftheHydrostaticForceforCurved Surfaces.............................................................. 216
2.4.7.1MagnitudeandLocationoftheHorizontalComponent ofForceonCurvedSurfaces.............................. 217
2.4.7.2MagnitudeandLocationoftheVerticalComponent ofForceonCurvedSurfaces.............................. 219
2.4.7.3MagnitudeandLocationoftheResultantForce onCurvedSurfaces....................................... 220
2.4.8MagnitudeandLocationoftheHydrostaticForceforSurfaces SubmergedinaMultilayeredFluid.................................
2.4.8.1MagnitudeandLocationoftheResultantForceActing inFirstFluid.............................................. 223
2.4.8.2MagnitudeandLocationoftheResultantForceActing inSecondFluid...........................................
2.4.8.3MagnitudeandLocationoftheResultantForceActing inMultilayerFluid........................................
2.5BuoyancyandStabilityofaFloatingoraNeutrallyBuoyantBody..........
2.5.1BuoyancyandArchimedesPrinciple................................
2.5.1.1MagnitudeandLocationoftheBuoyantForce...........
2.5.1.2BuoyancyofaCompletelySubmergedSinkingBody....
2.5.1.3BuoyancyofaCompletelySubmergedNeutrally BuoyantBody.............................................
2.5.1.4BuoyancyofaPartiallySubmergedFloatingBody.......
2.5.1.5TheRoleoftheBuoyantForceinPractical Applications..............................................
2.5.2StabilityofaFloatingoraNeutrallyBuoyantBody.................
2.5.2.1VerticalStabilityofaFloatingorNeutrally BuoyantBody.............................................
2.5.2.2RotationalStabilityofaFloatingorNeutrallyBuoyant Body......................................................
2.5.2.3RotationalStabilityofNeutrallyBuoyant/Suspended Bodies.....................................................
2.5.2.4RotationalStabilityofFloatingBodies....................
2.5.2.5MaximumDesignAngleofRotationforaTop-Heavy FloatingBody.............................................
2.5.2.6MetacentricHeightforaTop-HeavyFloatingBody......
2.5.2.7ComputationoftheMetacentricHeightandthe ResultingMoment(Restoring,Overturning,orZero) foraTop-HeavyFloatingBody...........................
3.3.1InternalversusExternalFlow.......................................
3.3.2PressureversusGravityFlow.......................................
3.3.3RealversusIdealFlow...............................................
3.3.3.1LaminarversusTurbulentFlow..........................
3.3.4CompressibleversusIncompressibleFlow..........................
3.3.5SpatiallyVaried(Nonuniform)versusSpatially
3.3.6UnsteadyversusSteadyFlow.......................................
3.3.7One-,Two-,andThree-DimensionalFlows..........................
3.4FlowVisualization/Geometry................................................
3.4.1PathLines............................................................
3.5DescribingandObservingtheFluidMotion..................................
3.5.1DefinitionofaFluidSystem:TheLagrangianPointofView........
3.5.1.1DefinitionoftheFluidParticleVelocityinthe LagrangianPointofView................................
3.5.2DefinitionofaControlVolume:TheEulerianPointofView........
3.5.2.1DefinitionoftheFluidParticleVelocityintheEulerian PointofView.............................................
3.5.2.2FixedversusMovingControlVolumeintheEulerian PointofView.............................................
3.5.3DerivingOneViewfromtheOther..................................
3.5.4ApplicationoftheLagrangianViewversusApplicationofthe EulerianView........................................................
3.6ThePhysicalLawsGoverningFluidinMotion...............................
3.6.1StatementoftheGoverningPhysicalLawsAssuming aFluidSystem.......................................................
3.6.1.1ExtensiveandIntensiveFluidProperties.................
3.6.2TheReynoldsTransportTheorem...................................
3.6.3StatementoftheAssumptionsMadefortheTypeofFluidFlow....
3.6.3.1TheAssumptionofSpatialDimensionality oftheFluidFlow.........................................
3.6.3.2TheModelingofTwo-DimensionalFlows................
3.6.3.3TheModelingofOne-DimensionalFlows................
3.6.3.4AComparisonofVelocityProfilesforVarious FlowTypes...............................................
3.7ConservationofMass:TheContinuityEquation.............................
3.7.1TheContinuityEquation............................................
3.7.1.1TheContinuityEquationforaFluidSystem.............
3.7.1.2TheContinuityEquationforaControlVolume..........
4.4.1Newton’sSecondLawofMotionforaFluidSystem................
4.4.2Newton’sSecondLawofMotionforaControlVolume.............
4.5TheEnergyEquationBasedonNewton’sSecondLawofMotion............
4.5.2.1DefinitionoftheTermsintheEnergyEquation..........
4.5.2.2PracticalAssumptionsMadefortheEnergyEquation...
4.5.2.3Application oftheEnergyEquationforPressure
4.5.3TheBernoulliEquation..............................................
4.5.3.1ApplicationsoftheBernoulliEquation...................
4.5.3.2ThePitot-StaticTube:DynamicPressureIsModeled byaPressureRise........................................
4.5.3.3IdealFlowMeters,IdealGradualChannel Contractions,andIdealFlowfromaTank:Dynamic PressureIsModeledbyaPressureDrop................. 382
4.5.3.4TheEnergyGradeLine(EGL)andtheHydraulicGrade Line(HGL)...............................................
4.5.3.5TheHydraulicGradeLine(HGL)andCavitation........
4.5.4ApplicationoftheEnergyEquationinComplementtothe MomentumEquation................................................
4.5.4.1ApplicationoftheGoverningEquations................. 392
4.5.4.2ModelingFlowResistance:ASubsetLevelApplication oftheGoverningEquations.............................. 401
4.5.5ApplicationoftheEnergyandMomentumEquationsforReal InternalFlow........................................................ 403
4.5.5.1EvaluationoftheMajorandMinorHead LossTerms................................................
4.5.5.2EvaluationofthePumpandTurbineHeadTerms.......
4.5.5.3ApplicationsoftheGoverningEquationsforReal InternalFlow............................................. 406
4.5.5.4ApplicationoftheEnergyEquationforReal PipeFlow................................................. 406
4.5.5.5ApplicationoftheEnergyEquationforRealOpen ChannelFlow............................................. 410
4.5.6ApplicationoftheEnergyandMomentumEquationsforIdeal InternalFlowandIdealFlowfromaTank.......................... 412
4.5.6.1EvaluationoftheActualDischarge....................... 413
4.5.6.2ApplicationsoftheGoverningEquationsforIdeal InternalFlowandIdealFlowfromaTank............... 414
4.5.6.3ApplicationoftheBernoulliEquationforIdeal PipeFlow................................................. 415
4.5.6.4ApplicationoftheBernoulliEquationforIdealOpen ChannelFlow............................................. 419
4.5.6.5ApplicationsoftheBernoulliEquationforIdeal FlowfromaTank......................................... 423
4.5.7ApplicationoftheEnergyandMomentumEquationsforIdeal ExternalFlow........................................................ 442
4.5.7.1EvaluationoftheDragForce............................. 442
4.5.7.2ApplicationsoftheGoverningEquationsforIdeal ExternalFlow............................................. 444
4.5.7.3ApplicationsoftheBernoulliEquationforIdeal ExternalFlow............................................. 444
4.5.8ApplicationoftheEnergyandMomentumEquationsfora HydraulicJump...................................................... 449
4.5.8.1ApplicationsoftheGoverningEquationsfora HydraulicJump.......................................... 450
4.5.8.2ApplicationoftheEnergyEquationfora HydraulicJump.......................................... 451
4.6ConservationofEnergyPrinciple:FirstLawofThermodynamics............
