Copyright by Dr. Zheyan Jin
Aerodynamics
Zheyan Jin
School of Aerospace Engineering and Applied Mechanics
Tongji University
Shanghai, China, 200092
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Infinite wings versus finite wings
Airfoil shapes are two-dimensional. We consider them to stretch
indefinitely along the span direction, and therefore the flow around them
must also be two-dimensional.
Airfoil data comes from airfoil shapes that completely span the wind
tunnel section, and this is a reasonable approximation to a 2D airfoil.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Infinite wings versus finite wings
Real wings are not infinite in span. They have a finite wing-span. As
we shall see, because real wings do not go on forever, the flow over
them is not two-dimensional. This has very important effects.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Infinite wings versus finite wings
Streamline over the bottom surface
cr
Wing span b
Low pressure
Front view
High pressure
ct
Wing tip
Top view
(planform)
V
Wing root
Streamline over the
top surface
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Infinite wings versus finite wings
The physical mechanism for generating lift on the wing is the existence of a
high pressure on the bottom surface and a low pressure on the top surface. As
a by-product of this pressure imbalance, the flow near the wing tips tends to
curl around the tips.
On the top surface, there is generally a spanwise component of the flow from
the tip toward the wing root. On the contrary, there is generally a spanwise
component of the flow from the root toward the tip on the bottom surface.
View from
above
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Infinite wings versus finite wings
The tendency of the flow to “leak” around the
wing tips has another important effect on the
aerodynamics of the wing. This flow establishes
a circulatory motion that trails downstream of
the wing; that is, a trailing vortex is created at
each wing tip.
These wing-tip vortices downstream of the
wing induce a small downward component of air
velocity in the neighborhood of the wing itself.
This downward component is called downwash,
denoted by the symbol w.
Condensation in the cores
of wingtip vortices from an
F-15E as it disengages from
a KC-10 Extender following
midair refueling.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Downwash
One reason that geese and pelicans fly in neat Vformation is that by maintaining a very precise
spacing, birds can get a slight upward boost form the
trailing vortex of the bird in front: the outer side of the
trailing vortex of the bird in front has an upward
component.
On course, the leader at the point of the V gets no
such benefit, which is undoubtedly why birds
exchange the lead position relatively often in such
flocks.
Canada Geese in V formation
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Downwash
Wingtip vortices can be harmful.
When small airplanes are thrown out
of control by trailing of very large ones.
The vortex system of a fixed wing is
quite simple, in practice consisting of
the bound vortex on the wing and a
trailing vortex off each wing tip that
gradually dissipates far behind the wing.
Wake turbulence and tip vortices
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Downwash
The downwash combines with the free
stream velocity V͚ to produce a local
relative wind which is canted downward
in the vicinity of each airfoil section of
the wing.
The angle between the chord line and
the direction of V͚ is the angle of attack
α. We now precisely define α as the
geometric angle of attack.
The local relative wind is inclined
below the direction of V͚ by the angle αi,
called the induced angle of attack.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Downwash
The presence of downwash has two
important effects on the local airfoil
section, as follows:
1. The angle of attack actually seen
by the local airfoil section is the angle
between the chord line and the local
relative wind. This angle is defined as
the effective angle of attack.
 eff     i
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Downwash
2. The local lift vector is aligned
perpendicular to the local relative wind,
and hence is inclined behind the vertical
by the angle αi.
Consequently, there is a component
of the local lift vector in the direction of
V͚ ; this drag is created by the presence
of downwash.
This drag is defined as induced drag.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Downwash
The tilting backward of the lift vector
shown in the right picture is one way
visualizing the physical generation of
induced drag. Two alternative ways are
as follows:
1. The three-dimensional flow induced
by the wing-tip vortices alters the
pressure distribution on the finite wing in
such a fashion that a net pressure
imbalance exists in the direction of V͚ .
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Downwash
2. The wing-tip vortices contain a large
amount of translational and rotational
kinetic energy. This energy is provided
by the aircraft engine.
In effect, the extra power provided by
the engine that goes into the vortices is
the extra power required from the
engine to overcome the induced drag.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Road Map
General features of finite-wing aerodynamics:
downwash, effective angle of attack, and
induced drag
Additional tools needed for finite wing analysis:
1. Curved vortex filament
2. Biot-Savart Law
3. Helmholz’s vortex theorems
Method of analysis
Prandtl’s classical
lifting-line theory
Modern numerical
lifting-line method
Lifting surface
theory
Modern vortex lattice
numerical method
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Differences in nomenclature
For the two-dimensional bodies:
Lift, drag, and moments per unit span have been denoted with primes. For
example, L’, D’, and M’.
The corresponding lift, drag, and moment coefficients have been denoted
by lower case letters, for example, cl, cd, and cm.
For the three-dimensional bodies:
Lift, drag, and moments on a complete three-dimensional body are given
without primes. For example, L, D, and M.
The corresponding lift, drag, and moment coefficients have been denoted
by capital letters, for example, CL, CD, and CM.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.1 Introduction: Downwash and Induced Drag
Drag
Total drag on a subsonic finite wing in real life
Induced drag Di
Skin friction drag Df
Pressure drag Dp
Profile drag
Profile drag coefficient:
cd 
D f  Dp
q S
Di
q S
Induced drag coefficient:
C D ,i 
Total drag coefficient:
C D  cd  C D ,i
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.2 The vortex filament, the Biot-Savart law, and Helmholtzˈs theorems
The Biot-Savart law
In general, a vortex filament can be
curved. The filament induces a flow field in
the surrounding space.
The velocity at point P, dV, induced by
a small directed segment dl of a curved
filament with strength  is
dV 
 dl  r
4 r 3
Vortex filament
of strength Г
dl
r
dV
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.2 The vortex filament, the Biot-Savart law, and Helmholtzˈs theorems
The Biot-Savart law

