Chapter 5

Incompressible Flow Finite Wings

5.1 Introduction

The properties of airfoils are the same as the

properties of a wing with infinite span,However,

all real airplanes have wings of finite span,

In the present chapter,we will apply our knowledge

of airfoil properties to the analysis for finite wings,

As we have mentioned in the previous chapter,the

analysis for the aerodynamics of wings is separated

in two steps,Now,we are going on the second step

in Prandtl’s philosophy of wing theory,

Question, why are the aerodynamic characteristics

of a wing any different from the properties of its

section? Airfoil can be respected as two-dimensional

body,but any finite wing is a three-dimensional body,

※ Attention should be paid for the flow pattern near

the wing tips,And try to understand the reason for

such a flow phenomena,

※ For general cases,there is a spanwise velocity

component on both top and bottom surface of the

wing,but their direction are different,

※ A trailing vortex is created at each wing tip,These

wing-tip vortices downstream of the wing induce a

small downward velocity in the neighborhood of the

wing itself,

※ The two vortices tend to drag the surrounding air

with them,and this secondary movment induces a

small component is called downwash(下洗）,

※ The downwash velocity combines with the

freestream velocity to produce a local relative wind

which is canted downward in the vicinity of each airfoil

section of the wing,

※ definition of induced angle of attack

※ Two important effects due to the downwash,

1 the angle of attack actually seen(or feel) by the local

section is the angle between the chord line and the

local relative wind,This angle is defined by

effa

ie f f ??? ??

2 The local lift vector is in the direction perpendicular

to the local relative wind,As a subsequence,there is a

drag created by the presence of downwash,

Conclusion,the presence of downwash over a finite

wing reduces the angle of attack that each section

effectively sees,and moreover,it creates a component

of drag ---- the induced drag, And this drag is not

produced by viscous friction,iD

Explanation of induced drag in physical sense

a wing-tip vortices destroy the net pressure balance

b the wing-tip vortices contain large amount of

translational and rotational energy,and this energy

serves no useful purpose,In effect,the extra power

should be provided by the engine to overcome the the

induced drag,

※ Road map and purpose of this chapter

※ Support points for our analysis

1,Curved vortex filament

2,Biot-Savart Law

3,Helmholtz’s vortex theorems

5.2 The Vortex Filament,The Biot-Savart

Law,And Helmholtz’s Theorems

The vortex filament

Biot-Savart Law

34

r

rld

Vd ?

???

??

?

?

Application of Biot-Savart Law

1,To a straight vortex filament of infinite length

2,To a semi-infinite vortex filament

Helmholtz’s Theorems

For inviscid and incompressible flows

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 the boundary of the fluid or form a closed

path,

Concept of lift distribution along the span of a

finite wing

1., lift per unit span at location )(

1yL? 1y

2,With the different location in the span direction,the

chord and attack angle may be different,that means

will be different,?,c

3,Concept of geometric twist,washout and washin,

has a distribution along the span direction

4,Concept of aerodynamic twist,

has a distribution along the span direction

?

0?L?

5,As there is different distribution of the chord of the

airfoil,geometric angle of attack,and zero lift angle of

attack,the lift per unit span at location,will

be different from

2y )( 2yL?

)( 1yL?

6,As the lift per unit span is proportional to the

circulation,so,the circulation is also a function of y

7,The lift distribution goes to zero at the wing tips,

8,The calculation of the lift distribution [or the

circulation distribution ] is one of the central

problems of the finite-wing theory,

)( yL?

)(y?

5.3 Prandtl’s Classical Lifting-line Theory

Importance of the Prandtl’s lifting-line theory

bound vortex and free vortex

Replacement of the finite wing with a bound vortex

Single horseshoe vortex

1,The bound vortex induces no velocity along itself

2,The two trailing vortices both contribute to the

induced velocity along the bound vortex,and their

contributions are in the downward direction,

3,The origin is taken at the center of the bound

vortex,

4,The induced velocity at any location of y along the

bound vortex by the two trailing vortices is,

)2(4)2(4

)(

ybyb

yw

?

??

?

???

??

The above equation can be reduced

)2(4)2(4

)(

ybyb

yw

?

??

?

???

??

?

22)2(4)( yb

byw

?

???

?

It approaches to infinite as y approaches the tips

Superposition of large number of horseshoe vortices,

1,Problems for the single horseshoe vortex

2,Superposition of a large number of horseshoe

vortices,each with a different length of the bound

vortex,but all the bound vortices coincident along a

single line,called the lifting line,

3,Description for the lifting line and trailing vortices

system,

4,The strength of each trailing vortices is equal to the

change in circulation along the lifting line

Combination of three horseshoe vortices,

infinite number of horseshoe vortices,

Case for a infinite number of horseshoe vortices,

1,Each single horseshoe vortex has a vanishingly

small strength ?d

2,For this case,the distribution of is continuous

along the lifting line,The value of the circulation at

the origin is,

)(y?

