Is there a formula for calculating lift coefficient based on the NACA airfoil?

Question

Are there any formulas that can be used to calculate this?

Answer 1

Thin airfoil theory can be used to calculate lift based on the shape of the mean camber line. This involves some integrals, but these have been worked out ahead of time for certain camber line families like the 4-digit one you are interested in.

$c_l = 2 \pi \alpha + \pi (A_1-2 A_0)$

$A_0 = \frac{1}{\pi} \int_0^\pi \frac{dy}{dx} d\theta$

$A_1 = \frac{2}{\pi} \int_0^\pi \frac{dy}{dx} \cos(\theta) d\theta$

$\frac{dy}{dx}$ Is the slope of the camber line.

We use the transformation:

$x = \frac{1}{2} (1-\cos( \theta))$

These integrals have been worked out for a 4-digit mean camber line (Houghton and Carpenter, 4th Ed, pp. 252).

$A_0 = \frac{m}{\pi p^2} [(2 p - 1) \theta_p + \sin(\theta_p) ]\\ + \frac{m}{\pi (1-p)^2} [(2 p - 1)(\pi - \theta_p) + \sin(\theta_p) ]$

$A_1 = \frac{2 m}{\pi p^2} [(2 p - 1) \sin(\theta_p) + \frac{1}{4} \sin(2 \theta_p) + \frac{\theta_p}{2}]\\ - \frac{2 m}{\pi (1-p)^2} [(2 p - 1) \sin(\theta_p) + \frac{1}{4} \sin(2 \theta_p) - \frac{\pi - \theta_p}{2}]$

For a 4412, $m=0.04$ and $p=0.4$. $\theta_p=\cos^{-1}(1-2*p)=1.3694$ radians. $A_0 = 0.00811$ and $A_1 = 0.1632$.

Pulling it all together gives:

$c_l = 0.46175 + 2 \pi \alpha$

Finite thickness and viscous effects will reduce the true value a bit from this, but it is pretty good. To do better than this, things get substantially more complex.