## 2 Numerical Methods and Validation  
In the transonic buffet simulation, detached eddy simulation (DES) [1; 47; 48; 49] and large eddy simulation (LES) [12] series turbulence models can predict three-dimensional buffet flow fields. However, it is not practical to compute various upstream conditions by using DES or LES turbulence models to determine buffet onset boundaries because these methods require a grid resolution that is much higher than that for URANS simulations, resulting in excessively longer computational times. Meanwhile, [PERSON] [50] conceived in their 2007 book that numerical simulations of this phenomenon based on URANS current codes are quite successful in predicting the magnitude and frequency of periodic shock motions during buffet. Thus, in two-dimensional studies of transonic buffet mechanisms and transonic buffet flow controls, URANS [8; 9; 13; 16; 24; 30; 32; 43; 44] turbulence models such as SA, SST, and so on are widely employed, and favorable results have been obtained. To include transition effects, the transition SA turbulence model [51] was employed in the present study.  
The discretization schemes usually employed in compressible Navier-Stokes equations enclosed by URANS turbulence models are HLLC [13; 32], Roe [14; 16], AUSM + P [8], and AUSP + up [9]. [PERSON] et al. [8] stated that [PERSON] and [PERSON] schemes were not considered in their work due to the schemes' poor shock treatment when investigating the stability of the flow, and AUSM+ series schemes were their choice. By improving the AUSM+ scheme for shock-stable and accurate hypersonic heating computations, [PERSON] and [PERSON] [52] developed an AUSM + up2 scheme for a wide spectrum of Mach numbers, including hypersonic heating, low speed flow, and 3D aerodynamic applications. This scheme is included in the open source code SU2 [53], and it was employed in the present study to solve the compressible Navier-Stokes equations.  
The airfoil profiles widely used in transonic buffet studies are NACA0012 [9; 13; 17; 21; 54] and OAT15A [1; 30; 55]. By referring to the well-known NACA0012 2-D airfoil shock-buffet test case by [PERSON] and [PERSON] [54], the NACA0012 airfoil was used in the present study. The steady-state numerical methods were validated by the wind tunnel test results from reference [54]. The free stream Mach number was 0.751, the Reynolds number was \(9.5\times 10^{6}\), and the incidence angle of the NACA0012 airfoil was \(1.99^{\circ}\). The chord length of the NACA0012 airfoil was set to be 1.0 m in present study. The computational domain size and the structured grid is shown in Figure 1. The mesh node numbers on both the upper and lower surfaces of the airfoil were set to be 281, respectively. The far-field boundary was placed at least 50 times that of the chord length. The first grid layer thickness normalized by the chord length was set to be \(1\times 10^{-6}\), corresponding to y\({}^{+}\approx 1\). The calculated surface pressure agreed well with test results, as plotted in Figure 2.  
To check out the unsteady flow simulation's capability of presenting numerical methods, an unsteady transonic flow around an NACA0012 airfoil was simulated at \(\alpha=6^{\circ}\), M\({}_{\infty}=0.751\), Re \(=1\times 10^{7}\). The time history of lift coefficient is compared with that conducted by [PERSON] [56], as shown in Figure 3. The results indicate that the computed result matched well that from the published literature. Thus, the validated numerical methods and the grid parameters were adopted in the following research.  
Figure 1: Grid around an airfoil.  
Figure 2: Comparison of numerical and test results.