0?

3,The collection of the trailing vortices becomes

continuous vortex sheet,

4,The vortex sheet is parallel to the free stream

direction,

Induced velocity at a given location along the lifting

line by an entire semi-infinite trailing vortex located

at y,

1,The strength of the trailing vortex at y is equal to

the change of in circulation along the lifting line,

? ?dydydd ???

2,Application of Biot-Savart law,induced velocity at

along the lifting line by the entire trailing vortex at

? ?

)(4 0 yy

dydyddw

?

???

?

?d

total change in circulation over the segment dy

0y

y

3,The total velocity w induced at by the entire

trailing vortex sheet,0y

? ??

? ?

?

??

2

2 0

0 )(4

1

)(

b

b yy

dydyd

yw

?

4,Pause for a while,and referring back to the text

book on page 328,

Induced angle of attack at the arbitrary spanwise

station

0y

??

?

?

??

?

? ?

?

?

?

V

yw

yi

)(

t a n)( 010?

For general cases,w is much smaller than,then

?V

?

??

V

ywy

i

)()( 0

0?

? ??

?? ?

?

?

2

2 0

0 )(4

1

)(

b

b

i yy

dydyd

V

y

?

?

? ??

? ?

??? 2

2 0

0 )(4

1)( b

b yy

dydydyw

?

? ??

?? ?

?

?

2

2 0

0 )(4

1

)(

b

b

i yy

dydyd

V

y

?

?

Induced angle of attack in terms of the circulation

distribution along the lifting line,

It is an important result in our process for the

derivation of the finite wing theory,

We have to review back to remember based on what

tools and approximations this result is achieved,

)(y?

Relation between the effective angle of attack and

the lift coefficient

? ? ? ?00000 )(2)( ?? ???? Le f fLe f fl yyac ?????

)()(

2

1

00

2 yVcycVL

l ???? ???? ??

?

)(

)(2

0

0

ycV

yc

l

?

??

Lift slope of arbitrary airfoil shape,

0a

? ? ? ?00000 )(2)( ?? ???? Le f fLe f fl yyac ?????

)(

)(2

0

0

ycV

yc

l

?

??

0

0

0

)(

)(

?

?

??? Le f f

ycV

y ?

?

?

0??? Le f f ???

? ??

??

?

? ?

????? 2

2 0

00

0

0

0 )(4

1)(

)(

)()( b

b

L yy

dydyd

V

y

ycV

yy

?

?

?

?

Fundamental equation of Prandtl’s lifting

line theory

It simply states that the geometric angle of attack is

equal to the sum of the effective angle plus the

induced angle of attack,

It is an integro-differential equation,

Why it is called as lifting line theory? What are the

known and what is the unknown inside this equation

for general cases?

After the solution is obtained from the

fundamental equation of the lifting line theory,three

aerodynamic characteristics can be achieved

)( y???

1,Lift distribution

)()( 00 yVyL ??? ???

2,Total lift

dyyVdyyLL

b

b

b

b ?? ?

??

?

????

2

2

2

2

)()( ?

Lift coefficient

dyy

SVSV

dyyV

Sq

L

C

b

b

b

b

L ?

?

????

?

??

?

??

?

??

2

2

2

2

2

)(

2

5.0

)(

?

?

3,Induced drag and induced drag coefficient

ii LD ?s i n???

?

ii LD ????

1??i?

Section induced drag

Total induced drag

dyyyVdyyyLD i

b

b

b

b

ii )()()()(

2

2

2

2

??? ??

?

??

?

????

Induced drag coefficient

dyyy

SVSq

D

C i

b

b

i

iD )()(

2 2

2

,??

???

???

it is obviously to see,in Prandtl’s lifting line theory,

the solution of is a key to obtain the

aerodynamic characteristics,

)( y???

FAQs for the lifting-line theory

1,Location for the lifting-line relative to the actual

wing geometry

2,Shape and orientation of the trailing vortex sheet

3,Is there any conflict between the boundary

condition respect to the airfoil surface and the induced

velocity on the lifting-line?

5.3.1 Elliptical Lift distribution

Consider a circulation distribution given by

2

0

2

1)( ?

?

?

?

?

?

????

b

y

y

That can be expressed in a more familiar form

2

0

2

2 2)( ???

?

?

?

?

???

b

y

y

It is an elliptical equation

Some features related to the above circulation

distribution

1,is the circulation at the origin,or to say,at the

root of the finite wing 0?

