BasicConceptsand Definitions
“Toworkintelligently”(OrvilleandWilburWright)“one needstoknowtheeffectsofvariationsincorporatedinthe surfaces. Thepressuresonsquaresaredifferentfromthose onrectangles,circles,triangles,orellipses. Theshapeof theedgealsomakesadifference.” from TheStructureofthePlane –MurielRukeyser
LEARNINGOBJECTIVES
• Reviewthefundamentalprinciplesoffluidmechanicsandthermodynamics requiredtoinvestigatetheaerodynamicsofairfoils,wings,andairplanes.
• Recalltheconceptsofunitsanddimensionandhowtheyareappliedtosolving andunderstandingengineeringproblems.
• Learnaboutthegeometricfeaturesofairfoils,wings,andairplanesandhowthe namesforthesefeaturesareusedinaerodynamicscommunications.
• Exploretheaerodynamicforcesandmomentsthatactonairfoils,wings,and airplanesandlearnhowwedescribetheseloadsquantitativelyindimensional formandascoefficients.
• Determinetheconditionsforlongitudinallystable,steady,levelaircraftflight.
• Reviewcontrol-volumeanalysisbyexaminingthemomentumtheoryofpropellers andhelicopterrotors.
• Learnthefundamentalsofhydrostaticsandwhenthetopicappliesto aerodynamics.
1.1 INTRODUCTION
Thestudyofaerodynamicsrequiresanumberofbasicdefinitions,includinganunambiguousnomenclatureandanunderstandingoftherelevantphysicalproperties, relatedmechanics,andappropriatemathematics.Ofcourse,thesenotionsarecommontootherdisciplines,anditisthepurposeofthischaptertoidentifyandexplain
AerodynamicsforEngineeringStudents.DOI: 10.1016/B978-0-08-100194-3.00001-8 Copyright©2017ElsevierLtd.Allrightsreserved.
thosethatarebasicandpertinenttoaerodynamicsandthataretobeusedintheremainderofthevolume.
1.1.1 BasicConcepts
Thistextisanintroductoryinvestigationofaerodynamicsforengineeringstudents.1 Hence,weareinterestedintheorytotheextentthatitcanbepracticallyappliedto solveengineeringproblemsrelatedtothedesignandanalysisofaerodynamicobjects.
Thedesignofvehiclessuchasairplaneshasadvancedtothelevelwherewerequirethewealthofexperiencegainedintheinvestigationofflightoverthepast100 years.Weplantoinvestigatethecleverapproximationsmadebythefewwholearned howtoapplymathematicalideasthatledtoproductivemethodsandusefulformulas topredictthedynamicalbehaviorof“aerodynamic”shapes.Weneedtolearnthe strengthsand,moreimportant,thelimitationsofthemethodologiesanddiscoveries thatcamebeforeus.
Althoughwehaveextensivearchivesofrecordedexperienceinaeronautics,there arestillmanyopportunitiesforadvancement.Forexample,significantadvancements canbeachievedinthestateoftheartindesignanalysis.Aswedevelopideasrelatedtothephysicsofflightandtheengineeringofflightvehicles,wewilllearn thestrengthsandlimitationsofexistingproceduresandexistingcomputationaltools (commerciallyavailableorotherwise).Wewilllearnhowairfoilsandwingsperform andhowweapproachthedesignsoftheseobjectsbyanalyticalprocedures.
Thefluidofprimaryinterestisair,whichisagasatstandardatmosphericconditions.Weassumethatthedynamicsoftheaircanbeeffectivelymodeledintermsof thecontinuumfluiddynamicsmodel2 incompressibleorsimple-compressiblefluid. Airisafluidwhoselocalthermodynamicstateweassumeisdescribedeitherby itsmassdensity ρ = constant,orbytheidealgaslaw.Inotherwords,weassume airbehavesaseitheranincompressibleorasimple-compressiblemedium,respectively.Theconceptsofacontinuum,anincompressiblesubstance,andasimplecompressiblegaswillbeelaboratedoninChapter 4.
The equationofstate,knownastheidealgaslaw,relatestwothermodynamic propertiestootherpropertiesand,inparticular,thepressure.Itis
p = ρRT (1.1) where p isthethermodynamicpressure, ρ ismassdensity, T isabsolute(thermodynamic)temperature,andthespecificgasconstantforairis R = 287J/(kgK)or
1 Ithaslongbeencommoninengineeringschoolsforanelementary,macroscopicthermodynamicscourse tobecompletedpriortoacompressible-flowcourse.Theportionsofthistextthatdiscusscompressible flowassumethatsuchacourseprecedesthisone,andthusthediscussionsassumesomeelementaryexperiencewithconceptssuchasinternalenergyandenthalpy.
2 Thatis,airapproximatedasacontinuousformofmatter,whichissufficientlyaccurateformostformsof flightpropelledbyair-breathingengines.
R = 1716ft-lb/(slug°R) 1 .Pressureandtemperaturearerelativelyeasytomeasure. Forexample,“standard”barometricpressureatsealevelis p = 101,325Pascals, whereaPascal(Pa)is1N/m2 .InImperialunitsthisis14.675psi,wherepsiislb/in2 and1psiisequalto6895Pa(notethat14.675psiisequalto2113.2lb/ft2 ).Thestandardtemperatureis288.15K(or15°C,whereabsolutezeroequalto 273.15°C isused).InImperialunitsthisis519°R(or59°F,whereabsolutezeroequalto 459.67°Fisused).Substitutingintotheidealgaslaw,wegetforthestandarddensity ρ = 1.225kg/m3 inSIunits(and ρ = 0.00237slugs/ft3 inImperialunits).This isthedensityofairatsealevelgiveninthetableofdataforatmosphericair;thetable forstandardatmosphericconditionsisprovidedin AppendixB.
