# On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil
## 4 Results and Discussions  
### Energy Exchange of the Buffet on the Prescribed Pitching Airfoil  
The OAT15A airfoil is mounted in a 1-DOF pitch operating at an initial angle of attack of \(\theta_{0}=3.5^{\circ}\), with conditions of the coming stream, far field pressure, etc. the same as the fixed OAT15A airfoil. The pivot point is fixed at \(x=0.45c,\ y=0\) of the airfoil around the mean shock location. The pitch motion of the airfoil is defined as follows:  
\[\theta(t)=-A_{\theta}sin\left(2\pi f_{p}t\right) \tag{12}\]  
where \(f_{p}\) denotes the excitation frequency of the pitch motion and \(A_{\theta}\) denotes the amplitude of the pitch oscillation. The positive direction of the pitch motion goes counterclockwise. [PERSON] et al. (2014) suggested that the amplitude of the pitch motion should not be too small; otherwise, the aerodynamic buffet response cannot be affected for airfoils in deep buffet conditions, yet near the buffet, onset small disturbances should have a noticeable influence. The experimental buffet \(AoA\) onset of the OAT15A airfoil at identical conditions with the test case in Section 3.1 is about \(3.2^{\circ}\)([PERSON] et al., 2015), which determines that \(A_{\theta}\) should be small enough to ensure the occurrence of buffet. Furthermore, to allow the lock-in and LCO to occur, the system's damping must approach zero so that the system is on the brink of instability. The pitch system in this study thus has no damping. According to the definition in Equation (5), the energy extraction from the flow by the airfoil, regardless of damping, is defined as follows:  
\[E^{*}=\int_{t_{n}}^{t_{n}+T}M\dot{\theta}dt \tag{13}\]  
where \(M=0.5\rho U_{\infty}^{2}c^{2}C_{m}\) is the aerodynamic moment about the elastic axis and \(T\) is the period of the pitch oscillation. In this section, the evolution of the energy extraction \(E^{*}\) in Equation (13) in a continuous form  
\[E(t)=\int_{0}^{t}M\dot{\theta}dt \tag{14}\]  
is also investigated. The prescribed pitch frequencies with various pitch amplitudes \(A_{\theta}\) are given in Table 2, and the pitch cases are all calculated after the same timestep of the buffet on the fixed airfoil, recording at least 50,000 steps each. Figure 6: \(AoA\) versus (**a**) \(C_{l}\) and (**b**) \(C_{m}\) for RSM simulation and AGARD CT5 case. In Figure 7, the time-accurate energy evolution \(E(t)\) from the flow and the moment coefficient \(C_{m}\) of \(A_{\phi}=0.3^{\circ}\) are plotted, as well as the energy extraction \(E^{*}\) of each cycle. Leaving out the first several transient cycles, the energy extraction \(E^{*}\) from Case 1 to Case 4 reveals a negative energy transfer from the flow to the airfoil. As the pitch frequency grows, the flow transfers positive energy to the airfoil in Case 5, where \(f_{p}/f_{b}=1.154\). The energy extraction evolution \(E(t)\) from Case 3 to Case 5 is defined as the linear stage in this paper, where the energy exchange within each cycle is approximately the same after the lift coefficients reach constant LCOs as time develops. From Case 1 and Case 2 and from Case 6 to Case 9, the energy evolution exhibits an oscillating characteristic compared with the former cases, referred to as the linear stage, where the exact energy exchange within each cycle varies from negative to positive, modulated by a secondary frequency lower than the excitation. The energy evolution in these cases is thus defined as the non-linear stage and seems unlikely to reach a stationary regime as time develops, implying a rather independent motion between the pitch excitation and the buffet. The oscillating behavior of the non-linear stage poses a challenge for determining the exact energy extraction over a pitch period. In this sense, the energy extraction in Equation (13) should be modified to address the non-linear stage of the energy evolution.