2,The circulation along the lifting line varies elliptically

with the distance along the span,according to Kutta-

Joukovski theorem,we have

2

0

2

1)()( ?

?

?

?

?

?

?????? ????

b

y

VyVyL ??

Hence,we are dealing with an elliptical lift distribution

3,As,the lift goes to zero at

the wing tips,0)2()2( ????? bb

The lift distribution given above is not obtained from

the lifting-line theory of finite wing,it is stipulated as

an elliptic distribution,

4,Aerodynamic characteristics of finite wing with such

an elliptic distribution

※ distribution of induced velocity

? ??

? ?

?

??

2

2 0

0 )(4

1

)(

b

b yy

dydyd

yw

?

? ? 21222

0

41

4

by

y

bdy

d

?

?

??

?

? ??? ??

?

?

2

2

0

21222

0

0

)(41

)(

b

b

dy

yyby

y

b

yw

?

Introducing a integral variable transformation

?? c o s

2

s i n

2

bdyby ???

? ????

0

0

0

0 )c o s( c o s

c o s

2

)(

?

?

??

?

?

? d

b

w

or

? ????

?

?

??

?

?

?

0 0

0

0 )c o s( c o s

c o s

2

)( d

b

w

0

0

0 0 s i n

s i n

c o sc o s

c o s

?

??

??

??? ndn ?

??

?

b

w

2

)( 00 ????

The downwash is constant over the span for an

elliptical lift distribution

※ induced angle of attack

??

????

bVV

w

i 2

0?

Note,As the wing span becomes infinite,both

downwash and induced angle of attack go to zero,

※ total lift on the wing due to the elliptical lift distr,

???? ???

?

??

?

?

???

2

2

21

2

2

0

4

1

b

b

dy

b

y

VL ?

? ?? s i n

2c o s2

bdyby ???

?????

?

4

s in

2 00

2

0

bVdbVL ????

???? ?

?? bV

L

??

?? 40

LSCVL

2

2

1

??? ?

as

?? bV

L

??

?? 40 ?

?b

SCV L??? 2

0

??

????

bVV

w

i 2

0?

?

?

??

bVb

SCV L

i 2

14

?

?

or

2b

SC L

i ?? ?

※ another expression of the induced angle of attack,

S

bAR 2? Aspect ratio

AR

C L

i ?? ?

The induced angle of attack is inversely proportional

to the aspect ratio for elliptical lift distribution

※ the induced drag coefficient

dyyy

SVSq

D

C i

b

b

i

iD )()(

2 2

2

,??

???

???

?

The induced angle of attack is constant

SV

bdb

SV

dyy

SV

C ii

b

b

i

iD

????

?????? ??

2

s i n

2

2)(2 0

0

20

2

2

,

?????? ?

??

?

b

SCV

AR

C

SV

bC LL

iD

?

?

?

?

??

?

?? 2

2,

AR

C L

i ?? ?

?b

SCV L??? 2

0

?

??

?

b

SCV

AR

C

SV

bC LL

iD

?

?

?

?

??

?

?? 4

2,

?

AR

CC L

iD ?

2

,?

※ two important explanations for the induced drag,

referring back to our textbook,

※ elliptical planform wing

1,If the finite wing has neither geometric twist nor

aerodynamic twist,as

.c o n s t??,.

0 co n s tL ???

and

.c o n s ti ??

then

.c o n s tie f f ??? ???

as the lift coefficient is given by

)( 00 ??? Le f fl ac ??

For thin airfoil,?2

0 ?a

so,must be constant along the span

lc

2,The lift per unit span is

lccqyL ??? )(

express the distribution of chord with lift distribution

lcq

yLyc

?

?

? )()(

It is clear to see that with constant dynamic pressure

and lift coefficient per unit span,and elliptical lift

distribution,the chord must vary elliptically along the

span,i.e.,the wing planform is elliptical

5.3.2 General Lift distribution

※ with the transformation

?c o s2by ??

2

0

2

1)( ?

?

?

?

?

?

????

b

y

y

?

?? s i n)( 0???

Elliptical distribution

※ general distribution

????

N

n nAbV

1

s i n2)( ??

The coefficient are unknowns,and they should

satisfy the fundamental equation of Prandtl’s lifting-

line theory,

nA

? ??

??

?

? ?

????? 2

2 0

00

0

0

0 )(4

1)(

)(

)()( b

b

L yy

dydyd

V

y

ycV

yy

?

?

?

?

The differential of the of the circulation distribution is

dy

d

nnAbV

dy

d

d

d

dy

d

N

n

?

?

?

? ??

?

?

?

?

1

c o s2

? ??

??

?

? ?

????? 2

2 0

00

0

0

0 )(4

1)(

)(

)()( b

b

L yy

dydyd

V

y

ycV

yy

?