Thethermodynamicpropertiesofpressure,temperature,anddensityareassumed tobethepropertiesofamass-pointparticleofairatalocation x = (x,y,z) in spaceataparticularinstantintime, t .Weassumethemeasurementvolumetobe sufficientlysmalltobeconsideredamathematicalpoint.Wealsoassumethatitis sufficientlylargesothatthesepropertieshavemeaningfromtheperspectiveofequilibriumthermodynamics.Andwefurtherassumethatthepropertiesarethesameas thosedescribedinacourseonclassicalequilibriumthermodynamics.Therefore,we assumethat localthermodynamicequilibrium prevailswithinthemass-pointparticleat x and t regardlessofhowfastthethermodynamicstatechangesastheparticle movesfromonelocationinspacetoanother.Thisisanacceptableassumptionforour macroscopicpurposesbecausemolecularprocessesaretypicallymuchfasterthan anychangesintheflowfieldweareinterestedinfromamacroscopicpointofview are.Inaddition,weinvokethe continuumhypothesis,withwhichweassumethat theairisacontinuousformofmatterratherthandiscretemolecules.Thuswecan defineallflowpropertiesascontinuousfunctionsofpositionandtimeandthatthese functionsaresmooth,thatis,theirderivativesarecontinuous.Thisallowsustoapply differentialintegralcalculustosolvepartialdifferentialequationsthatsuccessfully modeltheflowfieldsofinterestinthiscourse.Inotherwords,predictionsbasedon thetheoryreportedinthistexthavebeenexperimentallyverified.TheContinuum Hypothesisisvalidformostatmosphericflightbecausetherearesomanymolecules perunitvolume(approximately1019 cm 3 atsealevel)thatthemotionofanyindividualmoleculecannotbesensed.
Todevelopthetheory,thefundamentalprinciplesofclassicalmechanicsareassumed.Theyare
•Conservationofmass
•Newton’ssecondlawofmotion
•Firstlawofthermodynamics
•Secondlawofthermodynamics
Theprincipleofconservationofmassdefinesamass-pointparticle,whichisafixedmassparticle.Thustheprinciplealsodefinesmassdensity ρ ,whichismassperunit volume.Ifamass-pointparticleconservesmass,aswehavepostulated,thendensity changescanonlyoccurifthevolumeoftheparticlechanges,becausethedimension
ofmassdensityisM/L3 ,whereMismassandLislength.TheSIunitofdensityis thuskg/m3 .
Avehiclemovingthroughtheairorairinmotionaroundthevehicleareofcourse causesofourstudyofthetopicofaerodynamics.Itisnaturalforthestudentreading thistexttowishtogetstartedquicklyintoastudyofsuchmotion.Aerodynamics, andfluiddynamicsingeneral,arerichlynon-linearandthus,arerarelysimpleand quickstudies.However,thestudentwillfindoneimportantconceptinfluidmotion developedinSection 2.2.1 andthatconceptisarelationshipbetweenpressureand velocityknownasBernoulli’sequation.Itcanbewritteninseveralforms,buthere consideritthisway:
Heretheleftside, po ,isknownbythesynonyms“totalpressure”and“stagnation pressure.”Inmany,butnotall,ofthesimpleflowsastudentencounters,thistotal pressureisconserved—itisaconstant.Inthesecases,andalongastreamlineinsome morecomplexflows,thetwotermsontherighthandsidemustsumtoaconstant. Thefirsttermontherightisthestaticpressure,generallyjustcalledpressure.For pressuretobereduced,suchasoverthetopofanairfoilorwing,thesecondtermon therightsidemustbecomegreater.Becauseinlow-speedaerodynamicsthedensity isconstant,anyincreaseinthemagnitudeofthesecondtermiscausedbyanincrease inairvelocity, V .EvenwhenBernoulli’sequationisnotquantitativelycorrectfor acertainsituation,theenergyexchangebetweenstaticpressureandvelocityofthe flowexists.
Studentsshouldapplythisequationwithcarewhilelearning,insubsequentSectionsandChapters,theconditionsunderwhichBernoulli’sequationcanbeused properly.
Newton’ssecondlawdefinestheconceptofforceintermsofacceleration(F = ma ).Theaccelerationofamass-pointparticleisthechangeinitsvelocitywithrespecttoachangeintime.Letthevelocityvector V = (u,v,w);thisisthevelocity ofamass-pointparticleatapointinspace, x = (x,y,z),ataparticularinstantin time t .Theaccelerationofthismass-pointparticleis
Thisisknownasthesubstantialderivativeofthevelocityvector.Sinceweareinterestedinthepropertiesatfixedpointsinspaceinacoordinatesystemattachedtothe objectofinterest(i.e.,the“laboratory”coordinates),therearetwopartstomass-point particleacceleration.Thefirstisthelocalchangeinvelocitywithrespecttotime.The secondtakesintoaccounttheconvectiveaccelerationassociatedwithachangeinvelocityofthemass-pointparticlefromitslocationupstreamofthepointofinterestto theobservationpoint x attime t
Wewillalsobeinterestedinthespatialandtemporalchangesinanyproperty f ofamass-pointparticleoffluid.Thesechangesaredescribedbythesubstantial
derivativeasfollows:
Thisequationdescribesthechangesinanymaterialproperty f ofamasspointat aparticularlocationinspaceataparticularinstantintime.Thisisinalaboratory referenceframe,theso-called Eulerianviewpoint. Thenextstepinconceptualdevelopmentofatheoryistoconnectthechanges inflowpropertieswiththeforces,moments,andenergyexchangethatcausethese changestohappen.Themathematicalconceptspresentedandappliedinthisbook describethedynamicbehaviorofathermo-mechanicalfluid.Inotherwords,weneglectelectromagnetic,relativistic,andquantumeffectsondynamics.Wedothisby firstadoptingtheNewtoniansimple-compressibleviscousfluidmodelforrealfluids (e.g.,waterandair),whichisdescribedindetailinChapter 2.Moreover,wewill applythesimpler,yetquiteuseful,Euler’sperfectfluidmodel,alsodescribedin Chapter 2.Itisquitefortunatethatthelattermodelhassignificantpracticaluseinthe designanalysisofaerodynamicobjects.