4.6.1.1TotalEnergyforaFluidFlowSystem....................
4.6.1.2DimensionsandUnitsforEnergy........................
4.6.2EnergyTransferbyHeat.............................................
4.6.3EnergyTransferbyWork............................................
4.7TheEnergyEquationBasedontheFirstLawofThermodynamics...........
4.7.1TheEnergyEquationforaFluidSystem............................
4.7.2TheEnergyEquationforaControlVolume.........................
4.7.2.1TheEnergyEquationExpressedinPowerTerms........
4.7.2.2PumpandTurbineLossesAreModeledbyTheir Efficiencies................................................
4.7.2.3Pump,Turbine,andSystemEfficiencies..................
4.7.2.4TheEnergyEquationExpressedinEnergy
4.7.2.5DefinitionoftheTermsintheEnergyEquation..........
4.7.2.6PracticalAssumptionsMadefortheEnergyEquation...
4.7.3ApplicationoftheEnergyEquationinComplementtothe MomentumEquation................................................
4.7.4ApplicationoftheEnergyEquationforRealInternalFlow withaPumporaTurbine...........................................
4.7.4.1DimensionsandUnitsforthePumpandTurbine HeadTerms...............................................
4.7.4.2ApplicationsoftheEnergyEquationforReal InternalFlowwithaPump...............................
4.7.4.3ApplicationsoftheEnergyEquationforReal InternalFlowwithaTurbine.............................
5.2.1TheMomentumEquationforaFluidSystem:DifferentialForm oftheMomentumEquation.........................................
5.2.1.1DifferentialFormoftheMomentumEquation...........
5.2.1.2ApplicationoftheDifferentialFormoftheMomentum Equation.................................................. 559
5.2.2TheMomentumEquationforaControlVolume:Integral FormoftheMomentumEquation................................... 564
5.2.2.1FixedversusMovingControlVolumeinthe Eulerian(Integral)PointofView......................... 565
5.2.2.2ApplicationoftheIntegralFormoftheMomentum Equation.................................................. 566
5.3ApplicationoftheEnergyEquationinComplement totheMomentumEquation.................................................. 576
5.3.1ApplicationoftheGoverningEquations............................ 577
5.3.2ModelingFlowResistance:ASubsetLevelApplicationof theGoverningEquations............................................ 578
5.3.3ApplicationoftheEnergyandMomentumEquationsforReal InternalFlow........................................................ 580
5.3.3.1EvaluationoftheMajorandMinorHeadLossTerms.... 580
5.3.3.2EvaluationofthePumpandTurbineHeadTerms....... 583
5.3.3.3ApplicationsoftheGoverningEquationsforReal InternalFlow............................................. 583
5.3.3.4ApplicationsoftheGoverningEquationsforReal InternalFlow:DifferentialFluidElement................. 584
5.3.3.5ApplicationsoftheGoverningEquationsforReal InternalFlow:FiniteControlVolume.................... 592
5.3.4ApplicationoftheEnergyandMomentumEquationsforIdeal InternalFlowandIdealFlowfromaTank.......................... 598
5.3.4.1EvaluationoftheActualDischarge....................... 598
5.3.4.2ApplicationsoftheGoverningEquationsforIdeal InternalFlowandIdealFlowfromaTank............... 600
5.3.4.3ApplicationsoftheGoverningEquationsforIdeal PipeFlow................................................. 601
5.3.4.4ApplicationsoftheGoverningEquationsforIdeal OpenChannelFlow...................................... 602
5.3.4.5ApplicationsoftheGoverningEquationsforIdeal FlowfromaTank......................................... 605
5.3.5ApplicationoftheEnergyandMomentumEquationsfor IdealExternalFlow.................................................. 614
5.3.5.1EvaluationoftheDragForce............................. 614
5.3.5.2ApplicationsoftheGoverningEquationsforIdeal ExternalFlow............................................. 616
5.3.6ApplicationoftheEnergyandMomentumEquationsfora HydraulicJump...................................................... 618
5.3.6.1ApplicationsoftheGoverningEquationsfora HydraulicJump.......................................... 618
6.1.1ModelingFlowResistance:ASubsetLevelApplicationofthe GoverningEquations................................................
