[PDF] Aerodynamics

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Aerodynamics

AE 301

Contents

a.Importance of Aerodynamics, b.Aerodynamics: Classification and Practical

Objectives

c.Some Fundamental AerodynamicVariables d.Aerodynamic Forces and Moments e.Center of Pressure f.Dimensional Analysis: The BuckinghamPi

Theorem

g.Flow Similarity h.Fluid statics i.Types of Flow j.Applied aerodynamics

Importance of aerodynamics

ƒNaturalPhilosophy

ƒPhysics

ƒMechanics

ƒDynamics

ƒAerodynamics"

Importance of aerodynamics

ƒNavalpowerwasgoingtodependgreatlyonthespeedand maneuverabilityofships. ƒToincreasethespeedofaship,itisimportanttoreducethe resistancecreatedbythewaterflowaroundthehull. ƒSuddenly,thedragonshiphullsbecameanengineering problemofgreatinterest,thusgivingimpetustothestudyoffluid mechanics.

ƒOnAugust8,1588,ThegreatSpanish

Armadawasmethead-onbytheEnglish

fleetunderthecommandofSirFrancis

Drake.

ƒTheSpanishshipswerelargeandheavy,incontrast,the

Englishshipsweresmallerandlighter.

ƒEnglandwonthenavalwar.

Importance of aerodynamics

ƒIn1781,LeonhardEuler(17071783)pointedoutthe

physicalinconsistencyofmodelandmodifiedit. ƒTherapidriseintheimportanceofnavalarchitecture madefluiddynamicsanimportantscience,occupyingthe mindsofNewton,,andEuler,amongmany othersinEurope.

ƒin1687,IsaacNewton(16421727)publishedhis

famousinwhichtheentiresecondbook wasdevotedtofluidmechanics.

ƒIn1777,aseriesofexperimentswascarriedout

byJeanLeRond(17171783)inorder tomeasuretheresistanceofshipsincanals.

ƒIntheUS,sincetheir1901gliderwasofpoor

aerodynamicdesign,theWrightBrotherssetabout determiningwhatconstitutesgoodaerodynamicdesign.

ƒInthefallof1901,theydesignandbuildawindtunnel

poweredbyatwo-bladedfanconnectedtoagasoline engine. ƒTheaerodynamicdataaretakenlogicallyandcarefully.

ƒArmedwiththeirnewaerodynamicinformation,the

Wrightsdesignanewgliderinthespringof1902andflew

successfully. ƒThegoodaerodynamicswasvitaltotheultimatesuccess oftheWrightbrothersand,ofcourse,toallsubsequent successfulairplanedesignsuptothepresentday.Importance of aerodynamics

Aerodynamics: Classification and Objectives

ƒThewordisusedtodenoteeithera

liquidoragas.

ƒThe liquid and gas will change its shape

to conform to that of the container

ƒThemostfundamentaldistinctionbetweensolids,

liquids,andgasesisattheatomicandmolecularlevel.

ƒFluiddynamics is subdivided into three areas:

Hydrodynamicsflow of liquids

Gas dynamicsflow of gases

Aerodynamicsflow of air

Aerodynamics: Classification and Objectives

ƒAerodynamicsisanappliedsciencewithmany

practicalapplicationsinengineering.

ƒItisaimedatoneormoreofthefollowingpractical

objectives:

Thepredictionofforcesandmoments

on,andheattransferto,bodiesmoving throughafluid(usuallyair). lift,drag,andmomentsonairfoils,wings, fuselages,enginenacelles,andmost importantly,wholeairplaneconfigurations

Determination of flows moving internally

through ducts. theflowpropertiesinsiderocketandair- breathingjetenginesandtocalculatethe enginethrust

Some Fundamental AerodynamicVariables

ƒThefourofthemostfrequentlyusedwordsin

aerodynamics:and isthenormalforceperunitarea exertedonasurfaceduetothetimerateof changeofmomentumoftheliquid/gas moleculesimpactingon(orcrossing)that surface.

ƒPressureisaandcanhaveadifferent

valuefromonepointtoanotherinthefluid.

ƒItisascalarquantity,notavector,

ƒItisperpendiculartothesurface,

ƒItactsinward,istowardthesurface

Some Fundamental AerodynamicVariables

ƒAnotherimportantaerodynamicvariable

isdefinedasthemassperunit volume.

ƒItisascalarquantitythat

canvaryfrompointtopointinthefluid.

Some Fundamental AerodynamicVariables

ƒtakesonanimportantroleinhigh-speed

aerodynamics. ƒThetemperatureofagasisdirectlyproportionaltothe averagekineticenergyofthemoleculesofthefluid. ƒWecanqualitativelyvisualizeahigh-temperaturegasas oneinwhichthemoleculesandatomsarerandomly rattlingaboutathighspeeds. ƒTemperatureisalsoapointproperty,scalarquantity,which canvaryfrompointtopointinthegas.

Some Fundamental AerodynamicVariables

ƒTheprincipalfocusofaerodynamicsisfluidsinmotion.

Hence,flowvelocityisanextremelyimportant

consideration.

ƒVelocityisthetimerateofchangeofdisplacement.

ƒIncontrasttosolid,afluidisasubstance.

ƒForafluidinmotion,onepartofthefluidmaybe

travelingatadifferentvelocityfromanotherpart. :Thevelocityofaflowing gasatanyfixedpointinspaceisthe velocityofaninfinitesimallysmallfluid elementasitsweepsthrough.

ƒTheflowvelocityhasbothmagnitudeanddirection;

hence,itisavectorquantity,anditisapointproperty.

Some Fundamental AerodynamicVariables

ƒwenotethatfrictioncanplayaroleinternallyinaflow.

TheshearstressIJisthelimitingformofthemagnitude

ofthefrictionalforceperunitarea ƒConsidertwoadjacentfluidlayers,streamlines.Dueto differentvelocityvalues,therewillbeshearstresson thefluidsurfaces. ƒisdirectlyproportionaltovelocitydifference,and inverselyproportionaltoverticaldistance.

ƒTheconstantofproportionalityisdefinedasthe

,ȝ; itisafunctionofthe temperatureofthefluid. Units ƒTwoconsistentsetsofunitswillbeusedthroughoutthis course,SIunits(InternationalUnites)andthe

Englishengineeringsystemofunits.

ƒThebasicunitsofforce,mass,length,time,andabsolute temperatureinthesetwosystemsaregiveninTable1.1.

Aerodynamic forces and moments

ƒAtfirstglance,thegenerationoftheaerodynamicforce mayseemcomplex

ƒHowever,inallcases,theaerodynamicforcesand

momentsonthebodyareduetoonlytwobasic sources:

1.Pressuredistribution,

2.Shearstressdistributionoverthebodysurface

ƒpactsnormaltothesurface,

ƒIJactstangentialtothesurface.

Aerodynamic forces and moments

ƒTheneteffectoftheandIJdistributionsintegratedoverthe completebodysurfaceisaresultantaerodynamicforce andmomentonthebody.

ƒTheresultantcanbesplitinto

components,Normal&AxialorLift&

Dragforces.

Aerodynamic forces and moments

ƒTheflowfarawayfromthebodyis

calledtheandhenceis alsocalledthefreestreamvelocity. component of perpendicular to component of parallel to ƒThechordisthelineardistancefrom theleadingedgetothetrailingedgeof thebody.

ƒTheangleofattackĮisdefinedasthe

anglebetweenand.

ƒThegeometricalrelationbetweenthese

twosetsofcomponentsis,

Aerodynamic forces and moments

ƒWecanexamineinmoredetailtheintegrationofthe

pressureandshearstressdistributionstoobtainthe aerodynamicforcesandmoments. =⤨1 body surface, body surface,

ƒThetotalnormalandaxialforcesare

obtainedbyintegratingequationsfromtheleadingedge (LE)tothetrailingedge(TE):

ƒWecangetliftanddragforcesbasedontheprevious

equations;Aerodynamic forces and moments

Aerodynamic forces and moments

ƒTheaerodynamicmomentexertedonthebodydepends

onthepointaboutwhichmomentsaretaken.

ƒLetsconsidermomentstakenabouttheleadingedge.

ƒThemomentperunitspanabouttheleadingedgedueto

andIJontheelementalareaontheupperandlower surfaceare ƒByintegratingfromtheleadingtothetrailingedges,we obtainthepitchingmomentabouttheleadingedgeper unitspan

Aerodynamic forces and moments

ƒInEquations;ș,,andareknownfunctionsoffora

givenbodyshape. ƒAmajorgoaloftheoreticalorexperimentalaerodynamics istocalculateandIJforagivenbodyshapeand freestreamconditions.

ƒWegettheaerodynamicforcesandmomentsbasedon

them. ƒInaerodynamics,shapeisprobablythemostimportant factor.

ƒWemayeliminatethescaleoftheshapebydefining

somedimensionlesscoefficients.

Aerodynamic forces and moments

ƒLetȡandbethedensityandvelocity,respectively,in thefreestream,faraheadofthebody. ƒWedefineadimensionalquantitycalledthefreestream as

ƒInaddition,letbeareferenceareaandbea

referencelength.