BeforeweproceedtoChapter 2 andlookatthedevelopmentoftheequations ofmotionandthesimplificationswewillapplytopotentialflowsinChapters 5, 6, and 7,wereviewsomeusefulmathematicaltools,definethegeometryofthewing, andprovideanoverviewofwingperformanceinthenextthreesections,respectively.
AERODYNAMICSAROUNDUS
IsBernoulli’sEquationaSpring-MassSystem?
YouhaveseenthatBernoulli’sequationinaerodynamicsiswritten
wherethe 1 2 ρV 2 termiscalledthe“dynamicpressure”.Butatfirstyoulikelyare unsureofjustwhatthedynamicpressure,whichwecall q ,is.Staticpressureyou experienceclearlyinyourearsasyouascendordescendathousandfeetinairora coupleoffeetunderwater.Total,orstagnation,pressureyoumightconsiderasthe increasedstaticpressureyoufeelwhenyourhandisheldoutthecarwindowwith palmfacingforward.Dynamicpressureseemsfarlessphysicaltomoststudents. Recallanelementaryphysicsproblem,amassonafrictionlesssurfaceand attachedtoaspring:
Whenthemassispulledtoextendthespring,energyisstoredinthespring.If youreleasethemass,itmovestowardsthespring,convertingspringenergyinto
kineticenergy.Thesumofspringenergyandkineticenergyisconstant;ifthe energyisnotinthemotion,it’sinthespring.Prettysimple.
Considerpressure,whichyouarelikelyaccustomedtothinkingofasaforce perarea.Buttheunitsofforceperareaarealsounitsofenergypervolume.So Bernoulli’sequationisalsoadescriptionofhowthreeenergies(pervolume)are related.Whatthreeenergies?Becausedensityismasspervolume,dynamicpressureisakineticenergypervolume.Thisisaddedtoanotherenergypervolume, staticpressure,toequalstagnationpressure,whichisaconstantinmanyflows. Thus,staticpressureislikeaspringenergypervolume.Inthisview,Bernoulli’s equationtellsusthatanincompressiblesteadyflowislikeaspring-masssystem; iftheenergyisnotinthespring,(staticpressure)thenit’sinthemotion(kinetic energy).There’ssimplynootherplacefortheenergytogoinsuchasystem.
Bernoulli’slawsimplytellsusthemotionandcompressiontradeoffbackand forthinasteadyincompressibleflow,likeamass–springsystem.
1.2 UNITSANDDIMENSIONS
Measurementandcalculationrequireasystemofunitsinwhichquantitiesaremeasuredandexpressed.Aerospaceisaglobalindustry,andtobebestpreparedfora globalcareer,engineersneedtobeabletoworkinbothsystemsinusetoday.Even whenoneworksforacompanywithastrictstandardforuseofonesetofunits, customers,suppliers,andcontractorsmaybebetterversedinanother,anditisthe engineer’sjobtoefficientlyreconcilethevariousdocumentsorspecificationswithoutintroducingconversionerrors.Consider,too,thephysicsbehindtheunits.That is,oneknowsthatforlinearmotion,forceequalstheproductofmassandacceleration.Theunitsoneusesdonotchangethephysicsbutchangeonlyourquantitative descriptionsofthephysics.Whenconfusedaboutunits,focusontheprocessorstate beingdescribedandstepthroughtheanalysis,trackingunitstheentireway.
IntheUnitedStates,“Imperial”or“English”unitsremaincommon.Distance (withinthescaleofanaerodynamicdesign)isdescribedininchesorfeet.Massis describedbyeithertheslugorthepound-mass(lbm).Weightisdescribedbypounds (lb)orbytheequivalentunitwitharedundantname,thepound-force(lbf).Large distances—forexample,therangeofanaircraft—aredescribedinmilesornautical miles.Speedisfeetpersecond,milesperhour,orknots,whereoneknotisone nauticalmileperhour.Multimilliondollaraircraftarestillmarketedandsoldusing knotsandnauticalmiles(tryawebsearchon“777range”),sotheseunitsarenot obsolete.
Inotherpartsoftheworld,andinK-12educationintheUnitedStates,thedominantsystemofunitsistheSystèmeInternationald’Unités,commonlyabbreviatedas “SIunits.”Itisusedthroughoutthisbook,exceptinaveryfewplacesasspecially noted.
Itisessentialtodistinguishbetween“dimension”and“unit.”Forexample,the dimension“length”expressesthe qualitative conceptoflineardisplacement,ordistancebetweentwopoints,asanabstractidea,withoutreferencetoactualquantitative measurement.Theterm“unit”indicatesaspecifiedamountofaquantity.Thusameterisaunitoflength,beinganactual“amount”oflineardisplacement,andsoisa mile.Themeterandmilearedifferent units,sinceeachcontainsadifferent amount oflength,butbothdescribelengthandthereforeareidentical dimensions 3
Expressingthisinsymbolicform:
• x meters = [L](aquantityof x metershasthedimensionoflength)
• x miles = [L](aquantityof x mileshasthedimensionoflength)
• x meters = x miles(x milesand x metersareunequalquantitiesoflength)
•[x meters] = [x miles](thedimensionof x metersisthesameasthedimensionof x miles).