6.1.2DerivationoftheFlowResistanceEquationsandtheDrag Coefficients.......................................................... 645
6.1.2.1EmpiricalEvaluationoftheDragCoefficientsand ApplicationoftheFlowResistanceEquations............ 646
6.1.3ModelingtheFlowResistanceasaLossinPumpandTurbine EfficiencyinInternalFlow........................................... 646
6.2TypesofFlow................................................................. 647
6.2.1InternalFlowversusExternalFlow.................................. 647
6.2.2PipeFlowversusOpenChannelFlow............................... 647
6.2.3RealFlowversusIdealFlow......................................... 648
6.2.4IdealFlow........................................................... 649
6.2.4.1IgnoringtheFlowResistanceinIdealFlow.............. 649
6.2.4.2SubsequentlyModelingtheFlowResistancein IdealFlow................................................ 650
6.2.4.3ThePitot-StaticTube:DynamicPressureIsModeled byaPressureRise........................................ 651
6.2.4.4TheVenturiMeter:DynamicPressureIsModeled byaPressureDrop....................................... 653
6.2.5RealFlow............................................................ 654
6.2.5.1DirectlyandSubsequentlyModelingtheFlow ResistanceinRealFlow................................... 654
6.2.5.2DirectlyModelingtheFlowResistanceinRealFlow..... 655
6.2.5.3LaminarFlowversusTurbulentFlow.................... 656
6.2.5.4TheVelocityProfilesforLaminarandTurbulent InternalFlows............................................ 657
6.2.5.5DevelopingFlowversusDevelopedFlow................ 659
6.3ModelingtheFlowResistanceasaDragForceinExternalFlow............. 659
6.3.1EvaluationoftheDragForce........................................ 660
6.3.2ApplicationoftheBernoulliEquation:DerivationoftheIdeal Velocity.............................................................. 661
6.3.3ApplicationoftheMomentumEquationandDimensional Analysis:DerivationoftheActualVelocity,DragForceEquation, andtheDragCoefficient............................................. 661
6.3.4ModelingtheDragForceintheMomentumEquation.............. 662
6.3.4.1SubmergedStationaryBodyinaStationaryRealor IdealFluid................................................ 662
6.3.4.2SubmergedMovingBodyinaStationary IdealFluid................................................ 663
6.3.4.3SubmergedMovingBodyinaStationaryRealFluid..... 664
6.3.5SupplementingtheMomentumEquationwithDimensional Analysis:DerivationoftheDragForceandtheDragCoef ficient... 665
6.4ModelingtheFlowResistanceasaMajorHeadLossinInternalFlow....... 666
6.4.1EvaluationoftheMajorHeadLoss.................................. 667
6.4.2ApplicationoftheEnergyEquation:Derivationofthe HeadLoss........................................................... 668
6.4.3ApplicationoftheMomentumEquationsandDimensional Analysis:DerivationofthePressureDrop,ShearStress,andthe DragCoef ficient..................................................... 668
6.4.4ApplicationoftheGoverningEquationstoDerivethe MajorHeadLossEquationsforLaminarandTurbulentFlow...... 669
6.5LaminarandTurbulentInternalFlowCharacteristics........................ 671
6.5.1DeterminingtheVariationofShearStresswithRadialDistance forLaminarandTurbulentFlow.................................... 672
6.5.2TheVariationoftheVelocitywithTimeforLaminarand TurbulentFlow...................................................... 675
6.5.3TheVariationoftheVelocity,PressureDrop,andtheWallShear StresswithLengthofPipeforLaminarandTurbulentFlow: DevelopingversusDevelopedFlow................................. 675
6.5.3.1TheExtentofDevelopingFlowinthePipeLength....... 676
6.5.3.2TheVariationoftheVelocityProfilewith PipeLength............................................... 678
6.5.3.3TheVariationofthePressureDropandthe WallShearStresswithPipeLength...................... 679
6.5.3.4LaminarversusTurbulentFlow.......................... 680
6.6DerivationoftheMajorHeadLossEquationforLaminarFlow.............. 681
6.6.1ApplicationoftheIntegralFormoftheMomentum Equation:DerivingtheVelocityProfileforLaminarFlow.......... 681
6.6.2ApplicationoftheDifferentialformoftheContinuityEquation: DerivingPoiseuille’sLaw............................................ 683
6.6.3SubstitutingPoiseuille’sLawintotheIntegralFormoftheEnergy Equation:DerivingtheMajorHeadLossEquationforLaminar Flow(Poiseuille’sLaw).............................................. 684
6.6.4InterpretationofPoiseuille’sLawExpressedinTermsofthe PressureDrop........................................................ 685
6.7DerivationoftheMajorHeadLossEquationforTurbulentFlow............ 686
6.7.1ApplicationoftheIntegralMomentumEquation:Derivingan ExpressionforthePressureDrop.................................... 687
6.7.2ApplicationoftheDifferentialMomentumEquation:Interpreting theFrictionSlopeinTurbulentFlow................................ 687
6.7.3ApplicationofDimensionalAnalysis:EmpiricallyInterpreting FrictionSlope(EmpiricallyDerivinganExpressionforWallShear Stress)inTurbulentFlow............................................ 688
6.7.3.1DerivationofanEmpiricalExpressionfortheWall ShearStressasaFunctionoftheVelocity................ 689
6.7.3.2DerivationoftheChezyEquationandEvaluationofthe ChezyCoefficient, C ...................................... 690
6.7.3.3ApplicationoftheChezyEquation....................... 690
6.7.4SubstitutingtheChezyEquationintotheIntegralFormofthe EnergyEquation:DerivingtheMajorHeadLossEquationfor TurbulentFlow...................................................... 691
6.7.5TheDarcy–WeisbachEquation...................................... 692
6.7.5.1DerivationoftheDarcy–WeisbachFriction Coefficient, f .............................................. 692
6.7.5.2EvaluationoftheDarcy–WeisbachFriction Coefficient, f .............................................. 693
6.7.5.3TheDarcy–WeisbachHeadLossEquation............... 694
6.7.6Manning’sEquation................................................. 694
6.7.6.1DerivationandEvaluationoftheManning’ s RoughnessCoefficient, n ................................. 695
6.7.6.2Manning’sEquation...................................... 695
6.7.6.3Manning’ s Head Loss Equation.......................... 696
6.7.7TheHazen–WilliamsEquation...................................... 696
6.7.7.1TheHazen–WilliamsEquation........................... 696
6.7.7.2EvaluationoftheHazen–WilliamsEquation RoughnessCoefficient, Ch ................................ 697
6.7.7.3TheHazen–WilliamsHeadLoss Equation.................................................. 697
6.7.8TheRelationshipbetweentheDragCoef ficient, CD;theChezy Coefficient, C;theDarcy–WeisbachFrictionFactor, f;and Manning’sRoughnessCoef ficient, n ................................. 698
6.7.9AComparisonbetweenLaminarandTurbulentFlowUsingthe Darcy–WeisbachHeadLossEquation............................... 698
6.7.9.1TheDarcy–WeisbachHeadLossEquationandthe ReynoldsNumber, R forNoncircularPipes.............. 699
6.7.9.2EvaluatingtheDarcy–WeisbachFrictionCoef ficient, f forLaminarPipeFlow.................................. 700
6.7.9.3AComparisonbetweenLaminarandTurbulentFlow UsingtheDarcy–WeisbachHeadLossEquation......... 700
6.7.10DeterminingtheVelocityProfileforTurbulentFlow................ 702
6.7.10.1AComparisonbetweentheVelocityProfilesfor LaminarFlowandTurbulentFlow....................... 704
6.7.10.2TheRoleoftheBoundaryRoughnessinLaminarFlow andTurbulentFlow...................................... 705
6.7.11ApplicationoftheMajorHeadLossEquationforOpen ChannelFlow........................................................ 706
6.7.11.1ApplicationoftheChezyEquationforOpen ChannelFlow............................................. 708
6.8ModelingtheFlowResistanceasaMinorHeadLossinPipeFlow.......... 709
6.8.1EvaluationoftheMinorHeadLoss................................. 710
6.8.2ApplicationoftheEnergyEquation:Derivationofthe HeadLoss........................................................... 711
6.8.3ApplicationoftheMomentumEquationandDimensional Analysis:DerivationofthePressureDrop,ShearStress,andthe DragCoef ficient..................................................... 713
6.8.4AnalyticalDerivationoftheMinorHeadLossEquationdueto aSuddenPipeExpansion............................................ 714
6.8.5EmpiricalDerivationoftheMinorHeadLossEquationdueto PipeComponentsinGeneral........................................ 715
6.8.5.1ApplicationoftheEnergyEquation:Derivationofthe HeadLoss................................................ 715
6.8.5.2ApplicationoftheMomentumEquationand DimensionalAnalysis:DerivationofthePressure Drop,ShearStress,andtheDragCoefficient............. 716
6.8.5.3EvaluationoftheMinorHeadLoss Coefficient, k .............................................. 716
6.9ModelingtheFlowResistanceasaLossinFlowratein InternalFlow.................................................................. 718
6.9.1EvaluationoftheActualDischarge.................................. 719
6.9.2ApplicationoftheBernoulliEquation:DerivationoftheIdeal Velocity.............................................................. 720
6.9.3ApplicationoftheContinuityEquation:DerivationoftheIdeal DischargeandtheActualDischarge................................. 720
6.9.4ApplicationoftheMomentumEquationandDimensional Analysis:DerivationoftheActualVelocity,ActualArea,Actual Discharge,andtheDischargeCoef ficient............................ 721
6.9.4.1SupplementingtheMomentumEquationwith DimensionalAnalysis:Derivationofthe Reduced/ActualDischargeEquationfor PipeFlow................................................. 722
6.9.4.2SupplementingtheMomentumEquationwith DimensionalAnalysis:Derivationofthe Reduced/ActualDischargeEquationforOpen ChannelFlow.............................................
