Compare the zero-lift angle you found for the NACA 4412 a) in the laboratory1
, and b) in homework
assignment #3, to that predicted by thin-airfoil theory. The equations for the camber line of a
NACA 4-digit series airfoil are:
Zc(x) = m
p
2
(2px x
2
) for x < p (1)
Zc(x) = m
(1 p)
2
[(1 2p) + 2px x
2
] for x > p (2)
For these equations, m is the maximum ordinate of the camber line (the actual maximum camber)
expressed as a fraction of the chord, and p is the chordwise position of the maximum ordinate
expressed as a fraction of the chord. For example, for the NACA 4412, m = 0:04 and p = 0:4: Note
that, in the above equations, both Zc and x are made dimensionless by the airfoil chord. You may
wish to consult example 6.2 in the textbook to aid in solving this problem. Discuss the validity of
thin-airfoil theory for determining the lift-curve slope and the zero-lift angle of attack.
2. Use the thin-airfoil theory to Önd the center of pressure (as a function of lift coe¢ cient) and the
moment coe¢ cient about the aerodynamic center for the NACA 4412. Plot the center of pressure
location vs. c` (be sure to include negative values for lift coe¢ cient). Discuss your result.
3. In some situations, it is necessary for an airfoil to have a positive moment about the aerodynamic
center but to also generate positive lift at  = 0. This type of airfoil could have a camberline given
by a cubic equation:
Zc = kx(x 1)(x b)
where Zc and x have been made dimensionless by the airfoil chord and k and b are dimensionless
constants.
(a) Find the lower limit on b; that is, what is the minimum value for b that would make c` positive
at  = 0?
(b) Find the upper limit on b; that is, what is the maximum value for b that would make cma c
positive?
(c) For a value of b that gives c` = 0 at  = 0, plot the camberline. Discuss the similarities and
di§erences between this camberline and a ìnormalî camberline, and explain why this airfoil
has positive moment coe¢ cient about the aerodynamic center at  = 0. (Recall that an airfoil
with ìnormalîcamber, such as the NACA 4412, has negative moment about the aerodynamic
center.)
Note that your integration may be easier if you use the identity cos2
 =
1
2
(1 + cos 2).
4. Alter the MATLAB code you have been using throughout the semester for lift and drag calculations
to also calculate the moment coe¢ cient about the quarter-chord. Note that, for a “thick” airfoil,
pressure forces in both x and z directions produce a moment:
mc=4 =
Z
airfoil
(x xc=4
)p dx +
Z
airfoil
zp dz
Make the above equation dimensionless to Önd cmc=4
. Use the TACAA data for the NACA 4412 at
= 6

through  = 15
, and run your code to calculate and plot the cmc=4
vs. angle of attack.
1
If you are not taking AEE 361, use Z L =