# On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil
## 4 Results and Discussions
### Lock-In Boundaries Determination  
The energy map is firstly applied to seek the lock-in boundaries of the prescribed pitching airfoil. The energy extraction is modified to only take advantage of the pitching component of \(C_{m}\), which is noted as \(C_{m\_motion}\) and the equation goes as:  
\[E^{*}=\int_{t_{R}}^{t_{R}+T}0.5\rho U_{\infty}^{2}c^{2}C_{m\_motion}\dot{\theta}dt \tag{15}\]  
Figure 10: Relative phase \(\Phi\) between \(C_{m}\) and the angular velocity \(\dot{\theta}\) versus frequency ratio \(f_{p}/f_{b}\). Figure 9: Gain of the amplitude of the pitching component \(A_{motion}\) and the airfoil pitch amplitude \(A_{\theta}\). Further, Equation (15) can be transformed to a constant considering that the phase difference \(\Phi\) between the angular velocity and the FFT amplitude of the \(C_{m,motion}\) marked as \(A_{motion}\) are known from FFT analysis in 4.1. The final form of the modified energy extraction is  
\[ \begin{split} E^{*}=\int_{t_{n}}^{t_{n}+T}0.5\rho U_{\infty}^{2}c^{ 2}C_{m,motion}\dot{\theta}dt\\ =\int_{t_{n}}^{t_{n}+T}-0.5\rho U_{\infty}^{2}c^{2}A_{motion}cos \big{(}2\pi f_{p}t+\Phi\big{)}\times 2\pi f_{p}A_{\theta}cos\big{(}2\pi f_{p}t \big{)}dt=-0.5\pi\rho U_{\infty}^{2}c^{2}A_{motion}A_{\theta}cos(\Phi)\end{split}  \tag{16}\]  
where \(C_{m,motion}\) is approximated by \(A_{motion}cos(2\pi f_{p}t+\Phi)\). The modified energy map based on Equation (16) as a function of the airfoil pitch amplitude and the frequency ratio \(f_{p}/f_{b}\) is plotted in Figure 11, interpolated from 40 data points presented in Figures 9 and 10 through cubic spline, where the frequency responses of selected test cases are also marked. The positive contour level suggests that the airfoil absorbs energy from the flow and should develop an unstable flutter, while the negative contour level means that the flow extracts energy from the airfoil motion and the airfoil motion should decay over time in an unforced case. The zero-contour level is usually related to flutter instability. The sign of the modified energy map is determined by \(cos(\Phi)\) according to Equation (16), and two branches of \(E^{*}=0\) exist on the map (solid line), conforming to the transonic flutter boundary branches in [24]. We can see that for both upper and lower frequency bounds at \(E^{*}=0\), there are regions corresponding to the stable range where \(\frac{dE^{*}}{dA_{\theta}}<0\) allowing the airfoil to pitch sustainably without extra damping, and also regions corresponding to the unstable range for \(\frac{dE^{*}}{dA_{\theta}}>0\) where small perturbations can break the equilibrium. However, these zero-energy transfer boundaries do not indicate the lock-in boundaries, and they are not even close to the lock-in boundaries. According to [PERSON] et al. [15; 16], the lock-in is in essence 1-DOF flutter caused by the linear coupling between the structural mode and the fluid mode that leads to the instability of the structure. The zero energy exchange, on the other hand, is an usual indicator for the flutter boundary in stable flows [23; 24]. Such discrepancy suggests that the idea of the energy transfer between the flow and the airfoil may fail to explain lock-in phenomena in unstable flows like transonic buffets, for the exact energy extraction varies drastically within each pitching cycle under the influence of the buffet outside the lock-in regime. Figure 11: Modified energy map and the frequency response of the moment coefficient. Solid line: the flutter boundary at \(E^{*}=0\). Alternatively, we plot the normalized relative phase map defined by \(sin(\Phi)\) based on the data points shown in Figure 10 to avoid discontinuity, along with the frequency responses of the test cases marked in Figure 12. The phase shifts, where \(C_{m,motion}\) is exactly in phase with the angular velocity, generally outline the lock-in region except for onset at \(A_{\theta}\geq 0.4^{\circ}\).