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Introduction to Engineering Fluid Mechanics 1st Edition Marcel Escudier

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IntroductiontoEngineering FluidMechanics

Introduction toEngineering FluidMechanics

GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom

OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries ©MarcelEscudier2017

Themoralrightsoftheauthorhavebeenasserted FirstEditionpublishedin2017 Impression:1

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TomywifeAgnes,oursonStephen,andthememoryofmyMother andGrandmother

Preface

Afluidisamaterialsubstanceintheformofaliquid,agas,oravapour.Themostcommon examples,tobefoundinbotheverydaylifeandinengineeringapplications,arewater,air,and steam,thelatterbeingthevapourformofwater.Theflow(i.e.motion)offluidsisessentialto thefunctioningofawiderangeofmachinery,includingtheinternal-combustionengine,the gasturbine(whichincludestheturbojet,turbofan,turboshaft,andturbopropengines),wind andhydraulicturbines,pumps,compressors,rapidlyrotatingdiscs(asincomputerdrives), aircraft,spacecraft,roadvehicles,andmarinecraft.Thisbookisconcernedprimarilywith Newtonianfluids,suchaswaterandair,forwhichtheviscosityisindependentoftheflow.The quantitativeunderstandingoffluidflow,termedfluiddynamics,isbasedupontheapplication ofNewton’slawsofmotiontogetherwiththelawofmassconservation.Toanalysetheflow ofagasoravapour,forwhichthedensitychangesinresponsetopressurechanges(known ascompressiblefluids),itisalsonecessarytotakeintoaccountthelawsofthermodynamics, particularlythefirstlawintheformofthesteady-flowenergyequation.Thesubjectoffluid mechanicsencompassesbothfluidstaticsandfluiddynamics.Fluidstaticsconcernsthevariationofpressureinafluidatrest(aswillbeseeninChapter4,thislimitationneedstobestated moreprecisely),andisthebasisforasimplemodeloftheearth’satmosphere.

Thistextisaimedprimarilyatstudentsstudyingforadegreeinmechanicalengineering oranyotherbranchofengineeringwherefluidmechanicsisacoresubject.Aeronautical(or aerospace),chemical,andcivilengineeringarealldisciplineswherefluidmechanicsplaysan essentialrôle.Thatisnottosaythatfluidflowisofnosignificanceinotherareas,suchas biomedicalengineering.Thehumanbodyinvolvestheflowofseveraldifferentfluids,some quiteordinarysuchasairintherespiratorysystemandwater-likeurineintherenalsystem. Otherfluids,likebloodinthecirculatorysystem,andsynovialfluid,whichlubricatesthejoints, havecomplexnon-Newtonianproperties,asdomanysyntheticliquidssuchaspaint,slurries, andpastes.Abriefintroductiontotherheologyandflowcharacteristicsofnon-Newtonian liquidsisgiveninChapters2,15,and16.

Asindicatedinthetitle,thistextisintendedtointroduce thestudenttothesubjectof fluidmechanics.Itcoversthosetopicsnormallyencounteredinathree-yearmechanicalengineering-degreecourseorthefirstandsecondyearsofafour-yearmechanical-engineeringdegreecourse,aswellassometopicscoveredingreaterdetailinthefinalyears.Thefirstten chapterscovermaterialsuitableforafirst-yearcourseormoduleinfluidmechanics.Compressibleflow,flowthroughaxial-flowturbomachineryblading,internalviscousfluidflow, laminarboundarylayers,andturbulentflowarecoveredintheremainingeightchapters.There aremanyothertextbookswhichcoverasimilarrangeofmaterialasthistextbutoftenfrom amuchmoremathematicalpointofview.Mathematicsisessentialtotheanalysisoffluid flowbutcanbekepttoalevelwithinthecapabilityofthemajorityofstudents,asistheintentionherewheretheemphasisisonunderstandingthebasicphysics.Theanalysisofmany

flowsituationsrestsuponasmallnumberofbasicequationswhichencapsulatetheunderlyingphysics.Betweenthesefundamentalequationsandthefinalresults,whichcanbeapplied directlytothesolutionofengineeringproblems,canbequiteextensivemathematicalmanipulationanditisalltooeasytolosesightofthefinalaim.Abasicunderstandingofvectorsis requiredbutnotofvectoranalysis.Tensornotationandanalysisisalsonotrequiredandthe useofcalculusiskepttoaminimum.