?

?

?

?

?

??

?

?

???

??

??

?

d

nnA

nA

c

b

N

n

L

N

n ?

?

? ???? ?

0 0

1

00

10

0

)c o s( c o s

c o s

1

)(s i n

)(

2

)(

?

?? ??? ?

N

nL

N

n

nnAnA

c

b

1 0

0

00

10

0 s i n

s i n)(s i n

)(

2)(

?

????

??

??

?? ??? ?

N

nL

N

n

nnAnA

c

by

1 0

0

00

10

0 s i n

s i n)(s i n

)(

2)(

?

????

??

?

※ description of the equation above,what are the

known quantities and what is the unknown? Strategy

for solve the circulation distribution,

After the circulation distribution being solved

1,Lift coefficient

??? ??? ?

?

?

???

0

1

22

2

s i ns i n

2

)(

2

dnA

S

b

dyy

SV

C

N

n

b

b

L

ARA

S

bAC

L ?? 1

2

1 ??

The lift coefficient is only depends on the leading

coefficient of the Fourier series expansion,But,

should be solved together with the all coefficient,1

A

nA

2,Induced drag coefficient

? ?

?

?

?

?

?

?

?

?

?

?

??

?

?

?

?????

?

0

1

2

2

2

,

s in)(s in

2

)()(

2

dnA

S

b

dyyy

SV

C

i

N

n

i

b

b

iD

? ??

?? ?

?

?

2

2 0

0 )(4

1

)(

b

b

i yy

dydyd

V

y

?

?

?

??

N

ni

n

nA

1 0

0

0 s i n

s i n

)(

?

?

??

?

??

N

ni

n

nA

1

s i n

s i n

)(

?

?

??

can be replaced by

0? ?

? ?

?

?

?

?

?

?

?

?

?

?

??

?

?

?

?????

?

0

1

2

2

2

,

s in)(s in

2

)()(

2

dnA

S

b

dyyy

SV

C

i

N

n

i

b

b

iD

??

N

ni

nnA

1 s i n

s i n)(

?

???

? ?? ??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

???

0

11

2

,s ins in

2

dnnAnA

S

b

C

N

n

N

niD

?

? ?? ??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

???

0

11

2

,s ins in

2

dnnAnA

S

b

C

N

n

N

niD

?

?

?

?

???

km

kmdkm

2

0s i ns i n

0 ?

???

?

?

?? ???

?

?

?

?

?

?

?

N

n

N

niD nAARnAS

b

C

1

2

1

2

2

,2

2

?

?

?

?

?

?

?

?

?

?

????

?

?

?

?

?

?

?? ??

N

n

N

niD A

nA

ARAnAAARC

2

2

1

2

2

1

2

22

1,1??

ARA

S

bAC

L ?? 1

2

1 ??

? ?2

2

2

1 AR

C

A L

?

?

?

?

?

?

?

?

?

?

????

?

?

?

?

?

?

?? ??

N

n

N

niD A

nA

ARAnAAARC

2

2

1

2

2

1

2

22

1,1??

?

?

)1(1

2

2

2

1

22

,??? ???

?

?

?

?

?

?

?

?? ?

AR

C

A

nA

AR

C

C L

N

nL

iD

)1(1

2

2

2

1

22

,??? ???

?

?

?

?

?

?

?

?? ?

AR

C

A

nA

AR

C

C L

N

nL

iD

Where

0

2

2

1

2 ??? N

n AnA?

e A R

CC L

iD ?

2

,?

Where )1( ???e

The lift distribution which yields minimum induced

drag is the elliptical lift distribution

5.3.3 General Lift distribution

Effects of aspect ratio and on the induced drag ?

iD

The induced drag is inversely proportional to AR,but

how to verify it with experimental results?

Prandtl’s verification

※ total drag of a finite wing

e A R

CcC L

dD ?

2

??

※ drag coefficients of two wings with different aspect

ratio,assuming that the wings are at the same

1

2

1,e A R

CcC L

dD ???

LC

2

2

2,e A R

CcC L

dD ???

if the shape of the airfoil is the same for these two

wings,then the profile drag coefficients will be the

same,Moreover,the variation of e between thee wings

is only a few percent and can be ignored,

??

?

?

??

?

?

???

21

2

2,1,

11

ARARe

C

CC LDD

?

??

?

?

??

?

?

???

2

2

2,1,

1

5

1

ARe

C

CC LDD

?

0)( ad

dC

i

L ?

? ??

.)(0 co n s taC iL ??? ??

AR

C L

i ?? ?

.)(0 c o n s t

AR

CaC L

L ??? ??

ARa

aa

d

dC L

?? 0

0

1 ?

??

? ? )1(1 0

0

?? ??

?

