# On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil
## 4 Results and Discussions
### Harmonic Analysis  
The frequency contents of the energy extraction evolution \(E(t)\) and the moment coefficient \(C_{m}\) of for example \(A_{\theta}=0.3^{\circ}\) are also discussed in this section by the Fast-Fourier Transform (FFT) as Figure 8 demonstrates. From Case 3 to Case 5 both \(E(t)\) and \(C_{m}\) present a mono-frequency characteristic under 100 Hz: the buffet frequency locks onto the pitch motion frequency, and the buffet frequency vanishes. This synchronization of frequencies is the typical lock-in phenomenon reported by [PERSON] et al. [11], as they studied a transonic buffet on an oscillating NACA0012 airfoil and found that only when the airfoil oscillated at frequencies close enough to the buffet frequency would lock-in occur. The linear stage of the energy evolution from Case 3 to Case 5 results from the fact that the LCO moment of the airfoil and the motion possess the same frequency. In Case 1 and Case 2 and Case 6 to Case 9, however, \(E(t)\) and \(C_{m}\) display responses in the airfoil motion and the buffet frequency showing no existence of the lock-in phenomena. The buffet frequency lock-in phenomenon is reminiscent of the vortex-induced vibration of the flow past a cylinder, where most researchers owed the inducement to resonance [36; 37]. Figure 8: _Cont_. Figure 7: The energy evolution \(E(t)\) and the energy extraction \(E^{*}\) in each cycle of all cases at \(A_{\theta}=0.3^{\circ}\). In the following discussion, we lay emphasis on the pitching component of the moment coefficient \(C_{m}\), since it possesses a constant phase lag/lead with respect to the airfoil pitch motion, is one of the most dominant components in FFT, and is clear to determine its part of energy donation within a pitching period. While we have acquired the amplitude of the pitching component from FFT, the phase angle of the pitching component is also required. The phase difference to the angular velocity of the airfoil is also derived from the FFT. We have evaluated cases with a pitch amplitude of a wider range from \(A_{\theta}=0.1^{\circ}\) to \(A_{\theta}=0.65^{\circ}\). The gain model of the amplitude of the pitching motion component \(A_{motion}\) and the airfoil pitch amplitude \(A_{\theta}\) is given in Figure 9. The phase differences \(\Phi\) are presented in Figure 10. From Figure 10, it is clear that for a certain pitch amplitude, the relative phase \(\Phi\) curve undergoes two shifts from positive to negative, or vice versa, as the pitching frequency grows. Before the phase shifts, the value of the relative phase decreases as the pitch frequency increases, which conforms with [PERSON] et al. [12] and [PERSON] et al. [13] as they studied the prescribed pitch motion of the NACA0012 airfoil in deep buffet or pre-buffet conditions. The discrepancy between this study and theirs is that we study pitch motion with the mean \(AoA\) slightly exceeding the buffet onset, where small pitch amplitude should have a more significant influence on the lock-in phenomena compared with pitch motion in a deep buffet. In the previous study of [PERSON] and [PERSON] [13], the pitch amplitude of the airfoil \(A_{\theta}\) was \(0.002^{\circ}\) to ensure linear response at pre-buffet conditions, and they found damped resonance peaks for mean \(AoA\geq 4^{\circ}\). For smaller \(AoA\leq 3^{\circ}\), the response gain decays monotonically with increased pitch frequency. In this study, as Figure 9 shows, the resonance peak frequency reduces as \(A_{\theta}\) grows, which is also a trend of damped resonance peaks for larger pitch amplitudes above the buffet onset conditions. Also, as \(A_{\theta}\) continues to increase where the minimum \(AoA\) can be far below the buffet onset, it is possible to predict a monotonically decreasing behavior of the response gain, as [PERSON] and [PERSON] [13] suggested based on Figure 9, indicating the decline of the nonlinear effect that dominates the shock buffet oscillation. In deep buffet conditions,  
Figure 8: Frequency contents of all cases at \(A_{\theta}=0.3^{\circ}\).