Angle of attack and lift: is there a model?

Question

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

$$\text{Lift}\propto f(\alpha)$$

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

Answer 1

Your tennis ball model is called "impact theory" and has been postulated by people like Isaac Newton but turned out to be false. Only in high supersonic and hypersonic flow can it be used for first-order approximations. In subsonic flow local pressure gradients are far more important for lift creation than the momentum of air molecules.

The best you can do for an analytic description is to divide the flow regime into two parts, one with attached flow at small angles of attack and the other with fully separated flow on the suction side. The transition region between both has up to now stubbornly resisted all attempts at an analytic formulation.

Any equation for the attached flow part must include those parameters:

• angle of attack (obviously, listed only for completeness)

• Surface area S of the projection of the body at zero angle of attack.

• Flow speed v.

• Fluid density $\rho$

• Body aspect ratio AR (ratio of width to depth). The gradient for slender bodies is $\frac{\pi}{2}\cdot AR$, as in $$L = \frac{\rho}{2}\cdot v^2\cdot S \cdot\frac{\pi}{2}\cdot AR$$

• In case of flat bodies with a predominant extension orthogonal to the direction of movement, aka wings, more parameters need to be added:

  • Wing sweep
  • Wing thickness ratio
  • Flow Mach number (ratio of flow speed to speed of sound)
  • Flow Reynolds number (ratio of inertial to viscous forces)

I cannot come up with any simple way to model the maximum at 𝛼=15∘

Nor can anybody else on this planet; however, given enough sweep and a low aspect ratio, the maximum is shifted to much higher angles of attack. Determining the location of this maximum is only possible for some specific cases, anyway.

One of those has been formulated by Laitone: If the local Mach number reaches 1.581, lift stops to grow with angle of attack (E.V. Laitone: Local supersonic region on a body moving at subsonic speeds, Symposium Transsonicum 1964, International Union of Theoretical and Applied Mechanics (IUTAM)). Richard Eppler used the drag coefficient in his airfoil code to predict the maximum lift point with good success. Note that each could apply this only in a narrow field - a general rule to find the maximum still eludes us.