AERODYNAMICS I
LECTURE 6
AERODYNAMICS OF A WING
FUNDAMENTALS OF THE LIFTING-LINE
THEORY
AERODYNAMICS I
The Biot-Savart Law
The velocity induced by the singular vortex line with the
circulation  can be determined by means of the BiotSavart formula
dl  ( x  ξ )
υ( x)   
4 VL x  ξ 3
Special case – induction of the
straight vortex line:
dl  d e x , x  ξ  ( x   )e x  ye y
dl  ( x  ξ )  y d  e x  e x  y d  ez
2 3/2
x  ξ  ( x   )  y 
3
2
AERODYNAMICS I
From the Biot-Savart formula one gets

υ( x, y,0)  
 4
2

1

d  ez
2
2 3
[( x   )  y ]

y
where
2

1
s  (  x ) y 1
d 

2
2 3
ds  d  / y
y
[( x   )  y ]
y
2  x
2  x
y
y
1
1
s
1 x (1  s 2 )3 2 ds  y 1  s 2
y

1  x
y

1
x  1
x  2
 


2
2
2
2
y  ( x  1 )  y
( x  2 )  y 

Case 1 – induction of the infinite vortex line (equivalent to the 2D point vortex!)



x


x





υ( x, y,0) 
lim

ez 
ez
2
2
2
2


4 y
2 y
 (x   )  y
( x   )  y 

AERODYNAMICS I
Case 2 – induction of the semi-infinite vortex line segment   [0, )




x
x 
 
x
υ( x, y,0) 
lim 

 1 e z
 ez 
 2
2
2
2
2
2


4 y
4 y  x  y
( x   )  y 
 x  y

If x  0 then
υ(0, y,0) 

ez
4 y
AERODYNAMICS I
Flow past a finite-span wing – physical properties
AERODYNAMICS I
Lifting-line model of a finite-span wing
Flow past a wing is modeled by the superposition of the uniform free stream and the
velocity induced by a plane vortex sheet “pretending” to be the cortex wave behind the
wing.
The vortex sheet behind the wing is “woven” from continuum of infinitesimally weak
horseshoe vortices. These vortices are “attached” to the lifting line leading to a continuous
distribution of circulation along the wing span.
AERODYNAMICS I
The vortex sheet induces vorticity all around. The idea is to calculate the calculate the
velocity induced by this sheet on its front edge, i.e., along the lifting line. Next, it is
assumed that each infinitely thin slice of the wing generates the (differential)
contribution to the total aerodynamic force as it were a two-dimensional airfoil.
Each slice “senses” its individual direction of “free stream”, which results from the real
free stream vector V and the vertical (normal to the vortex sheet) velocity induces at
the lifting line in the point corresponding to the position of the wing slice.
According to the Biot-Savart formula, the infinitesimal contribution to the velocity induces
along the lifting line at the point (0, y0 ,0) is
 ( y)dy
dw  
4 ( y0  y)
The total velocity induces at this point is obtained by integration
1
w( y0 )  
4
b /2

b /2
 ( y)dy
y0  y
AERODYNAMICS I
Due to (generally) non-uniform distribution of the induced velocity along the wing
span, the effective angle of attack has an individual value of each wing section – see
figure below.
The direction of flow “sensed” by the
wing section at y  y0 is rotated
clockwise by the induces angle
i ( y0 )  atan[w( y0 ) V ]
For small angles …
b /2
w( y0 )
1
 ( y)dy
i ( y0 )  