z
y
Now apply the Biot-Savart law to a
straight vortex filament of infinite length.
dl
P
The velocity induced at P by the
entire vortex filament is:
V 



 dl  r
4 r 3
h
-

V
2h
The direction of the velocity is downward.
The magnitude of the velocity is given by:
r
x

V
4
sin 
 r 2 dl

V
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.2 The vortex filament, the Biot-Savart law, and Helmholtzˈs theorems
The Biot-Savart law

z
y
Let h be the perpendicular distance form
point P to the vortex filament. Then,
dl
P
h
sin
h
l
tan
r
dl  
r
x
h
-
h
d
sin 2

V
2h
Thus, the velocity induced at a given point P by an infinite, straight vortex
filament at a perpendicular distance h from P is simply:

V
4
 0

sin 
dl


d

sin


 r 2
4h 
2h

V
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.2 The vortex filament, the Biot-Savart law, and Helmholtzˈs theorems
Helmholtz’s vortex theorems:
The great German mathematician, physicist, and physician Hermann von
Helmholtz was the first to make use of the vortex filament concept in the analysis of
inviscid, incompressible flow.
Helmholtz’s 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 the boundaries of the
fluid or form a closed path.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.2 The vortex filament, the Biot-Savart law, and Helmholtzˈs theorems
Lift distribution:
Front view of wing
L'  L' ( y )   V ( y )
-b/2
b/2
y
Most finite wings have a variable chord, with the exception of a simple rectangular
wing.
Also, many wings are geometrically twisted so that α is different at different
spanwise locations- so-called geometric twist. If the tip is at a lower α than the root
the wing is said to have washout; if the tip is at a higher α than the root, the wing
has washin.
The wings on a number of modern airplanes have different airfoil sections along the
span, with different values of αL=0; this is called aerodynamic twist.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Prandtl reasoned as follows.
A vortex filament of strength Г that
is somehow bound to a fixed location
in a flow -a so-called bound vortex- will
experience a force Lˈ=ρV͚Г from the
Kutta-Joukowski theorem.
This bound vortex is in contrast to a
free vortex, which moves with the
same fluid elements throughout a flow.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Prandtl reasoned as follows.
Let us replace a finite wing of span b
with a bound vortex, extending from
y=-b/2 to y=b/2. Since a vortex filament
cannot end in the fluid, we assume the
vortex filament continues as two free
vortices trailing downstream from the
wing tips to infinity.
This vortex is in the shape of
horseshoe, and therefore is called a
horseshoe vortex.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
A single horseshoe vortex.
Consider the downwash w induced along
the bound vortex from –b/2 to b/2 by the
horseshoe vortex.
Bound
vortex
The two trailing vortices both contribute
to the induced velocity along the bound
vortex, and both contributions are in the
downward direction.
b/2
z y
-b/2
Trailing vortex
x
Trailing vortex
A single horseshoe vortex
The bound vortex induces no velocity along itself.
w( y )  

4(b / 2  y )