Theapproachtocertaintopicsmaybeunfamiliartosomelecturers.Aprimeexampleisdimensionalanalysis,whichwesuggestisapproachedusingthemathematicallysimplemethod ofsequentialeliminationofdimensions(Ipsen’smethod).TheauthorbelievesthatthistechniquehasclearpedagogicaladvantagesoverthemorewidelyusedRayleigh’sexponentmethod, whichcaneasilyleavethestudentwiththemistaken(andpotentiallydangerous)ideathatany physicalprocesscanberepresentedbyasimplepower-lawformula.Theimportanceofdimensionsanddimensionalanalysisisstressedthroughoutthebook.Theauthorhasalsofoundthat thedevelopmentofthelinearmomentumequationdescribedinChapter9ismorestraightforwardtopresenttostudentsthanitisviaReynoldstransporttheorem.Theapproachadopted hereshowsveryclearlytherelationshipwiththefamiliar F = ma formofNewton’ssecond lawofmotionandavoidstheneedtointroduceanentirelynewconceptwhichisultimately onlyasteppingstonetotheendresult.Thetreatmentofcompressibleflowisalsosubtlydifferentfrommosttextsinthat,forthemostpart,equationsaredevelopedinintegralrather thandifferentialform.Theanalysisofturbomachineryislimitedtoflowthroughtheblading ofaxial-flowmachinesandreliesheavilyonChapters3,10,and11.

‘Whydoweneedafluidmechanicstextbookcontaininglotsofequationsandalgebra,given thatcomputersoftwarepackages,suchasFLUENTandPHOENICS,arenowavailablewhich canperformveryaccuratecalculationsforawiderangeofflowsituations?’Toanswerthis questionweneedfirsttoconsiderwhatismeantbyaccurateinthiscontext.Thedescriptionof anyphysicalprocessorsituationhastobeintermsofequations.Inthecaseoffluidmechanics, thefullsetofgoverningequationsisextremelycomplex(non-linear,partialdifferentialequationscalledtheNavier-Stokesequations)andtosolvepracticalproblemswedealeitherwith simplified,orapproximate,equations.Typicalassumptionsarethatallfluidpropertiesremain constant,thatviscosity(theessentialpropertywhichidentifiesanymaterialasbeingafluid) playsnorole,thattheflowissteady(i.e.therearenochangeswithtimeatanygivenlocation withinthefluid),orthatfluidandflowpropertiesvaryonlyinthedirectionofflow(so-called one-dimensionalflow).ThederivationoftheNavier-Stokesequations,andtheaccompanying continuityequation,isthesubjectofChapter15.Exactanalyticalsolutionoftheseequations ispossibleonlyforahandfulofhighlysimplified,idealisedsituations,oftenfarremovedfrom therealworldofengineering.Althoughthesesolutionsarecertainlymathematicallyaccurate,duetothesimplificationsonwhichtheequationsarebasedtheycannotbesaidtobean accuraterepresentationofphysicalreality.Evennumericalsolutions,howevernumericallyaccurate,areoftenbaseduponsimplifiedversionsoftheNavier-Stokesequations.Inthecase ofturbulentflow,thetopicofChapter18,calculationsofpracticalinterestarebasedupon approximateequationswhichattempttomodelthecorrelationswhicharisewhentheNavierStokesequationsaretimeaveraged.Itisremarkablethatvaluableinformationaboutpractical engineeringproblemscanbeobtainedfromconsiderationsofsimplifiedequations,suchasthe

one-dimensionalequations,atminimalcostintermsofbothtimeandmoney.Whatisessential,however,isagoodphysicalunderstandingofbasicfluidmechanicsandaknowledgeof whatanycomputersoftwareshouldbebasedupon.Itistheaimofthistexttoprovidejustthat.

Alreadyinthisbrief Preface thenamesNavier,Newton,Rayleigh,Reynolds,andStokes haveappeared.InAppendix1weprovidebasicbiographicalinformationabouteachofthe scientistsandengineerswhosenamesappearinthisbookandindicatetheircontributionsto fluidmechanics.

Acknowledgements

Theauthorgratefullyacknowledgestheinfluenceofseveraloutstandingteachers,bothasa studentatImperialCollegeLondonandsubsequentlyasaResearchAssociateattheMassachusettsInstituteofTechnology.Myinterestin,andenjoymentof,fluidmechanicswassparked whenIwasanundergraduatebytheinspiringteachingofRobertTaylor.BrianSpalding,my PhDsupervisor,andBrianLaunderarenotonlyinternationallyrecognisedfortheirresearch contributionsbutwerealsoexcellentcommunicatorsandteachersfromwhomIbenefitted asapostgraduatestudent.AsaresearchassociateatMITIattendedlecturesandseminars byAscherH.Shapiro,JamesA.Fay,RonaldF.Probstein,andErikMollo-Christensen,allinspiringteachers.Finally,myfriendFernandoTavaresdePinhohasgivenfreelyofhistimeto answerwithinsightmanyquestionswhichhaveariseninthecourseofwritingthisbook.