V
4V b/2 y0  y
Clearly, an effective angle  eff   eff ( y0 ) .
AERODYNAMICS I
For small angles one can assume that the local lift coefficient changes linearly with the
(local) angle. Hence
cL ( y0 )  a[eff ( y0 )  0 ( y0 )]
Here, a denotes the slope of the lift characteristics for the wing section,  0 is the
angle of attack corresponding to the zero lift. Note that a  2 if the thin-airfoil
theory is used. Note also that – in general – the angle  0   0 ( y0 ) .
Next, we assume that the spanwise density of the lift force developed on the wing can
be computed from the Kutta-Joukovski formula, namely
L( y0 )  12 V2cL ( y0 )c( y0 )  V ( y0 )
where
c( y0 ) is local chord of the wing section
Hence, the local lift coefficient is
2 ( y0 )
cL ( y0 ) 
Vc( y0 )
AERODYNAMICS I
Assuming that a  2 , the local effective angle of attack is
 eff ( y0 ) 
 ( y0 )
Vc( y0 )
  0 ( y0 )
Finally, the sum of the two local angles:  eff ( y0 ) and i ( y0 ) is equal to the geometric
angle of attack  . If the wing has geometric twist, this angle also depends of the
spanwise location, i.e.,    ( y0 ) .
Hence, we have obtained the following integro-differential equation for the
spanwise distribution of the circulation
 ( y0 )
 ( y)dy   ( y )   ( y )
1
0
0 0
Vc( y0 ) 4V b/2 y0  y

b/2
AERODYNAMICS I
One this equation is solved, then the spanwise distribution of the circulation is known.
The lift force developed on the wing can be calculated as follows
L
b /2
b/2
L( y)dy  V   ( y)dy

b /2
b/2
b /2
The (global) lift coefficient is
L
2
CL 

 ( y)dy

q S V S b/2
The local contribution to the drag force is
Di  L sin i  Li
The induced drag force is equal
Di 
b /2
b/2
L( y)i ( y)dy  V   ( y)i ( y)dy

b /2
b/2
AERODYNAMICS I
Thus, the coefficient of the induced drag is equal
b /2
Di
2
CDi 

 ( y)i ( y)dy

q S V S b/2
Important case - elliptical distribution of the circulation
 ( y)   0 1 
Assume
meaning that
We have
Hence
 
2y
b
L( y)  V 0 1 
2
 
2y
b
2
4 0
y
 ( y)   2
b
1  4 y 2 / b2
1
w( y0 )  
4
b /2

b /2
 0 b/2
y dy
 2 
y0  y
 b b/2 ( y0  y) 1  4 y 2 / b2
 ( y)dy
AERODYNAMICS I
Let us apply the following change of coordinates
y  12 b cos , dy   12 b sin d
Thus
0 
0
cos
w(0 )  
d  

2 b 0 cos  cos0
2b
Conclusion: for the elliptical distribution of the circulation, the downwash
velocity is constant!
0
wi
i   
V 2bV
The induced angle is
The lift force
L  V 0
b/2

b/2

1  4 y 2 b2 dy  V 0 b2  sin 2  d  14 V 0b
Thus, the maximal circulation is
0
0 
4L
Vb
AERODYNAMICS I
L  12 V2 SCL
On the other hand
2V SCL
0 
b
Hence
and the induced angle is
0
2V SCL 1
SCL
i 

 2
2bV
 b 2bV  b
2
b

S
We define the aspect ratio of the wing
Then, the alternative form of the formula for the induced angle for the elliptical
distribution of vorticity is
i 
CL

AERODYNAMICS I
The coefficient of the induced drag is calculated as follows

2


i 0b  b CL 2V SCL
2i
2
i 0 b
CDi 
 ( y)dy 
sin  d 



V S b/2
V S 2 0
2V S
2V S   b
b /2
Thus, we have obtained the formula
CD 
i
CL2

Consider the wing with no geometrical or aerodynamic twist.
Then, both  and  0 are constant along the wing span. For the elliptical load
 i is also constant, hence the effective angle of attack  eff and
the lift coefficient cL  a ( eff   0 ) are also constant along the wing span.
distribution the angle
Since L( y)  cL qc( y) then
L( y)
c( y ) 
.
cL q
AERODYNAMICS I
Conclusion: the spanwise variation of the wing chord follows the variation of the
aerodynamic load. Hence, the planform of such wing is also elliptical!
AERODYNAMICS I
General lift distribution
y   12 b cos ,  [0, ]
Again, we use the transformation
The elliptic distribution is expressed now as
 ( )   0 1  cos2    0 sin

