3d effects

Chapter 4

AerodynamicPerformanceofFinite Wings

Reading:Sections7.1–7.3in Bertin. Muchof therestof thechapterfocuseson vortex latticemethods,which wewill only touchon briefly.

After completingthis unit, you will be able to explain the physicaldif-ferencesbetweeninfinite span(two-dimensional)and finite span(three-dimensional)wing performance.Specifically, you will beableto:

� Explainphysicallywhy lift is lowerfor afinite-spanwing thanacom-parableinfinite spanwing.

� Describethecauseof induceddragfor finite-spanwings.

� Usetheresultsof lifting line theoryto estimate3D wing performancefrom 2D airfoil data.

� Apply lifting line theoryto estimate3D wing performancefor arbi-traryplanform.

23

24

4.0.1 BasicResultsof Lifting Line Theory

Prandtl’s lifting line theoryfinds the spanwiselift distribution that makesdownwashandlift matchupeverywhere.

In thebestcase,thefinite wing gives

CL� Cl

1� 2

AR

CDi� C2

L

π � ARwhereAR � b2

S is thewing aspectratio.

4.0.2 Experimental Verification

SeeFigs7.9and7.10in Bertin.

4.1. BASIS OFLIFTING LINE THEORY 25

4.0.3 Roadmap

1. Developmentof lifting line theory

2. Bestcasescenario:theelliptical lift distribution

3. Realisticcases:variablechord,airfoil section,andangleof attack

4. Estimatingmaximumlift from experimentaldata

4.1 Basisof Lifting Line Theory

4.1.1 Finite Wing with Non-constant Cir culation Distri-bution

Γ(y)

x

yz

U 8

w(y)

4.1.2 DownwashVelocity

A singlevortex filamentof circulationdΓ locatedaty givesanetdownwashof

This is exactlyhalf of theinducedvelocity for adoubly-infiniteline vortex.Thetotal inducedvelocity is therefore

w � y1 � � 14π

� b2� b2

dΓ y dy

y � y1dy

4.1. BASIS OFLIFTING LINE THEORY 26

ε � y1 � �� w � y1 �U∞

� 14πU∞

� b2� b2

dΓ y dy

y � y1dy (4.1)

4.1.3 Induced Angle of Attack

U

U

w

8ε

4.1.4 Pick an Angle, Any Angle

Definitions:

α Geometricangleof attack(AOA): directionof freestreamflow

αL0 Zerolift angle:AOA for which2D sectiongiveszerolift; usuallyneg-ative

αe EffectiveAOA: α � ε; whenmultiplied by 2D lift-slopecurve a0 gives2D lift coefficient

ε InducedAOA: changein flow directiondueto 3D downwash;oppositesignaslift

4.2. APPLYING LIFTING LINE THEORY 27

α0l

αe

U 8

αε

Zero-lift flowdirection

Chord line

flow directionEquivalent 2D

Freestream flowdirection

αa

α

α00

C

α

2D lift curve

3D lift curve

a a

L0 α αe

l

ε

4.2 Applying Lifting Line Theory

Onceweknow circulationdistributionΓ y � , thelift perunit spanis easy:

l y �� ρU∞Γ y � (4.2)

Sois thetotal lift:

L � � b2� b2

l y � dy

� ρU∞ � b2� b2

Γ y � dy

Induceddragis relatedto inducedangleof attackandlocal lift by:

di y �� l y � ε y �� ρU∞Γ y � w y �U∞

(4.3)

andtotal induceddragis foundby integration:

D �� ρ � b2� b2

w y � Γ y � dy

4.3. SPECIALCASE:ELLIPTICAL DISTRIBUTION OFCIRCULATION28

4.3 SpecialCase: Elliptical Distrib ution of Cir -culation

Supposethatcirculationdistribution is elliptical:

Γ � y � � Γ0 1 � y2� b � 2� 2whereΓ0 is thecirculationat thewing symmetryplane.

Lift coefficient: If wewrite wing areaasS,

CL� 1

12ρU2

∞S

� b � 2� b � 2 ρU∞Γ � y � dy � Γ0πb2U∞S

Induced angleof attack: Substitutinginto Equation4.1,

ε � y1 � � Γ0

4πU∞

� b2� b2

ddy � 1 � y2 b � 2 2

y � y1dy

Wecanmakea trig substitutiony � � b2 cosφ andget

ε � φ0 � � Γ0

2πbU∞

� π

0

cosφcosφ0 � cosφ

dφ� Γ0

2bU∞� CL

π � AR4.3.1 Ideal RelationshipBetween2D and 3D Lift

For a2D wing with constantcrosssection,no twist, andelliptical lift distri-bution, the2D lift coefficient from thin airfoil theoryis

Cl� 2π � α � αL0 �

With elliptical lift distribution, thereis an inducedflow anglethat reducestheangleof attack

