# Can an airfoil have multiple aerodynamic centers?

Aerodynamic center is a point where the moment is independent of angle of attack. Can we have 2 or more such points? Does it necessarily lie inside the airfoil?

No, it's not possible to have them simultaneously. Here's a proof by contradiction:

Assume that there are two points on the airfoil that have constant pitching moment, located at $$x_0$$ and $$x_1$$, each having constant moments $$M_0$$ and $$M_1$$, respectively, and that $$\frac{\partial{M_0}}{{\partial\alpha}}=\frac{\partial{M_1}}{{\partial\alpha}}=0$$ (which make them aerodynamic centres).

Therefore, using the $$M_0$$, we can write the moment about any other point as:

$$M(x)=M_0 + L(x-x_0) \tag{1}$$

where $$L$$ is lift and is a function of the AOA, $$\alpha$$.

Using (1), we can try to calculate the moment at $$x_1$$, which gives:

$$M_1 = M(x_1)=M_0+L(x_1-x_0) \tag{2}$$

Differentiate (2) with respect to $$\alpha$$, and we get:

$$\frac{\partial M_1}{\partial \alpha} = \frac{\partial L}{\partial \alpha}(x_1 - x_0) \tag{3}$$

But (3) can't be zero since $$\frac{\partial L}{\partial \alpha}$$ is always non-zero, which contradicts with the assumption that $$\frac{\partial M_1}{\partial \alpha}=0$$.

However, if the lift is non-linear, then the AC will move around at different AOAs, each satisfying local linearity of the aerodynamic center definition.

• Actually, there is one for subsonic flow and one for supersonic flow, with a complicated transition at transsonic speed. But not two at the same flow condition. +1 – Peter Kämpf Jul 26 '20 at 15:51