Cheshire,August2016

3 Unitsofmeasurement,dimensions,anddimensional

3.2TheInternationalSystemofUnits(SI)

3.5Theprincipleofdimensionalconsistency(orhomogeneity)

3.6Dimensionalversusnon-dimensionalrepresentation

3.7Buckingham’s (pi)theorem

3.8Sequentialeliminationofdimensions(Ipsen’smethod)

3.9Rayleigh’sexponentmethod

3.14Similarityandscaling

3.15Scalingcomplications

3.16OtherReynolds-numberconsiderations

3.18SELF-ASSESSMENTPROBLEMS

4.2Pressurevariationinafluidatrest;thehydrostaticequation

4.3Pressurevariationinaconstant-densityfluidatrest

4.4Basicpressuremeasurement

4.5Mercurybarometer

4.6Piezometertube

4.7U-tubemanometer

4.8Effectofsurfacetension

4.9Inclined-tubemanometer

4.10Multiplefluidlayers

4.11Variable-densityfluid;stability

4.13Earth’satmosphere

4.14Pressurevariationinanacceleratingfluid

4.15SUMMARY

4.16SELF-ASSESSMENTPROBLEMS

5 Hydrostaticforceexertedonasubmergedsurface 124

5.1Resultantforceonabodyduetouniformsurfacepressure 124

5.2Verticalcomponentofthehydrostaticforceactingonasubmerged surface 126

5.3Archimedes’principleandbuoyancyforceonasubmergedbody133

5.4Hydrostaticforceactingonasubmergedverticalflatplate 137

5.5Hydrostaticforceactingonasubmergedcurvedsurface 143

5.6Stabilityofafully-submergedbody 147

5.7Stabilityofafreelyfloatingbodyandmetacentricheight 148

5.8SUMMARY 154

5.9SELF-ASSESSMENTPROBLEMS 154

6 Kinematicdescriptionoffluidsinmotion andapproximations

6.1Fluidparticles

6.2Steady-flowassumption

6.3Pathlines,streamlines,streamsurfaces,andstreamtubes 162

6.4No-slipconditionandtheboundarylayer 163

6.5Single-phaseflow 164

6.6Isothermal,incompressible,andadiabaticflow 164

6.7One-dimensionalflow 165

6.8One-dimensionalcontinuityequation(mass-conservationequation)166

6.9Averageflowvelocity V 170

6.10Flowofaconstant-densityfluid 171

6.11SUMMARY 172

6.12SELF-ASSESSMENTPROBLEMS 172

7 Bernoulli’sequation 174

7.1Netforceonanelementalsliceoffluidflowingthroughastreamtube174

7.2Accelerationofafluidslice 176

7.3Euler’sequation 178

7.4Bernoulli’sequation

7.5InterpretationsofBernoulli’sequation

7.6Pressurelossversuspressuredifference

7.7SUMMARY

7.8SELF-ASSESSMENTPROBLEMS

8 EngineeringapplicationsofBernoulli’sequation

8.1Wind-tunnelcontraction

8.2Venturi-tubeflowmeter

8.3Venturi-tubedesignandthecoefficientofdischarge CD

8.4OtherVenturi-tubeapplications

8.5Orifice-plateflowmeter

8.6Otherdifferential-pressureinlineflowmeters

8.7FormulaOneracingcar

8.8Pitottube

8.9Pitot-statictube

8.10Liquiddrainingfromatank

8.11Cavitationinliquidflows

8.12SUMMARY

9 Linearmomentumequationandhydrodynamicforces

9.1Problemunderconsideration

9.2Basiclinearmomentumequation

9.3Fluid-structureinteractionforce

9.5SUMMARY

10 Engineeringapplicationsofthelinearmomentum equation

10.1Forcerequiredtorestrainaconvergentnozzle

10.2Rocket-enginethrust

10.5Flowthroughasuddenenlargement

10.7Reactionforceonapipebend

10.8Reactionforceonapipejunction

10.9Flowthroughalinearcascadeofguidevanes

10.10Freejetimpingingonaninclinedflatsurface

10.11Peltonimpulsehydraulicturbine

11.1Introductoryremarks

11.3Bernoulli’sequationandotherrelationsforcompressible-gasflow279

11.4Subsonicflowandsupersonicflow

11.5MachwaveandMachangle

11.6Steady,one-dimensional,isentropic,perfect-gasflowthrough agraduallyconvergentduct 283

11.7Steady,one-dimensional,isentropic,perfect-gasflowthrougha convergent-divergentnozzle

