lower surface lifts act along the same vertical 
line. An increase in.lift on the symmetrical 
airfoil produces no change in this situation and 
the center of pressure remains fixed at the aero- 
dynamic center. 
The location of the aerodynamic center of an 
airfoil is not affected by camber, thickness, and 
angle of attack. In fact, two-dimensional in- 
compressible airfoil theory will predict the 
aerodynamic center at the 25 percent chord point 
for any airfoil regardless of camber, thickness, 
and angle of attack. Actual airfoils, which 
are subject to real fluid flow, may not have the 
lift due to angle of .attack concentrated at the 
exact 25 percent chord point. However, the 
actual location of the aerodynamic center for 
various sections is rarely forward of 23 percent 
or aft of 27 percent chord point. 
The moment about the aerodynamic center 
has its source in the relative pressure distribu- 
tion and requires application of the coefficient 
form of expression for proper evaluation. The 
moment about the aerodynamic center is ex- 
pressed by the following equation : 
where 
A&, = moment about the aerodynamic center, 
a.c., ft.-lbs. 
CMa.c,=coefbcient of moment about the a.c. 
q= dynamic pressure, psf 
S=wing area, sq ft. 
c=chord, ft. 
The moment coefficient used in this equation is 
the dimensionless ratio of the moment pressure 
to dynamic pressure moment and is a function 
c ML3.C. 
%.c. = p- 
of. the shape of the airfoil mean camber line. 
Figure 1.22 shows the moment coefficient, 
NAVWEPS O&601-80 
BASIC AERODYNAMICS 
C%C. versus lift coefficient for several repre-. 
sentative sections. The sign convention ap- 
plied to moment coefficients is that the nose-up 
moment is positive. 
The NACA Ooog airfoil is a symmettical sec- 
tion of 9 percent maximum thickness. Since 
the mean line of this airfoil has no camber, 
the coefhcient of moment about the aerody- 
namic center is zero, i.e., the c.p. is at the ac. 
The departure from zero cno.+ occurs only as the 
airfoil approaches maximum lift and the stall 
produces a moment change in the negative 
(nose-down) direction. The NACA 4412 and 
63,-412 sections have noticeable positive cam- 
ber which cause relatively large moments about 
the aerodynamic center. Notice that for each 
sectionshowninfrgure 1.22, the c,,,.... isconstant 
for all lift coefficients less than cl,-. 
The NACA 23012 airfoil is a very efficient 
conventional section which has been used on 
many airplanes. One of the features of the 
~section is a relatively high c& with only a 
small c,,,,,,; The pitching moment coefficients 1 
for this section are shown on figure 1.22 along 
with the effect of various type flaps added to 
the basic section. Large amounts of camber 
applied well aft on the chord cause large nega- 
tive moment coefficients. This fact is illus- 
trated by the large negative moment coefli- 
cients produced by the 30” deflection of a 25 
percent chord flap. 
me kc. is a quantity determined by the 
shape of the mean-camber line. Symmetrical 
airfoils have zero c,,,,. and the c.p. remains at 
the a.~. in unstalled flight. The airfoil with 
positive camber will have a negative c,,,~,~, 
which means the c.p. is behind the a.~. Since 
the c5.c. is constant in unstalled flight a certain 
relationship between lift coefficient and center 
of pressure can be evolved. An example of 
this relationship is shown in figure 1.22 for the 
NACA 63i-412 airfoil by a plot of c.p. versus 
c,. Note that at low lift coefficients the center 
of pressure is well aft-even past the trailing 
edge-and an increase in C~ moves the c.p, for- 
ward toward the a.~. The c.9. approaches the 
49 
Revised Jmuoy 1965