ƒThedimensionlessforceandmomentcoefficientsare

definedasfollows:

Liftcoefficient

DragcoefficientMoment coefficient

Aerodynamic forces and moments

Fortwo-dimensionalbodies,itisconventionaltodenote

theaerodynamiccoefficientsbylowercaseletters;for example, ƒTwoadditionaldimensionlessquantitiesofimmediate useare

ƒFrom the geometry

Aerodynamic forces and moments

ƒWeobtainthefollowingintegralformsfortheforceand momentcoefficients

ƒTheliftanddragcoefficientscanalsobeobtained:

Example 1.1

ƒConsiderthesupersonicflowovera5half-anglewedgeat zeroangleofattack,assketchedinfigure. ƒThefreestreamMachnumberaheadofthewedgeis2.0,and thefreestreampressureanddensityare101105N/m2and

1.23kg/m3,respectively.

ƒThepressuresontheupperandlowersurfacesofthewedge areconstantwithdistanceandequaltoeachother,namely, ==1.31105N/m2. ƒThepressureexertedonthebaseofthewedgeisequalto.

ƒTheshearstressvariesoverboththeupperandlower

surfacesasIJ=431 ƒThechordlength,,ofthewedgeis2m.Calculatethedrag coefficientforthewedge.

Example 1.1

Flow field picturePressure distribution

Shear stress distribution

Example 1.1

ƒWecancalculatethedragandthenobtainthedrag

coefficient.

ƒThedragcanbeobtainedfrom

Example 1.1

ƒAddingthepressureintegrals,andthenaddingtheshear stressintegrals,wehavefortotaldrag

Example 1.1

ƒNotethat,forthisratherslenderbody,butatasupersonic speed,mostofthedragispressuredrag. ƒThisistypicalofthedragofslendersupersonicbodies.

ƒThedragcoefficientisobtainedasfollows.

ƒThevelocityofthefreestreamistwicethesonicspeed, whichisgivenby

ƒMachNumber=V/a

Center of pressure

ƒWeseethatthenormalandaxialforcesonthebody

areduetotheloadsimposedbythe pressureandshearstressdistributions.

ƒMoreover,thesedistributedloadsgenerateamoment

abouttheleadingedge.

ƒIftheaerodynamicforceonabodyis

specifiedintermsofaresultantsingleforce,orits componentssuchasand,onthebody shouldthisresultantbeplaced? ƒTheansweristhattheresultantforceshouldbelocated onthebodysuchthatitproducesthesameeffectas thedistributedloads.

ƒThecomponentsoftheresultedaerodynamicforceR;

andmustbeplacedontheairfoilatsuchalocation togeneratethesamemomentabouttheleadingedge.

ƒIfisplacedonthechordline,thenmustbelocated

adistancecpdownstreamoftheleadingedgesuchthatCenter of pressure

ƒIncaseswheretheangleofattackofthebodyissmall,

sinĮ0andcosĮ1;hence,.Thus,Equation becomes

Center of pressure

ƒNotethatifmomentsweretakenaboutthecenterof

pressure,theintegratedeffectofthedistributedloads wouldbezero. ƒHence,analternatedefinitionofthecenterofpressure isthatpointonthebodyaboutwhichtheaerodynamic momentiszero. ෍ܯ ƒTodefinetheforce-and-momentsystem,theresultant forcecanbeplacedatpointonthebody,aslongas thevalueofthemomentaboutthatpointisalsogiven.

Example 1.4

ƒConsidertheDC-3A/C.Justoutboardoftheengine

nacelle,theairfoilchordlengthis15.4ft. ƒAtcruisingvelocity(188mi/h)atsealevel,themoments perunitspanatthisairfoillocationare4=1071ftlb/ft andLE3213.9ftlb/ft. ƒCalculatetheliftperunitspanandthelocationofthe centerofpressureontheairfoil.

Example 1.4

ƒFrom given equations;

ƒWe know that

Dimensional analysis: the Buckingham Pi theorem

ƒWhatphysicalquantitiesdeterminethevariationof

theseforcesandmoments?Theanswercanbefound fromthepowerfulmethodofdimensionalanalysis. ƒDimensionalanalysisisbasedontheobviousfactthat inanequationdealingwiththerealphysicalworld,each termmusthavethesamedimensions.

ƒForexample,if

isaphysicalrelation,thenȥ,Ș,ȗ,andijmusthavethe samedimensions.Otherwisewewouldbeadding applesandoranges.

ƒTheaboveequationcanbemadedimensionlessby

dividingbyanyoneoftheterms,say,ij:

Dimensional analysis: the Buckingham Pi theorem

ƒTheseideasareformallyembodiedintheBuckingham

pitheorem,statedbelowwithoutderivation.

ƒLetequalthenumberoffundamentaldimensions

requiredtodescribethephysicalvariables.

ƒLet12representphysicalvariablesinthe

physicalrelation;

ƒThen,thephysicalrelationmaybere-expressedasa

relationofdimensionlessproducts(calledߨ products),

ƒEachproductisadimensionlessproductofasetof

physicalvariablesplusoneotherphysicalvariable.

ƒLet12betheselectedsetofphysical

variables.ThenDimensional analysis: the Buckingham Pi theorem

ƒThechoiceoftherepeatingvariables,12

shouldbesuchthattheyincludeallthedimensions withtheminimumnumberusedintheproblem. ƒAlso,thedependentvariableshouldappearinonlyone oftheߨ

Dimensional analysis: the Buckingham Pi theorem

ƒConsiderabodyofgivenshapeatagivenangleofattack.

Theresultantaerodynamicforceis.

ƒOnaphysical,intuitivebasis,weexpecttodependon:

Freestreamvelocity.

Freestreamdensityȡ.

Viscosityofthefluid,bythefreestreamviscositycoefficientȝ.

The size of the body,

The compressibility of the fluid, by the freestream speed of sound, .

ƒInlightoftheabove,wecanusecommonsensetowrite

Dimensional analysis: the Buckingham Pi theorem

ƒEquationcanbewrittenintheformof

ƒFollowingtheBuckinghampitheorem,Thephysical

variablesandtheirdimensionsare

ƒSo,thefundamentaldimensionsare;

ƒPhysicalfactors;,andtherequireddimensions

Dimensional analysis: the Buckingham Pi theorem

ƒThenEquationcanbere-expressedintermsof

=63=3dimensionlessߨ of

ƒTheseproductsare

ƒWechoose,ȡ,csuchthattheyincludeallthe

dimensions()withtheminimumnumberused.

ƒLetsassumethat

where,,andareexponentstobefound.

Dimensional analysis: the Buckingham Pi theorem

ƒIndimensionalterms,equationis

ƒBecauseߨ

mustalsobedimensionless.

ƒThismeansthattheexponentsofmustaddtozero,

andsimilarlyfortheexponentsofand.Hence,

ƒSolvingtheaboveequations,wefindthat=1,=2,

and=2. ƒSubstitutingthesevaluesintoequation,wehaveDimensional analysis: the Buckingham Pi theorem

ƒWecanreplacewithanyreferenceareasuchasthe

planformareaofawing.

ƒMoreover,wecanmultiplyߨ

willstillbedimensionless.Thus

Homework

: findߨଶand ߨ

Dimensional analysis: the Buckingham Pi theorem

ƒBased on similar approach, we can find

ƒThedimensionlesscombinationofߨ

freestream ƒTheReynoldsnumberisphysicallyameasureoftheratio ofinertiaforcestoviscousforcesinaflowandisoneofthe mostpowerfulparametersinfluiddynamics.

ƒThedimensionlesscombinationofߨ

freestream ƒItisapowerfulparameterinthestudyofgasdynamics.

Dimensional analysis: the Buckingham Pi theorem

ƒTheresultsofourdimensionalanalysismaybe

organizedasfollows; ƒSincetheliftanddragarecomponentsoftheresultant force,corollariestoequationare

Dimensional analysis: the Buckingham Pi theorem

ƒKeepinmindthattheanalysiswasforagivenbodyshape atagivenangleofattackĮ.

ƒIfĮisallowedtovary,then,,andwillingeneral

dependonthevalueofĮ.

ƒHence,Equationscanbegeneralizedto

ƒWhichmeansifthedimensionlessparametersarethe

same,theliftcoefficientwillbethesameforthesame geometry,independentfromthescale

Flow similarity

ƒConsidertwodifferentflowfieldsovertwodifferent bodies.

ƒBydefinition,differentflowsareif:

Thestreamlinepatternsaregeometricallysimilar.

Thedistributionsof,,,andtheforce

coefficientsarethesame.

Whatarethetoensurethattwoflowsare

dynamicallysimilar?

ƒTheanswercomesfromtheresultsofthedimensional

analysis.Twoflowswillbedynamicallysimilarif:

Thebodiesandanyothersolidboundariesare

geometricallysimilarforbothflows.

Thesimilarityparametersarethesameforbothflows.

Flow similarity

ƒSofar,wehaveemphasizedtwoparameters,Reand.

ƒApplicabletomanyproblems,wecansaythatflows

overgeometricallysimilarbodiesatthesameMach andReynoldsnumbersaredynamicallysimilar.

ƒHencethelift,drag,andmomentcoefficientswillbe

identicalforthebodies. ƒThisisakeypointinthevalidityofwind-tunneltesting. ƒIfascalemodelofaflightvehicleistestedinawindtunnel,themeasuredlift,drag, andmomentcoefficientswillbethesameasforfreeflightaslongastheMachand Reynoldsnumbersofthewind-tunneltest-sectionflowarethesameasforthefree- flightcase.