4(b / 2  y )
The first term on the right-hand side is the contribution from the left trailing
vortex. The second term is the contribution from the right trailing vortex.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Prandtl’s Classical Lifting-Line Theory:
A single horseshoe vortex.
We can reduce the above equation to:

b
w( y )  
4 (b / 2) 2  y 2
Bound
vortex
b/2
z y
Trailing vortex
x
Trailing vortex
-b/2
A single horseshoe vortex
Note that w approaches -∞ as y approaches –b/2 or b/2.
The downwash distribution due to the single horseshoe vortex shown in above
figure does not realistically simulate that of a finite wing.
The downwash approaching an infinite value at the tips is especially disconcerting.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
d1  d2  d3
Let us superimpose a large number of
horseshoe vortices, each with a
different length of bound vortex, but with
all the bound vortices coincident along a
single line, called the lifting line.
b/2
F
d1
E
D
d1  d2
d1
C
B
The circulation varies along the line of
bound vortices.
∞
d2
∞
d3
∞
d3
∞
d2
d1
-b/2 A
∞
∞
Lifting line
Along AB and EF, the circulation is:
d1
Along BC and DE, the circulation is:
d1  d2
Along CD, the circulation is:
d1  d2  d3
Note that the strength of each trailing vortex is equal to the change in circulation along
the lifting line.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
y
0
Let us assume there are infinite number
of horseshoe vortices along the lifting line.
b/2
  (y)
z
V͚
There is continuous distribution of Г(y) along
the lifting line. The value of the circulation at
the origin is Г0。
y0
θ
x
dГ
-b/2
Lifting line
The vortex sheet is parallel to the direction of V͚ .
The total strength of the sheet integrated across the span is zero, because it
consists of pairs of trailing vortices of equal strength but in opposite direction.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
y
0
The velocity dw at y0 induced by the
entire semi-infinite trailing vortex located
at y is given by:
dw  
(d / dy )dy
4 ( y0  y )
  (y)
z
V͚
dw
y0
b/2
θ
dx
x
dГ
-b/2
Lifting line
The minus sign in the above equation is needed for consistency with the right
picture.
For the trailing vortex shown, the direction of dw at y0 is upward and hence is
a positive value, whereas Г is decreasing in the y direction, making dГ/dy a
negative quantity. The minus sign in the above equation makes the positive dw
consistent with the negative dГ/dy.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
The total velocity w induced at y0 by
the entire trailing vortex sheet is the
summation of the last equation over all
the vortex filaments:
w( y0 )  
1
4
(d / dy )dy
b / 2 y0  y
y
0
  (y)
z
V͚
dw
y0
b/2
θ
dx
x
dГ
b/2
-b/2
Lifting line
The value of the downwash at y0 due to all the trailing vortices.
Keep in mind that although we label w as downwash, w is treated as positive in
the upward direction in order to be consistent with the normal convention in an xyz
rectangular coordinate system.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Assume this section is located at the
arbitrary spanwise station y0. The
induced angle of attack αi is given by:
 i ( y0 )  tan 1 (
 w( y0 )
)
V
Generally, w is much smaller
than V͚ , and hence αi is a small
angle, on the order of a few
degrees at most.
 i ( y0 ) 
 w( y0 )
V
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
By substituting the above equation
into the downwash equation, we
obtain
 i ( y0 ) 
1
4V
(d / dy )dy
b / 2 y0  y
b/2
that is, an expression for the
induced angle of attack in terms
of the circulation distribution Г(y)
along the wing.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Consider again the effective angle of attack αeff.
Since the downwash varies across the span, then αeff is also variable;
αeff= αeff(y0). The lift coefficient for the airfoil section located at y=y0 is:
cl  a0 [ eff ( y0 )   L 0 ]  2 [ eff ( y0 )   L 0 ]
From the Kutta-Joukowski theorem, lift for the local airfoil section
located at y0 is
L 
1
 V2 c( y0 )cl   V ( y0 )
2
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
cl 
2( y0 )
V c( y0 )
Thus，
cl  2 [ eff ( y0 )   L 0 ]
  eff 
Since
 ( y0 )
  L 0
V c( y0 )
 eff     i
Finally, we obtain
 ( y0 ) 
 ( y0 )
1
  L  0 ( y0 ) 
V c( y0 )
4V
(d / dy ) dy
b / 2 y0  y
b/2
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
 ( y0 ) 
 ( y0 )
1
  L  0 ( y0 ) 
V c( y0 )
4V
(d / dy )dy
b / 2 y0  y
b/2
The 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.
αeff is expressed in terms of Г, and αi is expressed in terms of an
integral containing dГ/dy. Hence, the above equation is an integrodifferential equation, in which the only unknown is Г;
all the other quantities, α ,c, V͚ , and αL=0, are known for a finite wing of
given design at a given geometric angle of attack in a freestream with
given velocity.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
 ( y0 ) 
 ( y0 )
1
  L  0 ( y0 ) 
4V
V c( y0 )
(d / dy )dy
b / 2 y0  y
b/2
The solution Г=Г(y0) obtained from the above equation gives us the three
main aerodynamic characteristics of a finite wing, as follows:
1. The lift distribution is obtained from the Kutta-Joukowski theorem:
L' ( y0 )   V ( y0 )
2. The total lift is obtained by integrating the above equation over the span：
L
b/2
b / 2
L' ( y )dy
L   V 
b/2
b / 2
CL 
( y )dy
2
L

