## 3 Computational Methods
### Validation of Computational Methods  
Many structures of the pantograph could be treated as cylinders. The acoustic finite element method is first validated by a cylindrical flow calculation, as shown in Figure 8. The cylinder has a diameter of 10 mm, the length of the cylinder is \(\pi\)D, and the radius of the flow field is 15D. The trimmer meshing method is employed, and the first layer thickness is 0.0035 mm with a growth ratio of 1.1 in the wall-normal direction. The value of normalized wall-normal distance y+ is defined as:  
\[\mathrm{y}^{+}=\frac{u_{*}y}{\
u} \tag{10}\]  
where y is the distance from the wall, u, is the friction velocity (u\({}_{*}=\left(\frac{\tau_{\mathrm{w}}}{p}\right)^{1/2}\)), \(\tau_{\mathrm{w}}\) is wall shear stress, \(\rho\) is the air density and \(\
u\) is the air kinetic viscosity. The values of y+ are below 1 in downstream of the front facing stagnation line on the cylinder surface, which is adequate for the LES model. The cylindrical surface grid size is 0.25 mm. The total number of cells is about 6 million. Figure 8 shows the boundary conditions, namely, the pressure far-field boundary with a uniform inlet velocity of 72 m/s and a gauge pressure of 0 Pa, and the parallel plane (on the two ends of a cylinder) is set as periodic. Figure 8: Circular cylinder model domain and discretization. Figure 7: Medium sized computational mesh: (**a**) grid around pantograph; and (**b**) enlarged view of pantograph grid. The LES method is used to calculate the flow field. The pressure-velocity coupling method is addressed by the Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm. The discrete algebraic equation is solved by the Gauss-Seidel iteration technique. Figure 9 shows the generation, development, and detachment of the leeward vortices downstream of the cylinder. The vortices alternate axially in the downstream direction of the flow, forming a von Karman vortex street. It can be seen that the leeward side vortices are mainly generated when the airflow flowing from the cylindrical wall surface leaves the leeward side wall surface. The generation, development and shedding of the vortices lead to noise generation. The acoustic FEM is used to solve the far-field noise at an inflow Mach number is 0.21. In this study, Kato equation [22] is used to correct the SPL obtained from numerical simulations because the spanwise length of the LES computational domain is limited by the available computational resources. As a result, the simulation spanwise length, Lsim, of the geometry is shorter than the experimental spanwise length, Lexp, reported in [PERSON] et al. [23]. To compare the predicted SPL with experimental data, the value obtained from numerical simulations must be corrected. The SPL is monitored 185D above directly the cylinder. Figure 10 shows the validations of the aerodynamic results and SPL results of 185D. It can be seen that the pressure coefficient (Cp) and the corresponding Strouhal number (St) of the vortex shedding predicted by the present study agrees with the experimental results and other numerical simulation solutions obtained in [23; 24; 25; 26]. Therefore, it is reasonable to use the FEM for the far-field noise characteristic prediction in the following analysis. This paper seeks to show that the numerical method is able to model the evolution of the structures in the gap between the solid bodies. A comparison between downstream flow characteristics of the cylinder obtained by measurements and simulation using the same mesh level changes in the cylinder wake as used for the pantograph simulation is performed. As shown in Figure 11, the cylinder has a diameter of 0.05715 m, the separation distance, L, between the cylinders is 3.7D. The length is 3D. Figure 10: Validations of the Cp and SPL results at 185D (where f\({}_{0}\). = 1508 Hz). Figure 9: Vortex shedding on plane at 50% span(s\({}^{-1}\)). The trimmer meshing method is employed, the first layer thickness is 0.001 mm with a growth ratio of 1.2 in the wall-normal direction.