11.8Normalshockwaves

11.9Perfectlyexpanded,underexpanded,andoverexpandednozzleflow307 11.10SUMMARY

12.1Obliqueshockwaves

12.2Prandtl-Meyerexpansionfan(centredexpansionfan)

12.3Supersonicaerofoilsandshock-expansiontheory

13.3Isothermalpipeflowwithwallfriction

13.4Frictionlesspipeflowwithheatadditionorextraction:Rayleighflow353 13.5SUMMARY

14 Flowthroughaxial-flow-turbomachineryblading

14.1Turbomachinery(general)

14.2Dimensionalanalysisandbasicnon-dimensionalparameters

14.3Linearbladecascade:Geometryandnotation

14.4Incompressibleflowthroughalinearcascade

14.5Compressibleflowthroughalinearcascade

14.6Rotor-flowvelocitytriangles

14.7Euler’sturbomachineryequationforanaxial-flowrotor

14.8Compressibleflowthroughanaxialturbomachinestage

14.9Degreeofreaction Λ

15 Basicequationsofviscous-fluidflow

15.1EquationsofmotioninCartesian-coordinateform

15.2Equationsofmotionincylindrical-coordinateform

15.3Boundaryconditions

15.4Non-dimensionalformoftheNavier-Stokesandcontinuityequations405

15.5FlowofageneralisedNewtonianfluid

15.6SUMMARY

16 Internallaminarflow

16.1Generalremarks

16.2PoiseuilleflowofaNewtonianfluid,hydraulicdiameter,andPoiseuille number 412

16.3Poiseuilleflowthroughanaxisymmetriccylindricalduct 416

16.4CombinedplaneCouetteandPoiseuilleflowbetweeninfinite parallelplates:Couette-Poiseuilleflow 421

16.5Taylor-Couetteflow 427

16.6PoiseuilleflowofgeneralisedNewtonianfluidsbetween infiniteparallelplates 431

16.7Viscometerequations 438

16.8SUMMARY 442

16.9SELF-ASSESSMENTPROBLEMS 443

17 Laminarboundarylayers 445

17.1Introductoryremarks 445

17.2Two-dimensionallaminarboundary-layerequations 447

17.3Flat-platelaminarboundarylayer:Blasius’solution 451

17.4Wedge-flowlaminarboundarylayers:FalknerandSkan’sequation461

17.5vonKármán’smomentum-integralequation 468

17.6Profilemethodsofsolution 473

17.7Aerofoilliftinsubsonicflow 484

17.8SUMMARY 487

17.9SELF-ASSESSMENTPROBLEMS 488

18 Turbulentflow 490

18.1Transitionalandturbulentflow 490

18.2Reynoldsdecomposition,Reynoldsaveraging,andReynoldsstresses491

18.3Turbulent-kinetic-energyequationandReynolds-stressequation494

18.4Turbulencescales 496

18.5Turbulencemodelling 498

18.6Two-dimensionalturbulentboundarylayersandCouetteflow 499

18.7PlaneturbulentCouetteflowandtheLawoftheWall 499

18.8Fully-developedturbulentflowthroughasmoothcircularpipe506

18.9Surfaceroughness 508

18.10Fully-developedturbulentflowthrougharough-surfacecircularpipe509

18.11Minorlossesinpipesystems 511

18.12Momentum-integralequation 517

18.13Flat-plateboundarylayer 518

18.14Boundarylayerswithstreamwisepressuregradient 525

18.15Bluff-bodydrag 526

18.16SUMMARY 531

18.17SELF-ASSESSMENTPROBLEMS 532

Appendix1Principalcontributorstofluidmechanics 535

Appendix2Physicalpropertiesofselectedgasesandliquids,andotherdata 545

Appendix3Areas,centroidlocations,andsecondmomentsofarea forsomecommonshapes 553

Appendix4Differentialequationsforcompressiblepipeflow 556

Appendix5Roughnessheights

Notation

EachRoman,Greek,andmathematicalsymbolisfollowedbyitsmeaning,itsSIunit,andits dimension(s).

