## 2 Numerical Simulation Methodology
### Validation of the Methods  
Herein, we considered the lack of aerodynamic noise experiment data for real pantographs and the inevitable simplification of pantographs for numerical simulations. For the purpose of verifying the reliability of the computational method, a classical finite length cylindrical flow is selected for numerical simulation in this paper. The rod-airfoil experiment was carried out by [PERSON] et al. [27] and was introduced as a validation case in a previous study on pantograph noise [13]. The diameter of the rod (D) obtained for the experiment is 10 mm, and the spanwise length is 30 D. The incoming flow velocity is 72 m/s and the free flow turbulence intensity is set as 0.8%. The Reynolds number based on the diameter is about 4.8 \(\times\) 10\({}^{4}\), based on rod diameter. Figure 1 displays the computational domain for the validation. The size of 40 D in the stream-wise direction and 20 D in the corresponding normal direction is employed. In addition, considering the computational resources due to the large amount of mesh brought about by the experimental spanwise length, here the spanwise length is chosen as \(\pi\)D according to references [13; 28; 29; 30]. The computational mesh is shown in Figure 2. Two refinement zones are used to refine the volume in the wake flow region of the rod. To make y+ values meet the requirement and y+ \(<\) 1 in this paper; the first layer thickness is 0.004 mm and the growth ratio of 1.1 is given in the wall-normal direction. The radial grid near the wall is generated according to reference [13; 28]. The total number of grid cells is about 1 million. Moreover, the size of grid cells in the spanwise direction is about 1 mm and the spanwise direction is uniformly distributed with about 33 grid cells. In a previous study, the authors of [31] found that the two-point pressure correlation length is about 3 D, indicating that the spanwise space in this study is physically reasonable. To be consistent with the study by [PERSON] et al. [13], the velocity inlet and pressure outlet boundary conditions are prescribed. The incoming free flow velocity is 72 m/s, which is consistent with the experiment. Of note, both the velocity inlet and pressure far-field conditions are compared here and the results show few discrepancies. The latter is used in the following simulations of the pantograph. In addition, the turbulence intensity is set to 0.8%. For the spanwise and normal directions, symmetry boundary conditions are used. Contrarily to the periodic condition, the symmetry condition only makes the velocity components in the z-direction disappear and does not correlate the aerodynamic flow field variables in the existing planes. In addition, the two-point spanwise pressure correlation on the separate rod was in good agreement with the experimental results, which is very important for acoustic calculations and cannot be acquired with the periodic condition. Therefore, the symmetry condition should be a better choice here [32]. Figure 1: Schematic diagram of the computational domain for the validation. Figure 2: Schematic diagram of the computational mesh for the validation. Table 1 presents the comparison of the drag coefficients. Of note, the average drag coefficient and its fluctuating values are in good agreement with those reference experimental results, indicating the rationality of the computation method in this study. Herein, we considered the effect of spanwise correlation, according to reference [35]. At the sound monitoring point, which is located at a distance of 185 D in the normal direction of the flow, the far-field SPL versus the Strouhal number is displayed in Figure 3. It is shown that the base level of the spectrum is captured. Specifically, the acoustic peak of nearly 90 dB appears at St = 0.197, indicating the shedding frequency of Karman vortex. Meanwhile, the second and third modes can also be accurately predicted by the present numerical simulations.