ε � CL

πAR

4.3. SPECIALCASE:ELLIPTICAL DISTRIBUTION OFCIRCULATION29

sothelift coefficientat eachwing sectionis

l � y �12ρU2

∞c � y � � 2π � α � αL0 � ε �which is constant!And thereforeequalto CL. So

CL� 2π � α � αL0 � CL

πAR �CL � 1 � 2

AR � � 2π � α � αL0 �CL

� Cl

1�

2� AR4.3.2 Induced Drag for Elliptic Lift Distrib ution

Induced angleof attack again: We cannow relateε to CL for an ellipticlift distribution:

ε � Γ0

2bU∞� 1

2bU∞

CL2U∞Sπb� CL

πAR

whereAR � b2

S

Induced drag: Becauseε is constantfor this case,

Di� Lε

CDi� CLε� C2

L

πAR

4.3.3 Example

Wing characteristics:� Elliptical planform� Elliptical lift distribution� Aspectratio of 6

4.4. GENERAL LIFTING LINE THEORY 30� Span12m� Wing loading900N � m2� Flying @ 150km� hr atsealevel

Find: � Wing area� Total lift� Induceddrag� Power to overcomeinduceddrag

4.4 GeneralLifting Line Theory

Sectionlift coefficientCl in termsof circulation:

Cl � y � � l � y �12ρU2

∞c� ρU∞Γ � y �

12ρU2

∞c� 2Γ � y �

U∞c

Also canwrite in termsof effectiveangleof attackandlift-curveslope:

Cl � y � � � dCl

dα � 2D� αe � α0l �

4.4. GENERAL LIFTING LINE THEORY 31

Equatingthesetwo andreplacingαe� α � ε and � dCl

dα � 2D� a0,

2Γ � y �U∞c � y � � a0 � α � y � � ε � y � � α0l � y ��

Five thingscanvaryalongthespanhere:� circulationΓ� chordc (taper)� geometricangleof attackα (twist)� downwashangleε� zerolift angleof attackα0l (aerodynamictwist)

Restrictions:� Smallsweep� Not-too-smallaspectratio

If restrictionsdon’t hold, thensomethinglike a vortex latticemethodis thenext-simplestchoice.

4.4.1 GeneralCir culation Distrib utions

If y � � b2 cosφ, theelliptical circulationdistribution is

Γ � φ � � Γ0sinφ

Supposethatweuseageneralsineseriesinstead:

Γ � φ � � 2bU∞∞

∑n � 1

An sinnφ

Canrelatedownwashto circulation:

ε � � wU∞

� � 14πU∞

� b � 2� b � 2 dΓ � dyy � y1

dy� ∑nAn sinnφsinφ


4.4.2 GeneralLifting Line Results

4bU∞ ∑An sinnφU∞c � y � � a0 � α � y � � ∑nAn sinnφ

sinφ� α0l � y � �

c � φ � a0

4b� α � φ � � α0l � φ �� ∑An sinnφ � c � φ � a0

4b∑nAn sinnφ

sinφc � φ � a0

4bsinφ � α � φ � � α0l � φ �� ∑An sinnφ � n

c � φ � a0

4b�

sinφ �After a fair bit of manipulation,

CL� πA1 � AR (4.4)

CDi� π � AR∑nA2

n� C2

L

π � AR � 1� ∞

∑n � 2

n � An

A1 � 2 (4.5)

4.4.3 What TheseMode ShapesLook Lik e

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1

Mag

nitu

de o

f sin

(n φ

)

2y/b

N=1N=2N=3N=4N=5

4.4.4 What Do You Do With This Mess?

c � φ � a0

4bsinφ � α � φ � � α0l � φ �� ∑An sinnφ � c � φ � a0

4b�

sinφ � (4.6)


What do weknow?� Distributionof chordc� Distributionof geometricangleof attack(includingtwist) α� Distributionof zerolift angleof attackαL0� Distributionof lift curveslopefor wing sectionsa0 (doesn’t varymuchin practice)

What do weneedto know?� CoefficientsAn

How do wefind those?� Decidehow many termsto keepin theseries(termsthroughA8

is generallyenough)� Pick asmany valuesof φ over thewing asAn (or half-wing, forcaseswithout roll)� EvaluateEq.4.6for eachφ to getasetof lineareqnsfor theAn

4.4.5 Example: RectangularWing

Given: � Aspectratio = 6� No camber(αL0� 0 everywhere)� No twist (α constant)� Cl

� 2πα

Find: (asfunctionsof α)� Lift coefficient� Induceddragcoefficient

4.4. GENERAL LIFTING LINE THEORY 34� Comparewith elliptical planform


4.4.5.1 Results: Cir culation Distrib ution for RectangularWing

-0.05

0

0.05

0.1

0.15

0.2

0.25

-1 -0.5 0 0.5 1

Mag

nitu

de o

f sin

(n φ

)