q S V S

b/2
b / 2
( y )dy
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
3. The induced drag per unit span is:
D i'  L' sin  i
Since αi is small, this relation becomes
D i'  L'  i
The total induced drag is obtained by integrating the above equation
over the span
b/2
Di  
b / 2
L' ( y ) i ( y )dy
Di   V 
b/2
b / 2
C D ,i
( y ) i ( y )dy
D
2
 i 
q S V S

b/2
b / 2
( y ) i ( y )dy
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Consider a circulation distribution given by:
Note:
( y )  0 1  (
2y 2
)
b
1. Г0 is the circulation at the origin.
2. The circulation varies elliptically with distance y along the span; hence, it
is designated as an elliptical circulation distribution. Since
L' ( y )   V ( y )
we also have
L' ( y )   V 0 1  (
2y 2
)
b
Hence, we are dealing with an elliptical lift distribution.
3. (b / 2)  ( b / 2)  0
Thus, the circulation, hence lift, properly goes to zero at the wing tips.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
What are the aerodynamic properties of a finite wing with such an elliptic
lift distribution?
First, let us calculate the downwash.
4
d
y
  20
dy
b (1  4 y 2 / b 2 )1/ 2
Substitute this into the down wash equation, we obtain,
w( y0 ) 
0
b 2
y
b / 2 (1  4 y 2 / b 2 )1/ 2 ( y0  y) dy
b/2
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
What are the aerodynamic properties of a finite wing with such an elliptic
lift distribution?
The integral can be evaluated easily by making the substitution
y
Hence,
b
cos 
2
b
dy   sin  d
2
w( 0 )  
0
2b
w( 0 )  
0
2b
0



0
cos 
d
cos  0  cos 
cos 
d
cos   cos  0
0
2b
This states that the downwash is constant over the span for an elliptical lift
distribution.
w( 0 )  
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
What are the aerodynamic properties of a finite wing with such an
elliptic lift distribution?
The induced angle of attack
i  

w
 0
V 2bV
The induced angle of attack is also constant over the span for an elliptical lift
distribution.
Note that both the downwash and induced angle of attack go to zero as the
wing span becomes infinite- which is consistent with our previous discussions
on airfoil theory.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
What are the aerodynamic properties of a finite wing with such an elliptic
lift distribution?
A more useful expression for αi can be obtained as follows.
4 y 2 1/ 2
L   V 0  (1  2 ) dy
b / 2
b
b/2
Again use the transformation
y  (b / 2) cos 
b  2
b
L   V 0  sin d   V 0 
2 0
4
4L
0 
 V b
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
What are the aerodynamic properties of a finite wing with such an elliptic
lift distribution?
1
However,
L   V2 SC L
2
2V SC L
b
2V SC
1
SC
 2L
i   L
b 2bV b 
0 
An important geometric property of a finite wing is the aspect ratio, denoted
by AR and defined as
AR  b 2 / S
The induced angle of attack:
i 
CL
AR
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
The induced drag coefficient
C D ,i 
2 i
V S

b/2
b / 2
( y )dy 
C D ,i 
Or,
C D ,i
b
2 i 0 b  2
 i 0b
sin
d



V S 2 0
2V S
C L 2V SC L
)
b
2V S AR
(
C L2

AR
The above equation states that the induced drag coefficient is directly
proportional to the square of the lift coefficient.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
The induced drag coefficient
C D ,i
C L2

AR
First property:
The dependence of induced drag coefficient on the lift is not surprising, for
the following reason.
The induced drag is a consequence of the presence of the wing-tip vortices,
which in turn are produced by the difference in pressure between the lower
and upper wing surfaces.
The lift is produced by this same pressure difference.
Hence, induced drag is intimately related to the production of lift on a finite
wing; indeed, induced drag is frequently called drag due to lift.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
The induced drag coefficient
C D ,i
C L2

