# On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil
## 1 Introduction
It occurs when the excitation/natural frequency and the fixed airfoil buffet frequency are close, about the ratio of 1, with sufficient moving amplitudes. In the lock-in region, the lift coefficient oscillates more drastically than that of the buffet on the fixed airfoil. [PERSON] and [PERSON] [9] discovered that the airfoil aerodynamic coefficients resonated tremendously when the NACA 64A010 airfoil was moving at a certain pitch frequency in their experiments, and [PERSON] [10] found similar resonances when the OAT15A airfoil flaps were in harmonic vibration at an excitation frequency near the buffet frequency. Given a prescribed oscillating motion of the airfoil, [PERSON] and fellows [11; 12; 13; 14] conducted simulations on the NACA0012 lock-in phenomena in deep buffet conditions and owed the LCO of the system to the phase lag/lead between the lift and the excitation movement, and their further works on aeroelastic airfoils also supported this point of view. [PERSON] et al. [15; 16] established a linear reduced-order model (ROM) to study the frequency lock-in phenomena on an aeroelastic  
Figure 1: [PERSON]’s self-sustained oscillation model. airfoil and found that the coupling of one structural mode and one fluid mode leads to frequency shift. [PERSON] et al. [17] carried out experiments on the prescribed oscillating airfoils and discovered a reduced level of shock oscillation and that the lock-in is only related to the major changes in the flow conditions and phase lagged by the time it takes until the altered pressure waves generated in the trailing-edge region reach the shock, which proves that [PERSON]'s model is still fit for the lock-in phenomenon. Note that in spite of considerable numerical simulations that deprive [PERSON]'s model of its feasibility in the transonic buffet, experimental results from [PERSON] et al. [17; 18] and [PERSON] et al. [19; 20; 21] all confirm that the buffet is driven by an acoustic feedback loop. Recently, energy maps have been adopted for flow-induced pitching airfoil or flutter analysis [22; 23; 24]. These studies focus on fabricating the energy extraction as a function of pitch oscillation's amplitudes and frequencies, or as a function of pitch oscillation's amplitudes and \(Ma\), which provides convenience in quantitatively determining the flutter boundary. The flutter means the instability of the structure under the interaction of the flow, where the LCOs of the aerodynamic coefficients manifest. In the transonic flutter of one degree of freedom (1-DOF) for pitch and two degrees of freedom (pitch and heave), the lower and upper boundaries are pinned down for the equilibrium energy transfer from the flow, where the instantaneous moment is in phase with the airfoil's angular velocity near the mean \(AoA\)[24]. But for the transonic buffet lock-in, where there is not only a phase difference between the moment and the pitch motion but also unstable characteristics of the flow, the boundary can be altered due to the complex shock/boundary interaction, which has not yet been illustrated from the perspective of the energy exchange. The utilization of the energy maps or energy extraction in the present study may give new insights into the mechanism of the lock-in phenomenon. This study aims to analyze the energy exchange between the airfoil and the flow when buffet lock-in occurs under given pitch oscillations near the buffet onset, where small pitch amplitudes should have significant impacts, and to try to relate the lock-in boundaries to the pitching component of the moment or of the flow field. The paper is structured as follows: The instruction is followed by the theoretical background of the energy map used in this study in Section 2 and the numerical set-up and validation in Section 3. Section 4 exhibits and explains the results in detail through the modified energy map proposed in this study and the relative phase map. The concluding remarks are given in Section 5.