Lower-caseRomansymbols

a acceleration m/s2 L/T2

c bladechordlength mL

c concentration kg/m3 M/L3

c soundspeed m/sL/T

c wettedperimeter mL

cf skin-frictioncoefficient

c0 speedoflightinvacuum m/sL/T

d diameter mL

e energy JML2 /T2

˙ exx extensionalstrainratein x-direction 1/s1/T

f non-dimensionalvelocity

fx bodyforceperunitmassactinginthe x-directionm/s2 L/T2

fD Darcyfrictionfactor

fF Fanningfrictionfactor

fF averageFanningfrictionfactor

g accelerationduetogravity m/s2 L/T2

g0 accelerationduetogravityatsealevel(z = z =0)m/s2 L/T2

h height mL

h spacingofparallelplates mL

h specificenthalpy

kJ/kgL2 /T2

h0 specificstagnationenthalpy kJ/kgL2 /T2

h0,REL relativestagnationenthalpy kJ/kgL2 /T2

i angleofincidence ◦ orrad–

j numberofindependentdimensions

k numberofnon-dimensionalgroups

k radiusofgyration mL

k specificturbulentkineticenergy m2 /s2 L2 /T2

k time-averagedspecificturbulentkineticenergym2 /s2 L2 /T2

kB Boltzmannconstant J/KML2 /T2 K

l length mL

lK Kolmogorovlengthscale mL

lM mixinglength mL

m mass kgM

m wedge-flowexponent – –

mA addedmass kgM

˙ m massflowrate kg/sM/T

n amountofsubstance kmolM

n numberofphysicalquantities – –

n power-lawexponentinpower-lawviscositymodel– –

p staticpressure PaM/LT2

pG gaugepressure PaM/LT2

pH hydrostaticpressure PaM/LT2

pREF referencepressure PaM/LT2

pT totalpressure PaM/LT2

pV vapourpressure PaM/LT2

p0 stagnationpressure PaM/LT2

p0,REL relativestagnationpressure PaM/LT2

p averagestaticpressure PaM/LT2

p fluctuatingcomponentofstaticpressure PaM/LT2

p intermediatestaticpressure PaM/LT2

p∗ non-dimensionalstaticpressure – –

˙ q heattransferrate WML2 /T3

˙ q heattransferrateperunitlength W/LML/T3

r radialdistance m L

s arclength m L

s cascade-bladespacing(orpitch) m L

s distancealongastreamline m L

s specificentropy m2 /s2 · KL2 /T2 θ

s0 specificstagnationentropy m2 /s2 · KL2 /T2 θ

t elapsedtime s T

t temperature ◦ C θ

t non-dimensionaltime – –

t ∗ non-dimensionaltime – –

u specificinternalenergy kJ/kgL2 /T2

u velocitycomponentin x-direction m/sL/T

u time-averagedvalueofvelocitycomponent u m/sL/T

u fluctuatingcomponentofvelocitycomponent u m/sL/T

u∗ non-dimensionalvalueofvelocitycomponent u

u+ velocitycomponent u normalisedby uτ

uP velocityofplasticplug m/sL/T

u0 centrelinevelocity m/sL/T

uτ frictionvelocity m/sL/T

v specificvolume m3 /kgL3 /M

v velocitycomponentin y-or r -direction m/sL/T

v time-averagedvalueofvelocitycomponent v m/sL/T

v fluctuatingcomponentofvelocitycomponent v m/sL/T

v+ velocitycomponent v normalisedby uτ

vK Kolmogorovvelocityscale m/sL/T

w specificweight N/m3 M/L2 T2

w velocitycomponentin z -or θ -direction m/sL/T

w time-averagedvalueofvelocitycomponent w m/sL/T

w fluctuatingcomponentofvelocitycomponent w m/sL/T

w+ velocitycomponent w normalisedby uτ

x distancealongorparalleltoasurface/streamwisedistancemL

X length mL

y distancenormaltoasurface mL

y+ distance y normalisedby uτ and ν

z bladeheight(orlength) mL

z depth(i.e.distancemeasuredverticallydownwards)mL

z height(i.e.distancemeasuredverticallyupwards) mL

z geometricaltitude mL

zG geopotentialaltitude mL

zC depthofcentroid mL

zP depthofcentreofpressure mL

Upper-caseRomansymbols

A cross-sectionalarea

choking(orsonic)area

AE nozzleexitarea m2 L2

AT nozzlethroatarea m2 L2

B barometric(oratmospheric)pressureorexternalpressurebarM/LT2

B log-lawconstant – –

Bi Binghamnumber – –