Generalization:
 ( )  2Vb An sin n
n1
d   d  d  2V b d
 dy
dy
d dy
We will need the derivative …

nAn cos n

n1
The central equation of the lifting-line theory takes the form



2b
1
cos n
 (0 )   0 (0 ) 
An sin n0 
d 
 nAn
 c(0 ) n1
 n1  0 cos  cos0 





AERODYNAMICS I

The Glauert integral appears again
 sin n 0
cos n
d


0 cos  cos0
sin  0
Hence, the main equation is transformed to the algebraic form
sin n 0
2b 
1 
 ( 0 )   0 ( 0 ) 
A
sin
n


nA


0
 c(0 ) n1 n
 n1 n sin  0
In order to find approximate solution, we first truncate the infinite series …
sin n0
2b N
1 N
 (0 )   L0 (0 ) 
An sin n 0   nAn

 c(0 ) n1
 n1
sin  0
… and make use of the collocation method, i.e., require fulfillment of this equation at
N different point m [0,  ], m  1,.., N .
This way, the linear algebraic system is obtained for the unknown coefficients
{ A1 , A2 ,..., AN }, which can be solved, e.g., by the Gauss Elimination Method.
AERODYNAMICS I
Once  ( ) is known, one can calculate all aerodynamic characteristics of the wing.
We have
b/2

2
2b2 N
CL 
 ( y)dy 
An  sin n sin d


V S b/2
S n1 0
We use the orthogonality of the Fourier modes

0
 2 , n  1
sin n sin  d  
 0 , n 1
and obtain the formula
2
b
CL   A1  A1
S
We see that only the first coefficient of the Fourier series is needed to calculate the
lift force coefficient!
AERODYNAMICS I
The calculation of the induced drag is more complicated …
We have
b/2
2
2b
CDi 

(
y
)

(
y
)
dy

i
V S b/2
S
2
N

i ( )sin  An sin n  d
 n1

0


We need expression for the induced angle, namely
 
 N
sin n0
1
 ( y)dy 1 N
cos n
i 

nA
d


nA
  n
n  0


4V b/2 y0  y  n1
sin 0
 cos  cos0
 n1
b /2
Hence, the formula for the CDi can be transformed as follows

N

2b2
sin k   N
CDi 
sin   kAk
   An sin n  d 

S 0
sin    n1
 k 1


2b
S
2
N

kAk An  sin k sin n d

k ,n1
0
AERODYNAMICS I
Again, using the orthogonality property of the Fourier modes

 0 , k  m
sin k sin n d   1
 2  , k  m
0

the formula for the induced drag coefficient simplifies to the form
N
N
N

An2 
2b2  N 2
2
2
2
2
CDi 
nAn   nAn  ( A1   nAn )   A1 1   n 2 

S 2 n1
n1
n 2
 n2 A1 
We can write shortly
CL2
CL2
CDi 
(1   ) 

e
An2
where    n 2 and e  (1   )1 (Oswald aerodynamic efficiency parameter).
n2 A1
N
Note that   0, hence
CDi
elliptic wing
 CDi
(optimality!)
any wing
AERODYNAMICS I
The most famous airplane with the elliptical wing:
Trapezoidal wings are easier to construct and to build. Theoretically, they are nearly as
good as elliptic ones if only the taper ratio (i.e., ctip / croot ) is near the optimal value. In
the wide range of aspect ratios, (   4 10 ), the smallest values of  are achieved
when the taper ratio is close to 0.3. Other factors, like the stall pattern also matters!
AERODYNAMICS I
LEFT: Spanwise lift distribution for trapezoidal wing with different taper ratios.
RIGHT: Stall patterns for different planforms.
AERODYNAMICS I
Reduction of the lift slope
The finite span not only leads to the appearance of the induced drag – it also changes
(reduces) the slope of the “lift vs angle of attack” characteristic.
Denote:
dc
a  L - slope of the lift characteristic for the 2D wing section (equivalent to    )
d
dC
a  L - slope of the lift characteristic for the 3D wing.
d
Due to appearance of the
induced angle, the 3D wing
achieves the same value of
the lift coefficient at larger
geometric angle of attack –
see figure.
We have …
CL  a (  i )  const
AERODYNAMICS I
Hence, for the elliptic wing
Thus
For other planforms …
CL  a ( 
CL

)  const
dCL
a
 a 
d
1  a 
a 
a
1  (a )(1   )
Correction factor  typically ranges from 0.05 to 0.25. The value of this factor can be
expressed by the coefficients { A1, A2 ,..., AN }