Coecients and supersonic ows
1 Denitions
R = Resultant force that acts on the wing (N)
N = Component of R acting perpendicular to the chord line (N)
A = Component of R acting tangential to the chord line (N)
L = Lift - component of R acting perpendicular to the relative wind (N)
D = Drag - component of R acting parallel to the relative wind (N)
= Angle of attack (usually the angle between the chord line and the relative wind) (deg)
M = Mach number (dimensionless)
Re = Reynolds number (dimensionless)
c = Chord - the length of the chord line (line between the leading edge and the trailing edge) (m)
c
l
= Lift coecient for innite wings (dimensionless)
= Air density (kg/m
3
)
V = Air velocity (m/s)
S = Wing area (m
2
)
q = Dynamic air pressure, dened as q =
1
2
V
2
(Pa = N/m
2
)
c
d
= Drag coecient for innite wings (dimensionless)
c
m
= Moment coecient for innite wings (dimensionless)
C
p
= Pressure coecient (dimensionless)
C
p,l
= Pressure coecient on the lower side of the wing (dimensionless)
C
p,u
= Pressure coecient on the upper side of the wing (dimensionless)
c
n
= Normal force coecient for innite wings (dimensionless)
c
a
= Axial force coecient for innite wings (dimensionless)
M
cr
= Critical Mach number - Mach number at which shock waves start occurring (dimensionless)
= Mach angle - Angle between direction of ight and shock wave boundary (rad/deg)
2 Airfoil nomenclature
During ight, a resultant force R acts on the wing. This force can be resolved into two forces in multiple
ways. R can be resolved in N and A, where N is the component perpendicular (normal) to the chord,
and A is the component tangential (axial) to the chord. However, R can also be resolved in a component
L (lift) perpendicular to the relative wind, and a component D (drag) parallel to the relative wind. These
forces have the following relation:
L = N cos Asin (2.1)
D = N sin +Acos (2.2)
1
3 Applying dimensional analysis to innite wings
Using dimensional analysis it can be found that:
L = Z
V
2
S
_
1
M
_
e
_
1
Re
c
_
f
Where Z is a constant (as long as remains constant, because if changes, also Z changes). So if we
dene the lift coecient c
l
such that:
c
l
2
= Z
_
1
M
_
e
_
1
Re
c
_
f
(3.1)
Then we see that:
L =
1
2
V
2
Sc
l
= q
Sc
l
(3.2)
And now we see that the lift coecient is also equal to:
c
l
=
L
q
S
(3.3)
Doing the same steps for the drag and the moment, will give:
D = q
Sc
d
(3.4)
M = q
Sc c
m
(3.5)
And nally we summarize all the equations:
c
l
=
L
q
S
c
d
=
D
q
S
c
m
=
M
q
Sc
(3.6)
c
l
= f
1
(, M
, Re) c
d
= f
2
(, M
, Re) c
m
= f
3
(, M
, Re) (3.7)
Where f
1
, f
2
and f
3
are functions. This is to emphasize that the coecients depend on the parameters
noted in brackets.
4 Bending coecients
As was already mentioned in the previous paragraph, the moment coecient is dened as:
c
mx
=
M
x
q
Sc
(4.1)
Where x can be any distance from the leading edge of the wing (so 0 x c) and M is the moment
acting on that point. Note that there is now an extra variable c in the equation, where there was none in
the denition for the force coecients. If the chord c wouldnt be present, c
m
wouldnt be dimensionless.
There are two specic points concerning moments which are often used in aerodynamics. The rst one
is the center of pressure (dp in short). This is where there is no bending moment. The position of this
point usually changes if the angle of attack changes. The second point is the aerodynamic center (ac).
This is the point where the bending moment stays constant as the angle of attack changes. Its position,
which is almost always around the quarter chord position (
xac
c
0.25), doesnt change if the angle of
attack changes. So in formula:
c
m
dp
= 0
dc
mac
d
= 0 (4.2)
2
So the moment coecient is dierent on dierent positions on an airfoil. But there exists a relationship
between those moment coecients. And, using simple statics, it can be shown that:
c
m
Q
1
c
m
Q
2
= c
n
_
x
Q1
c
x
Q2
c
_
(4.3)
Do remember that this formula only applies for constant angle of attack, since otherwise variables like
c
n
change. However, the fact that
xac
c
remains constant for dierent angles of attack can be used.