AR
Second property:
CDi is inversely proportional to aspect ratio.
To reduce the induced drag, we want a finite wing with the highest possible
aspect ratio.
Unfortunately, the design of very high aspect ratio wings with sufficient
structural strength is difficult.
Therefore, the aspect ratio of a conventional aircraft is a compromise
between conflicting aerodynamic and structural requirements.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
Third property:
Consider a wing with no geometric twist (i.e., α is constant along the span)
and no aerodynamic twist (i.e., αL=0 is constant along the span).
The local section lift coefficient cl is given by:
c l  a 0  eff   L 0 
Assuming that a0 is the same for each section, cl must be constant along
the span. The lift per unit span is given by
L' ( y )
L ( y )  q ccl  c( y ) 
q cl
'
For an elliptic lift distribution, the chord must vary elliptically along the span;
that is, for the condition given above, the wing planform is elliptical.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical Lift Distribution:
Illustration of the related quantities: an elliptic lift distribution,
elliptic planform, and constant downwash
Although an elliptical lift distribution may appear to be a restricted, isolated case,
in reality it gives a reasonable approximation for the induced drag coefficient for
an arbitrary finite wing.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Elliptical lift distribution:
Supermarine Spitfire
Supermarine Spitfire
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
Consider the transformation:
b
y   cos 
2
where the coordinate in the spanwise direction is now given by θ, with 0<= θ
<=π.
In terms of θ, the elliptic lift distribution
( y )  0 1  (
can be written as
( )  0 sin 
2y 2
)
b
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
This equation hints that a Fourier sine series would be an appropriate
expression for the general circulation distribution along an arbitrary finite wing.
N
( )  2bV  An sin n
1
where as many terms N in the series can be taken as we desire for accuracy.
An must satisfy the fundamental equation of Prandtl’s lifting-line theory.
N
d d d
d

 2bV  nAn cos n
dy d dy
dy
1
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
Substituting the above equations into the angle of attack equation, we obtain
N
sin n 0
2b N


A
n
nA
 ( 0 ) 
sin


(

)
 n
1 n sin 
0
0
L 0
c( 0 ) 1
0
This equation is evaluated at a given spanwise location; hence, θ0 is specified. In
turn, b, c(θ0), and αL=0(θ0), are known quantities from the geometry and airfoil
section of the finite wing. The only unknowns in the above equation are the An’s.
Hence, written at a given spanwise location, the above equation is one algebraic
equation with N unknowns, A1,A2,…, AN.
However, let us choose N different spanwise stations, and let us evaluate the
above equation at each of these N stations. We then obtain a system of N
independent algebraic equations with N unknowns, namely, A1, A2,…, AN.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
Now that Г(θ) is known.
2
CL 
V S
2b 2
b / 2 ( y)dy  S
b/2
N

 An  sin n sin d
1
0
 / 2 (n  1）

sin
sin
n


d


0
 0 (n  1)
b2
 A1AR
C L  A1
S

Note that CL depends only on the leading coefficient of the Fourier series expansion.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
The induced drag is obtained by
C D ,i
2

V S
2b 2
b / 2 ( y) i ( y)dy  S
b/2


0
 N

  An sin n  i ( ) sin d
 1

The induced angle of attack is obtained by
1
 i ( y0 ) 
4V

(d / dy )dy 1 N
cos n

A
b / 2 y 0 - y  1 n 0 cos  cos  0 d
b/2


0
sinn 0
cos n
d 
cos   cos  0
sin  0
N
Thus,
 i ( 0 )   nAn
1
sin n 0
sin  0
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
N
 i ( 0 )   nAn
1
sin n 0
sin  0
In the above equation, θ0 is simply a dummy variable which ranges from 0 to π
across the span of the wing; it can therefore be replaced by θ, and the above
equation can be written as:
N
 i ( )   nAn
1
sin n
sin 
Substitute this equation to the drag coefficient equation, we have
C D ,i
2b 2

S

N
N
 ( A sin n )( nA sin n )d
0
n
1
n
1
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:


0
 0( m  k )
sin m sin k  
 / 2(m  k )
In the drag coefficient equation, the mixed product terms involving unequal
subscripts are equal to zero. Hence,
C D ,i
N
2b 2 N
2 

( nAn )  AR  nAn2
S 1
2
1
N
N
C D ,i  AR ( A   nA )  ARA [1   n(
2
1
2
2
n
2
1
2
An 2
) ]
A1
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
Thus,
C D ,i
C 2L