6.9.5AComparisonoftheVelocityProfilesforIdealandRealFlows....
6.10ModelingtheFlowResistanceasaLossinPumpandTurbineEfficiency inInternalFlow...............................................................
6.10.1EvaluationoftheEfficiencyofPumpsandTurbines................
6.10.1.1SupplementingtheMomentumEquationwith DimensionalAnalysis:DerivationoftheEfficiency ofPumpsandTurbines...................................
7.DimensionalAnalysis
7.1Introduction...................................................................
7.1.1TheRoleofDimensionalAnalysisintheModelingof FluidFlow...........................................................
7.1.2TheRoleofDimensionalAnalysisintheEmpiricalModelingof FlowResistanceinRealFluidFlow..................................
7.1.2.1ModelingofFlowResistanceinRealFluidFlow.........
7.1.2.2UsingDimensionalAnalysisintheEmpiricalModeling ofFlowResistanceinRealFluidFlow....................
7.1.3FlowTypesandDimensionalAnalysis..............................
7.1.4InternalFlowversusExternalFlow..................................
7.1.5ModelingFlowResistance:ASubsetLevelApplicationofthe GoverningEquations................................................
7.1.6SupplementingtheMomentumTheorywithDimensional Analysis..............................................................
7.1.7DerivationoftheFlowResistanceEquationsandtheDrag Coefficients..........................................................
7.1.7.1EmpiricalEvaluationoftheDragCoefficientsand ApplicationoftheFlowResistanceEquations............ 757
7.1.8DerivationoftheEfficiencyofPumpsandTurbines................ 757
7.2DimensionalAnalysisofFluidFlow.......................................... 757
7.2.1DynamicForcesActingonaFluidElement......................... 758
7.2.2Two-DimensionalSystems........................................... 759
7.2.3DerivingaFunctionalRelationship /DimensionlessNumbers....... 759
7.2.4MainPiTerms....................................................... 761
7.2.5TheDefinitionofNewPiTerms..................................... 762
7.2.6GuidelinesintheDerivationoftheFlowResistance EquationsandtheDragCoefficients................................ 762
7.2.7TheDefinitionoftheDragCoefficient............................... 769
7.2.8GuidelinesintheDerivationoftheEfficiencyofPumps andTurbines......................................................... 770
7.2.9TheDefinitionofthePump(orTurbine)Efficiency . ................ 773
7.2.10SpecificGuidelinesandSummaryintheApplicationof DimensionalAnalysisforExampleProblemsandEnd-ofChapterProblems....................................................
7.3ModelingtheFlowReistanceasaDragForceinExternalFlow.............. 776
7.3.1EvaluationoftheDragForce........................................ 776
7.3.2ApplicationoftheBernoulliEquation:DerivationoftheIdeal Velocity.............................................................. 777
7.3.3ApplicationoftheMomentumEquationandDimensional Analysis:DerivationoftheActualVelocity,DragForce Equation,andtheDragCoef ficient.................................. 777
7.3.3.1ApplicationofDimensionalAnalysis:Derivationofthe DragForceandtheDragCoef ficient..................... 778
7.3.3.2ApplicationofDimensionalAnalysistoDerivethe DragForceandDragCoef ficientforMoreSpecific AssumptionsofExternalFlow............................ 787
7.3.4DerivationoftheLiftForceandtheLiftCoef ficient................. 791
7.4ModelingtheFlowResistanceasaMajorHeadLossinInternalFlow....... 795
7.4.1EvaluationoftheMajorHeadLoss.................................. 796
7.4.2ApplicationoftheEnergyEquation:DerivationoftheHead Loss.................................................................. 797
7.4.3ApplicationoftheMomentumEquationsandDimensional Analysis:DerivationofthePressureDrop,ShearStress, andtheDragCoefficient............................................. 797
7.4.4DerivationoftheMajorHeadLossEquationforLaminarFlow..... 799
7.4.5DerivationoftheMajorHeadLossEquationforTurbulentFlow... 799
7.4.5.1ApplicationofDimensionalAnalysis:Derivation oftheWallShearStressandDragCoefficientin TurbulentFlow........................................... 800
7.4.5.2DerivationoftheChezyEquationandEvaluationof theChezyCoefficient, C 804
7.4.5.3SubstitutingtheChezyEquationintotheEnergy Equation:DerivingtheMajorHeadLossEquation forTurbulentFlow....................................... 805
7.4.6TheDarcy–WeisbachEquation...................................... 806
7.4.6.1DerivationoftheDarcy–WeisbachFriction Coefficient, f 806
7.4.6.2EvaluationoftheDarcy–WeisbachFriction Coefficient, f .............................................. 808
7.4.6.3TheDarcy–WeisbachHeadLossEquation............... 808
7.4.6.4EvaluatingtheDarcy–WeisbachFriction Coefficient, f forLaminarPipeFlow...................... 809
7.4.7Manning’sEquation................................................. 809
7.4.7.1Derivation andEvaluationoftheManning’ s RoughnessCoefficient, n ................................. 809
7.4.7.2Manning’sEquation...................................... 810
7.4.7.3Manning’sHeadLossEquation.......................... 811
7.4.8ApplicationoftheMajorHeadLossEquationforOpen ChannelFlow........................................................ 811
7.4.8.1InterpretationoftheResultsofDimensionalAnalysis forOpenChannelFlow................................... 813
7.4.8.2ApplicationoftheChezyEquationforOpen ChannelFlow............................................. 814
7.4.8.3ApplicationoftheDarcy–WeisbachEquationfor OpenChannelFlow...................................... 815
7.4.8.4ApplicationofManning’sEquationforOpen ChannelFlow............................................. 815
7.4.9ApplicationofDimensionalAnalysistoDerivetheWallShear StressandDragCoefficientforMoreSpecificAssumptionsof InternalFlow........................................................ 816
7.5ModelingtheFlowResistanceasaMinorHeadLossinPipeFlow.......... 820
7.5.1EvaluationoftheMinorHeadLoss................................. 820
7.5.2ApplicationoftheEnergyEquation:Derivationofthe HeadLoss........................................................... 821
7.5.3ApplicationoftheMomentumEquationandDimensional Analysis:DerivationofthePressureDrop,ShearStress,andthe DragCoef ficient..................................................... 822
7.5.4AnalyticalDerivationoftheMinorHeadLossEquation duetoaSuddenPipeExpansion.................................... 823
7.5.5EmpiricalDerivationoftheMinorHeadLossEquation duetoPipeComponentsinGeneral................................. 824
