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
