At high angles of attack, the empirical formulae for the lift and moment coefficients for airfoils pitching at high frequencies are given in terms of maximum dynamic moment coefficient \(\left(C_{M} \text { max }\right)_{\mathrm{DYN}}\) and the normal force coefficient \(\Delta C_{n v}\) due to vortex as follows.

\[
\left(C_{M \max }\right)_{\mathrm{DNN}}=-0.75 \Delta C_{n v}, \quad \Delta C_{n r}=1.5 \pi \sin ^{2}\left(\alpha_{v s}\right)_{\mathrm{eff}}
\]
and \(\left(\alpha_{v s}\right)_{\text {eff }}=\alpha_{o}+\Delta \theta \sin \left[(\omega t)_{v s}+0.45 k\right]\).
Here, \((\omega t)_{v s}\) and \(\Delta \theta\) is the pitch amplitude:
if \(\omega \Delta \theta \cos (\omega t)_{v s}<0.02\) then: \((\omega \mathrm{t})_{v s}=2 \tan ^{-}\left[\frac{\cos (0.995)}{1.5 k+\sin (0.995)+\left(\alpha_{0}-\alpha_{s}\right)}\right]\)
\[
x\left\{\sqrt{1+\frac{(1.5 k+\sin (0.995))^{2}-\left(\left(\alpha_{0}-\alpha_{s}\right) / \Delta \theta\right)^{2}}{\cos ^{2}(0.995)}}\right\}
\]
and if, \(\omega \Delta \theta \cos (\omega t)_{v s}>0.02\) then: \((\omega t)_{v s}=0.995+\sin ^{-}\left(\frac{\left.0.995+\alpha_{s}-\alpha_{o}\right)}{\Delta \theta}\right)\).
Wherein: \(\alpha_{s}\) is the static stall angle, \(\alpha_{o}\) is the average amplitude for the angle of attack. Using these formulae, find the normal force and the moment coefficients for NACA0012 airfoil, whose separation angle is \(14.5^{\circ}\) pitching with \(k=0.25\) at \(15^{\circ}\) average angle of attack with \(10^{\circ}\) pitch amplitude.