Center of Pressure
We recall that the center of pressure is the point about which the total moment is zero. To find the location of the center of pressure, move the couple to that point, as shown in Fig. 5.24, where ac is the aerodynamic center and the center of pressure, cp, is located a distance Ax downstream. For equilibrium, the total moment about the center of pressure must be zero. Thus,
L’Ax + Member = 0 ^ WAx + (CJcamber(qc2) = 0
AX (Cm )camber = _ П *2 ~ A1
c CI 4 _ CL
Because the aerodynamic center was already found at the quarter-chord point, the nondimensional distance from the leading edge to the center of pressure is given by:
xcp _ 1 _П (A2 A) _ 1 _ ^mac
c 4 4 C. 4 C.
Notice that the location of the center of pressure changes with lift. Also, for small values of lift coefficient, the center of pressure may be downstream of the airfoil. Referring to Fig. 5.24, this is because the moment due to camber is a constant, depending only on geometry, so that as the L’ decreases, the lever arm Ax must become increasingly larger to balance the constant moment. The fixed aerodynamic center, then, is a more convenient reference point than the center of pressure, and the load system on an airfoil is described most conveniently by a lift force and a constant moment, both acting at the aerodynamic center.
This discussion concludes the mathematical development of the thin-airfoil theory. Important properties of arbitrary thin airfoils now can be evaluated with relative ease. Also, useful physical insight into the role of camber can be gained.
For example, consider a simple thin airfoil as shown in Fig. 5.25.
We let the maximum camber, H, be fixed and let the chordwise location of the maximum camber, L, vary. Applying the thin-airfoil solutions developed herein, we find that as the location of (fixed) maximum camber ratio H/c moves aft from 25 to 95 percent chord, the angle of zero lift (and, hence, the lift coefficient at an angle of attack) increases dramatically. The magnitude of the zero lift angle increases by about a factor of 5. A similar result is observed regarding the moment coefficient about the aerodynamic center as the maximum camber moves aft. This says, for instance, that flaps and aerodynamic controls that act to change the camber of a wing section should be located near the trailing edge where there is maximum sensitivity to changes in camber.