1.2.1 FundamentalDimensionsandUnits
Therearefivefundamentaldimensionsintermsofwhichthedimensionsofallother physicalquantitiesmaybeexpressed.Theyaremass[M],length[L],time[T],temperature[θ ],andcharge.4 (Chargeisnotusedinthistextsoisnotdiscussedfurther.)
Aconsistentsetofunitsisformedbyspecifyingaunitofparticularvalueforeachof thesedimensions.Inaeronauticalengineeringtheacceptedunitsare,respectively,the kilogram,themeter,thesecond,andtheKelvinordegreeCelsius.Theseareidentical withtheunitsofthesamenamesincommonuseandaredefinedbyinternational agreement.
Itisconvenientandconventionaltorepresentthenamesoftheseunitsbyabbreviations:
kgforkilogram,slugsforslugs,andlbmforpound-mass mformeterandftforfeet sforsecond
◦ CfordegreeCelsiusand ◦ FfordegreeFahrenheit KforKelvinandRforRankine(butalsoforthespecificgasconstant)
ThedegreeCelsiusisoneone-hundredthpartofthetemperatureriseinvolved whenpurewateratfreezingtemperatureisheatedtoboilingtemperatureatstandard
3 Quiteoften“dimension”appearsintheform“adimensionof8meters,”meaningaspecifiedlength. Thisisthuscloselyrelatedtotheengineer’s“unit,”andimplieslinearextensiononly.Anothercommon exampleoftheuseof“dimension”isin“three-dimensionalgeometry,”implyingthreelinearextensionsin differentdirections.Referencesinlaterchapterstotwo-dimensionalflow,forexample,illustratethis.The meaningheremustnotbeconfusedwitheitheroftheseuses.
4 Someauthoritiesexpresstemperatureintermsoflengthandtime.Thisintroducescomplicationsthatare brieflyconsideredinSection 1.3.8
pressure.IntheCelsiusscale,purewateratstandardpressurefreezesat0 ◦ C(32◦ F) andboilsat100 ◦ C(212◦ F).
TheunitKelvin(K)isidenticalinsizetothedegreeCelsius(◦ C),buttheKelvin scaleoftemperatureismeasuredfromtheabsolutezerooftemperature,whichis approximately 273 ◦ C.ThusatemperatureinKisequaltoatemperaturein ◦ Cplus 273.15.Similarly,degreesRankineequals ◦ Fplus459.69.
1.2.2 FractionsandMultiples
Sometimes,thefundamentalunitsjustdefinedareinconvenientlylargeorinconvenientlysmallforaparticularcase.Ifso,thequantitycanbeexpressedasafraction ormultipleofthefundamentalunit.Suchmultiplesandfractionsaredenotedbya prefixappendedtotheunitsymbol.Theprefixesmostusedinaerodynamicsare:
M(mega)—1million k(kilo)—1thousand m(milli)—1-thousandthpart µ(micro)—1-millionthpart n(nano)—1-billionthpart
Thus
1MW = 1,000,000W
1mm = 0.001m 1µm = 0.001mm
Aprefixattachedtoaunitmakesanewunitso,forexample, 1mm2 = 1(mm)2 = 10 6 m2
Forsomepurposes,thehourortheminutecanbeusedastheunitoftime.
ForImperialunits,everydayscientificnotationisusedratherthansuffixesorprefixes.Oneexceptionisstressorpressureofthousandsofpoundspersquareinch, knownaskpsi.Additionally,lengthmayswitchfromfeettoinchesormiles.Itis commontousefractionalinches,butthestudentengineerneedstobeawarethat theimpliedprecisioninafractionincreasesrapidly.Forexample,1/2 = 0.5,but 1/32 = 0.03125.
1.2.3 UnitsofOtherPhysicalQuantities
Havingdefinedthefourfundamentaldimensionsandtheirunits,itispossibleto establishunitsofallotherphysicalquantities(see Table1.1).Speed,forexample,is definedasthedistancetraveledinunittime.ItthereforehasthedimensionLT 1 and ismeasuredinmeterspersecond(ms 1 ).Itissometimesdesirabletousekilometers
Table1.1 UnitsandDimensions
Quantity
Dimension Unit(abbreviation)
LengthLMeter(m)orfeet(ft)
MassMKilogram(kg)orslugorpound-mass(lbm)
TimeTSecond(s)
Temperature θ
AreaL2
VolumeL3
SpeedLT 1
AccelerationLT 2
DegreeCelsius(◦ C)orFahrenheit(◦ F)orKelvin(K)or Rankine(R)
Squaremeter(m2 )orsquarefoot(ft2 )
Cubicmeter(m3 )orcubicfoot(ft3 )
Meterspersecond(ms 1 )orfeetpersecond(fts 1 )
Meterspersecondpersecond(ms 2 )orfeetpersecond squared(fts 2 )
Angle1Radianordegree(◦ )(radianisexpressedasaratioand isthereforedimensionless)
AngularvelocityT 1
AngularaccelerationT 2
FrequencyT 1
DensityML 3
ForceMLT 2
StressML 1 T 2
Radianspersecond(s 1 )
Radianspersecondpersecond(s 2 )
Cyclespersecond,Hertz(s 1 ,Hz)
Kilogramspercubicmeter(kgm 3 )orslugspercubic foot(slugft 3 )orpound-masspercubicfoot(lbmft 3 )
Newton(N)orpound(lb)
NewtonspersquaremeterorPascal(Nm 2 orPa)or poundspersquareinch(psi)orpoundspersquarefoot (psf)
Strain1None(expressedasanondimensionalratio)
PressureML 1 T 2
EnergyworkML2 T 2
PowerML2 T 3
MomentML2 T 2
AbsoluteviscosityML 1 T 1
KinematicviscosityL2 T 1
BulkelasticityML 1 T 2
NewtonspersquaremeterorPascal(Nm 2 orPa)or poundspersquareinch(psi)orpoundspersquarefoot (psf)
Joule(J)orfoot-pounds(ftlb)
Watt(W)orhorsepower(Hp)
Newtonmeter(Nm)orfoot-pounds,(ftlb)
KilogramspermeterpersecondorPoiseuilles (kgm 1 s 1 orPI)orslugsperfootpersecond (slugft 1 s 1 )
Meterssquaredpersecond(m2 s 1 )orfeetsquaredper second(ft2 s 1 )
NewtonspersquaremeterorPascal(Nm 2 orPa)or poundspersquareinch(psi)orpoundspersquarefoot (psf).