7.5.5.1ApplicationofDimensionalAnalysis:Derivation ofthePressureDropandDragCoef ficientforPipe ComponentsinGeneral.................................. 824
7.5.5.2SubstitutingthePressureDropintotheEnergy Equation:DerivingtheMinorHeadLossEquationfor PipeComponentsinGeneral............................. 831
7.5.5.3EvaluationoftheMinorHeadLossCoef ficient, k ........ 831
7.5.5.4ApplicationofDimensionalAnalysistoDerivethe PressureDropandDragCoefficientforMoreSpecific AssumptionsforPipeFlowComponents................. 833
7.6ModelingtheFlowResistanceasaLossinFlowrateinInternalFlow........ 838
7.6.1EvaluationoftheActualDischarge.................................. 839
7.6.2ApplicationoftheBernoulliEquation:DerivationoftheIdeal Velocity.............................................................. 840
7.6.3ApplicationoftheContinuityEquation:DerivationoftheIdeal DischargeandtheActualDischarge................................. 840
7.6.4ApplicationoftheMomentumEquationandDimensional Analysis:DerivationoftheActualVelocity,ActualArea,Actual Discharge,andtheDischargeCoef ficient............................ 841
7.6.4.1ApplicationofDimensionalAnalysis:Derivation oftheReduced/ActualDischargeandDragCoef ficient forPipeFlowMeasuringDevices........................ 842
7.6.4.2ApplicationofDimensionalAnalysis:Derivationofthe Reduced/ActualDischargeandDragCoefficientfor OpenChannelFlow-MeasuringDevices................. 854
7.6.4.3ApplicationofDimensionalAnalysistoDerivethe Reduced/ActualDischargeandDragCoefficientfor MoreSpecificAssumptionsforOpenChannel Flow-MeasuringDevices................................. 861
7.7Modeling the FlowResistanceasaLossinPumpandTurbine EfficiencyinInternalFlow.................................................... 866
7.7.1EvaluationoftheEfficiencyofPumpsandTurbines................ 867
7.7.1.1ApplicationofDimensionalAnalysis:Derivationofthe EfficiencyofPumpsandTurbines........................ 867
7.7.1.2ApplicationofDimensionalAnalysistoDerivethe EfficiencyofPumpsandTurbinesforMoreSpeci fic FlowAssumptions........................................ 878
7.8ExperimentalFormulationofTheoreticalEquations.......................... 893 End-of-ChapterProblems............................................................ 901 8.PipeFlow
8.1Introduction................................................................... 919
8.2ApplicationoftheEulerian(Integral)versusLagrangian(Differential) FormsoftheGoverningEquations........................................... 920
8.2.1Eulerian(Integral)ApproachforPipeFlowProblems............... 920
8.2.2Lagrangian(Differential)ApproachforPipeFlowProblems........ 921
8.3ModelingFlowResistanceinPipeFlow:ASubsetLevelApplication oftheGoverningEquations................................................... 921
8.4ApplicationoftheGoverningEquationsinPipeFlow........................ 922
8.4.1ApplicationoftheGoverningEquationsforRealPipeFlow........ 923
8.4.1.1IntegralApproachforRealPipeFlowProblems......... 923
8.4.1.2EvaluationoftheMajorandMinorHead LossTerms................................................ 923
8.4.1.3EvaluationofthePumpandTurbineHeadTerms....... 926
8.4.1.4DifferentialApproachforRealPipeFlow Problems.................................................. 926
8.4.1.5ApplicationsoftheGoverningEquationsforReal PipeFlowProblems...................................... 927
8.4.2ApplicationoftheGoverningEquationsforIdealPipeFlow....... 927
8.4.2.1EvaluationoftheActualDischarge....................... 928
8.4.2.2ApplicationsoftheGoverningEquationsfor IdealPipeFlow........................................... 929
8.5SinglePipes:MajorHeadLossinRealPipeFlow............................ 929
8.5.1EvaluationoftheMajorHeadLossTerminthe EnergyEquation..................................................... 930
8.5.1.1LaminarPipeFlow:Poiseuille’sLaw..................... 930
8.5.1.2TurbulentPipeFlow:TheChezyEquation............... 934
8.5.2TurbulentPipeFlowResistanceEquationsandTheirRoughness Coefficients.......................................................... 937
8.5.2.1AComparisonoftheThreeStandardEmpiricalFlow ResistanceCoefficients.................................... 938
8.5.2.2AComparisonbetweenManning’sand Hazen–WilliamsRoughnessCoefficients................. 939
8.5.2.3TurbulentPipeFlowResistanceEquations............... 939
8.5.3TheDarcy–WeisbachFrictionCoefficient, f andthe Darcy–WeisbachEquation........................................... 941
8.5.3.1Evaluation oftheDarcy–WeisbachFriction Coefficient, f 942
8.5.3.2ApplicationoftheDarcy–WeisbachEquation............ 944
8.5.4Manning’sRoughnessCoef ficient, n andManning’ s Equation............................................................. 952
8.5.4.1ApplicationofManning’sEquation...................... 954
8.5.5TheHazen–WilliamsRoughnessCoefficient, Ch andthe Hazen–WilliamsEquation........................................... 957
8.5.5.1ApplicationoftheHazen–WilliamsEquation............ 959
8.6PipeswithComponents:MinorHeadLossesandReactionForces inRealPipeFlow............................................................. 962
8.6.1EvaluationoftheMinorHeadLossTerminthe EnergyEquation..................................................... 963
8.6.1.1EvaluationoftheMinorHeadLoss...................... 963
8.6.1.2DerivationoftheMinorHeadLossCoef ficient, k 965
8.6.1.3EvaluationoftheMinorHeadLossCoef ficient, k ........ 965
8.6.2AlternativeModelingoftheMinorHeadLossTerminthe EnergyEquation..................................................... 966
8.6.3EvaluationoftheMinorHeadLossduetoPipeComponents....... 967
8.6.4MinorLossesinValvesandFittings(Tees,Unions,Elbows, andBends)........................................................... 968
8.6.4.1Valves..................................................... 969
8.6.4.2Fittings(Tees,Unions,Elbows,andBends)............... 974
8.6.5MinorLossesinEntrances,Exits,Contractions,andExpansions.... 979
8.6.5.1PipeEntrances............................................ 980
8.6.5.2PipeExists................................................ 983
8.6.5.3SuddenPipeExpansions................................. 984
8.6.5.4SuddenPipeContractions................................ 989
8.6.5.5GradualPipeExpansions................................. 990
8.6.5.6GradualPipeContractions............................... 995
8.6.6ReactionForcesonPipeComponents............................... 998
8.7PipeSystems:MajorandMinorHeadLossesinRealPipeFlow............ 1000
8.7.1SinglePipes......................................................... 1001
8.7.2PipeswithComponents............................................ 1001
8.7.3PipeswithaPumporaTurbine.................................... 1004
