How do i know there is always an aerodynamic center?

[DaNiEl!]

hi
i'm still learning these basic concepts of airplane stability, specifically the aerodynamic center, the point about which the pitching moment does not vary with AOA. i just wanted to know if there is a theoretical basis for this, or if it is just experimental. is there some sort of "mathematical" proof for the fact that there exists such a point on all airfoils?

 

[caslav.ilic]

Depends on what you mean by "theoretical basis". If the flow is assumed inviscid, incompressible, irrotational, i.e. representable by linear potential equation, and the airfoil something akin to a thin plate, then, if I remember correctly, it mathematically drops down to existence of such a point.

On the other hand, if you think realism, full Navier-Stokes[*] equations, then there is no such point in general. For example, think of an airfoil at near-transonic Mach number at zero AOA; it's AC is at ~25% chord, but, as the AOA increases, the flow will become transonic, shock forming on the upper surface, which will very much change the AC and everything else.

So, physically, I'd say AC ranks as much theoretical as the $C_l = a_0 (\alpha - \alpha_0)$ relation. However, in stability and other analysis concerning time-dependent, oscillatory effects, frequently the concept of "linearization about a point" is invoked. So you could also think of AC as property of the current AOA, as if the flow were behaving "linearly" for small perturbations of AOA around the current. This leads to mention of "moving AC", which by itself sounds as an oxymoron.

* Not even the existence of solution of NS equations has been proven, hence any aerodynamics result for NS-governed flow cannot be considered theoretical in the mathematics/physics sense.

--
Chusslove Illich (Часлав Илић)