2y/b

N=1N=3N=5N=7

Total

4.4.6 GroupExample: A RectangularWing with Washout

Given: � Aspectratio = 6� No camber(αL0� 0 everywhere)� Washoutof 3! at tip (lineardistribution in y)� Angle of attackα � 5o� Cl

� 2πα for all sections� Useonly A1, A3, andA5 andthreepoints(at π6, π

3, π2)

Find: � Lift coefficient

4.4. GENERAL LIFTING LINE THEORY 36� Induceddragcoefficient

4.4.6.1 Converting Cir culation to Local Lift Coefficient

UsingLLT, wecalculateAn in expansionof Γ. To getlift:

Cl � φ � � ρU∞Γ � φ �12ρU2

∞c � φ �� 2c � φ � U∞

2bU∞ ∑An sinnφ� 4bc � φ � ∑An sinnφ

Sofor thewashoutexample,atα � 5! :Cl � φ � � 4b

cbb� A1sinφ � A3sin3φ � A5sin5φ �

4.5. ESTIMATING MAXIMUM LIFT COEFFICIENT 37� 4AR � A1sinφ � A3sin3φ � A5sin5φ �� 0 " 3738sinφ � 0 " 0205sin3φ � 0 " 0173sin5φ

4.4.6.2 Results: Lift Distrib ution for Wing with Washout

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-1 -0.5 0 0.5 1

C

# l

2y/b

N=1N=3N=5

Total

4.5 Estimating Maximum Lift Coefficient

Needto find αstall, soneedCl $ φ % α & :Cl $ φ &�' 4b

c $ φ & ∑An sinnφ

with An $ α & . For thewashoutexample,

Cl ( φ )+* 4AR ( A1sinφ , A3sin3φ , A5sin5φ )* ( 5 - 755α . 0 - 1285) sinφ , ( 0 - 6715α . 0 - 0791) sin3φ, ( 0 - 0954α , 0 - 0090) sin5φ

In this case, thingsareeasy, becausemaxsectionlift coefficient is at mid-span / φ ' π

2 0 . SomaxCl is

Cl 1max $ α &�' 5 2 178α 3 0 2 0404

4.6. BREAKDOWN OFLIFTING LINE THEORY 38

If thewing sectionhasaCl at stall (from experiment)of 1.6,then

α � 0 " 317rad� 18" 2!4.5.1 Group Example: Max Lift (RectangularWing)

For therectangularwing examplewithoutwashout,andCl 4max� 1 " 55,find:� Stall angleof attack� Maximumwing lift coefficient.

Wing Stall

SeeFigure7.17and7.18in Bertin.

4.6 Breakdown of Lifting Line Theory

SweptDelta

Low Aspect Ratio

4.7. BASIC FLIGHT MECHANICS 39

4.7 BasicFlight Mechanics

4.7.1 Forceson an Air plane in Flight

L

W

TD

4.7.2 Steady, Level Flight

Forcesarein equilibrium,so:

L � W � CLρ∞U2

∞2

S

T � D � CDρ∞U2

∞2

S

where

L is lift

W weight

T thrust

D drag

CL lift coefficient

CD dragcoefficient

S wing planformarea

4.7.2.1 Minimizing Drag

Dragcomesin two mainflavors: parasitedraganddragdueto lift


CD� CDP

� C2L

π � AR � 1 � δ �For agivenweightandflight condition,CL is fixed,sominimizeCD � CL:

CD

CL

� CDP

CL

� CL

π � AR � 1 � δ �Needd CD � CL

dCL

� 0:

d � CD � CL �dCL

� � CDP

C2L

� 1π � AR � 1 � δ � � 0

Sotheparasitedragcoefficientmustbe

CDP� C2

L

π � AR � 1 � δ �which is thesameasthedragdueto lift!

4.7.2.2 Flight Speedfor Minimum Power

Power in level flight5 � DU∞� � CDP

� C2L

π � AR � 1 � δ � � ρ∞U3∞

2S� � CDP

� C2L

π � AR � 1 � δ � � ρ∞2� 2W

CLρS � 3� 2S

� 2W 3

ρ∞S � CDP

C3� 2L

� C1� 2L

π � AR � 1 � δ � Differentiateto find theoptimalCL:

C2L� 3πCDP � AR� 1 � δ � � 3CL 6 L 7 D 8 max


4.7.3 Climb Performance

For climb atanangleγ,

L � W cosγT � D

�W sinγ

For smallangleof climb, cosγ 9 1, and

sinγ � T � DW

9 TW� D

L

Rateof climb is

RC � U∞ sinγ � U∞ � T � D �W� enginepower � dragpower

W

3d effects

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