perhourorknots(nauticalmilesperhour;see AppendixD)asunitsofspeed;care mustbeexercisedtoavoiderrorsofconsistency.
Tofindthedimensionsandunitsofmorecomplexquantities,weusetheprinciple of dimensionalhomogeneity.Thissimplymeansthat,inanyvalidphysicalequation, thedimensionsofbothsidesmustbethesame.Thus,forexample,if(mass)n appears ontheleft-handsideoftheequation,itmustalsoappearontheright-handside; similarlyforlength,time,andtemperature.
Thus,tofindthedimensionsofforce,weuseNewton’ssecondlawofmotion
Force = mass × acceleration whereaccelerationisspeed ÷ time.Expresseddimensionally,thisis
Writingintheappropriateunits,itisseenthataforceismeasuredinunitsofkgms 2 . Since,however,theunitofforceisgiventhenameNewton(abbreviatedusuallytoN), itfollowsthat
1N = 1kgms 2
Itshouldbenotedthattherecanbeconfusionbetweentheuseofmbothfor “milli”andfor“meter.”Thisisavoidedbyuseofaspace.Thusmsdenotesmillisecondwhilemsdenotestheproductofmeterandsecond.
Theconceptofdimensionformsthebasisofdimensionalanalysis,whichisused todevelopimportantandfundamentalphysicallaws.Itstreatmentispostponedto Section 1.5
1.2.4 ImperialUnits
Engineersinsomepartsoftheworld,theUnitedStatesinparticular,useasetofunits basedontheImperialsystems5 inwhichthefundamentalunitsare
Mass—slug
Length—foot
Time—second
Temperature—degreeFahrenheitorRankine
5 SincemanyvaluabletextsandpapersexistusingImperialunits,thisbookcontains,as AppendixD, atableoffactorsforconvertingfromtheImperialtotheSIsystem.
1.3 RELEVANTPROPERTIES
Anyfluidthatwewishtodescribeexistsinsomestateofmatter.Forexample,ifwe areworkingwithaflowofnitrogen,isitgaseousnitrogenorliquidnitrogen?For whateverthestatethefluidisin,weneedacollectionof“tools”tousetodescribe thethermodynamicstateofthefluidatapoint,overtime,andthroughoutafield. Anunambiguousdescriptionofthethermodynamicstateofthefluidisimportantof coursetoamathematicalmodelofaflowandisvitaltoeffectiveengineeringcommunication.Thusinthissectionwedevelopthetoolstousetoformtheseunambiguous descriptions.
1.3.1 FormsofMatter
Mattermayexistinthreeprincipalforms—solid,liquid,orgas—correspondingin thatordertodecreasingrigidityofthebondsbetweenthemoleculesthemattercomprises.Aspecialformofagas,a plasma,haspropertiesdifferentfromthoseofa normalgas;althoughbelongingtothethirdgroup,itcanberegardedjustifiablyas aseparate,distinctformofmatterthatisrelevanttothehighest-speedaerodynamics suchasflowsoverspacecraftreenteringtheatmosphere.
Inasolidtheintermolecularbondsareveryrigid,maintainingthemoleculesin whatisvirtuallyafixedspatialrelationship.Thusasolidhasafixedvolumeand shape.Thisisseenclearlyincrystals,inwhichthemoleculesoratomsarearranged inadefinite,uniformpattern,givingallcrystalsofthatsubstancethesamegeometric shape.
Aliquidhasweakerbondsbetweenitsmolecules.Thedistancesbetweenthe moleculesarefairlyrigidlycontrolled,butthearrangementinspaceisfree.Therefore,liquidhasacloselydefinedvolumebutnodefiniteshape,andmayaccommodate itselftotheshapeofitscontainerwithinthelimitsimposedbyitsvolume.
Agashasveryweakbondingbetweenthemoleculesandthereforehasneither definiteshapenordefinitevolume,butratherwillfillthevesselcontainingit.
Aplasmaisaspecialformofgasinwhichtheatomsareionized—thatis,they havelostorgainedoneormoreelectronsandthereforehaveanelectricalcharge. Anyelectronsthathavebeenstrippedfromtheatomsarewanderingfreewithinthe plasmaandhaveanegativeelectricalcharge.Ifthenumberofionizedatomsandfree electronsissuchthatthetotalpositiveandnegativechargesareapproximatelyequal, sothatthegasasawholehaslittleornocharge,itistermedaplasma.Inastronautics plasmaisofparticularinterestforthereentryofrockets,satellites,andspacevehicles intotheatmosphere.
