= Aerodynamic center
The \b[aerodynamic center] is the reference point about which an <airfoil>'s pitching <torque> is approximately independent of <angle of attack> over a stated flow regime. More generally the defining test uses the pitching-moment coefficient at fixed flow conditions. The <aerodynamic center> simplifies replacing a distributed <pressure> field by an aerodynamic <force> and a nearly constant residual <torque>. It differs from the <center of pressure (fluid mechanics)>, where the resultant's pitching <torque> is zero and whose position can move as <angle of attack> changes.<Aerodynamic Center by NASA Glenn Research Center>[Angle of Attack and Aerodynamic Center]{ref}
A thin-<airfoil> model makes the distinction quantitative. Let $c$ be the <airfoil chord>, $S$ a reference <area> and $q_\infty=\rho V^2/2$ the <dynamic pressure>. Define $C_L=L/(q_\infty S)$ and $C_m=M/(q_\infty Sc)$. Take positive $x$ aft from the leading edge and positive pitching <torque> nose-up. In a chord-line lift model, moving the reference point gives
$$
C_m(x)=C_m(0)+\frac{x}{c}C_L.
$$
Consequently, when $\partial C_L/\partial\alpha\ne0$,
$$
\boxed{\frac{x_{\rm ac}}c=-\frac{\partial C_m(0)/\partial\alpha}{\partial C_L/\partial\alpha}.}
$$
Here $\alpha$ is <angle of attack>; differentiating holds the other test conditions fixed. A vanishing lift slope or strongly changing slopes defeats this simple inference.
For an original model choose $C_m(0)=-0.05-C_L/4$. Its <aerodynamic center> is $x_{\rm ac}=c/4$, with $C_m(x_{\rm ac})=-0.05$. In contrast, setting the pitching <torque> to zero gives
$$
\frac{x_{\rm cp}}c=\frac14+\frac{0.05}{C_L}.
$$
The <center of pressure (fluid mechanics)> moves from $0.35c$ to $0.30c$ as $C_L$ increases from $0.5$ to $1$. With $c=1\,\mathrm m$ and $q_\infty S=1000\,\mathrm N$, the <aerodynamic center>'s residual pitching <torque> remains $-50\,\mathrm{N\,m}$ while <lift> doubles. At zero <lift> the nonzero pure <torque> cannot be represented by a finite lift application point.
A <wind tunnel> test can measure <lift> and pitching <torque> about a fixed support while stepping <angle of attack>. The measured slopes determine the inferred <aerodynamic center>; approximately flat moment data about that point test the model. NASA describes the quarter-chord position as characteristic of most low-speed <airfoils>. Flow separation, compressibility and three-dimensional wing geometry require their own assessment rather than transferring this location unchanged.<Aerodynamic Center by NASA Glenn Research Center>[Aerodynamic Center and Wing Shape]{ref}
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