Example 1.6

ƒConsideraBoeing747airlinercruisingatavelocityof

550mi/hatastandardaltitudeof38,000ft.

ƒCalculatetherequiredvelocityandpressureofthetest airstreaminthewindtunnelsuchthattheliftanddrag coefficientsmeasuredforthewind-tunnelmodelarethe sameasforfreeflight.

ƒAssumethatbothȝandareproportionaltoT1/2.

ƒThefreestreampressureandtemperature

are432.6lb/ft2and390R,respectively.

ƒAone-fiftiethscalemodelofthe747is

testedinawindtunnelwherethe temperatureis430R.

Example 1.6

ƒLetsubscripts1and2denotethefree-flightandwind

tunnelconditions,respectively.

ƒFor1=2and1=2,thewindtunnelflowmustbe

dynamicallysimilartofreeflight.

ƒForthistohold,1=2andRe1=Re2:

ƒHence,

Example 1.6

ƒWe have

ƒWe know that

ƒSo,

ƒTheequationofstateforaperfectgasis=ȡ,

whereisthespecificgasconstant.Thus,

Flow similarity

ƒInExample1.6,thewind-tunnelteststreammustbe

pressurizedfaraboveatmosphericpressureinorderto simulatetheproperfree-flightReynoldsnumber. ƒMachnumbersimulationisachievedinonewindtunnel, andReynoldsnumbersimulationinanothertunnel.

ƒTheresultsfrombothtunnelsarethenanalyzedand

correlatedtoobtainreasonablevaluesforand appropriateforfreeflight.

ƒHowever,moststandardsubsonicwind

tunnelsarenotpressurizedassuch, becauseofthelargeextrafinancialcost involved.

ƒToday,forthemostpart,wedonot

attempttosimulatealltheparameters simultaneously.

Example 1.7

ƒConsideranexecutivejettransportCessna560

CitationV.

ƒTheairplaneiscruisingatavelocityof492mphatan

altitudeof33,000ft,wheretheambientairdensityis

7.9656104slug/ft3.

ƒTheweightandwingplanformareasoftheairplane

are15,000lband342.6ft2,respectively. ƒThedragcoefficientatcruiseis0.015.Calculatethelift coefficientandthelift-to-dragratioatcruise.

Example 1.7

ƒIfithasastallingspeedatsealevelof100mphatthe

maximumtake-offweightof15,900lb.

ƒTheambientairdensityatstandardsealevelis

0002377slug/ft3.

ƒCalculatethevalueofthemaximumliftcoefficientforthe airplane.

Example 1.7

ƒToconvertbetweenmphandft/s,itisusefulto

rememberthat88ft/s=60mph.

ƒWecansaythatliftmustbeequaltoweightforlevel

flight;So,

ƒThelift-to-dragratio(fines)

ƒOnceagainwehavetouseconsistentunits,so

Example 1.7

Fluid statics

ƒInaerodynamics,weareconcernedaboutfluidsin

motion,andtheresultingforcesandmomentsonbodies duetosuchmotion. ƒHowever,inthissection,weconsiderthespecialcase offluidmotion(i.e.,). ƒAbodyimmersedinafluidwillstillexperienceaforce evenifthereisnorelativemotionbetweenthebodyand thefluid.

ƒLetusseewhy.

Fluid statics

ƒConsiderastagnantfluidabovetheplane.The

verticaldirectionisgivenby. ƒConsideraninfinitesimallysmallfluidelementwithsides oflength,,and.

ƒTherearetwotypesofforcesactingonthisfluid

element:pressureforcesandthegravityforce.

ƒConsiderforcesinthedirection.

Fluid statics

ƒLettingupwardforcebepositive,wehave

ƒSincethefluidelementisstationary(inequilibrium),the sumoftheforcesexertedonitmustbezero:

ƒItiscalledthe

ƒItisadifferentialequationwhichrelatesthechangein pressureinafluidwithachangeinverticalheight.

Fluid statics

ƒEquationgovernsthevariationofatmospheric

propertiesasafunctionofaltitudeintheairaboveus.

ƒItisalsousedtoestimatethepropertiesofother

planetaryatmospheressuchasforVenus,Mars,and

Jupiter.

ƒLetthefluidbealiquid,forwhichwecanassumeȡis constant.

ƒWehave

ƒAsimpleapplicationisthecalculationofthepressure distributiononthewallsofacontainerholdingaliquid, andopentotheatmosphereatthetop.Fluid statics

ƒNotethatthepressureisafunctionofandthat

increaseswithdepthbelowthesurface.

Fluid statics

ƒAnothersimpleandverycommonapplicationisthe

liquidfilledU-tubemanometerusedformeasuring pressuredifferences.

ƒNotethatthepressureonthesamelevelwillbethe

sameinfluid.

ƒWestatedthatasolidbodyimmersedinafluidwill

experienceaforceevenifthereisnorelativemotion betweenthebodyandthefluid. ƒWearenowinapositiontoderiveanexpressionforthis force,henceforthcalledtheFluid statics

ƒWeseethattheverticalforceonthe

bodyduetothepressuredistribution overthesurfaceis

ƒForsimplicity,considerarectangular

bodyofunitwidth(1),length,and height12

ƒweobtainthebuoyancyforce.

Fluid statics

ƒConsiderthephysicalmeaningoftheintegralin

Equation.

ƒItistheweightoftotalvolumeoffluid;

ƒTherefore,wecanstatesinwordsthat

the well-known

ƒThedensityofliquidsisusuallyseveralordersof

magnitudelargerthanthedensityofgases.

ƒForwaterȡ=103kg/m3,forairȡ=1.23kg/m3).

ƒTherefore,agivenbodywillexperienceabuoyancy

forceathousandtimesgreaterinwaterthaninair.

Example 1.9

ƒAhot-airballoonwithaninflateddiameterof30ftiscarrying aweightof800lb,whichincludestheweightofthehotair insidetheballoon.

ƒCalculate;

ƒitsupwardaccelerationatsealeveltheinstantthe

restrainingropesarereleased.

ƒthemaximumaltitudeitcanachieve.

ƒAssumethatthevariationofdensityinthestandard

atmosphereisgivenby wherehisthealtitudeinfeetandȡisinslug/ft3.

Example 1.9

ƒThenetupwardforceatsealevelis

ƒMassvalue(W/g);

ƒHence

ƒThemaximumaltitudeoccurswhen=

Example 1.9

ƒFromthegivenvariationofȡwithaltitude,,

Example 1.11

ƒConsideraU-tubemercurymanometeroriented

vertically.

ƒOneendiscompletelysealedwithatotalvacuumabove

thecolumnofmercury.

ƒTheotherendisopentotheatmospherewherethe

atmosphericpressureisthatforstandardsealevel.

ƒWhatisthedisplacementheightofthe

mercuryincentimeters,andinwhich endisthemercurycolumnthehighest?

ƒThedensityofmercuryis1.36104

kg/m3.

Example 1.11

ƒConsiderthesealedendwiththetotalvacuumtobeon

theleft,where=0.

ƒWehave

mercury

Example 1.10

ƒShowhowthestandardaltitudetablesareconstructed withtheuseoftheHydrostaticequation.

ƒWeknowthat

ƒAlsowehavetheequationofstateforaperfectgas

ƒLetsdividethem

ƒWeknowtherelationshipbetweenaltitudeand

temperature:

ƒTherefore;

ƒFromsealeveltoanaltitudeof11km,thestandardaltitudeisbasedon alinearvariationoftemperaturewithaltitude,,wheredecreasesat arateof6.5Kperkilometer(thelapserate).

ƒLetsintegratetheequationfromsealevelwherethe

standardvaluesofpressureandtemperaturearedenoted byand,respectively,Example 1.10

ƒAtsealevel,thestandardpressure,

density,andtemperatureare1.01325

105N/m2,1.2250kg/m3,and288.16

K,respectively.

Types of flow

ƒAnunderstandingofaerodynamics,likethatofanyother physicalscience,isobtainedthrougha- approach. ƒAnexampleofthisprocessisthewaythatdifferenttypes ofaerodynamicflowsarecategorizedandvisualized.

ƒAsaresult,astudyofaerodynamicshasevolvedintoa

studyofnumerousanddistincttypesofflow;fromthe simplestflowtothemostcomplexone

Types of flow

ƒConsidertheflowoverabody,say,forexample,acircular cylinderofdiameter. ƒAlso,considerthefluidtoconsistofindividualmolecules, whicharemovingaboutinrandommotion.

ƒThemeandistancethatamoleculetravelsbetween

collisionswithneighboringmoleculesisdefinedasthe Ȝ. ƒIfȜisordersofmagnitudesmallerthanthescaleofthe bodymeasuredby,thentheflowappearstothebodyas acontinuoussubstance.

ƒSuchflowiscalled.:ܭ

ௗͲǤͲͳ

Types of flow

ƒTheotherextremeiswhereȜisonthesameorderasthe bodyscale. ƒThegasmoleculesarespacedsofarapart(relativeto) thatcollisionswiththebodysurfaceoccuronlyinfrequently. ƒThebodysurfacecanfeeldistinctlyeachmolecularimpact.

ƒSuchflowiscalled.

