Introduction to Lifting Line theory

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MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Introduction to Lifting Line Theory

April 11, 2011

Mechanical and Aerospace Engineering DepartmentFlorida Institute of Technology

D. R. Kirk


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NECESSARY TOOL

Return to vortex filament, which in general maybe curved

General treatment accomplished with Biot-Savart Law

34 r

rdl

dV

Electromechanical Analogy:Think of vortex filament as a wire carrying an electrical current I

The magnetic field strength, dB, induced at point P by segment dl is:

3

4 r

rdlIdB


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EXAMPLE APPLICATIONS

h

V

4

h

V

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


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BIOT-SAVART LAW


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EXAMPLE APPLICATIONS

Case 1:


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HELMHOLTZS VORTEX THEOREMS

1. 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 pathNote: Statement that vortex lines do not end in the fluid is kinematic, due to

definition of vorticity, w, (orxin Anderson) and totally general

We will use Helmholtzs vortex theorems for calculation of lift distribution which

will provide expressions for induced drag

L=L(y)=rV(y)


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CONSEQUENCE: ENGINE INLET VORTEX


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CHAPTER 4: AIRFOILEach is a vortex line

One each vortex line 1=constant

Strength can vary from line to line

Along airfoil, g=g(s)

Integrations done:

Leading edge to

Trailing edge

z/c

x/c

Side viewEntire airfoil has

14 7


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CHAPTER 5: WINGS
http://www.airliners.net/open.file?id=790618&size=L&sok=JURER%20%20%28ZNGPU%20%28nvepensg%2Cnveyvar%2Ccynpr%2Ccubgb_qngr%2Cpbhagel%2Cerznex%2Ccubgbtencure%2Crznvy%2Clrne%2Cert%2Cnvepensg_trarevp%2Cpa%2Cpbqr%29%20NTNVAFG%20%28%27%2B%22777%22%27%20VA%20OBBYRNA%20ZBQR%29%29%20%20beqre%20ol%20cubgb_vq%20QRFP&photo_nr=341


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PRANDTLS 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)

Helmholtzs 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


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Trailing vorticies induce velocity along bound vortex with both contributions in

downward direction (w is in negative z-direction)

22

2

4

24

24

4

yb

b

yw

yb

yb

yw

hV

Contribution from left trailing vortex

(trailing fromb/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)


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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=constant

We need a way to let =(y)


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Example shown here will use 3 horseshoe vorticies

d1


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d1

d2


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d1

d2d3


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Represent wing by a large number of horseshoe vorticies, each with differentlength 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

d2d3


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Infinite number of horseshoe vorticies superimposed along lifting line

Now have a continuous distribution such that = (y), at origin = 0

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

Total strength integrated across sheet of wing is zero


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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

dydy

d

dw

04


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Total velocity w induced at y0 by entire trailing vortex sheet can be found by

integrating fromb/2 to b/2:

2

2 0

04

1b

b

dyyy

dy

d

yw

Equation gives value of

downwash at y0 due to

all trailing vorticies


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SUMMARY SO FAR

Weve 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 aeff

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


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FINITE WING DOWNWASH

Recall: Wing tip vortices induce a downward component of air velocity near wing

by dragging surrounding air with them

2

20

04

1 b

b

i dyyy

dy

d

Vy

a

ai

V

ywy

Vywy

i

i

0

0

010 tan

a

a

Equation for induced angle of attack

along finite wing in terms of(y)


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EFFECTIVE ANGLE OF ATTACK, aeff, EXPRESSION

0

0

0

00

0

0

00

2

00000

0

2

2

2

1

2

Leff

Leffl

l

l

LeffLeffl

effeff

ycV

y

yc

ycV

yc

yVcycVL

yyac

y

a

a

aa

rr

aaaa

aa aeffseen locally by airfoilRecall lift coefficient

expression (Ref, EQ: 4.60)

a0 = lift slope = 2

Definition of lift coefficient

and Kutta-Joukowski

Related both expressions

Solve foraeff


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COMBINE RESULTS FOR GOVERNING EQUATION

2

20

0

0

0

0

2

2

0

0

0

0

0

4

1

4

1

b

b

L

ieff

b

b

i

Leff

dyyy

dy

d

VycV

yy

dyyy

dy

d

Vy

ycV

y

a

a

aaa

a

a

a

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


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PRANDTLS LIFTING LINE EQUATION

Fundamental Equation of Prandtls Lifting Line Theory

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

attack plus induced angle of attack

Mathematically: a = aeff+ ai

Only unknown is (y)

V, c, a, aL=0 are known for a finite wing of given design at a given a

Solution gives (y0), whereb/2 y0 b/2 along span

2

20

0

0

00

4

1b

b

L dyyy

dy

d

VycV

yy

a

a


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WHAT DO WE GET OUT OF THIS EQUATION?

1. Lift distribution

2. Total Lift and Lift Coefficient

3. Induced Drag

dyyy

SVSq

DC

dyyyVdyyyLD

LD

dyySVSq

LC

dyyVL

dyyLL

yVyL

b

b

ii

iD

i

b

b

i

b

b

i

iii

b

b

L

b

b

b

b

2

2

,

2

2

2

2

2

2

2

2

2

2

00

2

2

a

ara

a

r

r


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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

ARC

LiD

Li

a

2

,

SPECIAL SOLUTION
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SPECIAL SOLUTION:


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

r

SPECIAL SOLUTION


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SPECIAL SOLUTION:


Elliptic distribution

Equation for downwash

Coordinate transformation q

See reference for integral

bVV

w

bw

db

w

db

dyb

y

dy

yy

b

y

y

byw

by

y

bdy

d

i

b

b

2

2

coscos

cos

2

sin2

;cos2

41

41

4

0

0

0

0 0

0

0

2

20

21

2

22

00

2

22

0

a

q

qqq

q

q

qqq

Downwash is constant over span for an elliptical lift distribution

Induced angle of attack is constant along span

Note: w and ai 0 as b

SPECIAL SOLUTION


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SPECIAL SOLUTION:


AR

CC

dyySV

C

AR

C

S

b

AR

b

SC

bVdy

b

yVL

LiD

b

b

iiD

Li

Li

b

b

a

a

a

rr

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 more

useful expression forai

Combine L definition for elliptic

profile with previous result forai

Define AR because it

occurs frequently

Useful expression forai

Calculate CD,i

SUMMARY TOTAL DRAG ON SUBSONIC WING


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SUMMARY: TOTAL DRAG ON SUBSONIC WING

eAR

CcSq

DcC

DDDDDDD

Lprofiled

iprofiledD

inducedprofile

inducedpressurefriction

2

,,

Also called drag due to lift

Profile Drag

Profile Drag coefficient

relatively constant withM at subsonic speeds

Look up

(Infinite Wing)

May be calculated from

Inviscid theory:

Lifting line theory


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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

WHAT IS NEXT?


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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

Introduction to Lifting Line theory

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