# On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil
## 1 Introduction  
The transonic buffet refers to the non-linear phenomenon that the shock moves back and forth in a specific period on the suction side of the airfoil under the interaction between the shock and the boundary layer (Type II) or on both of the airfoil surfaces (Type I), which gives rise to the drastic fluctuation of the lift and the moment. Various physics mechanisms, such as the shock wave, K-H instability, shock-induced intermittent flow separation, the generation and propagation of sound, and so on, are involved [1]. When aircraft go through constant low-frequency buffet conditions combining specific upstream Mach numbers (\(Ma\)) with angles of attack (\(AoA\)), the fatigue life of the structures can be reduced, which poses risks to the performance of aircraft. As a consequence, it is of great importance to study the mechanisms of transonic buffets in aerospace. [PERSON] observed three types of transonic buffet phenomenon [2]. Type A oscillation refers to a small and sinusoidal motion of shock on the upper surface of an airfoil; Type B oscillation refers to a larger motion of the shock with the shock vanishing at some particular phases during the downstream excursion; and Type C oscillation refers to the behavior that compression waves accumulate somewhere downstream so as to form a shock, while on the other hand, the shock gradually evolves into a weak shock or compression wave until leaving the leading edge of the airfoil when the shock travels upstream. Experimental and numerical results show that the main difference among these three types of transonic buffets is due to Reynolds number (\(Re\)) effects. There are mainly two theories to shed light on the buffet phenomena, despite the fact that the mechanisms behind the buffet are still not fully understood. [PERSON] [3, 4] proposed a self-sustained oscillation model, as Figure 1 illustrates, that defines a close loop in whichperturbations emanating from the shock travel downstream at the speed of \(u_{p}\). When they reach the trailing edge, upward Kutta waves propagate within the subsonic area above the separated shear layer at a speed of \(u_{c}^{\prime}\) until they reach the shock. According to [PERSON], the estimated buffet frequency can be expressed as Equation (1):  
\[f_{b}\ =\bigg{(}\frac{c-x_{s}}{u_{c}^{\prime}}+\frac{c-x_{s}}{u_{p}}\bigg{)}^{ -1}. \tag{1}\]  
\(f_{b}\) refers to the buffet frequency, \(x_{s}\) refers to the mean location of the shock on the upper surface, \(c\) refers to the chord, and \((\cdot)^{\prime}\) denotes the velocity propagating upward. \(u_{p}\) and \(u_{c}^{\prime}\) can be derived using various approaches like the cross-correlation method, phase shift, wavenumber-frequency spectrum, and so on. However, some researchers argue that [PERSON]'s model may not be suitable. [PERSON] et al. [5] pointed out that the path between the trailing edge and the shock outside the boundary layer is not strictly necessary for the instability, even though the acoustic waves can be superimposed on the buffet phenomena based on unsteady Reynolds-averaged Navier-Stokes equations simulations (URANS) locally filtered with a selective frequency damping (SFD) technique. [PERSON] et al. [6] found that the wave propagation speed does not match their simulations and that these waves are associated with vortex shedding, which plays no major role in buffet. In contrast to [PERSON]'s model [3; 4], [PERSON] et al. [7; 8] established that the transonic buffet is actually a global linear instability. The global instability linearizes the Reynolds averaged Navier-Stokes equations (RANS) to study the most unstable eigenmode of the flow field and determine the buffet onset, which is consistent with a supercritical Hopf bifurcation. As a peculiar case of buffet, the lock-in phenomenon is generally defined as the synchronization of the buffet frequency with the airfoil excitation/natural frequency.