Can an airfoil have multiple aerodynamic centers?

Question

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?

Answer 1

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.