CD coefficientofdischarge

CD dragcoefficient

CF averagefrictionfactor

CL liftcoefficient

CP pressurecoefficient –

CP specificheatatconstantpressure

CV specificheatatconstantvolume m2 /s2 · KL2 /T2 θ

D diameter m L

D drag(ordragforce) N ML/T2

D meandiameter m L

DH hydraulicdiameter m L

DT nozzlethroatdiameter m L

D dragforceperunitlengthofsurface N/mM/T2

E energyreleased J ML2 /T2

E Young’smodulus PaM/LT2

Eu Eulernumber

F force NML/T2

F non-dimensionalstreamfunction

FB buoyancyforce NML/T2

Fθ functioninThwaites’method

Fr Froudenumber

G massvelocity kg/m2 sM/L2 T

G shearmodulus(fluid) PaM/LT2

G modulusofrigidity(solid) PaM/LT2

H heightordepth mL

H horizontalcomponentofforce NML/T2

H boundary-layershapefactor

He Hedstromnumber

H12 boundary-layershapefactor

I secondmomentofarea m4 L4

IC secondmomentofareaaboutanaxisthroughthem4 L4 area’scentroid

Ixy productofinertia m4 L4

K bulkmodulusofelasticity PaM/LT2

K consistencyindexinpower-lawviscositymodelPa · sn M/LT2–n

K losscoefficient

K turbomachinestagnation-pressurelosscoefficient––

Kn Knudsennumber

1/K compressibility 1/PaLT2 /M

L length mL

L lift(orliftforce) NML/T2

L∗ chokinglength mL

M Machnumber – –

M molarmass kg/kmol–

M momentum kg m/sML/T

M molecularweight kg/kmol–

MREL relativeMachnumber

˙ M momentumflowrate kg m/s2 ML/T2

˙ M momentumflowrateperunitwidthofductkg/s2 M/T2

MGmetacentricheight mL

N molecularnumberdensity 1/m3 1/L3

N numberofmolecules

N rotationalspeed rps1/T

NA Avogadronumber 1/kmol1/M

NP turbomachinepower-specificspeed

NS turbomachinespecificspeed

P piezometricpressure PaM/LT2

P power W ML2 /T3

Po Poiseuillenumber – –

Pr Prandtlnumber – –

˙ Q volumetricflowrate m3 /sL3 /T

˙ Q volumetricflowrateperunitwidth m2 /sL2 /T

R radius m L

R reactionforce N ML/T2

R resultantforce N ML/T2

R specificgasconstant m2 /s2 KL2 /T2 θ

RE meanradiusoftheearth m L

RH hydraulicradius m L

RI innerradiusofannulus m L

RO outerradiusofannulus m L

R molargasconstant(universalgasconstant)kJ/kmol KL2 /T2 θ

Re Reynoldsnumber – –

Rex Reynoldsnumberbaseduponlength x

Reδ Reynoldsnumberbaseduponlength δ

ReC criticalReynoldsnumber – –

ReD Reynoldsnumberbaseduponpipediameter– –

ReH Reynoldsnumberbaseduponhydraulicdiameter– –

Rep Reynoldsnumberbaseduponplasticviscosity

S fluid-structureinteractionforce N ML/T2

St Strouhalnumber – –

T absolutetemperature

T skin-frictioncoefficient= θτS /μU∞

T surface-tensionforce

T thrust(orthrustforce) N ML/T2

T timeinterval s T

T torque N · mML2 /T2

T0 stagnation(ortotal)temperature

T0,REL relativestagnationtemperature

Ta Taylornumber – –

U free-streamvelocity m/s L/T

U0 scalingvelocity m/s L/T

U∞ free-streamvelocity m/s L/T

V velocity m/s L/T

V verticalcomponentofforce

VB buoyancyforce

VD verticallydownwardsforce

VU verticallyupwardsforce

ML/T2

ML/T2

ML/T2

ML/T2

V∞ terminalvelocity m/s L/T

V average(bulk-mean)velocity m/s L/T

V non-dimensionalvelocity – –

V + averagevelocity V normalisedby uτ

V volume m3 L3

VC criticalvolumeforvalidityofcontinuumhypothesism3 L3

VD displacedvolume m3 L3

VS submergedvolume m3 L3

V∞ y-directionvelocityatedgeofboundarylayerm/sL/T

W relativevelocity m/sL/T

W weight NML/T2

W width mL

W work JML2 /T2

˙ W rateofworkinput(powerinput) WML2 /T3

We Webernumber

X length mL

Y boundary-layerthickness mL

Y surfacetension N/mM/T2

Z depthofliquid mL

Lower-caseGreeksymbols (Englishwordinparentheses)