ARa

aa

Incompressible Flow Finite Wings

5.1 Introduction

The properties of airfoils are the same as the

properties of a wing with infinite span,However,

all real airplanes have wings of finite span,

In the present chapter,we will apply our knowledge

of airfoil properties to the analysis for finite wings,

As we have mentioned in the previous chapter,the

analysis for the aerodynamics of wings is separated

in two steps,Now,we are going on the second step

in Prandtl’s philosophy of wing theory,

Question, why are the aerodynamic characteristics

of a wing any different from the properties of its

section? Airfoil can be respected as two-dimensional

body,but any finite wing is a three-dimensional body,

※ Attention should be paid for the flow pattern near

the wing tips,And try to understand the reason for

such a flow phenomena,

※ For general cases,there is a spanwise velocity

component on both top and bottom surface of the

wing,but their direction are different,

※ A trailing vortex is created at each wing tip,These

wing-tip vortices downstream of the wing induce a

small downward velocity in the neighborhood of the

wing itself,

※ The two vortices tend to drag the surrounding air

with them,and this secondary movment induces a

small component is called downwash(下洗）,

※ The downwash velocity combines with the

freestream velocity to produce a local relative wind

which is canted downward in the vicinity of each airfoil

section of the wing,

※ definition of induced angle of attack

※ Two important effects due to the downwash,

1 the angle of attack actually seen(or feel) by the local

section is the angle between the chord line and the

local relative wind,This angle is defined by

effa

ie f f ??? ??

2 The local lift vector is in the direction perpendicular

to the local relative wind,As a subsequence,there is a

drag created by the presence of downwash,

Conclusion,the presence of downwash over a finite

wing reduces the angle of attack that each section

effectively sees,and moreover,it creates a component

of drag ---- the induced drag, And this drag is not

produced by viscous friction,iD

Explanation of induced drag in physical sense

a wing-tip vortices destroy the net pressure balance

b the wing-tip vortices contain large amount of

translational and rotational energy,and this energy

serves no useful purpose,In effect,the extra power

should be provided by the engine to overcome the the

induced drag,

※ Road map and purpose of this chapter

※ Support points for our analysis

1,Curved vortex filament

2,Biot-Savart Law

3,Helmholtz’s vortex theorems

5.2 The Vortex Filament,The Biot-Savart

Law,And Helmholtz’s Theorems

The vortex filament

Biot-Savart Law

34

r

rld

Vd ?

???

??

?

?

Application of Biot-Savart Law

1,To a straight vortex filament of infinite length

2,To a semi-infinite vortex filament

Helmholtz’s Theorems

For inviscid and incompressible flows

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 the boundary of the fluid or form a closed

path,

Concept of lift distribution along the span of a

finite wing

1., lift per unit span at location )(

1yL? 1y

2,With the different location in the span direction,the

chord and attack angle may be different,that means

will be different,?,c

3,Concept of geometric twist,washout and washin,

has a distribution along the span direction

4,Concept of aerodynamic twist,

has a distribution along the span direction

?

0?L?

5,As there is different distribution of the chord of the

airfoil,geometric angle of attack,and zero lift angle of

attack,the lift per unit span at location,will

be different from

2y )( 2yL?

)( 1yL?

6,As the lift per unit span is proportional to the

circulation,so,the circulation is also a function of y

7,The lift distribution goes to zero at the wing tips,

8,The calculation of the lift distribution [or the

circulation distribution ] is one of the central

problems of the finite-wing theory,

)( yL?

)(y?

5.3 Prandtl’s Classical Lifting-line Theory

Importance of the Prandtl’s lifting-line theory

bound vortex and free vortex

Replacement of the finite wing with a bound vortex

Single horseshoe vortex

1,The bound vortex induces no velocity along itself

2,The two trailing vortices both contribute to the

induced velocity along the bound vortex,and their

contributions are in the downward direction,

3,The origin is taken at the center of the bound

vortex,

4,The induced velocity at any location of y along the

bound vortex by the two trailing vortices is,

)2(4)2(4

)(

ybyb

yw

?

??

?

???

??

The above equation can be reduced

)2(4)2(4

)(

ybyb

yw

?

??

?

???

??

?

22)2(4)( yb

byw

?

???

?

It approaches to infinite as y approaches the tips

Superposition of large number of horseshoe vortices,

1,Problems for the single horseshoe vortex

2,Superposition of a large number of horseshoe

vortices,each with a different length of the bound

vortex,but all the bound vortices coincident along a

single line,called the lifting line,

3,Description for the lifting line and trailing vortices

system,

4,The strength of each trailing vortices is equal to the

change in circulation along the lifting line

Combination of three horseshoe vortices,

infinite number of horseshoe vortices,

Case for a infinite number of horseshoe vortices,

1,Each single horseshoe vortex has a vanishingly

small strength ?d

2,For this case,the distribution of is continuous

along the lifting line,The value of the circulation at

the origin is,

)(y?

