1

MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Introduction to Lifting Line Theory

April 11, 2011

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

2

NECESSARY TOOL• Return to vortex filament, which in general maybe curved

• General treatment accomplished with Biot-Savart Law

34 r

rdldV

Electromechanical Analogy:Think of vortex filament as a wire carrying an electrical current IThe magnetic field strength, dB, induced at point P by segment dl is:

34 r

rdlIdB

3

EXAMPLE APPLICATIONS

hV

4

hV

2

• Case 1: Biot-Savart Law applied between ± ∞

• Case 2: Biot-Savart Law applied between fixed point A and ∞ 34 r

rdldV

Case 1 Case 2

4

BIOT-SAVART LAW

5

EXAMPLE APPLICATIONS

Case 1:

6

HELMHOLTZ’S VORTEX THEOREMS1. The strength of a vortex filament is constant along its length

2. A vortex filament cannot end in a fluid; it must extend to boundaries of fluid (which can be ± ∞) or form a closed path

Note: Statement that “vortex lines do not end in the fluid” is kinematic, due to definition of vorticity, , (or in Anderson) and totally general

• We will use Helmholtz’s vortex theorems for calculation of lift distribution which will provide expressions for induced drag

L’=L’(y)=∞V∞(y)

7

CONSEQUENCE: ENGINE INLET VORTEX

8

CHAPTER 4: AIRFOILEach is a vortex lineOne each vortex line =constantStrength can vary from line to lineAlong airfoil, =(s)

Integrations done:Leading edge toTrailing edge

z/c

x/c

Side viewEntire airfoil has

14 7

9

CHAPTER 5: WINGS

10

PRANDTL’S LIFTING LINE THEORY

• Replace finite wing (span = b) with bound vortex filament extending from y = -b/2 to y = b/2 and origin located at center of bound vortex (center of wing)

• Helmholtz’s vorticity theorem: A vortex filament cannot end in a fluid

– Filament continues as two free vorticies trailing from wing tips to infinity

– This is called a ‘Horseshoe Vortex’

11

PRANDTL’S LIFTING LINE THEORY• Trailing vorticies induce velocity along bound vortex with both contributions in

downward direction (w is in negative z-direction)

2

2

2

4

24

24

4

yb

byw

yb

yb

yw

hV

Contribution from left trailing vortex(trailing from –b/2)

Contribution from right trailing vortex(trailing from b/2)

• This has problems: It does not simulate downwash distribution of a real finite wing

• Problem is that as y → ±b/2, w → ∞

• Physical basis for solution: Finite wing is not represented by uniform single bound vortex filament, but rather has a distribution of (y)

12

PRANDTL’S LIFTING LINE THEORY

• Represent wing by a large number of horseshoe vorticies, each with different length of bound vortex, but with all bound vorticies coincident along a single line

– This line is called the Lifting Line

• Circulation, , varies along line of bound vorticies

• Also have a series of trailing vorticies distributed over span

– Strength of each trailing vortex = change in circulation along lifting line

Instead of =constantWe need a way to let =(y)

13

PRANDTL’S LIFTING LINE THEORY

• Example shown here will use 3 horseshoe vorticies

d1

14

PRANDTL’S LIFTING LINE THEORY

d1

d2

15

PRANDTL’S LIFTING LINE THEORY

d1

d2

d3

16

PRANDTL’S LIFTING LINE THEORY

• Represent wing by a large number of horseshoe vorticies, each with different length of bound vortex, but with all bound vorticies coincident along a single line– This line is called the Lifting Line

• Circulation, , varies along line of bound vorticies• Also have a series of trailing vorticies distributed over span

– Strength of each trailing vortex = change in circulation along lifting line

• Example shown here uses 3 horseshoe vorticies→ Consider infinite number of horseshoe vorticies superimposed on lifting line

d1

d2

d3

17

PRANDTL’S LIFTING LINE THEORY

• Infinite number of horseshoe vorticies superimposed along lifting line

– Now have a continuous distribution such that = (y), at origin =

• Trailing vorticies are now a continuous vortex sheet (parallel to V∞)

– Total strength integrated across sheet of wing is zero

18

PRANDTL’S LIFTING LINE THEORY

• Consider arbitrary location y0 along lifting line

• Segment dx will induce velocity at y0 given by Biot-Savart law

• Velocity dw at y0 induced by semi-infinite trailing vortex at y is:

• Circulation at y is (y)

• Change in circulation over dy is d = (d/dy)dy

• Strength of trailing vortex at y = d along lifting line

yy

dydyd

dw

04

19

PRANDTL’S LIFTING LINE THEORY

• Total velocity w induced at y0 by entire trailing vortex sheet can be found by integrating from –b/2 to b/2:

2

20

0 4

1b

b

dyyy

dyd

yw

Equation gives value ofdownwash at y0 due toall trailing vorticies

20

SUMMARY SO FAR• We’ve done a lot of theory so far, what have we accomplished?

• We have replaced a finite wing with a mathematical model

– We did same thing with a 2-D airfoil

– Mathematical model is called a Lifting Line

– Circulation (y) varies continuously along lifting line

– Obtained an expression for downwash, w, below the lifting line

• We want is an expression so we can calculate (y) for finite wing (WHY?)