5 Prandtl-Glauert Rule
Lets dene a new dimensionless coecient to indicate the pressure over a wing. We dene the pressure
coecient C
p
as follows:
C
p
=
p p
=
p p
1
2
V
2
(5.1)
The pressure coecient can be plotted for a certain airfoil. Suppose the pressure coecient at a point
on an airfoil at low speeds (M
0) is measured. If the air velocity increases, also the absolute value
of the pressure coecient increases (negative C
p
get even more negative). This is, according to the
Prandtl-Glauert rule, approximately equal to:
C
p
=
C
p,0
_
1 M
2
(5.2)
Where C
p,0
is the pressure coecient at low speeds. For M
< 0.3 compressibility eects dont need to
be taken into account, and for M
> 0.7 this formula loses its accuracy. Therefore this formula is only
really applicable for 0.3 < M
< 0.7.
Now lets dene the normal force coecient and the axial force coecient for unit length (for just 1m of
the wing) in the same way as the lift coecient:
c
n
=
N
q
c
c
a
=
A
q
c
(5.3)
It can then be shown that the normal force coecient is also equal to the following integral:
c
n
=
_
1
0
(C
p,l
C
p,u
)d
x
c
(5.4)
Combining this with equation 2.1 results in:
c
l
= c
n
cos c
a
sin (5.5)
Most aircrafts have their cruising angle of attack at < 5
. And for such small angles of attack, sin 0
and cos 1. So c
l
c
n
, and then equation 5.4 can also be used to calculate the lift coecient.
But for this lift coecient, the Prandtl-Glauert rule can also be applied. If c
l,0
is the lift coecient for
low air velocities (M
< 0.3), then the lift coecient at higher Mach numbers is:
c
l
=
c
l,0
_
1 M
2
(5.6)
3
6 Critical pressure coecient
Using the denition of the pressure coecient, the denition of the dynamic pressure, the formula for the
speed of sound, the isentropic ow relations and a few assumptions, the following formula can be derived:
C
p
=
2
M
2
_
_
1 +
1
2
( 1)M
2
1 +
1
2
( 1)M
2
_
1
1
_
(6.1)
So for a certain atmosphere and a constant free-stream Mach number, the pressure coecient only
depends on the local Mach number M on the wing. To nd the critical pressure coecient, we should
ll in M = 1. This results in:
C
p
=
2
M
2
_
_
2 + ( 1)M
2
+ 1
_
1
1
_
(6.2)
So the critical pressure coecient only depends on the Mach number, which is quite an interesting thing.
To nd the critical pressure coecient, the Prandtl-Glauert rule should be used. If M
cr
is the critical
Mach number, and C
p,0
is the lowest pressure coecient on the wing for low air velocities, then the
following formula applies:
C
p,0
_
1 M
2
cr
= C
p
=
2
M
2
cr
_
_
2 + ( 1)M
2
cr
+ 1
_
1
1
_
(6.3)
7 Supersonic ight
To reduce the critical Mach number M
cr
, a plane can be equipped with swept wings. Because these wings
have a certain angle with respect to the airow, the speed of the air relative to the leading edge of the
wing is reduced. If is the angle between the leading edge of the wing, and the line perpendicular to the
direction of ight (so for normal planes = 0), then the new critical Mach number is
Mcr
cos
. However,
to use the component of the air velocity normal to the leading edge of the swept wings is not always
accurate. However, the following formula is correct:
M
cr,normal
< M
cr,swept
<
M
cr,normal
cos
(7.1)
But if supersonic ight does happen, shock waves occur. These shock waves have a certain angle with
respect to the direction of ight. For at plates, this angle can easily be calculated:
= arcsin
1
M
(7.2)
4