8.7.4PipesinSeries...................................................... 1009
8.7.5PipesinParallel..................................................... 1017
8.7.6BranchingPipes..................................................... 1030
8.7.6.1BranchingPipesConnectedtoThreeReservoirs........ 1031
8.7.6.2BranchingPipesConnectedtoaWaterSupply SourceunderPressure................................... 1044
8.7.7PipesinaLoop..................................................... 1055
8.7.8PipeNetworks...................................................... 1063
8.7.8.1ContinuityPrinciple..................................... 1063
8.7.8.2EnergyPrinciple......................................... 1064
8.7.8.3MomentumPrinciple.................................... 1067
8.7.8.4SummaryofGoverningEquationsforPipe Networks................................................ 1067
8.8PipeFlowMeasurementandControlDevices:ActualFlowrate inIdealFlowMeters......................................................... 1078
8.8.1EvaluationoftheActualFlowrateforIdealFlowMeters inPipeFlow........................................................ 1079
8.8.1.1DerivationoftheDischargeCoefficient, Cd .............. 1080
8.8.1.2EvaluationoftheDischargeCoef ficient, Cd ............. 1081
8.8.2EvaluationoftheActualFlowrateforaPitot-StaticTube.......... 1081
8.8.3EvaluationoftheActualFlowrateforOrifice,Nozzle, andVenturiMeters................................................. 1086
8.8.3.1EvaluationoftheActualFlowrateforanOrifice, aNozzle,oraVenturiMeter............................ 1088
8.8.3.2ActualFlowrateforanOrificeMeter.................... 1090
8.8.3.3ActualFlowrateforaNozzleMeter..................... 1093
8.8.3.4ActualFlowrateforaVenturiMeter..................... 1097
8.8.3.5EvaluationoftheMinorHeadLossduetoanOrifice, aNozzle,oraVenturiMeter............................ 1100 End-of-ChapterProblems........................................................... 1101 9.OpenChannelFlow
9.1Introduction..................................................................
9.2ApplicationoftheEulerian(Integral)versusLagrangian(Differential) FormsoftheGoverningEquations.......................................... 1116
9.2.1Eulerian(Integral)ApproachforOpenChannel FlowProblems...................................................... 1117
9.2.2Lagrangian(Differential)ApproachforOpenChannelFlow Problems............................................................
9.3TheOccurrenceofaMajorHeadLossinOpenChannelFlow.............. 1118
9.4ModelingFlowResistanceinOpenChannelFlow:ASubsetLevel ApplicationoftheGoverningEquations..................................... 1118
9.5ApplicationoftheGoverningEquationsinOpenChannelFlow............ 1119
9.5.1ApplicationoftheGoverningEquationsforRealOpen ChannelFlow....................................................... 1120
9.5.1.1IntegralApproachforRealOpenChannelFlow Problems................................................. 1120
9.5.1.2EvaluationoftheMajorHeadLossTerm............... 1120
9.5.1.3EvaluationofthePumpandTurbineHeadTerms...... 1122
9.5.1.4DifferentialApproachforRealOpenChannelFlow Problems................................................. 1122
9.5.1.5ApplicationsoftheGoverningEquationsfor RealOpenChannelFlow................................ 1123
9.5.2ApplicationoftheGoverningEquationsforIdealOpen ChannelFlow....................................................... 1123
9.5.2.1EvaluationoftheActualDischarge...................... 1124
9.5.2.2ApplicationsoftheGoverningEquationsforIdeal OpenChannelFlow..................................... 1126
9.5.3ApplicationoftheGoverningEquationsforaHydraulicJump.... 1126
9.6MajorHeadLossduetoFlowResistanceinRealOpenChannelFlow...... 1126
9.6.1UniformversusNonuniformOpenChannelFlow................. 1127
9.6.2EvaluationoftheMajorHeadLossTermintheEnergy Equation............................................................ 1128
9.6.3TheChezyEquationandEvaluationoftheChezy Coefficient, C ....................................................... 1128
9.6.4TurbulentChannelFlowResistanceEquationsandTheir RoughnessCoef ficients............................................. 1129
9.6.5Manning’sEquationandEvaluationofManning’sRoughness Coefficient, n ........................................................ 1129
9.7FlowType(State)andFlowRegime......................................... 1131
9.7.1TheRoleandSignificanceofUniformFlow........................ 1131
9.7.2TheDefinitionofFlowRegimesforUniformFlow................. 1132
9.7.3TheOccurrenceofNonuniformFlowandChangesinthe FlowRegime........................................................ 1132
9.8TransitionsandControlsinOpenChannelFlow............................ 1132
9.8.1TheDefinitionofaControlinOpenChannelFlow................ 1133
9.8.1.1TheOccurrenceofCriticalFlowatControls............. 1134
9.8.2TheDefinitionofaFlow-MeasuringDeviceinOpen ChannelFlow....................................................... 1135
9.9EnergyConceptsinOpenChannelFlow.................................... 1135
9.9.1HydrostaticPressureDistribution.................................. 1136
9.9.2DeviationfromaHydrostaticPressureDistribution............... 1137
9.9.2.1Sharp-CrestedWeirandFreeOverfall................... 1138
9.9.3SpecificEnergyinOpenChannelFlow............................. 1139
9.9.4SpecificEnergyforRectangularChannelCrossSections........... 1140
9.9.4.1TheSpeci ficEnergyCurveforRectangularChannel Sections.................................................. 1140
9.9.4.2DerivationofCriticalFlowforRectangularChannel Sections.................................................. 1141
9.9.4.3TheDepth–DischargeCurveforRectangularCross Sections.................................................. 1143
9.9.5SpecificEnergyforNonrectangularChannelCrossSections....... 1145
9.9.5.1DerivationofCriticalFlowforNonrectangular ChannelSections......................................... 1146
9.9.6SubcriticalandSupercriticalFlow.................................. 1147
9.9.7AnalysisoftheOccurrenceofCriticalFlowatControls............ 1148
9.9.7.1CriticalFlowatControlsduetoan Abrupt/MaximumVerticalConstriction................ 1149
9.9.7.2CriticalFlowatControlsduetoanAbrupt/Maximum HorizontalConstriction.................................. 1153
9.9.7.3CriticalFlowatControlsduetoaChangein ChannelBottomSlope................................... 1156
9.9.8AnalysisoftheOccurrenceofUniformFlowasa Control.............................................................. 1159
9.10MomentumConceptsinOpenChannelFlow............................... 1162