[1   ]
AR
A
   n n
2
 A1
N
where



2
Note that δ ≥0;hence, the factor 1+ δ in the above equation is either
greater than 1 or at least equal to 1. Let us define a span efficiency factor,
e, as e=1/(1+ δ).
C D ,i
C L2

eAR
where e ≤1. Note that δ =0 and e =1 for the elliptical lift distribution. Hence,
the lift distribution which yields minimum induced drag is the elliptical lift
distribution.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
General Lift Distribution:
Recall that for a wing with no
aerodynamic twist and no
geometric twist, an elliptical lift
distribution is generated by a wing
with an elliptical planform.
However, elliptic planforms are
more expensive to manufacture
than, say, a simple rectangular
wing.
On the other hand, a rectangular
wing generates a lift distribution
far from optimum. A compromise
is the tapered wing.
Elliptic wing
Rectangular wing
Tapered wing
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Effect of Aspect Ratio:
C D ,i
C 2L

[1   ]
AR
Note that the induced drag
coefficient for a finite wing with a
general lift distribution is inversely
proportional to the aspect ratio.
Note that AR, which typically varies
from 6 to 22 for standard subsonic
airplanes and sailplanes, has a
much stronger effect on CD,i than
the value of δ.
Induced drag factor δ as a function
of taper ratio.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Effect of Aspect Ratio:
C D ,i
C 2L

[1   ]
AR
Hence, the primary design factor for
minimizing induced drag is not the
closeness to an elliptical lift
distribution, but rather, the ability to
make the aspect ratio as large as
possible.
Induced drag factor δ as a function
of taper ratio.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
CL
Infinite
wing
a0
Effect of Aspect Ratio:
There are two primary differences between airfoil
and finite-wing properties.
 eff     i
1. A finite wing generates induced drag.
2. The lift slope is not the same.
CL
dC L
 0
d (   i )
Finite
elliptic
wing
a
C L   0 (   i )  const
C L   0 ( 
CL
)  const
AR
a0
dC L
a
d
1  a0 / AR

 L0
For a finite wing of general planform, the left
equation is slightly modified
a
a
0
1  a0 / AR (1   )
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Example 1:
Consider a finite wing with an aspect ratio of 8 and a taper ratio of 0.8.
The airfoil section is thin and symmetric. Calculate the lift and induced
drag coefficients for the wing when its angle of attack is 5°. Assume
that δ=τ.
Solution:
From a figure in the textbook, we can obtain
  0.055
Assume a0=2πfor thin airfoil theory
a
a0
2

 4.97rad 1
 1  ） 1  2 / 8（
 1.055）
1  a0 / AR（
Since the airfoil is symmetric, aL=0=0°. Thus,
C L  a  0.0867 deg ree 1 (50 )  0.4335
C D ,i
2
1  0.055）
C L2
（0.4335）
（
(1   ) 

 0.00789
8
AR
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Example 2:
Consider a rectangular wing with an aspect ratio of 6, and induced drag factor
δ=0.055, and a zero lift angle of attack of -2°. At an angle of attack of 3.4°, the
induced drag coefficient for this wing is 0.01. Calculate the induced drag for a
similar wing at the same angle of attack, but with an aspect ratio of 10. Assume
that the induced factors for drag and the lift slope, δ and τ, respectively, are
equal to each other. Also, for AR=10, δ=0.105.
Solution:
Firstly, let us calculate CL for the wing with aspect ratio 6.
C 2L 
Hence,
ARCD ,i 6 0.001

 0.1787
1 
1  0.055
C L  0.423
The lift slope of this wing is therefore
dC L
0.423

 0.078 / deg ree  4.485 / rad
d 3.4 0  ( 2 0 )
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Example 2:
Consider a rectangular wing with an aspect ratio of 6, and induced drag factor
δ=0.055, and a zero lift angle of attack of -2°. At an angle of attack of 3.4°, the
induced drag coefficient for this wing is 0.01. Calculate the induced drag for a
similar wing at the same angle of attack, but with an aspect ratio of 10. Assume
that the induced factors for drag and the lift slope, δ and τ, respectively, are
equal to each other. Also, for AR=10, δ=0.105.
Solution:
The lift slope for the airfoil can be obtained by
a0
dC L
a
 1  ）
d
1  a0 / AR（
4.485 
a0
a0