α (alpha)angleofattack

α absoluteflowangle

α conicalgapangle

◦ orrad–

◦ orrad–

◦ orrad–

α non-dimensionalconstantinBlasius’equation––

α constantinshock-structureanalysis m2 /sL2 /T

β (beta)obliqueshockangle

β relative-flowangle

β wedgeangle

γ (gamma)ratioofspecificheats

˙

γ shearrate

˙

◦ orrad–

◦ orrad–

◦ orrad–

1/s1/T

γxy shearratecorrespondingto τxy 1/s1/T

δ (delta)angleofdeflectionordeviation

◦ orrad–

δ boundary-layerthickness mL

δ radialgapwidth mL

δ A elementofarea m2 L2

δ F elementofforce NML/T2

δ h infinitesimalheightdifference mL

δ H elementofhorizontalforce NML/T2

δ m elementofmass kgM

δ p infinitesimalchangeordifferenceinpressurePaM/LT2

δ s infinitesimalchangeofdistance mL

δ t infinitesimalchangeintime sT

δ V elementofverticalforce NML/T2

δ V elementofvolume m3 L3

δ W elementofweight NML/T2

δ x elementofstreamwiseor x-directiondistance mL

δ y elementofdistancenormaltoasurfaceor y-directionmL distance

δ z elementofdepthor z -directiondistance mL

δ z elementofheight mL

δ ∗ boundary-layerdisplacementthickness mL

δSUB thicknessofviscoussublayer mL

δ1 boundary-layerdisplacementthickness mL

δ2 boundary-layermomentum-deficitthickness mL (epsilon)turbulentkineticenergydissipationrate m2 /s3 L2 /T3 upwashordownwashangle

◦ orrad–

ε (epsilon)eccentricity mL

ε non-dimensionalannulargapwith

ε surface-roughnessheight mL

ε + surface-roughnessheightnormalisedby uτ and ν

η (eta)dynamicviscosity

η boundary-layersimilarityvariable

θ (theta)angle

Pa · sM/LT

◦ orrad–

θ boundary-layermomentum-deficitthickness mL

θ contactangle

θ turningangle

˙ θ angularvelocity

◦ –

◦ –

rad/s1/T

θ angularacceleration rad/s2 1/T2

κ (kappa)lapserate K/m θ /L

κ vonKármán’sconstant

κ wavenumber 1/m1/L

λ (lamda)timeconstant sT

λ pressure-gradientparameter

λ Pohlhausen’spressure-gradientparameter

λ wavelengthofturbulence mL

λP Poiseuille-flowpressure-gradientparameter

λθ boundary-layerpressure-gradientparameter

μ (mu)dynamicviscosity

Pa · sM/LT

μ Machangle ◦ orrad–

μEFF effectiveviscosity

Pa · sM/LT

μP viscosityofplasticplug Pa sM/LT

μT eddyviscosity Pa · sM/LT

μ∞ infinite-shear-rateviscosity

ν (nu)kinematicviscosity

ν Prandtl-Meyerfunction

νT kinematiceddyviscosity

Pa sM/LT

m2 /sL2 /T

◦ –

m2 /sL2 /T

ξ (xi)bladestaggerangle

ξ non-dimensionaldistance

ξ turbomachineenthalpy-losscoefficient

ξP non-dimensionalradiusofplasticplug

◦ orrad–

ρ (rho)density kg/m3 M/L3

σ (sigma)densityratio

σ relativedensity

σ surfacetension N/mM/T2

σxx normalstressin x-direction PaM/LT2

τ (tau)characteristictime sT

τ shearstress PaM/LT2

τK Kolmogorovtimescale sT

τS surfaceshearstress PaM/LT2

τS averagesurfaceshearstress PaM/LT2

τY yieldstress PaM/LT2

τxy shearstressactingin y-direction PaM/LT2

φ (phi)angle

φ bladecamberangle

φ turbomachineflowcoefficient

χ (chi)bladeangle

χ boundary-layerscalefactor

◦ orrad–

◦ orrad–

◦ orrad–

ψ (psi)streamfunction 1/s1/T

ψ hydraulicmachinepressure-changecoefficient––

ω (omega)angularvelocity rad/s1/T

Upper-caseGreeksymbols

Γ (gamma)circulation m2 /sL2 /T

Γ lapserate ◦ C/km θ /L

ΓAD adiabaticlapserate ◦ C/km θ /L (delta)finitechangeordifference scalinglength mL

S shockthickness mL

p finitepressuredifference PaM/LT2

p0 reductioninstagnationpressure PaM/LT2

Z finitedepthdifference mL

ρ densitydifference kg/m3 M/L3

Θ (theta)dilation 1/s1/T

˜

Θ ratio θ /δ ,where θ =boundary-layer momentum-deficitthickness

Λ (lamda)degreeofreaction

Λ molecularmeanfreepath mL (pi)non-dimensionalgroup

Π shockstrength

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grace, her form all of beauty to me who opposite sat and was watching her dextrous fingers.