1.3.2 Fluids
Afluidisaliquidoragas.Thefieldofsciencewecall“FluidDynamics”thusencompassesbothaerodynamicsandhydrodynamics.Equationsofmotionforafluiddo notdependonthatfluidbeingliquidorgas,buttheequationofstatewilldiffer.The basicfeatureofafluidisthatitcanflow—thisistheessenceofanydefinitionofit.
However,flowappliestosubstancesthatarenottruefluids—forexampleafinepowderpiledonaslopingsurfacewillflow.Forexample,flourpouredinacolumnonto aflatsurfacewillformaroughlyconicalpile,withalargeangleofrepose,whereas water,whichisatruefluid,pouredontoahorizontalsurfacewillspreaduniformly overit.Equally,apowdermaybeheapedinaspoonorbowl,whereasaliquidwill alwaysformalevelsurface.Anydefinitionofafluidmustallowforthesefacts,soa fluidmaybedefinedas“mattercapableofflowing,andeitherfindingitsownlevel (ifaliquid),orfillingthewholeofitscontainer(ifagas).”Oncewerestrictourselves toanidealgas,suchasforsteady,levelatmosphericflight,distinctionsbetweenair asa“Newtonianfluid”andfineparticulatesareclear.ANewtonianfluidisonein whichshearstressisproportionaltorateofshearingstrain;thisisneverfoundin particulates.
Experimentshowsthatanextremelyfinepowder,inwhichtheparticlesarenot muchlargerthanmolecularsize,findsitsownlevelandmaythuscomeunderthe commondefinitionofaliquid.Also,aphenomenonwellknowninthetransportof sands,gravels,andsoforth,isthatthesesubstancesfindtheirownleveliftheyare agitatedbyvibrationorthepassageofairjetsthroughtheparticles.Thesearespecial cases,however,anddonotdetractfromtheauthorityofthedefinitionofafluidasa substancethatflowsor(tautologically)thatpossessesfluidity.
1.3.3 Pressure
Atanypointinafluid,whetherliquidorgas,thereisapressure.Ifabodyisplacedin afluid,itssurfaceisbombardedbyalargenumberofmoleculesmovingatrandom. Undernormalconditionsthecollisionsonasmallareaofsurfacearesofrequentthat theycannotbedistinguishedasindividualimpactsbutappearasasteadyforceonthe area.The intensity ofthis“molecularbombardment”isits staticpressure. Veryfrequentlythestaticpressureisreferredtosimplyaspressure.Theterm static israthermisleadingasitdoesnotimplythatthefluidisatrest.
Forlargebodiesmovingoratrestinthefluid(e.g.,air),thepressureisnotuniform overthesurface,andthisgivesriseto aerodynamic or aerostaticforce,respectively.
Sinceapressureisforceperunitarea,ithasthedimensions
[Force]÷[area]=[MLT 2 ]÷[L2 ]=[ML 1 T 2 ] (1.5) andisexpressedinunitsofNewtonspersquaremeterorinPascals(Nm 2 orPa). Pressureisalsocommonlyspecifiedinpoundspersquareinch(psi)orpoundsper squarefoot(psf).Itcanalsobeofusetoconsidertheaboveequationmultipliedby lengthoverlength:
[Force]∗[Length]÷ ([Area]∗[Length]) =[ML2 T 2 ]÷[L3 ]=[Energy]÷[Volume]
Thus,besidesthemostcommonviewofitasaforceperarea,pressurealsohasunits ofenergypervolume.
FIGURE1.1
Fictitioussystemsoftangentialforcesinstaticfluid.
PressureinFluidatRest
Considerasmallcubicelementcontainingfluidatrestinalargerbulkoffluidalso atrest.Thefacesofthecube,assumedconceptuallytobemadeofsomethinflexible material,aresubjecttocontinualbombardmentbythemoleculesofthefluidandthus experiencea force.Theforceonanyfacemayberesolvedintotwocomponents,one actingperpendiculartothefaceandtheotheralongit(i.e.,tangentialtoit).Considerthetangentialcomponentsonly;therearethreesignificantlydifferentpossible arrangements(Fig.1.1).System(a)wouldcausetheelementtorotate,andthusthe fluidwouldnotbeatrest;system(b)wouldcausetheelementtomove(upwardand totherightforthecaseshown),and,onceagain,thefluidwouldnotbeatrest.Since afluidcannotresistshearstressbutonlyrateofchangeinshearstrain(Sections 1.3.6 and 2.8.2),system(c)wouldcausetheelementtodistort,thedegreeofdistortion increasingwithtime,andthefluidwouldnotremainatrest.Theconclusionisthata fluidatrestcannotsustaintangentialstresses.
Pascal’sLaw
Considertherightprismoflength δz inthedirectionintothepageandcross-section ABC,theangleABCbeingarightangle(Fig.1.2).Theprismisconstructedofmaterialofthesamedensityasthefluidinwhichtheprismfloatsatrestwiththeface BChorizontal.
Pressures p1 , p2 ,and p3 actonthefacesshownand,asjustproved,actinthe directionperpendiculartotherespectiveface.Otherpressuresactontheendfacesof theprism,butareignoredinthepresentproblem.Inadditiontothesepressures,the weight W oftheprismactsverticallydownward.Considertheforcesactingonthe wedgethatisinequilibriumandatrest.