– Calculate Lift, L (Kutta-Joukowski theorem)

– Calculate CL

– Calculate eff

– Calculate Induced Drag, CD,i (drag due to lift)

21

FINITE WING DOWNWASH• Recall: Wing tip vortices induce a downward component of air velocity near

wing by dragging surrounding air with them

2

20

0 4

1b

bi dy

yy

dyd

Vy

i

V

ywy

V

ywy

i

i

00

010 tan

Equation for induced angle of attackalong finite wing in terms of (y)

22

EFFECTIVE ANGLE OF ATTACK, eff, EXPRESSION

0

0

0

00

0

0

002

00000

0

2

22

1

2

Leff

Leffl

l

l

LeffLeffl

effeff

ycV

y

yc

ycV

yc

yVcycVL

yyac

y

eff seen locally by airfoilRecall lift coefficientexpression (Ref, EQ: 4.60)a0 = lift slope = 2

Definition of lift coefficient and Kutta-Joukowski

Related both expressions

Solve for eff

23

COMBINE RESULTS FOR GOVERNING EQUATION

2

20

00

00

2

20

0

00

0

4

1

4

1

b

bL

ieff

b

bi

Leff

dyyy

dyd

VycV

yy

dyyy

dyd

Vy

ycV

y

Effective angle of attack(from previous slide)

Induced angle of attack(from two slides back)

Geometric angle of attack = Effective angle of attack + Induced angle of attack

24

PRANDTL’S LIFTING LINE EQUATION

• Fundamental Equation of Prandtl’s Lifting Line Theory

– In Words: Geometric angle of attack is equal to sum of effective angle of attack plus induced angle of attack

– Mathematically: = eff + i

• Only unknown is (y)

– V∞, c, , L=0 are known for a finite wing of given design at a given a

– Solution gives (y0), where –b/2 ≤ y0 ≤ b/2 along span

2

20

00

00 4

1b

bL dy

yy

dyd

VycV

yy

25

WHAT DO WE GET OUT OF THIS EQUATION?

1. Lift distribution

2. Total Lift and Lift Coefficient

3. Induced Drag

dyyySVSq

DC

dyyyVdyyyLD

LD

dyySVSq

LC

dyyVL

dyyLL

yVyL

b

bi

iiD

i

b

bi

b

bi

iii

b

bL

b

b

b

b

2

2

,

2

2

2

2

2

2

2

2

2

2

00

2

2

26

ELLIPTICAL LIFT DISTRIBUTION• For a wing with same airfoil shape across span and no twist, an elliptical

lift distribution is characteristic of an elliptical wing planform

AR

CC

AR

C

LiD

Li

2

,

27

SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION

Points to Note:

1. At origin (y=0) =0

2. Circulation varies elliptically with distance y along span

3. At wing tips (-b/2)=(b/2)=0

– Circulation and Lift → 0 at wing tips

2

0

2

0

21

21

b

yVyL

b

yy

28

SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION

Elliptic distribution

Equation for downwash

Coordinate transformation →

See reference for integral

bVV

wb

w

db

w

db

dyb

y

dy

yyby

y

byw

by

y

bdy

d

i

b

b

2

2

coscos

cos

2

sin2

;cos2

41

41

4

0

00

0 0

00

2

20

21

2

22

00

2

220

Downwash is constant over span for an elliptical lift distribution

Induced angle of attack is constant along spanNote: w and i → 0 as b → ∞

29

SPECIAL SOLUTION:ELLIPTICAL LIFT DISTRIBUTION

AR

CC

dyySV

C

AR

CS

bAR

b

SC

bVdy

b

yVL

LiD

b

b

iiD

Li

Li

b

b

2

,

2

2

,

2

2

0

2

2

21

2

2

0

2

4

41

CD,i is directly proportional to square of CL

Also called ‘Drag due to Lift’

We can develop a moreuseful expression for i

Combine L definition for elliptic profile with previous result for i

Define AR because itoccurs frequently

Useful expression for i

Calculate CD,i

30

SUMMARY: TOTAL DRAG ON SUBSONIC WING

eAR

Cc

Sq

DcC

DDD

DDDD

Lprofiled

iprofiledD

inducedprofile

inducedpressurefriction

2

,,

Also called drag due to lift

Profile DragProfile Drag coefficient relatively constant with M∞ at subsonic speeds

Look up(Infinite Wing)

May be calculated fromInviscid theory:Lifting line theory

31

SUMMARY• Induced drag is price you pay for generation of lift

• CD,i proportional to CL2

– Airplane on take-off or landing, induced drag major component

– Significant at cruise (15-25% of total drag)

• CD,i inversely proportional to AR

– Desire high AR to reduce induced drag

– Compromise between structures and aerodynamics

– AR important tool as designer (more control than span efficiency, e)

• For an elliptic lift distribution, chord must vary elliptically along span

– Wing planform is elliptical

– Elliptical lift distribution gives good approximation for arbitrary finite wing through use of span efficiency factor, e

32

WHAT IS NEXT?• Lots of theory in these slides → Reinforce ideas with relevant examples

• We have considered special case of elliptic lift distribution

• Next step: develop expression for general lift distribution for arbitrary wing shape

– How to calculate span efficiency factor, e

– Further implications of AR and wing taper

– Swept wings and delta wings

New A380:Wing is tapered and swept