ƒFormannedflight,vehiclessuchasthespaceshuttle

encounterfreemolecularflowattheextremeouteredgeof theatmosphere. ƒTheairdensityissolowthatȜbecomesontheorderofthe shuttlesize.:ܭ ௗ̱ͳ

Types of flow

ƒAflowthatisassumedtoinvolvenofriction,thermal

conduction,ordiffusioniscalledan. ƒIncontrast,aflowthatisassumedtoinvolvefriction, thermalconduction,ordiffusioniscalled

ƒInviscidflowsdonottrulyexistinnature.

ƒHowever,therearemanypracticalaerodynamicflows

wheretheinfluenceoftransportphenomenaissmall, andwecantheflowasbeinginviscid. ! no viscosity !

Types of flow

ƒAflowinwhichthedensityȡisconstantiscalled

. ƒIncontrast,aflowwherethedensityisvariableiscalled .

ƒAllflowsarecompressibleinnature.

ƒHowever,thereareanumberofaerodynamicproblems

thatcanbemodeledasbeingincompressible. ƒForexample,theflowofhomogeneousliquidsistreated asincompressible.

ƒAlso,theflowofgasesatalowMachnumberis

essentiallyincompressible;for03.

Types of flow

ƒIfisthelocalMachnumberatanarbitrarypointina

flowfield,thenbydefinitiontheflowislocally: if1 if~1 if1

Types of flow

ƒifissubsonicbutisnearunity,theflowcanbecome

locallysupersonic1.

ƒTheflowfieldsarecharacterizedbymixedsubsonic-

supersonicflows.Hence,suchflowfieldsarecalled .

Types of flow

ƒHypersonicaerodynamicsreceivedagreatdealof

attentionduringtheperiod19551970.

ƒBecauseatmosphericentryvehiclesencounterthe

atmosphereatMachnumbersbetween25(ICBMs)and

36(theApollolunarreturnvehicle).

ƒMathematicallyspeakingforsteady

flows,

ƒUnsteadyornon-steadyflowisone

wherethepropertiesdodependontime.Types of flow ƒTheflowfeaturesincludingvelocity,pressureandother propertiesoffluidflowcanbefunctionsofspaceandtime. where݌isanypropertylikepressure,velocityordensity. ƒIfaflowissuchthatthepropertiesateverypointintheflow donotdependupontime,itiscalledasteadyflow.݌݂ሺݔǡݕǡݖǡݐሻ unsteadysteady

Types of flow

ƒAflowfieldisbestcharacterizedbyitsvelocity

distribution. ƒAflowissaidtobeone-,two-,orthree-dimensionalif theflowvelocityvariesinone,two,orthreedimensions, respectively.

ƒInnature,everyflowis3D.

ƒHowever,thevariationofvelocityincertaindirections canbesmallrelativetothevariationinotherdirections andcanbeignored.

1D3D2D

Types of flow

ƒAflowfieldcanalsobecharacterizedbyitsflow

pattern.

ƒinwhichthestreamlinesaresmoothand

regularandafluidelementmovessmoothlyalonga streamline.

ƒinwhichthestreamlinesbreakupand

afluidelementmovesinarandom,irregular,and tortuousfashion.

Applied aerodynamics

ƒThemainpurposeistopresentknowledgeandtoshow

itsapplicationsinpractice.

QuestionWhataresometypicaldragcoefficientsfor

variousaerodynamicconfigurations?

ƒSomebasicvaluesareshowninFigure.

ƒThedimensionalanalysisprovedthat=Re.

ƒThedrag-coefficientvaluesareforlowspeeds,

essentiallyincompressibleflow;therefore,theMach numberdoesnotcomeintothepicture.

Applied aerodynamics

Applied aerodynamics

ƒBluntbody=abodywheremostofthe

dragispressuredrag(formdrag)

ƒStreamlinedbody=abodywheremost

ofthedragisskinfrictiondrag

Applied aerodynamics

ƒThebreakdownofvarioussourcesofdragona

late1930sairplane,theSeverskyXP-41

Applied aerodynamics

ƒTheaircraftT38isatasmallnegativeangleof

attacksuchthattheliftiszero,hencethein

Figureiscalledthe.

ƒNotethatthevalueofisrelativelyconstantfrom

=01toabout0.86.Why?

Applied aerodynamics

ƒVariationofsectionliftcoefficient

foraNACA63-210airfoil.

ƒRe=3106.

ƒNoflapdeflection.

ƒVariationofliftcoefficientwithangleof

attackfortheT-38.

ƒThreecurvesareshowncorresponding

tothreedifferentflapdeflections.

FreestreamMachnumberis0.4.

Example 1.12

ƒNotethatthedatagiveninFigureapplyforthespecific conditionwhere=0.15.

ƒThewingplanformareaandthegrossweightoftheP-

35are220ft2and5599lb,respectively.

ƒCalculatethehorsepowerrequiredfortheP-35toflyin steadylevelflightwith=0.15atstandardsealevel.

ƒConsidertheSeverskyP-35shownin

Figure.

ƒAssumethatthedragbreakdowngiven

fortheXP-41appliesalsototheP-35.

Example 1.12

ƒFrombasicmechanics,ifisaforceexertedonabody

movingwithavelocity,thepowergeneratedbythis systemis=ā.

ƒWhenandareinthesamedirection,thenthedot

productbecomes=whereandarethescalar magnitudesofforceandvelocity,respectively.

ƒWhentheairplaneisinsteadylevelflight(no

acceleration)thethrustobtainedfromtheengineexactly counteractsthedrag,i.e.,=.

ƒHencethepowerrequiredfortheairplanetoflyata

givenvelocityis

Example 1.12

ƒToobtain,wenotethatinsteadylevelflighttheweight isexactlybalancedbytheaerodynamiclift,

ƒwehave

ƒSolvingEq.forwehave

ƒAtstandardsealevel,ȡ=0002377slug/ft3.Also,

=220ft2,=5599lb,and=015.

ƒHence,fromEq.wehave

ƒTocompletethecalculationofpowerrequired,weneed thevalueof.

ƒThedrag,

ƒTherequiredpower

ƒNotethat1horsepoweris550ftlb/s.Thus,in

horsepower,

Example 1.12

from the table

ƒThefirstpersontodefineanduseaerodynamicforce

coefficientswasOttoLilienthal,thefamousGerman aviationpioneerattheendofthenineteenthcentury.

ƒBytheendofWorldWarI,LudwigPrandtlatGottingen

UniversityinGermanyestablishedthenomenclaturefor

theaerodynamicforcethatisacceptedasstandard today;Historical notes whereistheforce,istheareaofthe surface,isthedynamicpressure,and isa

Questions

Questions

volkan.pehlivanoglu@ieu.edu.tr

Aerodynamics

AE 301

Contents;

a.Review of vector relations, b.Models of fluid, c.Continuity equation, d.Momentum equation, e.Energy equation, f.Substantial derivative, g.Flow patterns, h.Velocity, vorticity, strain, i.Circulation, j.Flow functions, k.How do we solve the equations?

Introductions

ƒThe principle is most important, not the detail.

Introductions

ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ

ƒunitvector/

ƒ ƒ

ƒscalarproduct

ƒ

ƒvectorproduct

ƒ ƒ ƒ ƒ ƒ ƒ ƒ

ƒcartesian coordinate system

position vector

ƒcylindrical coordinate system

ƒtransformation

ƒspherical coordinate system

ƒ ƒ tscalarfield ƒ ƒ vectorfield ƒ ƒ ƒ

ƒLT
áU
áV
ä

ƒ(x,y)

p ƒ pthemost

ƒdirectionofthegradientp.

ƒÏp

p

ƒÏp

ƒÏp(x,y).

s s ƒp s p ƒ

ƒdivergenceÏā

ƒ ƒ ƒ

ƒÏāƒ

ƒ

ƒÏ

ƒ ƒ ƒ ƒ

ƒÏ

ƒ ƒC ab

ƒds

ƒds

ƒlineintegralC

ab ƒC counterclockwise C ƒS C

ƒPdS

ƒdS

ƒsurfaceintegralS

ƒSclosed

ƒ8ȡ

ƒvolumeintegralVȡ

ƒ ƒ V ƒ ƒS C ƒC S theorem: ƒV S ƒ divergencetheorem ƒp gradienttheorem; ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ finite

ƒcontrolvolumeV

controlsurfaceS ƒ ƒ dV ƒ ƒ ƒ ƒ ƒ

ƒkinetictheory.

ƒ ƒ

ƒflowfield.