Resolvingforceshorizontally,
FIGURE1.2 PrismforPascal’sLaw.
thatis,
Resolvingforcesvertically, p3 δxδy p2 (δx sec α)δy cos α W = 0(1.7)
Now W = ρg(δx)2 tan αδy/2
Therefore,substitutingthisinEq. (1.7) anddividingby δxδy ,
Ifnowtheprismisimaginedtobecomeinfinitelysmall,sothat δx → 0,thethird termtendstozero,leaving p3 p2 = 0(1.8)
Thus,finally,
Havingbecomeinfinitesimallysmall,theprismisineffectapoint,sothisanalysis showsthat,atapoint,thethreepressuresconsideredareequal.Inaddition,theangle α ispurelyarbitraryandcantakeanyvalue,whilethewholeprismcanberotated throughacompletecircleaboutaverticalaxiswithoutaffectingtheresult.Itmaybe
concluded,then,thatthepressureactingatapointinafluidatrestissensedthesame byvarioussurfacesinalldirections.
1.3.4 Temperature
Inanyformofmatterthemoleculesareinmotionrelativetoeachother.Ingasesthe motionisrandommovementofmagnituderangingfromapproximately60nmunder normalconditionstosometensofmillimetersatverylowpressures.Thedistanceof freemovementofamoleculeofgasisthedistanceitcantravelbeforecollidingwith anothermoleculeorthewallsofthecontainer.Themeanvalueofthisdistanceforall moleculesinagasiscalledthelengthofthemeanmolecularfreepath.
Byvirtueofthismotion,themoleculespossesskineticenergy,andthisenergyis sensedasthe temperature ofthesolid,liquid,orgas.Inthecaseofagasinmotion itiscalledthestatictemperatureor,moreusually,justthetemperature.Temperature hasthedimension[θ ]andtheunitsK, ◦ C, ◦ F,or ◦ R(Section 1.2).Inpracticallyall calculationsinaerodynamics,temperatureismeasuredinKor ◦ R(i.e.,measured fromabsolutezero).
1.3.5 Density
Thedensityofamaterialisameasureofthemass(amount)ofthematerialcontained inagivenvolume.Inafluidthedensitymayvaryfrompointtopoint.Considerthe fluidcontainedinasmallregionofvolume δV centeredatsomepointinthefluid, andletthemassoffluidwithinthissphericalregionbe δm.Thenthedensityofthe fluid, ρ ,atthepointonwhichthevolumeiscenteredisdefinedby
ThedimensionsofdensityarethusML 3 ,anddensityismeasuredinunitsof kilogrampercubicmeter(kgm 3 ).Atstandardtemperatureandpressure(288K, 101,325Nm 2 ),thedensityofdryairis1.2256kgm 3 or0.0023781slugft 3
Difficultiesariseinrigorouslyapplyingthedefinitiontoarealfluidcomposedof discretemolecules,sincethevolume,whentakentothelimit,eitherwillorwillnot containpartofamolecule.Ifitdoescontainamolecule,thevalueobtainedforthe densitywillbefictitiouslyhigh.Ifitdoesnotcontainamolecule,theresultantvalue willbezero.Thisdifficultyisgenerallyavoidedintherangeoftemperaturesand pressuresnormallyencounteredinaerodynamicsbecausethemolecularnatureofa gasmayformanypurposes—infact,fornearlyeveryterrestrialflightapplication—be ignoredandtheassumptionmadethatthefluidisacontinuum—thatis,itdoesnot consistofdiscreteparticles.This“continuumassumption”sufficesbecausethemean freepathofthemolecularmotionismuchlessthanthesmallestlengthscaleonthe vehicleforalmosteveryatmosphericflightregime.
FIGURE1.3
Simpleflowgeometrytocreateauniformshear.
1.3.6
Viscosity
Viscosityisregardedasthetendencyofafluidtoresistslidingbetweenlayersor, morerigorously(asexplainedlater)arateofchangeinshearstrain.Thereisvery littleresistancetothemovementofaknifebladeedge-onthroughair,buttoproduce thesamemotionthroughthickoilrequiresmuchmoreeffort.Thisisbecausethe viscosityofoilishighcomparedwiththatofair.
DynamicViscosity
Thedynamic(moreproperly,coefficientofdynamic,orabsolute)viscosityisadirect measureofthemagnitudeoftheviscosityofafluid.Considertwoparallelflatplates placedadistance h apart,withthespacebetweenthemfilledwithfluid(see Fig.1.3). Oneplateisheldfixed,andtheotherismovedinitsownplaneataspeed V .Thefluid immediatelyadjacenttoeachplatewillmovewithit(i.e.,thereisnoslipofthefluid pastthesurface).Thusthefluidincontactwiththelowerplatewillbeatrestwhile thatincontactwiththeupperplatewillbemovingwithspeed V .Betweentheplates thespeedofthefluidwillvarylinearly,asshownin Fig.1.3,intheabsenceofother influences.Asadirectresultofviscosity,aforce F hastobeappliedtoeachplateto maintainthemotion,thefluidtendingtoretardthemovingplateanddragthefixed platetotheright.Iftheareaoffluidincontactwitheachplateis A,theshearstress is F/A.Therateofshearstraincausedbytheupperplateslidingovertheloweris V/h.
ThesequantitiesareconnectedbyNewton’sequation,whichservestodefinethe dynamicviscosity μ:
andtheunitsof μ arethereforekgm 1 s 1 ;intheSIsystemthenamePoiseuille (Pl)hasbeengiventothiscombinationoffundamentalunits.At0 ◦ C(273K)the dynamicviscosityfordryairis1.714 × 10 5 kgm 1 s 1 .
NotethatwhiletherelationshipofEq. (1.11) withconstant μ appliesnicelyto aerodynamics,itdoesnotapplytoallfluids.Foranimportantclassoffluids,which includesblood,someoils,andsomepaints, μ isnotconstantbutisafunctionof V/h—thatis,therateatwhichthefluidisshearing.Numerousclassesof“nonNewtonianfluids”areimportantinfieldsoutsideofaerodynamics,andtheeager studentcanexplorethesebestwithgoodknowledgeofNewtonianfluidbehavioras discussedinthisbook.