 1  0.055） 1  0.056a0
1  a0 /  6（
Solving for a0, we find that this yields a0=5.989/rad. Since the second wing (with
AR=10) has the same airfoil section, then a0 is the same. The lift slope of the
second wing is given by
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.3 Prandtlˈs Classical Lifting-Line Theory
Example 2:
Consider a rectangular wing with an aspect ratio of 6, and induced drag factor
δ=0.055, and a zero lift angle of attack of -2°. At an angle of attack of 3.4°, the
induced drag coefficient for this wing is 0.01. Calculate the induced drag for a
similar wing at the same angle of attack, but with an aspect ratio of 10. Assume
that the induced factors for drag and the lift slope, δ and τ, respectively, are
equal to each other. Also, for AR=10, δ=0.105.
Solution:
For AR=10
a
a0
5.989

 4.95 / rad
 1  ） 1  5.989 /  10（
 1  0.105）
1  a0 / AR（
The lift coefficient for the second wing is therefore
CL  a (   L 0 )  0.086[3.40  (20）
]  0.464
C D ,i
2
C L2
（0.464）（
1  0.105）

 0.0076
(1   ) 
AR
10
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.4 A numerical nonlinear lifting-line method
Consider the most general case of a finite wing of given planform and geometric
twist, with different airfoil sections at different spanwise stations. Assume that we
have experimental data for the lift curves of the airfoil sections, including the
nonlinear regime. A numerical iterative solution for the finite-wing properties can
be obtained as follows:
1 2 3
n
k k+1
1: Divide the wing into a number of
spanwise stations. Here k+1 stations
are shown, with n designating any
specific station.
Δy
y
2: For the given wing at a given α, assume the lift distribution along the span; that
is, assume values for Г at all the stations Г1, Г2,…., Гn,…, Гk+1. An elliptical
lift distribution is satisfactory for such an assumed distribution.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.4 A numerical nonlinear lifting-line method
3:
With this assumed variation of Г，
calculate the induced angle of attack
αi
1 b / 2 (d / dy )dy
 i ( yn ) 
4V b / 2 yn  y
1 2 3
n
k k+1
Δy
y
The integral is evaluated numerically.
By using Simpson’s rule,
1 y
 i ( yn ) 
4V 3
k
(d / dy ) j 1
j  2, 4,6
yn  y j 1