ƒ

ƒfixed

ƒ ƒV BC ƒB

ƒdS

ƒmassflowA

A6I

ƒnetoutS

S ƒ ƒ ƒ

ƒincreasevolume

ƒdecreasethe

volume ƒ

ƒcontinuityequation

ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ pȡ ƒ

Bodyforces:

V

Surfaceforces:

S ƒ volume ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ dS(ȡā)

ƒdS

ƒ S ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒx ƒ ƒ ƒ

ƒyz

Navier-Stokes

ƒ ƒ ƒ

Eulerequations

ƒ

Navier-StokesequationsEuler

ƒ ƒ ƒ

ƒabcdefghia

ƒz ƒ

ƒabhi

abhi pp

ƒuux

uuf(y) ƒ ƒ ƒ ƒ ƒ

ƒuu

xx D

ƒabhi

p ƒ ƒ aibh ƒy ƒz

ƒdSdy()

ƒ ƒ ƒ ƒu ƒ ƒ

ƒȡ

ƒ ƒ ƒ ƒ

ƒsystem

ƒ ƒ e

ƒsurroundings

ƒįq

ƒį

ƒ

ƒde

ƒ ƒ B B B ƒ

ƒtimerate

power ƒ ƒ 6M ƒ ƒ ƒ ƒ ƒ Sis ƒ is ƒ ƒ ƒe ƒ V ƒ eV

ƒtotalenergy

ƒB

ƒdS

ƒ ƒ ƒV ƒB ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒt ƒt ƒ

ƒȡȡy,z,t)

ƒtt

ƒ

ƒ(ȡȡ)/(tt)average

ƒtt

ƒsubstantial

derivativeD/Dt

ƒDȡ/Dtinstantaneous

givenfluidelement

ƒȡ

fixed ƒ ƒ ƒ ƒ

ƒD/Dt

ƒtlocalderivative,

ƒāÏconvectivederivative,

ƒ ƒ ƒ ƒ ƒ ƒx ƒ ƒ ƒ ƒ ƒ ƒ A. ƒA A

ƒpathlineA

ƒstreamline

ƒ ƒ ƒ

ƒstreakline

ƒ ƒ xyz

ƒis

ƒ ƒ ƒ ƒ ƒ ƒ

ƒșșș

ƒ

ƒșșș

ƒABAC

dș/dtdș/dt ƒ AB AC

ƒȦz

ƒxyȦ

ƒ

ƒvorticity,

ȟ

ƒÏ

ƒ

Rotationalflow

Irrotational flow

ƒ ƒ

ƒABAC

ț ƒ ț

ƒtț

ttț ț

ƒstrainț

decreasingț ƒ

ƒİxy

ƒ

ƒyzzx

ƒbyu=y/(x2+y2)and

v=/(x2+y2). ƒ (,) ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ circulation ƒ ƒ ƒ ƒ ƒC ƒ C ƒ ƒ ƒ C ƒ ƒ ƒ ƒ ƒC C.

ƒirrotationalÏ

Cī ƒ ƒ C

Stheorem:

ƒ

ƒuv

ƒ ƒ ƒ ƒ

ƒuvxy

c ƒ %2streamfunction ƒ ƒ ƒn

ƒǻn

y ǻy x ǻx ƒ n yx ƒ %2 L %2:T
áU; ƒ ƒ ƒ %2 ƒ ȡ ƒ ƒ ƒ

ƒĭ

ƒ

ƒirrotational

ĭ ĭ

ƒĭvelocitypotential

ƒ ƒ ƒ ƒ

ĭȥ

ĭ ȥ ƒ

ijpotentialflowsequipotential linestream line

ƒ ĭ ȥ ƒ ƒ ƒ ƒ ƒ

ƒsolved

ƒ solution ƒ pȡ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ

ƒdiscretized,

ƒ ȡu xt ƒ xi ƒt tǻt ƒ ǻ

ȡǻ

ƒ ƒ ƒ

ƒȡt

uxȡx i volkan.pehlivanoglu@ieu.edu.tr

Aerodynamics

AE 301

Contents;Inviscid, Incompressible Flow

a., b.Incompressible Flow in a Duct, c.Pitot Tube: Measurement ofAirspeed, d.Pressure Coefficient, e.Condition on Velocity, f.Equation, g.Uniform Flow, h.Source Flow, i.Doublet Flow, j.Vortex Flow, k.The Kutta-Joukowski Theorem, l.Applied Aerodynamics, m.Historical Note

Introduction

Albert F. Zahm, 1912

߲ ߘ ȉ Ȟ Ȱ

ȲIntroduction

Partial

Delgradient -Nabla

Cross Dot Gamma Phi Psi

Infinity

lambdaλnomenclature Ȧ

Introduction

ƒInviscid,incompressibleflowActually,suchflowis amythontwoaccounts. ƒFirst,inreallifethereisalwaysfriction.Innature thereis,strictlyspeaking,noinviscidflow.

ƒSecond,everyflowiscompressible.Innaturethere

is,strictlyspeaking,noincompressibleflow.

ƒThereareawholehostofaerodynamicapplications

thataretobeinginviscidand incompressible. Wrightbrothers on December 17, 1903.ƒBymakingthatassumptionandwe obtainamazinglyaccurateresults.

ƒFromanaerodynamicpointofview,atairvelocities

between0and300mi/htheairdensityremains essentiallyconstant,varyingbyonlyafewpercent.

ƒHence,theaerodynamicsofthefamilyofairplanes

spanningtheperiodbetween1903and1940couldbe describedby. ƒToday,wearestillveryinterestedinincompressible aerodynamicsbecausemostmoderngeneralaviation aircraftstillflyatspeedsbelow300mi/h.Introduction

19031940

1985

ƒTheearlypartoftheeighteenthcenturyItwasatthis

timethattherelationbetweenpressureandvelocityin aninviscid,incompressibleflowwasfirstunderstood.

ƒequationrelatesvelocityandpressurefrom

onepointtoanotherinaninviscid,incompressibleflow.

ƒConsiderthecomponentofthemomentumequation

andcontinuityequation; ƒForaninviscidflowwithnobodyforces,thisequation becomes

ƒForsteadyflow,=0

ƒMultiplyEquationby

ƒConsidertheflowalongastreamlineinthree-

dimensionalspace.Theequationofastreamlineis givenbyEquations

ƒSubstitutingthemintopreviousequation,

ƒThedifferentialofis

ƒThisisexactlytheterminparenthesesinEquation.

Hence,

ƒor

ƒInasimilarfashion,startingfromthe

componentsofthemomentumequation,wehave

ƒAddingEquationsyields

ƒHowever,

ƒand

ƒSubstitutingthemintopreviousone,wehave

ƒEquationiscalled

ƒItrelatesthechangeinvelocityalongastreamline

tothechangeinpressurealongthesame streamline.

ƒInsuchacase,ȡ=constant,andcanbeeasily

integratedbetweenanytwopoints1and2alonga streamline.

ƒEquationiscanalsobewrittenas

ƒThephysicalsignificanceofequationis

obviousfromEquations.

ƒNamely,

ƒequationisalsoarelationformechanical

energyinanincompressibleflow.

ƒItstatesthattheworkdoneonafluidbypressure

forcesisequaltothechangeinkineticenergyofthe flow.

ƒIndeed,equationcanbederivedfromthe

generalenergyequation

ƒThestrategyforsolvingmostproblemsininviscid,

incompressibleflowisasfollows:

Obtainthevelocityfieldfromthegoverningequations

appropriateforaninviscid,incompressibleflow.

Oncethevelocityfieldisknown,obtainthe

correspondingpressurefieldfrom equation.

Example 3.1

ƒConsideranairfoilinaflowatstandardsealevel

conditionswithafreestreamvelocityof50m/s.

ƒAtagivenpointontheairfoil,thepressureis0.9105

N/m2.Calculatethevelocityatthispoint.

ƒAtstandardsealevelconditions,

Example 3.2

ƒConsidertheinviscid,incompressibleflowofairalonga streamline.

ƒTheairdensityalongthestreamlineis0.002377

slug/ft3,whichisstandardatmosphericdensityatsea level. ƒAtpoint1onthestreamline,thepressureandvelocity are2116lb/ft2and10ft/s,respectively.

ƒFurtherdownstream,atpoint2onthestreamline,the

velocityis190ft/s.

ƒCalculatethepressureatpoint2.

ƒWhatcanyousayabouttherelativechangein

pressurefrompoint1topoint2comparedtothe correspondingchangeinvelocity?

Example 3.2

ƒFromEquation

ƒOnlya2percentdecreaseinthepressurecreatesa

1900percentincreaseintheflowvelocity.

ƒThisisanexampleofageneralcharacteristicoflow-

speedflows.

ƒOnlyasmallbarometricchangefromonelocationto

anothercancreateastrongwind.

Incompressible flow in a duct:

theVenturi and low-speed wind tunnel

ƒConsidertheflowthroughaduct.Ingeneral,theflow

throughsuchaductisthree-dimensional.

ƒHowever,inmanyapplications,thevariationofarea

=ismoderate.

ƒForsuchcasesitisreasonabletoassumethatthe

flow-fieldpropertiesareuniformacrossanycross section,andhencevaryonlyinthedirection.

Incompressible flow in a duct

ƒConsidertheintegralformofthecontinuityequation

ƒForsteadyflow,thisbecomes

ƒApplyEquationtotheduct,wherethecontrolvolumeis boundedby1ontheleft,2ontheright,andthe upperandlowerwallsoftheduct.Hence, ƒAlongthewalls,theflowvelocityistangenttothewall, anddSisperpendiculartothewall,

Incompressible flow in a duct

ƒWehave

ƒSubstitutingEquationsintomainequation,weobtain

ƒEquationisthequasi-one-dimensionalcontinuity

equation. ƒItappliestobothcompressibleandincompressibleflow.

ƒItstatesthatthevolumeflow(cubicmetersper

second)throughtheductisconstant.

ƒWeseethatiftheareadecreasesalongtheflow

(convergentduct),thevelocityincreases. ƒConversely,iftheareaincreases(divergentduct),the velocitydecreases.ƒConsiderflowonly,whereȡ=constant.Incompressible flow in a duct

Incompressible flow in a duct

ƒMoreover,fromequationweseethatwhen

thevelocityincreasesinaconvergentduct,thepressure decreases. ƒConversely,whenthevelocitydecreasesinadivergent duct,thepressureincreases.

