The relation between the angle of attack $\alpha$ of an airplane's wing and the lift $F$ generated by it is quite complex. If I treat air particles as tennis balls hitting the wing at the front, then we know that
the number of hits per unit time $\propto$ the area of the wing facing the front $\propto$ $\sin \alpha$ (given that the airspeed is constant).
On the other hand, if we resolve forces exerted by each air molecule on the wing,
Lift per unit area $\propto$ Force exerted by the molecules on that area $\times \cos \alpha$; Force exerted by the molecules $\propto$ component of air velocity perpendicular to the wing $\propto$ airspeed $\times \sin \alpha$.
Conclusion: Lift $\propto$ $\sin \alpha\cos\alpha \propto\sin 2\alpha$.
Now if we look at the experimental results in the graph below, there is a maximum at $\alpha=\pi/4$, as expected. The model I describe above works well in the post-stall region(Large angle of attacks). However, I cannot come up with any simple way to model the maximum at $\alpha=15^\circ$.
How can I find an equation to model the behavior of small $\alpha$? Can anyone give me something in the form
where $f$ is a function of $\alpha$ to describe the lift for small $\alpha$?
(I don't expect the equation to be in any way simple...)
Please don't give me a general and lengthy verbal description of why there is a peak at $\alpha=15^\circ$. There are already plenty of posts explaining this on the internet. I am looking forward to seeing an equation.
The type of wing I am considering is just simple blades that have (almost) uniform thickness and is (almost) planar. (on the rotor of a fan, for example)
Any help is greatly appreciated.
Image taken from http://www.aerospaceweb.org/question/airfoils/q0150b.shtml