Thin Airfoil Theory Results

The basic equations derived from thin airfoil theory are repeated below:

Several important results are derived from these expressions and are described in the following sections.

Ideal angle of attack

The constants An just depend on the airfoil shape -- except for A0 which depends also on the angle of attack. The A0 term is a strange term in the Fourier series for g since it leads to a singularity at the leading edge (g -> infinity). Thus, there is one angle of attack, called the ideal angle of attack, at which A0 = 0 and the vorticity goes to 0 at the leading edge. This angle is called the ideal angle of attack.

Of course in real flows, the vorticity would not become infinite (why?), but the concept of ideal angle of attack is still important, identifying the flow conditions for which leading edge pressure peaks are avoided.

Lift curve slope

The rate of change of lift coefficient with angle of attack, dCL/da can be inferred from the expressions above.

The result, that CL changes by 2p per radian change of angle of attack (.1096/deg) is not far from the measured slope for many airfoils. The effects of thickness and viscosity which are ignored here cancel each other out to some extent with the result that most airfoils have a lift curve slope within 10% of the 2p value given by thin airfoil theory.

Aerodynamic center and pitching moment at L = 0

The expression for pitching moment coefficient measured about the leading edge is given above. If we measure the moment about another reference center at a position x0/c, the expression becomes:

Note that if we choose the point x0 = 0.25c, then the lift dependence drops out and the moment coefficient measured about this point is independent of the angle of attack*. The point about which dCm/dCL = 0 is called the aerodynamic center and according to thin airfoil theory it is the quarter chord point of the airfoil. Experiments show this to be quite close.

Results for a Simple Example, Approximate General Results

The parabolic camber meanline is used as an example of thin airfoil theory. Results for this case serve as useful first approximations for any thin cambered airfoil.

Assume: z(x) = 4 h x (1-x)


Thin airfoil theory can also be used to estimate the effect of flap deflection on airfoil lift and moment. It also provides an estimate of the hinge moments vs. the deflection angle and the angle of attack. This is a good problem to work on your own.

The results are:
DCl = [2 (p-q) + 2 sin q] d
DCmo = -d/2 sin q (1 + cos q) = -2 d xf1/2 (1-xf)3/2

where:

Because of the effects of viscosity, these results tend to overestimate, to some extent, the lift and moment due to flap deflections.

*The Cm about this point, denoted Cma.c., is therefore equal to the Cm at zero lift. Cm0, as it is written, is also independent of the moment reference location and so is a particularly useful quantity:
Cm0 = Cma.c.