9.10.1TheMomentumFunction........................................... 1162
9.10.1.1The UseofControlsintheFormationofa HydraulicJump......................................... 1163
9.10.1.2DefinitionoftheMomentumFunctionfor RectangularChannelSections........................... 1164
9.10.1.3TheMomentumFunctionCurveforRectangular ChannelSections......................................... 1166
9.10.1.4DefinitionoftheMomentumFunctionfor NonrectangularChannelSections....................... 1170
9.11GeometricPropertiesofSomeCommonChannelSections.................. 1171
9.11.1GeometricPropertiesofRectangularChannelSections............. 1172
9.11.2GeometricPropertiesofTrapezoidalChannelSections............. 1172
9.11.3GeometricPropertiesofTriangularChannelSections.............. 1174
9.11.4GeometricPropertiesofPartiallyFilledCircular ChannelSections.................................................... 1175
9.12FlowDepthandReactionForceforShortChannelTransitionsinOpen ChannelFlow:IdealFlow................................................... 1176
9.12.1FlowDepthforGradualChannelTransitions:NotControls....... 1176
9.12.1.1FlowDepthforaGradualUpwardStep................ 1177
9.12.1.2FlowDepthforaGradualDecreaseinChannel Width.................................................... 1180
9.12.2FlowDepthforAbruptChannelTransitions:Controls............. 1187
9.12.2.1FlowDepthforanAbruptUpwardStep................ 1187
9.12.2.2FlowDepthforanAbruptDecreasein ChannelWidth.......................................... 1192
9.12.2.3FlowDepthforTypicalFlow-MeasuringDevices....... 1196
9.12.3ReactionForceonOpenChannelFlowStructures /Controls (AbruptFlowTransitionsandFlow-MeasuringDevices).......... 1204
9.13FlowDepthandMajorHeadLossforaHydraulicJumpinOpen ChannelFlow................................................................ 1209
9.13.1CriticalFlowattheHydraulicJump................................ 1210
9.13.2DerivationoftheHydraulicJumpEquations:Rectangular ChannelSections.................................................... 1211
9.13.3NumericalSolutionforaHydraulicJump:Nonrectangular ChannelSections.................................................... 1214
9.13.4ComputationoftheMajorHeadLossduetoaHydraulicJump... 1215
9.14FlowDepthandMajorHeadLossinUniformOpenChannelFlow: RealFlow.................................................................... 1221
9.15FlowDepthandMajorHeadLossinNonuniformOpenChannel Flow:RealFlow.............................................................. 1228
9.16ActualFlowrateinFlowMeasrurementandControlDevicesin OpenChannelFlow:IdealFlowMeters..................................... 1253
9.16.1Flow-MeasuringDevicesinOpenChannelFlow................... 1255
9.16.1.1IdealFlowMeters....................................... 1255
9.16.1.2CriticalDepthMeters.................................... 1255
9.16.2ControlsServingasFlowMeasurementDevicesinOpen ChannelFlow....................................................... 1256
9.16.2.1DeterminationoftheDepth–DischargeRelationship forControlsServingasFlow-MeasuringDevices....... 1256
9.16.2.2DeviationsfromtheAssumptionsforCriticalFlow inControlsServingasFlow-MeasuringDevices........ 1257
9.16.2.3AppliedDepth–DischargeRelationshipforIdeal versusCriticalFlowMeters.............................. 1258
9.16.3ActualFlowrateforIdealFlowMeters............................. 1258
9.16.4ActualFlowrateforCriticalFlowMeters........................... 1259
9.16.5AComparisonbetweenIdealFlowMetersandCritical DepthMeters....................................................... 1260
9.16.5.1AdvantagesofCriticalDepthMeters.................... 1261
9.16.5.2DisadvantagesofCriticalDepthMeters................. 1261
9.16.5.3IdealFlowMeters:TypicalFlow-Measuring Devices.................................................. 1261
9.17EvaluationoftheActualFlowrateforIdealFlowMetersinOpen ChannelFlow................................................................ 1262
9.17.1EvaluationoftheActualFlowrateforIdealFlowMeters.......... 1263
9.17.1.1ApplicationoftheBernoulliEquation................... 1263
9.17.1.2ApplicationoftheContinuityEquation,the MomentumEquation,andDimensionalAnalysis....... 1264
9.17.1.3DerivationoftheDischargeCoefficient, Cd .............. 1265
9.17.1.4EvaluationoftheDischargeCoef ficient, Cd ............. 1266
9.17.2SluiceGates,Weirs,Spillways,VenturiFlumes,andContracted OpeningsinOpenChannelFlow................................... 1266
9.17.3EvaluationoftheActualFlowrateforaSluiceGate................ 1267
9.17.4EvaluationoftheActualFlowrateforaSharp-CrestedWeir....... 1272
9.17.4.1RectangularSharp-CrestedWeir........................ 1275
9.17.4.2TriangularSharp-CrestedWeir.......................... 1278
9.17.4.3AComparisonbetweenaRectangularandaTriangular Sharp-CrestedWeir...................................... 1282
9.17.5EvaluationoftheActualFlowrateforaSpillway.................. 1283
9.17.6EvaluationoftheActualFlowrateforaBroad-CrestedWeir....... 1285
9.17.7EvaluationoftheActualFlowratefortheParshallFlume (VenturiFlume)..................................................... 1291
9.17.8EvaluationoftheActualFlowrateforaContracted Opening............................................................
10.1.1OccurrenceandIllustrationoftheDragForceand theLiftForce........................................................
10.2ApplicationoftheEulerian(Integral)versusLagrangian (Differential)FormsoftheGoverningEquations............................ 1308
10.3ModelingFlowResistanceinExternalFlow:ASubsetLevel ApplicationoftheGoverningEquations..................................... 1308
10.4ApplicationoftheGoverningEquationsinExternalFlow.................. 1309
10.4.1ApplicationoftheGoverningEquationsforIdeal ExternalFlow....................................................... 1309
10.4.1.1EvaluationoftheDragForce............................ 1310
10.5TheDragForceandtheLiftForceinExternalFlow......................... 1311
10.5.1EvaluationoftheDragForceinExternalFlow..................... 1312
10.5.1.1ModelingtheDragForceintheMomentum Equation................................................. 1313
10.5.1.2DerivationoftheDragCoefficient, CD 1314
10.5.2DeterminationoftheDragCoef ficient, CD ......................... 1315
10.5.3EvaluationoftheDragCoef ficient, CD ............................. 1315
10.5.3.1TheRoleoftheVelocityofFlow......................... 1316
10.5.3.2TheRoleoftheShapeoftheBody....................... 1316
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countenance at the dinner-table, readily follows and fully comprehends the topics of conversation carried on by his English masters.