4
(d / dy ) j
yn  y j

(d / dy ) j 1
yn  y j 1
where Δy is the distance between stations.
4:
Using αi from step 3, obtain the effective angle of attack αeff at each station
form
 eff ( yn )     i ( yn )
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.4 A numerical nonlinear lifting-line method
5:
6:
With the distribution of αeff calculated from step 4, obtain the section lift
coefficient (cl)n at each station. These values are read from the known lift
curve for the airfoil.
From (cl)n obtained in step 5, a new circulation distribution is calculated from
the Kutta-Joukowski theorem and the definition of lift coefficient:
L' ( yn )   V ( yn ) 
1
 V2 cn (cl ) n
2
1
( yn )  V cn (cl ) n
2
where cn is the local section chord. Keep in mind that in all the above
steps, n ranges from 1 to k+1.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.4 A numerical nonlinear lifting-line method
7:
The new distribution of Г obtained in step 6 is compared with the values
that were initially fed into step 3. If the results from step 6 do not agree with
the input to step 3, then a new input is generated. If the previous input to
step 3 is designated as Гold and the result of step 6 is designated as Гnew,
then the new input to step 3 is determined from
input  old  D (new  old )
where D is a damping factor for the iterations.
8:
Steps 3 to 7 are repeated a sufficient number of cycles until Гnew and Гold
agree at each spanwise station to within acceptable accuracy.
9:
From the converged Г(y), the lift and induced drag coefficients are obtained.
The integrations in these equation can again be carried out by Simpson’s
rule.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.4 A numerical nonlinear lifting-line method
Lift coefficient versus angle of attack
The numerical lifting-line solution at high angle of attack agrees with the
experiment to within 20 percent, and much closer for many cases.
Therefore, such solutions give reasonable preliminary engineering
results for the high-angle-of-attack poststall region.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.4 A numerical nonlinear lifting-line method
where D is a damping factor for the iterations.
Surface oil flow pattern on a stalled, finite rectangular wing
with a Clark Y-14 airfoil section.
At high angle of attack, there is a strong spanwise flow, in combination with
mushroom-shaped flow separation regions.
Clearly, the basic assumptions of lifting-line theory, classical or numerical,
cannot properly account for such three-dimensional flows.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Supersonic airplanes usually have highly swept wings.
A special case of swept wings is those aircraft with a triangular planform - called delta
wings.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Question:
Since delta-winged aircraft are highspeed vehicles, why are we discussing
this topic in the present chapter, which
deals with the low-speed,
incompressible flow over finite wings?
Answer:
All high-speed aircraft fly at low speeds
for takeoff and landing;
Moreover, in most cases, these aircraft spend the vast majority of their flight time
at subsonic speeds, using their supersonic capability for short “supersonic dashes,”
depending on their mission.
Therefore, the low-speed aerodynamics of delta wings has been a subject of much
serious study over the past years.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Variants:
There are several variants of the basic
delta wing used on modern aircraft:
(a) Simple delta
(b) Cropped delta
(c) Notched delta
(d) Double delta
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Leading-edge vortices over the top surface of a delta wing at angle of attack. The
vortices are made visible by dye streaks in water flow.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
The flow field in the crossflow plane above a delta wing at angle of attack, showing
the two primary leading-edge vortices. The vortices are made visible by small air
bubbles in water.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Pressure over the bottom surface:
The spanwise variation of pressure over
the bottom surface is essentially
constant and higher than the free stream
pressure.
Pressure over the top surface:
The spanwise variation in the midsection
of the wing is essentially constant and
lower than the freestream pressure.
However, near the leading edge the
static pressure drops considerably.
The leading-edge vortices are literally
creating a strong “suction” on the top
surface near the leading edge.
Schematic of the spanwise pressure coefficient
distribution across a delta wing.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
The suction effect of the leading-edge vortices enhances the lift; for this reason,
the lift coefficient curve for a delta wing exhibits an increase in CL for values of
α at which conventional wing planforms would be stalled.
Note the following characteristics:
1. The lift slope is small, on the order
of 0.05/degree.
2. The lift continues to increase to
large values of α; the stalling angle of
attack is on the order of 35°. The net
result is a reasonable value of CL,max,
on the order of 1.3.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Leading edge vortex flap:
The direction of the suction due to
the leading-edge vortices is now
modified.
Since the pressure is low over this
frontal area, the net drag can
decrease.
Schematic of the spanwise pressure coefficient
distribution over the top of a delta wing as
modified by leading-edge vortex flaps.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.5 The Delta Wing
Vortex breakdown
The primary vortices begin
to fall apart somewhere
along the length of the
vortex when a delta wing is
at a high enough angle of
attack.
Vortical flow over a 70 degree delta wing at an angle of
attack of 30 degrees
In summary, the delta wing is a common planform for supersonic aircraft. The lowspeed aerodynamics of these wings are quite different from a conventional planform.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.6 Ludwig Prandtl - Father of modern aerodynamics
Ludwig Prandtl was a German scientist. He was a
pioneer in the development of rigorous systematic
mathematical analyses which he used to underlay the
science of aerodynamics, which have come to form the
basis of the applied science of aeronautical engineering.
Major contributions:
1. Thin airfoil theory.
2. Finite-wing theory.
3. Boundary-layer concept.
4. Compressibility corrections.
5. Supersonic shock and expansion-wave theory.
Notable students:
Hubert Ludwieg, Hermann Schlichting,
Theodore von Kármán, Reinhold Rudenberg
Ludwig Prandtl (1875-1953)
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.7 Summary
Downwash
The wing-tip vortices from a finite wing induce a downwash which
reduces the angle of attack effectively seen by a local airfoil section:
 eff     i
In turn, the presence of downwash results in a component of drag
defined as the induced drag Di.
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.7 Summary
Vortex Filament
Vortex sheets and vortex filaments are useful in modeling the
aerodynamics of finite wings.
The velocity induced by a directed segment dl of a vortex filament is
given by the Biot-Savart law:
dV 
 dl  r
4 r 3
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.7 Summary
Prandtl’s classical lifting-line theory
In Prandtl’s classical lifting-line theory, the finite wing is replaced by a
single spanwise lifting line along which the circulation Г(y) varies.
A system of vortices trails downstream from the lifting line, which
induces a downwash at the lifting line.
The circulation distribution is determined from the fundamental equation
 ( y0 )
1
  L  0 ( y0 ) 
 ( y0 ) 
4V
V c( y0 )
(d / dy )dy
b / 2 y0  y
b/2
Copyright by Dr. Zheyan Jin
Chapter 5 Incompressible Flow Over Finite Wings
5.7 Summary
Prandtl’s classical lifting-line theory
Results from classical lifting-line theory:
Elliptic wing:
Downwash is constant:
General wing:
0
2b
C
i  L
AR
C L2
C D ,i 
AR
a0
a
1  a0 / AR
w
C L2
C L2
C D ,i 
（1  ）
AR
eAR
a0
a
1  (a0 / AR)(1   )
Copyright by Dr. Zheyan Jin