The manufacture of flax into linen material was ever felt to be of vast importance, and was encouraged by legislation from earliest colonial days, but it received a fresh impulse in New England through the immigration of about one hundred Irish families from Londonderry. They settled in New Hampshire on the Merrimac about 1719. They spun and wove by hand, but with far more skill than prevailed among those English settlers who had already become Americans. They established a manufactory according to Irish methods, and attempts at a similar establishment were made in Boston. There was much public excitement over spinning. Women, rich as well as poor, appeared on Boston Common with their wheels, thus making spinning a popular holiday recreation. A brick building was erected as a spinning-school, and a tax was placed in 1737 to support it. But this was not an industrial success, the excitement died out, the public spinning-school lost its ephemeral popularity, and the wheel became again simply a domestic duty and pride.

For many years after this, housewives had everywhere flax and hemp to spin and weave in their homes, and the preparation of these staples seems to us to-day a monumental labor. On almost every farm might be seen a patch of the pretty flax, ripening for the hard work of pulling, rippling, rotting, breaking, swingling, and combing, which all had to be done before it came to the women’s hands for spinning. The seed was sown broad-cast, and allowed to grow till the bobs or bolls were ripe. The flax was then pulled and spread neatly in rows to dry. This work could be done by boys. Then men whipped or threshed or rippled out all the seed to use for meal; afterwards the flax stalks were allowed to lie for some time in water until the shives were thoroughly rotten, when they were cleaned and once more thoroughly dried and tied in bundles. Then came work for strong men, to break the flax on the ponderous flaxbreak, to get out the hard “hexe” or “bun,” and to swingle it with a swingle knife, which was somewhat like a wooden dagger. Active men could swingle forty pounds a day on the swingling-board. It was then hetchelled or combed or hackled by the housewife, and thus the rough tow was

gotten out, when it was straightened and made ready for the spruce distaff, round which it was finally wrapped. The hatchelling was tedious work and irritating to the lungs, for the air was filled with the fluffy particles which penetrated everywhere. The thread was then spun on a “little wheel.” It was thought that to spin two double skeins of linen, or four double skeins of tow, or to weave six yards of linen, was a good day’s work. For a week’s work a girl received fifty cents and “her keep.” She thus got less than a cent and a half a yard for weaving. The skeins of linen thread went through many tedious processes of washing and bleaching before being ready for weaving; and after the cloth was woven it was “bucked” in a strong lye, time and time again, and washed out an equal number of times. Then it was “belted” with a maple beetle on a smooth, flat stone; then washed and spread out to bleach in the pure sunlight. Sometimes the thread, after being spun and woven, had been washed and belted a score of times ere it was deemed white and soft enough to use. The little girls could spin the “swingling tow” into coarse twine, and the older ones make “all tow” and “tow and linen” and “harden” stuffs to sell.

To show the various duties attending the manufacture of these domestic textiles by a Boston woman of intelligence and social standing, as late as 1788, let me quote a few entries from the diary of the wife of Col. John May:—

A large kettle of yarn to attend upon. Lucretia and self rinse our through many waters, get out, dry, attend to, bring in, do up and sort 110 score of yarn, this with baking and ironing.

Went to hackling flax.

Rose early to help Ruth warp and put a piece in the loom.

Baking and hackling yarn. A long web of tow to whiten and weave.

The wringing out of this linen yarn was most exhausting, and the rinsing in various waters was no simple matter in those days, for the water did not conveniently run into the houses through pipes and

conduits, but had to be laboriously carried in pailfuls from a pump, or more frequently raised in a bucket from a well.

I am always touched, when handling the homespun linens of olden times, with a sense that the vitality and strength of those enduring women, through the many tedious and exhausting processes which they had bestowed, were woven into the warp and woof with the flax, and gave to the old webs of linen their permanence and their beautiful texture. How firm they are, and how lustrous! And how exquisitely quaint and fine are their designs; sometimes even Scriptural designs and lessons are woven into them. They are, indeed, a beautiful expression of old-time home and farm life. With their close-woven, honest threads runs this finer beauty, which may be impalpable and imperceptible to a stranger, but which to me is real and ever-present, and puts me truly in touch with the life of my forbears. But, alas, it is through intuition we must learn of this oldtime home life, for it has vanished from our sight, and much that is beautiful and good has vanished with it.

The associations of the kitchen fireside that linger in the hearts of those who are now old can find no counterpart in our domestic surroundings to-day. The welcome cheer of the open fire, which graced and beautified even the humblest room, is lost forever with the close gatherings of the family, the household occupations, the homespun industries which formed and imprinted in the mind of every child the picture of a home.

Transcriber’s Notes

Minor punctuation errors have been silently corrected.

Page 100: “take the the case” changed to “take the case”

Page 162: “promply sailed” changed to “promptly sailed”

Page 302: “was was set outside” changed to “was set outside”

Spelling and punctuation quoted from original sources has been left as-is.

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