ƒThevelocityincreasesintheconvergent

portionoftheduct,reachingamaximumvalue

V2attheminimumareaoftheduct.

ƒThisminimumareaiscalledthethroat.

ƒAtthethroat,thepressurereachesaminimum

valuep2.

ƒInanapplicationclosertoaerodynamics,a

venturicanbeusedtomeasureairspeeds.a

Incompressible flow in a duct

ƒFromequation,

ƒFromthecontinuityequation

ƒweobtain

ƒHistoricallythefirstpractical

airspeedindicatoronanairplane wasaventuriusedbytheFrench

CaptainA.EteveinJanuary1911

Incompressible flow in a duct

ƒAnotherapplicationofincompressibleflowinaductis thelow-speedwindtunnel. ƒTosimulateactualflightintheatmospheredatesbackto

1871,whenFrancisWenhaminEnglandbuiltandused

thefirstwindtunnelinhistory.

ƒInessence,alow-speedwindtunnelisalargeventuri

wheretheairflowisdrivenbyafanconnectedtosome typeofmotordrive.

ƒThewind-tunnelfanbladesaresimilartoairplane

propellersandaredesignedtodrawtheairflowthrough thetunnelcircuit.

ƒThewindtunnelmaybeopencircuit,orthewindtunnel

maybeclosedcircuit.

Incompressible flow in a duct

A full-scale wind tunnel, Langley-VA A small-scale wind tunnel ƒTheairisdrawninthefrontdirectlyfrom

theatmosphereandexhaustedoutthe back,againdirectlytotheatmosphere.

Incompressible flow in a duct

NASAAmesResearchCenter,MountainView,

California-USA

Builtintheearly1980's,the80-by120-footisanopen

circuittunnel.Airisdrawnfromthehuge360-footwide,

130-foothighairintake,passesthroughthe120-foot

wide,80-foothightestsectionandthenisexpelledto theatmosphere.Themaximumairspeedthroughthe testsectionis115mph.Powerisderivedfromsix40- footdiameterfanblades,eachmotorratedat23,500 hp.The80-by120-foottunneliscapableoftesting aircraftaslargeasaBoeing737.Thewindtunnel beganregularoperationsin1987.

Thelargest wind tunnel in the World.

Incompressible flow in a duct

NASAAmesResearchCenter,Mountain

View,California-USA

Incompressible flow in a duct

ƒTheairfromtheexhaustisreturneddirectlytothefrontof thetunnelviaaclosedductformingaloop

Size:9.1mhighx9.1mwidex24mlong

Maximumwindspeed:55m/s(200km/h)

TheNationalResearchCouncil(NRC),

Ottowa,Canada

Incompressible flow in a duct

ƒFromthecontinuityequation,thetest-

sectionairvelocityis

ƒThevelocityattheexitofthediffuseris

ƒThepressureatvariouslocationsinthe

windtunnelis ƒThebasicfactorthatcontrolstheairvelocityinthetestsectionof agivenlow-speedwindtunnelisthepressuredifference12

Incompressible flow in a duct

ƒThetest-sectionvelocity2isgovernedbythepressure difference12 ƒThefandrivingthewind-tunnelflowcreatesthispressure differencebydoingworkontheair.

ƒInlow-speedwindtunnels,amethodofmeasuringthe

pressuredifference12isbymeansofamanometer.

Denote the weight per unit volume by

Example 3.3

ƒConsideraventuriwithathroat-to-inletarearatioof0.8 mountedinaflowatstandardsealevelconditions.

ƒIfthepressuredifferencebetweentheinletandthe

throatis7lb/ft2,calculatethevelocityoftheflowatthe inlet. ƒAtstandardsealevelconditions,ȡ=0002377slug/ft3.

Hence,

Example 3.4

ƒConsideralow-speedsubsonicwindtunnelwitha12/1

contractionratioforthenozzle.

ƒTheflowinthetestsectionisatstandardsealevel

conditionswithavelocityof50m/s.

ƒCalculatetheheightdifferenceinaU-tubemercury

manometerwithonesideconnectedtothenozzleinlet andtheothertothetestsection.

ƒAtstandardsealevel,ȡ=123kg/m3.FromEquation

ƒThedensityofliquidmercuryis1.36104kg/m3.Hence,

Example 3.4

ƒConsideramodelofanairplanemountedinasubsonic

windtunnel.Thewind-tunnelnozzlehasa12-to-1 contractionratio. ƒThemaximumliftcoefficientoftheairplanemodelis1.3.

Thewingplanformareaofthemodelis6ft2.

ƒTheliftismeasuredwithamechanicalbalancethatis

ratedatamaximumforceof1000lb. ƒCalculatethemaximumpressuredifferenceallowable betweenthewind-tunnelsettlingchamberandthetest section.

ƒStandardsealeveldensityinthetestsection,

ȡ0.002377slug/ft3.

Example 3.4

ƒMaximumliftoccurswhenthemodelisatits

maximumliftcoefficient.

ƒThefreestreamvelocityatwhichthisoccursis

obtainedfrom

ƒFromEquation

Pitot tube

ƒIn1732,theFrenchmanHenriPitotwasbusytryingto

measuretheflowvelocityoftheSeineRiverinParis.

ƒHeusedhisowninvention,themost

commondeviceformeasuringflightvelocitiesof airplanes.

ƒConsideraflowwithpressure1movingwithvelocity

1.

ƒPressureisclearlyrelatedto

themotionofthemolecules, randombutinalldirections.

ƒNowimaginethatyouhoponafluidelementoftheflow

andridewithitatthevelocity1.

ƒThegasmolecules,becauseoftheirrandommotion,

willstillbumpintoyou,andyouwillfeelthepressure1ofthegas.

ƒWenowgivethispressureaspecificname:the

pressure.

ƒStaticpressureisameasureofthepurelyrandom

motionofmoleculesinthegas.

ƒItisthepressureyoufeelwhenyouridealongwiththe

gasatthelocalflowvelocity.Pitot tube

ƒFurthermore,consideraboundaryoftheflow,suchas

awall,whereasmallholeisdrilledperpendicularto thesurface.

ƒTheplaneoftheholeisparalleltotheflow.

ƒBecausetheflowmovesovertheopening,the

pressurefeltatpointisdueonlytotherandom motionofthemolecules.Pitot tube

ƒThatis,atpoint,thestaticpressureis

measured.

ƒSuchasmallholeinthesurfaceis

calledaora .

Pitot tube

ƒIncontrast,considerthataPitottubeisnowinsertedinto theflow,withanopenendfacingdirectlyintotheflow.

ƒThatis,theplaneoftheopeningofthetubeis

perpendiculartotheflow. ƒTheotherendofthePitottubeisconnectedtoapressure gage,suchaspoint

ƒThePitottubeisclosedatpoint.

Pitot tube

ƒForthefirstfewmillisecondsafterthePitottubeis

insertedintotheflow,thegaswillrushintotheopenend andwillfillthetube.

ƒHowever,thetubeisclosedatpoint;thereisnoplace

forthegastogo. ƒHenceafterabriefperiodofadjustment,thegasinside thetubewillstagnate;thatis,thegasvelocityinsidethe tubewillgotozero.

ƒIndeed,thegaswilleventuallypileupandstagnate

insidethetube,includingattheopenmouth atpoint.

ƒHence,pointattheopenfaceofthePitottubeisa

stagnationpoint,where=0

Pitot tube

ƒFromequationweknowthepressure

increasesasthevelocitydecreases.Hence,1

ƒThepressureatastagnationpointiscalledthe

pressure,orpressure,denotedby0.

Hence,atpoint=0

ƒThepressuregageatpointreads0.

ƒThismeasurement,inconjunctionwithameasurement

ofthestaticpressure1atpointyieldsthe differencebetweentotalandstaticpressure,01

ƒThispressurethatallowsthecalculationof

1viaequation.

ƒItispossibletocombinethemeasurementof

bothtotalandstaticpressureinone instrument,a

Pitot tube

ƒViaequation.

ƒEquationallowsthecalculationofvelocity

simplyfromthemeasureddifference betweentotalandstaticpressure. dynamic pressure

ƒItisimportanttorepeatthatequationholds

forincompressibleflowonly.

Pitot tube

ƒThediameterofthetubeisdenotedby.

ƒAnumberofstaticpressuretapsarearrayedradially

aroundthecircumferenceofthetube.

ƒThelocationshouldbefrom8to16downstreamofthe

nose,andatleast16aheadofthedownstreamsupport stem.

DESIGN BOX

Example 3.9

ƒConsidertheP-35aircraftcruisingatastandardaltitude of4km.

ƒThepressuresensedbythePitottubeonitsrightwing

is6.7104N/m2.

ƒAtwhatvelocityistheP-35flying?

ƒAtastandardaltitudeof4km,thefreestreamstatic

pressureanddensityare6.166104N/m2and0.81935 kg/m3,respectively.

ƒThePitottubemeasuresthetotalpressureof6.7104

N/m2.FromEquation

Pressurecoefficient

ƒPressure,byitself,isadimensionalquantity

ƒHowever,weestablishedtheusefulnessofcertain

dimensionlessparameterssuchasRe,

ƒItmakessense,therefore,thatadimensionless

pressurewouldalsofinduseinaerodynamics.