0?

3,The collection of the trailing vortices becomes

continuous vortex sheet,

4,The vortex sheet is parallel to the free stream

direction,

Induced velocity at a given location along the lifting

line by an entire semi-infinite trailing vortex located

at y,

1,The strength of the trailing vortex at y is equal to

the change of in circulation along the lifting line,

? ?dydydd ???

2,Application of Biot-Savart law,induced velocity at

along the lifting line by the entire trailing vortex at

? ?

)(4 0 yy

dydyddw

?

???

?

?d

total change in circulation over the segment dy

0y

y

3,The total velocity w induced at by the entire

trailing vortex sheet,0y

? ??

? ?

?

??

2

2 0

0 )(4

1

)(

b

b yy

dydyd

yw

?

4,Pause for a while,and referring back to the text

book on page 328,

Induced angle of attack at the arbitrary spanwise

station

0y

??

?

?

??

?

? ?

?

?

?

V

yw

yi

)(

t a n)( 010?

For general cases,w is much smaller than,then

?V

?

??

V

ywy

i

)()( 0

0?

? ??

?? ?

?

?

2

2 0

0 )(4

1

)(

b

b

i yy

dydyd

V

y

?

?

? ??

? ?

??? 2

2 0

0 )(4

1)( b

b yy

dydydyw

?

? ??

?? ?

?

?

2

2 0

0 )(4

1

)(

b

b

i yy

dydyd

V

y

?

?

Induced angle of attack in terms of the circulation

distribution along the lifting line,

It is an important result in our process for the

derivation of the finite wing theory,

We have to review back to remember based on what

tools and approximations this result is achieved,

)(y?

Relation between the effective angle of attack and

the lift coefficient

? ? ? ?00000 )(2)( ?? ???? Le f fLe f fl yyac ?????

)()(

2

1

00

2 yVcycVL

l ???? ???? ??

?

)(

)(2

0

0

ycV

yc

l

?

??

Lift slope of arbitrary airfoil shape,

0a

? ? ? ?00000 )(2)( ?? ???? Le f fLe f fl yyac ?????

)(

)(2

0

0

ycV

yc

l

?

??

0

0

0

)(

)(

?

?

??? Le f f

ycV

y ?

?

?

0??? Le f f ???

? ??

??

?

? ?

????? 2

2 0

00

0

0

0 )(4

1)(

)(

)()( b

b

L yy

dydyd

V

y

ycV

yy

?

?

?

?

Fundamental equation of Prandtl’s lifting

line theory

It simply states that the geometric angle of attack is

equal to the sum of the effective angle plus the

induced angle of attack,

It is an integro-differential equation,

Why it is called as lifting line theory? What are the

known and what is the unknown inside this equation

for general cases?

After the solution is obtained from the

fundamental equation of the lifting line theory,three

aerodynamic characteristics can be achieved

)( y???

1,Lift distribution

)()( 00 yVyL ??? ???

2,Total lift

dyyVdyyLL

b

b

b

b ?? ?

??

?

????

2

2

2

2

)()( ?

Lift coefficient

dyy

SVSV

dyyV

Sq

L

C

b

b

b

b

L ?

?

????

?

??

?

??

?

??

2

2

2

2

2

)(

2

5.0

)(

?

?

3,Induced drag and induced drag coefficient

ii LD ?s i n???

?

ii LD ????

1??i?

Section induced drag

Total induced drag

dyyyVdyyyLD i

b

b

b

b

ii )()()()(

2

2

2

2

??? ??

?

??

?

????

Induced drag coefficient

dyyy

SVSq

D

C i

b

b

i

iD )()(

2 2

2

,??

???

???

it is obviously to see,in Prandtl’s lifting line theory,

the solution of is a key to obtain the

aerodynamic characteristics,

)( y???

FAQs for the lifting-line theory

1,Location for the lifting-line relative to the actual

wing geometry

2,Shape and orientation of the trailing vortex sheet

3,Is there any conflict between the boundary

condition respect to the airfoil surface and the induced

velocity on the lifting-line?

5.3.1 Elliptical Lift distribution

Consider a circulation distribution given by

2

0

2

1)( ?

?

?

?

?

?

????

b

y

y

That can be expressed in a more familiar form

2

0

2

2 2)( ???

?

?

?

?

???

b

y

y

It is an elliptical equation

Some features related to the above circulation

distribution

1,is the circulation at the origin,or to say,at the

root of the finite wing 0?