The duties of the khitmutghar commence at daylight, when he puts in an appearance bearing the morning cup of tea. Unless otherwise ordered, he is only expected to be present, properly dressed, at each meal. One of his most important duties is to be able to cook fairly well when called upon to do so, more especially when his master may move into camp either on the march or on a shooting expedition. Then he is expected to show his powers in the culinary art; and, generally speaking, Mohammedan cooks acquit themselves admirably in this respect. They are especially clever at making omelettes, soufflets, and such-like. It may be here mentioned, however, by way of warning to the uninitiated in such matters, that the native method of preparing a meal is not always too nice to our ideas, so that it is well to avoid visiting the cooking-tent immediately before dinner, or not improbably you will there see something or other going on not calculated to give one an appetite.
Next in our list comes the bheestie or water-carrier, also of the Mohammedan religion; but altogether a less troublesome mortal to deal with. Generally speaking, the Bengal bheestie is a good, willing, hard-working servant, seldom giving trouble or requiring reproof. His chief duties are to supply the house and stables with fresh water from the best well in the neighbourhood. It is the special duty of the bheestie to keep the chatties or earthen jars of the bathroom filled with water. Where a garden is kept up—and in hot climates there is nothing so refreshing to the eye as a few flowers and bright-green shrubs around the house—it is the duty of the bheestie to assist the native gardener in watering the plants. He also, morning and evening, sprinkles with water the flooring of the verandas, footpaths, and dusty roads in the vicinity of his master’s abode. This has the effect of laying the dust and cooling the air—no slight boon to exhausted Europeans during the terrible months of April and May, just before the first rainfall.
The dhobie or washerman is another important individual in the Anglo-Indian establishment. The great majority of dhobies are
Hindus; but in Eastern Bengal, Mohammedan dhobies are often to be met with. Though given to assuming airs and importance, the dhobie is of low caste, generally speaking; a mild inoffensive being, plying his trade industriously, and giving little trouble to his master. There is a proverbial saying that obtains among the Hindus which pronounces a dhobie as untrustworthy; but in reality he is no worse than his brethren in this respect. The dhobie is one of the first to bestir himself in the early morning, and accompanied by a small bail or bullock, carrying his bundle of clothes, he may be seen making his way in the direction of some tank or distant pool on the river-bank. On reaching the scene of operations, he strips himself of superfluous clothing, girds up his loins, and proceeds to business. Soon the air resounds with the heavy thwacks of some article of raiment, which, twisted into a small compass, the dhobie again and again whirls round his head, and brings down upon a flat piece of wood or stone placed on the margin of the water. Each blow is accompanied by a grunt from the operator, as if to give an additional impetus to the stroke. This somewhat rough treatment is liable to wear out fine linen all too soon, and to make buttons fly; but considering that the dhobie has no mangle to assist him, nor any of the ordinary appliances of a laundry, and, generally speaking, only a small smoky hovel— probably filled to overflowing with his wife and numerous children— wherein to complete his work, it is astonishing how well he acquits himself of his task; the well-starched, snow-white shirt-fronts bearing witness to his skill and painstaking. Unless articles of clothing are plainly marked, the dhobie has a tiresome habit of sewing coloured pieces of cotton into the corners of every shirt and handkerchief, to distinguish them from others, which practice has anything but a beautifying effect. The dhobie considers himself so far independent that he need only appear at stated times, to receive or make over his master’s clothes from the hands of the bearer. He will never take service as an indoor servant in the house of a European.
The duties of the mehter, sweeper or ‘knight of the broom,’ are so commonplace as to require only a brief notice. He is always of low caste; and though often addressed as ‘jemadar’ by the other servants, he is always looked down upon, more especially for his habit of eating or drinking anything left from the table of his master. It
is his special duty to take charge of and feed his master’s dogs. He supplies them with food at a fixed rate, takes them in the early morning for a bathe in the nearest tank, and towards sunset, produces for inspection, in separate iron dishes, the food which he has provided for each one of his charges.
It is amusing to observe how well-bred English dogs despise and turn up their noses at their native attendant, permitting the latter to lead them about and wash them when necessary without a growl of disapproval, but at the same time clearly showing by their outward bearing that no familiarities will be permitted.
Next we come to the syce or native groom; and in a stable where a valuable Arab horse has to be cared for, he is a most important personage. A really good, trustworthy syce is nowadays seldom to be met with. There are Mohammedan syces throughout Northern India; but the great majority are Hindus of low caste. The duties of the syce are, to groom and feed the horse he is put in charge of—a separate syce is necessary for each one of the horses comprising a stable—to be ready to accompany his master to the parade-ground, the band-stand, or for wherever he may be bound; and to keep the latter in sight and follow him any distance, no matter at what pace the sahib may choose to ride. It is astonishing what powers of endurance these native grooms display in this respect; for however far the distance or quick the gallop, he is seldom left far behind, and nearly always makes his appearance soon after his master draws rein.
A Bengal syce worthy of the name can hardly in any country in the world be surpassed at his work. He is a most excellent groom; and by means of hand-rubbing—which he often practises for hours together—he brings out the muscles and sinews of a horse till they are as tough and hard as iron. It is a good custom to inspect daily the allowance of corn or grain provided by the syce for his charge, as not unfrequently dishonest grooms steal a portion of it and grind it for their own food.
The ‘grass-cutter,’ the last in our list, is a humble individual, who, as his title tells us, supplies grass for the horse to which he is attached.
Hay is seldom seen in India; but horses thrive well on a particular kind of soft green grass, which the grass-cutter cuts, or rather digs up with a small iron instrument called a koorpah. It is well, every now and again, to examine the quality and quantity of the grass supplied for each horse, or else lazy individuals will likely enough bring in coarse hard stuff quite unfit for the purpose. In large stations, a grass-cutter who performs his work properly has often to walk many miles before reaching a spot where soft tender grass is procurable. The grass-cutter is under the immediate orders of the syce, and usually receives four rupees a month as pay for his services.
In conclusion, it may be mentioned, that one of the most important rules in the young Englishman’s household should be that each native servant regularly receives his pay on a certain date in each month. Without this being steadily acted up to, matters never work smoothly in an establishment, but will cause constant bickerings. Whereas, when paid regularly, and treated with kindness and forbearance, these poor people speedily become attached to their master, and exert themselves to meet with his approval.
J. H. B.
