# On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil
## 3 Numerical Setup and Validation
### Numerical Verification  
The time-averaged pressure coefficient \(C_{p}\) and the root-mean-square error (RMSE) of the suction side pressure derived by the RSM and the SST are compared with the experiment in Figure 4. The trend and slope of \(C_{p}\) from the RSM match well with the experiment, while the RMS slightly underestimates the shock motion for the simulated upmost position of the shock, which is downward of the upmost position measured in the experiment, as Figure 4a exhibits. The mean shock location of the RMS is at \(0.49c\), close to the experimental mean location at \(0.45c\). The magnitude of pressure fluctuation is well predicted by the RSM compared with the experiment. On the other hand, however, though the SST yields the shock oscillation, the SST presents an inferior performance as the \(C_{p}\) curve deviates a lot from the experiment and the shock motion is much overestimated, as shown in Figure 4b. The magnitude of pressure fluctuations is also underestimated by the SST. The shape of both the \(C_{p}\) and the pressure RMSE cannot be seen as similar to those in the experiment, probably because the SST may generate another flow separation phenomenon that sustains the shock oscillation. Compared with the literature [27; 28], the inferior performance of the SST model in this study may result from the different resolution of the grid around the shock, and on the other hand, from the different spatial discretization precision (second-order MUSCL in [27] and third-order MUSCL in this paper), since the buffet simulation is highly sensitive to the grid resolution [29; 30] and the numerical discretization scheme [31; 32]. To regenerate a mesh setup that meets all requirements of SST and incorporates third-order spatial discretization precision is out of the scope of our discussion. Therefore, the RSM is adopted in the further investigation of this paper. Figure 3: Computational domain of the test case: (a) the whole domain; (b) grids around the airfoil. Apart from the aforementioned verification, the grid independence on the OAT15A airfoil buffet is also investigated using a coarse mesh and a fine mesh compared with the medium size mesh. The numerical results are presented in Table 1. All meshes are tested with the same turbulence model and the same timestep size. With a coarse mesh, the calculated buffet frequency is higher than the experimental and medium or fine mesh, and the amplitude of the lift coefficient \(C_{l}\) is much smaller than other mesh tests. The simulated buffet frequencies from the medium and fine mesh are very close to the experimental result, and the amplitude of \(C_{l}\) are similar too, indicating that the accuracy of the medium and fine mesh is sufficient for the calculation. Considering the balance between accuracy and efficiency, the medium size mesh is used for the following sections. Figure 5 shows the transient flow velocity contours and streamlines in the vicinity of the airfoil derived by the RSM within a complete buffet period. Phase 1 and phase 5 mark the most upstream and downstream positions during the oscillation of the shock wave, respectively. In the downstream excursion (phase 1 to phase 5), the shock gradually strengthens, and the separation shear layer grows thinner and gradually attaches to the airfoil surface. When the shock travels from the downstream to the upstream (phase 5 to phase 8), the shock weakens and the shear layer becomes thick again, making the separation significant. Figure 5 demonstrates that the RSM used in this study is able to reproduce the oscillation of the shock on the upper surface of the airfoil. However, the URANS simulation used in this study is not capable of capturing the vortex shedding or delicate structures within the shear layer. To study the development of the turbulent structures and their influence upon the separation and the wake in the buffet, one may refer to the approaches on the prolate spheroid and bluff bodies by means of large-eddy simulation [33; 34; 35].