2,The circulation along the lifting line varies elliptically

with the distance along the span,according to Kutta-

Joukovski theorem,we have

2

0

2

1)()( ?

?

?

?

?

?

?????? ????

b

y

VyVyL ??

Hence,we are dealing with an elliptical lift distribution

3,As,the lift goes to zero at

the wing tips,0)2()2( ????? bb

The lift distribution given above is not obtained from

the lifting-line theory of finite wing,it is stipulated as

an elliptic distribution,

4,Aerodynamic characteristics of finite wing with such

an elliptic distribution

※ distribution of induced velocity

? ??

? ?

?

??

2

2 0

0 )(4

1

)(

b

b yy

dydyd

yw

?

? ? 21222

0

41

4

by

y

bdy

d

?

?

??

?

? ??? ??

?

?

2

2

0

21222

0

0

)(41

)(

b

b

dy

yyby

y

b

yw

?

Introducing a integral variable transformation

?? c o s

2

s i n

2

bdyby ???

? ????

0

0

0

0 )c o s( c o s

c o s

2

)(

?

?

??

?

?

? d

b

w

or

? ????

?

?

??

?

?

?

0 0

0

0 )c o s( c o s

c o s

2

)( d

b

w

0

0

0 0 s i n

s i n

c o sc o s

c o s

?

??

??

??? ndn ?

??

?

b

w

2

)( 00 ????

The downwash is constant over the span for an

elliptical lift distribution

※ induced angle of attack

??

????

bVV

w

i 2

0?

Note,As the wing span becomes infinite,both

downwash and induced angle of attack go to zero,

※ total lift on the wing due to the elliptical lift distr,

???? ???

?

??

?

?

???

2

2

21

2

2

0

4

1

b

b

dy

b

y

VL ?

? ?? s i n

2c o s2

bdyby ???

?????

?

4

s in

2 00

2

0

bVdbVL ????

???? ?

?? bV

L

??

?? 40

LSCVL

2

2

1

??? ?

as

?? bV

L

??

?? 40 ?

?b

SCV L??? 2

0

??

????

bVV

w

i 2

0?

?

?

??

bVb

SCV L

i 2

14

?

?

or

2b

SC L

i ?? ?

※ another expression of the induced angle of attack,

S

bAR 2? Aspect ratio

AR

C L

i ?? ?

The induced angle of attack is inversely proportional

to the aspect ratio for elliptical lift distribution

※ the induced drag coefficient

dyyy

SVSq

D

C i

b

b

i

iD )()(

2 2

2

,??

???

???

?

The induced angle of attack is constant

SV

bdb

SV

dyy

SV

C ii

b

b

i

iD

????

?????? ??

2

s i n

2

2)(2 0

0

20

2

2

,

?????? ?

??

?

b

SCV

AR

C

SV

bC LL

iD

?

?

?

?

??

?

?? 2

2,

AR

C L

i ?? ?

?b

SCV L??? 2

0

?

??

?

b

SCV

AR

C

SV

bC LL

iD

?

?

?

?

??

?

?? 4

2,

?

AR

CC L

iD ?

2

,?

※ two important explanations for the induced drag,

referring back to our textbook,

※ elliptical planform wing

1,If the finite wing has neither geometric twist nor

aerodynamic twist,as

.c o n s t??,.

0 co n s tL ???

and

.c o n s ti ??

then

.c o n s tie f f ??? ???

as the lift coefficient is given by

)( 00 ??? Le f fl ac ??

For thin airfoil,?2

0 ?a

so,must be constant along the span

lc

2,The lift per unit span is

lccqyL ??? )(

express the distribution of chord with lift distribution

lcq

yLyc

?

?

? )()(

It is clear to see that with constant dynamic pressure

and lift coefficient per unit span,and elliptical lift

distribution,the chord must vary elliptically along the

span,i.e.,the wing planform is elliptical

5.3.2 General Lift distribution

※ with the transformation

?c o s2by ??

2

0

2

1)( ?

?

?

?

?

?

????

b

y

y

?

?? s i n)( 0???

Elliptical distribution

※ general distribution

????

N

n nAbV

1

s i n2)( ??

The coefficient are unknowns,and they should

satisfy the fundamental equation of Prandtl’s lifting-

line theory,

nA

? ??

??

?

? ?

????? 2

2 0

00

0

0

0 )(4

1)(

)(

)()( b

b

L yy

dydyd

V

y

ycV

yy

?

?

?

?

The differential of the of the circulation distribution is

dy

d

nnAbV

dy

d

d

d

dy

d

N

n

?

?

?

? ??

?

?

?

?

1

c o s2

? ??

??

?

? ?

????? 2

2 0

00

0

0

0 )(4

1)(

)(

)()( b

b

L yy

dydyd

V

y

ycV

yy

?

?