ƒSuchaquantityistheand

definedas ƒThepressurecoefficientisanothersimilarityparameter.

Pressurecoefficient

ƒForcanbeexpressedinterms

ofvelocityonly.

ƒFromequation,

ƒFinally,

ƒThepressurecoefficientatastagnationpoint(where =0)inanincompressibleflowisalwaysequalto1.0.

ƒAlso,keepinmindthatinregionsoftheflowwhere

orwillbeanegativevalue.

Example3.12

ƒConsidertheairplanemodelinwindtunnel.

ƒThepressurecoefficientwhichoccursatacertainpoint ontheairfoilsurfaceis53. ƒAssuminginviscid,incompressibleflow,calculatethe velocityatthispointwhen

ƒ()=80ft/s,

ƒ()=300ft/s.

ƒTheanswergiveninpart()ofExample3.12isnotcorrect.

ƒWhy?Thespeedofsoundatstandardsealevelis1117

ft/s

Condition on velocity forIncompressible flow

ƒFromthecontinuityequation,

ƒForincompressibleflow,ȡ=constant.Hence,

ƒRecallthat׏

thevolumeofamovingfluidelementperunitvolume.

Governingequationforirrotational,

incompressibleflow:Laplace'sequation

ƒForanincompressibleflow

ƒForanirrotationalflowwehaveseenthatavelocity

potentialĭcanbedefinedsuchthat ƒforaflowthatisbothincompressibleandirrotational,

ƒEquationisandoneofthemost

famousandextensivelystudiedequationsin mathematicalphysics.

ƒLaplace'sequationisasecond-orderpartial

differentialequationnamedafterPierre-

SimonLaplacewhofirststudieditsproperties.

Laplace'sequation

ƒequationiswrittenbelowintermsofthe

threecommonorthogonalcoordinatesystems

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

Laplace'sequation

ƒWecanshowthatthestreamfunctionalsosatisfiesthe equation. ƒRecallthat,foratwo-dimensionalincompressibleflow, astreamfunctionȥcanbedefinedsuchthat,

ƒFromtheirrotationalitycondition

Laplace'sequation

ƒNotethatequationisasecond-orderlinear

partialdifferentialequation.

ƒThefactthatitisisparticularlyimportant.

ƒBecausethesumofanyparticularsolutionsofalinear differentialequationisalsoasolutionoftheequation. ƒForexample,ifĭ1ĭ2ĭ3ĭrepresentseparate solutionsofEquation,thenthesum is also a solution of Equation.

ƒWeconcludethatacomplicatedflowpatterncanbe

synthesizedbyaddingtogetheranumberofelementary flowsthatareirrotationalandincompressible.Superposition principle

Laplace'sequation

ƒBythesameequation,namely,׏

weobtaindifferentflowsforthedifferentbodies?

ƒTheanswerisfoundinthe.

ƒAlthoughthegoverningequationforthedifferentflows isthesame,theboundaryconditionsfortheequation mustconformtothedifferentgeometricshapes andhenceyielddifferentflow-fieldsolutions. ƒBoundaryconditionsarethereforeofvitalconcernin aerodynamicanalysis.

ƒTherefore,twosetsofboundary

conditionsapplyasfollows.Laplace'sequation

Infinity Boundary Conditions

Laplace'sequation

Wall Boundary Conditions

ƒThevelocityvectormustbetothesurface.

ƒIfwearedealingwithȥratherthanĭ,thenthewall boundaryconditionis

ƒThebodysurfaceisastreamlineoftheflow.

ƒThegeneralapproachtothesolutionofirrotational, incompressibleflowscanbesummarizedasfollows:

ƒSolveequationforĭorȥalongwiththe

properboundaryconditions.

ƒObtaintheflowvelocityfromV=׏

and=ȥ ƒObtainthepressurefromequationLaplace'sequation

1749-1827

Uniform flow

ƒConsiderauniformflowwithvelocityoriented

inthepositivedirection.

ƒIntegrating1stEquationwithrespectto,we

have

ƒIntegrating2ndEquationwithrespectto,we

have

ƒBycomparingtheseequations,

the velocity potential for a uniform flow

Uniform flow

ƒConsidertheincompressiblestreamfunctionȥ.We have ƒIntegratingEquationswithrespectto,yandcomparing theresults,weobtain the stream function for auniform flow

ƒEquationscanbeexpressedintermsofpolar

coordinates,

Uniform flow

ƒConsiderthecirculationinauniformflow.

ƒEquationistrueforanyarbitraryclosedcurveinthe

uniformflow.

Source flow

ƒConsideratwo-dimensional,incompressibleflowwhere allthestreamlinesarestraightlinesemanatingfroma centralpoint

ƒLetthevelocityalongeachofthestreamlinesvary

inverselywithdistancefrompoint.

ƒSuchaflowiscalleda.

Source flow

ƒConsideratwo-dimensional,incompressibleflowwhere allthestreamlinesaredirectedtheoriginpoint ƒTheflowvelocityvariesinverselywithdistancefrom point. ƒIndeed,asinkflowissimplyanegativesourceflow.ƒthe origin is a

Source flow

ƒBydefinition,

whereisconstant.

ƒThevalueoftheconstantisrelated

tothevolumeflowfromthesource

ƒDenotethisvolumeflowrateperunitlengthas

ƒSo,theradialvelocity,

Ȧdefines the

source strength,

Source flow

ƒThevelocitypotentialforasourcecanbeobtainedas

follows.

ƒIntegratingEquationwithrespectto,wehave

ƒIntegratingEquationwithrespecttoș,wehave

ƒComparingEquations,weseethat

Source flow

ƒThestreamfunctioncanbeobtainedasfollows.

ƒToevaluatethecirculationforsourceflow,recallthe ׏

Combination of a uniform flowwith a

source and sink

ƒConsiderapolarcoordinatesystemwithasourceof

strengthȦlocatedattheorigin. ƒSuperimposeonthisflowauniformstreamwithvelocity movingfromlefttoright, ƒThestreamfunctionfortheresultingflowisthesumof

Equations

Combination of a uniform flowwith a

source and sink

ƒThevelocityfieldisobtainedbydifferentiating

Equation

ƒNotethat,consistentwiththelinearnatureof

equation,

ƒnotonlycanweaddthevaluesofĭorȥtoobtain

morecomplexsolutions, ƒwecanaddtheirderivatives,thatis,thevelocities, aswell.

ƒThestagnationpointsintheflowcanbeobtainedby

settingEquationsequaltozero one stagnation point exists

Combination of a uniform flowwith a

source and sink

ƒIfthecoordinatesofthestagnationpointatare

substitutedintoEquation,weobtain

ƒThisisahalf-bodythatstretchestoinfinityinthe

downstreamdirection(i.e.,thebodyisnotclosed).

ƒHowever,ifwetakeasinkofequalstrengthasthe

sourceandaddittotheflowdownstreamofpoint, thentheresultingbodyshapewillbeclosed.

ƒLetusexaminethisflowinmoredetail.

Combination of a uniform flowwith a

source and sink ƒConsiderapolarcoordinatesystemwithasourceandsink placedadistancetotheleftandrightoftheorigin, respectively.

ƒThestrengthsofthesourceandsinkareȁandȁ,

respectively(equalandopposite). ƒInaddition,superimposeauniformstreamwithvelocity, asshowninFigure.

ƒThestreamfunctionforthecombinedflowatanypoint

withcoordinatesșisobtainedfromEquationsCombination of a uniform flowwith a source and sink ƒNotefromthegeometrythatș1andș2arefunctionsof,

ș,and.

Scottish engineerW. J. M. Rankine.

a .

Combination of a uniform flowwith a

source and sink

ƒBysetting=0,twostagnationpointsarefound,

namely,pointsand ƒThestagnationstreamlineisgivenbyȥ=0,thatis,

ƒTheregioninsidetheovalcanbereplacedbyasolid

bodywiththeshapegivenbyȥ=0.

ƒTheregionoutsidetheovalcanbeinterpretedasthe

inviscid,potential(irrotational),incompressibleflowover thesolidbody.

Combination of a uniform flowwith a

source and sink

Doublet flow

ƒThereisaspecialcaseofasource-sinkpairthatleads toasingularitycalleda.

ƒConsiderasourceofstrengthȁandasinkofequal

strengthȁseparatedbyadistance,thestream functionis

ƒThegeometryyields

Doublet flow

ƒLetthedistanceapproachzerowhiletheabsolute

magnitudesofthestrengthsofthesourceandsink increaseinsuchafashionthattheproductȁremains constant.

ƒSubstitutingșequation,wehave

ƒThestrengthofthedoubletisdenotedbyț

ƒVelocitypotentialforadoubletisgivenby

Doublet flow

ƒThestreamlinesofadoubletflowareobtainedfrom

ƒAcirclewithadiameteronthevertical

axisandwiththecenterlocated2 directlyabovetheorigin. ƒThedirection of flow is out of the origin tothe left and back into the origin from the right.

ƒByconvention,wedesignatethedirectionofthe

doubletbyanarrowdrawnfromthesinktothesource.

