Chapter IV. Handout #3
Thin Airfoil Theory- Terminology and Definitions
An airfoil is defined by first drawing a “mean” camber line. The straight line that joins the leading and trailing ends
of the mean camber line is called the chord line. The length of the chord line is called chord, and given the symbol
‘c’.To the mean camber line, a thickness distribution is added in a direction normal to the camber line to produce the
final airfoil shape. Equal amounts of thickness are added above the camber line, and below the camber line.
An airfoil with no camber (i.e. a flat straight line for camber) is a symmetric airfoil.
The angle that a freestream makes with the chord line is called the angle of attack.
Forces along the x- and y- axes: The forces and moments acting on an airfoil may be computed as follows.
y
x
Let Y be the pressure force per unit span (i.e. per unit distance normal to the plane of the paper) along the y- axis.
This force may be computed as:
Y
 pdx   pdx    p
Lower
Surface
upper
 plower dx
Upper
Surface
If we subtract off a constant value p from the upper and lower side pressure values, then the force will not change.
Thus,
Y    pupper  p     plower  p  dx
We can non-dimensionalize the above dimensional form by dividing the pressure by the dynamic pressure, and the
distances by the chord c. Then,
CY 
Y
1
V2 c
2

 p
upper

 p     plower  p   x
x
d   C p ,upper  C p ,lower d
1
c
c
V2
2
We can likewise find the component of force acting along the x- axis. This force and its non-dimensional form are
given below:
 dY 
 dY 
X   pupper 
dx   plower 
dx


 dx  upper
 dx  lower
CX 

 x
 dY 
 dY 
  C p ,upper 
 C p ,lower 


d
1
dx  upper
dx  lower  c
2




V c
2
X
Lift and Drag: The X and Y- forces act along the x- and y- axes, respectively. Lift is defined as the component of
pressure force that is normal to the freestream direction, and drag is defined as the component of pressure force
along the freestream direction. If the airfoil was initially located so that the chord line is along the x- axis, then the
angle between the freestream direction and the x- axis is the angle of attack .
y, Y
L
D
x,X
Freestream
Direction
Lift and drag are related to the X- and Y- forces as follows:
L  Y cos   X sin 
D  X cos   Y sin 
C l  CY cos   C X sin 
C d  C X cos   CY sin 
The quantities Cl and Cd are called the lift, and drag coefficients, respectively. By convention, the lower case
subscripts are used in 2-D flows, while upper case subscripts are used to denote lift and drag coefficients of
three-dimensional configurations such as wings.
Pitching Moment: We can also define the pitching moment about any point on the chord line. Nose up moment is
considered positive. About a general point on the x- axis whose co-ordinates are given by (a,0), the pitching moment
per unit span is given in dimensional form by:
M    plower  pupper ( x  a )dx
The non-dimensional form is given by:
 x a x
C m   C p ,lower  C p ,upper   d
c c c
While the pitching moment can be defined about any point in space, it is customary to compute the pitching moment
M and the pitching moment coefficient Cm about the quarter chord, i.e. a location 25%c downstream of the leading
edge.
Center of Pressure: the center of pressure is defined as the point about which the pitching moment is zero. As the
flow conditions change (example, angle of attack  changes), the center of pressure will change.
Aerodynamic Center: The aerodynamic pressure is defined as the point where the pitching moment (or the pitching
moment coefficient) is independent of . That is, if we computed the pitching moment about the aerodynamic
center,
M C m

0


The thin airfoil theory to be covered will yield the following results:
C l  2    0 
C l
 2

C d  0 (d' Alembert' s Paradox)
C m
 0 about the quarter - chord point.

The quantity 0 is called the angle of zero lift, since lift is zero at =0. In real flows, Cl, Cd and Cm will differ from
our theory (hopefully, only slightly) due to viscous effects.
As may be expected, symmetric airfoils will have zero lift at zero angle of attack. Thus, 0 is zero for
symmetric airfoils. For Cambered airfoils 0 can have positive or negative, depending on whether they have a
positive camber (camber line is convex, i.e. curved up) or negative camber (camber line is concave, i.e. curved
down).
Airfoil Geometry Naming Conventions: The following is an excerpt
from: the web site: http://www.desktopaero.com/appliedaero/appliedaero.html
To respect the copyright , and to acknowledge the considerable effort the author of the web site has made in
preparing the material, this excerpt is printed in italics.
Airfoil geometry can be characterized by the coordinates of the upper and lower surface. It is often summarized by a
few parameters such as: maximum thickness, maximum camber, position of max thickness, position of max camber,
and nose radius. One can generate a reasonable airfoil section given these parameters. This was done by Eastman
Jacobs in the early 1930's to create a family of airfoils known as the NACA Sections.
Example of a NACA 4-Digit Series: Consider the airfoil NACA 4412. The first digit gives maximum camber in % of
chord, the second digit gives in tenth of a chord where the maximum camber occurs, and the last two digits give the
maximum thickness in %chord.
4
max camber
in % chord
4
12
position
max thickness
of max camber
in % of chord
in 1/10 of c
After the 4-digit sections came the 5-digit sections such as the famous NACA 23012. These sections had the same
thickness distribution, but used a camber line with more curvature near the nose. A cubic was faired into a straight
line for the 5-digit sections.
NACA 5-Digit Series:
2
30
12
approx max
position
max thickness
camber
of max camber
in % of chord
divided by 1/50 of c
The 6-series of NACA airfoils departed from this simply-defined family. These sections were generated from a more
or less prescribed pressure distribution and were meant to achieve some laminar flow.
NACA 6-Digit Series:
6
3,
2
2
12
Sixlocation
half width
ideal Cl
max thickness
Series of min Cp
of low drag
in tenths in % of chord
in 1/10 chord bucket in 1/10 of Cl
After the six-series sections, airfoil design became much more specialized for the particular application. Airfoils
with good transonic performance, good maximum lift capability, very thick sections, very low drag sections are now
designed for each use. Often a wing design begins with the definition of several airfoil sections and then the entire
geometry is modified based on its 3-dimensional characteristics.