KinematicViscosity
Thecoefficientofkinematicviscosity(or,morecommonly,thekinematicviscosity) isaconvenientnumericalforminwhichthemagnitudeoftheviscosityofafluidis oftenexpressed.Itisformedbycombiningthedensity ρ andthedynamicviscosity μ accordingtotheequation
andhasthedimensionsL2 T 1 andtheunitsm2 s 1 .Itmayberegardedasameasure oftherelativemagnitudesoffluidviscosityandinertiaandhasthepracticaladvantage,incalculations,ofreplacingtwovaluesrepresenting μ and ρ withasinglevalue.
1.3.7 SpeedofSoundandBulkElasticity
Bulkelasticityisameasureofhowmuchafluid(orsolid)willbecompressedbythe applicationofexternalpressure.Ifacertainsmallvolume V offluidissubjectedto ariseinpressure δp ,thisreducesthevolumebyanamount δV .Inotherwords,it producesavolumetricstrainof δV/V .Accordingly,bulkelasticityisdefinedas
Thevolumetricstrainistheratiooftwovolumesandisevidentlydimensionless,so thedimensionsof K arethesameasthoseforpressure:ML 1 T 2 .TheSIunitis Nm 2 (orPa)andtheImperialunitispsi.Whenwrittenintermsofdensityoftheair ratherthanvolume,Eq. (1.12) becomes
Thepropagationofsoundwavesinvolvesalternatingcompressionandexpansion ofthemedium.Accordingly,bulkelasticityiscloselyrelatedtothespeedofsound a
asfollows:
Letthemassofthesmallvolumeoffluidbe M ;thenbydefinitionthedensity ρ = M/V .Bydifferentiatingthisdefinition,keeping M constant,weobtain
Therefore,combiningthiswithEqs. (1.13) and (1.14),itcanbeseenthat
Thepropagationofsoundinaperfectgasisregardedasalosslessprocess;thatis, noenergyislostandthewaveprocesslacksheattransfertoorfromthesurrounding fluid.Accordingly(seethepassageon Entropy tocome),thepressureanddensityare relatedbyEq. (1.34),soforaperfectgas,forwhich P = ρRT ,
where γ istheratioofthespecificheatsand R isthespecificgasconstantforthatgas. Eq. (1.16) istheformulanormallyusedtodeterminethespeedofsoundingasesfor aerodynamicsapplications.Thisformulacanalsobederivedfromone-dimensional waveequationforcompressibleflow,suchasinChapter 4
1.3.8 ThermodynamicProperties
Heat,likework,isaformofenergytransfer.Consequently,ithasthesamedimensionsasenergy(i.e.,ML2 T 2 )andismeasuredinJoules(J)orfoot-pounds(ft-lb).
SpecificHeat
Thespecificheatofamaterialistheamountnecessarytoraisethetemperatureof aunitmassofthematerialbyonedegree.ThusithasthedimensionsL2 T 2 θ 1 andismeasuredinSIunitsofJkg 1 K 1 .Imperialunitsofft-lbslug 1 ◦ F 1 or ft-lbslug 1 ◦ R 1 aremostcommon.
Therearecountlesswaysinwhichgasmaybeheated.Twoimportantanddistinctwaysareatconstantvolumeandatconstantpressure.Thesedefineimportant thermodynamicpropertiesofthegas.
SpecificHeatatConstantVolume
Ifaunitmassofthegasisenclosedinacylindersealedbyapiston,andthepiston islockedinposition,thevolumeofthegascannotchange.Itisassumedthatthe
cylinderandpistondonotreceiveanyoftheheat.Thespecificheatofthegasunder theseconditionsisthespecificheatatconstantvolume cV .Fordryairatnormal aerodynamictemperatures, cV = 718Jkg 1 K 1 = 4290ft-lbslug 1 °R 1 .
Internalenergy(e )isameasureofthekineticenergyofthemoleculesthatmake upthegas,so
internalenergyperunitmass e = cV T
ormoregenerally
SpecificHeatatConstantPressure
Assumethatthepistonjustreferredtoisnowfreedandactedonbyaconstantforce. Thepressureofthegasisthatnecessarytoresisttheforceandisthereforeconstant aswell.Theapplicationofheattothegascausesitstemperaturetorise,whichleads toanincreaseinitsvolumeinordertomaintaintheconstantpressure.Thusthe gasdoesmechanicalworkagainsttheforce,soitisnecessarytosupplytheheat requiredtoincreaseitstemperature(asinthecaseatconstantvolume)aswellasheat equivalenttothemechanicalworkdoneagainsttheforce.Thistotalamountiscalled thespecificheatatconstantpressure cp andisdefinedasthatamountrequiredtoraise thetemperatureofaunitmassofthegasbyonedegree,thepressureofthegasbeing keptconstantwhileheating.Therefore, cp isalwaysgreaterthan cV .Fordryairat normalaerodynamictemperatures, cp = 1005Jkg 1 K 1 = 6006ft-lbslug 1 °R 1 . Thesumofinternalenergyperunitmassandpressureenergyperunitmassis knownas enthalpy (h perunitmass)(discussedmomentarily).Thus
ormoregenerally
RatioofSpecificHeats
Theratioofspecificheatsisapropertyimportantinhigh-speedflowsandisdefined bytheequation
(Thevalueof γ forairdependsonthetemperature,butformuchofpracticalaerodynamicsitmayberegardedasconstantatabout1.403.Thisvalueisofteninturn approximatedto γ = 1.4,whichisinfactthetheoreticalvalueforanidealdiatomic gas.)