?

?

?

?

??

?

?

???

??

??

?

d

nnA

nA

c

b

N

n

L

N

n ?

?

? ???? ?

0 0

1

00

10

0

)c o s( c o s

c o s

1

)(s i n

)(

2

)(

?

?? ??? ?

N

nL

N

n

nnAnA

c

b

1 0

0

00

10

0 s i n

s i n)(s i n

)(

2)(

?

????

??

??

?? ??? ?

N

nL

N

n

nnAnA

c

by

1 0

0

00

10

0 s i n

s i n)(s i n

)(

2)(

?

????

??

?

※ description of the equation above,what are the

known quantities and what is the unknown? Strategy

for solve the circulation distribution,

After the circulation distribution being solved

1,Lift coefficient

??? ??? ?

?

?

???

0

1

22

2

s i ns i n

2

)(

2

dnA

S

b

dyy

SV

C

N

n

b

b

L

ARA

S

bAC

L ?? 1

2

1 ??

The lift coefficient is only depends on the leading

coefficient of the Fourier series expansion,But,

should be solved together with the all coefficient,1

A

nA

2,Induced drag coefficient

? ?

?

?

?

?

?

?

?

?

?

?

??

?

?

?

?????

?

0

1

2

2

2

,

s in)(s in

2

)()(

2

dnA

S

b

dyyy

SV

C

i

N

n

i

b

b

iD

? ??

?? ?

?

?

2

2 0

0 )(4

1

)(

b

b

i yy

dydyd

V

y

?

?

?

??

N

ni

n

nA

1 0

0

0 s i n

s i n

)(

?

?

??

?

??

N

ni

n

nA

1

s i n

s i n

)(

?

?

??

can be replaced by

0? ?

? ?

?

?

?

?

?

?

?

?

?

?

??

?

?

?

?????

?

0

1

2

2

2

,

s in)(s in

2

)()(

2

dnA

S

b

dyyy

SV

C

i

N

n

i

b

b

iD

??

N

ni

nnA

1 s i n

s i n)(

?

???

? ?? ??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

???

0

11

2

,s ins in

2

dnnAnA

S

b

C

N

n

N

niD

?

? ?? ??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

???

0

11

2

,s ins in

2

dnnAnA

S

b

C

N

n

N

niD

?

?

?

?

???

km

kmdkm

2

0s i ns i n

0 ?

???

?

?

?? ???

?

?

?

?

?

?

?

N

n

N

niD nAARnAS

b

C

1

2

1

2

2

,2

2

?

?

?

?

?

?

?

?

?

?

????

?

?

?

?

?

?

?? ??

N

n

N

niD A

nA

ARAnAAARC

2

2

1

2

2

1

2

22

1,1??

ARA

S

bAC

L ?? 1

2

1 ??

? ?2

2

2

1 AR

C

A L

?

?

?

?

?

?

?

?

?

?

????

?

?

?

?

?

?

?? ??

N

n

N

niD A

nA

ARAnAAARC

2

2

1

2

2

1

2

22

1,1??

?

?

)1(1

2

2

2

1

22

,??? ???

?

?

?

?

?

?

?

?? ?

AR

C

A

nA

AR

C

C L

N

nL

iD

)1(1

2

2

2

1

22

,??? ???

?

?

?

?

?

?

?

?? ?

AR

C

A

nA

AR

C

C L

N

nL

iD

Where

0

2

2

1

2 ??? N

n AnA?

e A R

CC L

iD ?

2

,?

Where )1( ???e

The lift distribution which yields minimum induced

drag is the elliptical lift distribution

5.3.3 General Lift distribution

Effects of aspect ratio and on the induced drag ?

iD

The induced drag is inversely proportional to AR,but

how to verify it with experimental results?

Prandtl’s verification

※ total drag of a finite wing

e A R

CcC L

dD ?

2

??

※ drag coefficients of two wings with different aspect

ratio,assuming that the wings are at the same

1

2

1,e A R

CcC L

dD ???

LC

2

2

2,e A R

CcC L

dD ???

if the shape of the airfoil is the same for these two

wings,then the profile drag coefficients will be the

same,Moreover,the variation of e between thee wings

is only a few percent and can be ignored,

??

?

?

??

?

?

???

21

2

2,1,

11

ARARe

C

CC LDD

?

??

?

?

??

?

?

???

2

2

2,1,

1

5

1

ARe

C

CC LDD

?

0)( ad

dC

i

L ?

? ??

.)(0 co n s taC iL ??? ??

AR

C L

i ?? ?

.)(0 c o n s t

AR

CaC L

L ??? ??

ARa

aa

d

dC L

?? 0

0

1 ?

??

? ? )1(1 0

0

?? ??

?

ARa

aa