Nonlifting flow overA circular cylinder

ƒConsidertheadditionofauniformflowwithvelocity

andadoubletofstrengthț,asshowninFigure

ƒThestreamfunctionforthecombinedflowis

ƒLet

ƒThenEquationcanbewrittenasNonlifting flow overA circular cylinder ƒEquationisthestreamfunctionforauniformflow-doublet combination. ƒItisalsothestreamfunctionfortheflowoveracircular cylinderofradius

ƒThevelocityfieldisobtainedby

ƒThestagnationpoints,

Nonlifting flow overA circular cylinder

ƒTheequationofthisstreamlinewhereitgoesthroughthe stagnationpoints ƒConsequently,theinviscidirrotational,incompressible flowoveracircularcylinderofradiuscanbe synthesizedbyauniformflowandadoublet; ƒThevelocitydistributiononthesurfaceofthecylinderis givenby

Nonlifting flow overA circular cylinder

ƒThepressurecoefficientisgivenby

ƒThepressurecoefficientdistributionoverthesurfaceis sketched

Nonlifting flow overA circular cylinder

ƒClearly,thepressuredistributionoverthetophalfofthe cylinderisequaltothepressuredistributionoverthe bottomhalf.

ƒHencetheliftmustbezero.

ƒClearly,thepressuredistributionsoverthefrontandrear halvesarethesame.

ƒHencethedragistheoreticallyzero.

Example 3.13

ƒConsiderthenonliftingflowoveracircularcylinder. ƒCalculatethelocationsonthesurfaceofthecylinder wherethesurfacepressureequalsthefreestream pressure.

Example 3.14

ƒInthenonliftingflowoveracircularcylinder,consider theinfinitesimallysmallfluidelementsmovingalong thesurfaceofthecylinder.

ƒCalculatetheangularlocationsoverthesurface

wheretheaccelerationofthefluidelementsarea localmaximumandminimum.

ƒTheradiusofthecylinderis1mandthefreestream

flowvelocityis50m/s,calculatethevaluesofthe localmaximumandminimumaccelerations.

Example 3.14

ƒThelocalvelocityofthefluidelementsonthesurface

ƒTheaccelerationofthefluidelementsis

ƒIncrementaldistanceonthecylindersurfacesubtended byșis

Example 3.14

ƒTofindtheșlocationsatwhichtheaccelerationisa maximumorminimum,differentiateEquationwith respecttoș,andsettheresultequaltozero.

ƒThevaluesofthelocalflowaccelerationateachone

oftheselocationsarerespectively

Tremendously

large accelerations

Vortex flow

ƒConsideraflowwhereallthestreamlinesare

concentriccirclesaboutagivenpoint,assketchedin

Figure;

ƒLetthevelocityalonganygivencircular

streamlinebeconstant.

ƒButletitvaryfromonestreamlineto

anotherinverselywithdistancefromthe commoncenter.

ƒSuchaflowiscalleda.

Vortex flow

ƒToevaluatetheconstant,takethecirculationarounda givencircularstreamlineofradius ƒTherefore,forvortexflow,Equationdemonstratesthat thecirculationtakenaboutallstreamlinesisthesame value,namely,

ƒīiscalledtheofthevortexflow.

ƒWestatedearlierthatvortexflowisirrotationalexceptat theorigin.

ƒTheorigin,=0,isasingularpointintheflowfield.

Vortex flow

ƒThevelocitypotentialforvortexflow

ƒThestreamfunctionisdeterminedinasimilarmanner

Summary for elementary flows

Lifting over a cylinder

ƒConsidertheflowsynthesizedbytheadditionofthe

nonliftingflowoveracylinderandavortexofstrength

ī,asshowninFigure

Lifting over a cylinder

ƒNotethatthestreamlinesarenolongersymmetrical

aboutthehorizontalaxisthroughpoint ƒThatthecylinderwillexperiencearesultingfinitenormal force.

ƒHowever,thestreamlinesaresymmetricalaboutthe

verticalaxisthrough

ƒAsaresultthedragwillbezero.

ƒThevelocityfieldcanbeobtainedby

Lifting over a cylinder

ƒTolocatethestagnationpointsintheflow,set

ƒFromtheabovediscussion,īisclearlyaparameterthat canbechosenfreely.

ƒThereisnosinglevalueofīthattheflowover

acircularcylinder;rather,thecirculationcanbeany value. ƒTherefore,fortheincompressibleflowoveracircular cylinder,thereareaninfinitenumberofpossible potentialflowsolutions,correspondingtotheinfinite choicesforvaluesofī.

ƒThisstatementisnotlimitedtoflowovercircular

cylinders,butrather,itisageneralstatementthatholds fortheincompressiblepotentialflowoverallsmooth two-dimensionalbodies.Lifting over a cylinder

Lifting over a cylinder

ƒThevelocityonthesurfaceofthecylinderisgivenby

(=

ƒThepressurecoefficientis

ƒThedragcoefficientcdisgivenby

Lifting over a cylinder

ƒNotingthatwe immediately obtain

ƒTheliftonthecylindercanbeevaluatedinasimilar

manner

Lifting over a cylinder

ƒNotingthatwe immediately obtain

ƒFromthedefinitionofcl,theliftperunitspanLcanbe obtainedfrom ƒItstatesthattheliftperunitspanisdirectlyproportional tocirculation.It is called the Kutta-Joukowski theorem

Lifting over a cylinder

Norotation

RotationRotation

ƒThispressureimbalance

createsanetupwardforce, thatis,afinitelift. the Magnuseffect

Example 3.18

ƒConsidertheliftingflowoveracircularcylinderwitha diameterof0.5m.

ƒThefreestreamvelocityis25m/s,andthemaximum

velocityonthesurfaceofthecylinderis75m/s.

ƒThefreestreamconditionsarethoseforastandard

altitudeof3km.

ƒCalculatetheliftperunitspanonthecylinder.

ƒFromAppendix,atanaltitudeof3km,ȡ=0.90926

kg/m3. ƒThemaximumvelocityoccursatthetopofthecylinder, whereș=90ƕ,

Example 3.18

ƒFromEquation

ƒRecallingoursignconventionthatīispositiveinthe clockwisedirection,andșisnegativeintheclockwise direction

ƒFromEquation,theliftperunitspanis

The Kutta-Joukowski theoremand the generation of lift ƒConsidertheincompressibleflowoveranairfoilsection, assketchedinFigure.

ƒLetcurvebeanycurveintheflow

theairfoil.

ƒIftheairfoilisproducinglift,thevelocity

fieldaroundtheairfoilwillbesuchthat thelineintegralofvelocityaroundwill befinite,thatis,thecirculation

ƒInturn,theliftperunitspanon

theairfoilwillbegivenbythe The Kutta-Joukowski theoremand the generation of lift ƒThetheoreticalanalysisofliftontwo-dimensionalbodies inincompressible,inviscidflowfocusesonthecalculation ofthecirculationaboutthebody.

ƒOnceisobtained,thentheliftperunitspanfollows

directlyfromtheKutta-Joukowskitheorem.

Nonlifting flows over arbitrarybodies:

the numerical sourcepanel method

ƒRecallthatwehavealreadydealtwiththenonlifting

flows:aRankineovalandboththenonliftingandthe liftingflowsoveracircularcylinder.

ƒWeaddedourelementaryflowsincertainwaysand

discoveredthatthedividingstreamlinesturnedouttofit theshapesofsuchspecialbodies.

ƒHowever,thisindirectmethodcanhardlybeusedina

practicalsenseforbodiesofarbitraryshape.

ƒForexample,considertheairfoil.Doweknowin

advancethecorrectcombinationofelementaryflowsto synthesizetheflowoverthisspecifiedbody? No

ƒWhatwewantisadirectmethod.

ƒThatis,letustheshapeofanarbitrarybody

andforthedistributionofsingularitieswhich producetheflowoverthegivenbody.

ƒThepurposeofthissectionistopresentsuchadirect

method,limitedforthepresenttononliftingflows.Nonlifting flows over arbitrarybodies: the numerical sourcepanel method

Nonlifting flows over arbitrarybodies:

the numerical sourcepanel method ƒThepresenttechniqueiscalledthesourcepanelmethod. ƒNowimaginethatwehaveaninfinitenumberofsuchline sourcessidebyside,wherethestrengthofeachline sourceisinfinitesimallysmall.

ƒTheseside-by-sidelinesourcesforma

ƒLetbethedistancemeasuredalongthesourcesheetin

theedgeview.

ƒDefineȜ=Ȝtobethe

.

ƒThesmallsectionofthesourcesheet

canbetreatedasadistinctsourceof strengthȜ

Nonlifting flows over arbitrarybodies:

the numerical sourcepanel method

ƒNowconsiderpointintheflow,locatedadistance

from

ƒThecartesiancoordinatesofare.

ƒThesmallsectionofthesourcesheetofstrengthȜ

inducesaninfinitesimallysmallpotentialijatpoint.

ƒFromEquation,ĭisgivenby

ƒThecompletevelocitypotentialatpoint,inducedbythe entiresourcesheetfromto,isobtainedbyintegrating

Equation

Nonlifting flows over arbitrarybodies:

the numerical sourcepanel method ƒNotethat,ingeneral,Ȝcanchangefrompositiveto negativealongthesheet.

ƒThatis,thesheetisreallyacombinationofline

sourcesandlinesinks. ƒNext,consideragivenbodyofarbitraryshapeinaflow withfreestreamvelocityasshowninFigure.

ƒThecombinedactionoftheuniformflowandthesource

sheetmakestheairfoilsurfaceastreamlineoftheflow.

ƒOurp

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