## 3 Numerical Algorithms
### Algorithm and Mesh Validation  
To validate the prediction accuracy of the numerical algorithms, a wind tunnel experiment was carried out on a simplified train model. The wind tunnel is a low-speed open wind tunnel with an 8 \(\times\) 6 m cross section in Mianyang, Sichuan Province. The turbulence intensity of the incoming flow is 0.2%. The train model and the test section of the wind tunnel are shown in Figure 4. The train model is fixed with the subgrade by the force-measuring balances, which are mounted on the geometry center of each carriage. They are placed on a turntable, which could be used to adjust the angle of the train model and the incoming flow. As shown in Figure 4, the test model is a 1:8 scaled model with windshields and bogies included, and takes the subgrade and tracks into consideration. The scaled model takes a length of 4 m, a height of 0.4375 m, and a width of 0.4225 m. The velocity of the incoming flow is 60 m/s. The Reynolds number is 1.28 \(\times\) 106 taking the height of the model as the characteristic length. Meanwhile, in order to determine a proper mesh configuration for the current analysis, two mesh configurations are proposed on the wind tunnel model, just as Table 1 shows. Without special notification the aerodynamic loads on the train are all evaluated in the form of dimensionless coefficients as follows:  
\[\text{Pressure coefficient}\ (\text{Cp}):\ C_{p}=P/0.5\rho V^{2} \tag{7}\]  
\[\text{Drag coefficient}\ (\text{Cd}):\ C_{di}=F_{di}/0.5\rho V^{2}S \tag{8}\]  
\[\text{Side force coefficient}\ (\text{Cs}):\ C_{si}=F_{si}/0.5\rho V^{2}S \tag{9}\]  
\[\text{Lift coefficient}\ (\text{CI}):\ C_{li}=F_{li}/0.5\rho V^{2}S \tag{10}\]  
where \(P\) denotes the relative pressure, \(\rho\) is the air density, and \(V\) is the train velocity as well as the speed of incoming flow in the tunnel experiment. \(S\) represents the maximum cross section area, which is 10 m\({}^{2}\) in this study. The order \(i\), ranging from 1 to 8, represents the number of each carriage. When \(w\) is used it represents the whole train. When \(i\) equals 0, it denotes the streamlined shape of the head, while 9 the streamlined shape of the tail. The pressure and forces on the train in the wind tunnel experiment are measured by means of steady tests, so the time-averaged computational results are taken for comparisons. In Figure 5, plots of pressure coefficients (Cp) of the head and the tail profile are presented, with computation and experiments included. It could be found that results from both coarse mesh and fine mesh are in accordance with the experiment, while the two are consistent in the head and fine mesh performs better compared to the experiment than coarse mesh around the tail, indicating that the adopted numerical algorithms could obtain the flow data around the train precisely. \begin{table}
\begin{tabular}{c c c c} \hline  & **Boundary Layers** & **Minimum Size of a Cell** & **Number of Cells** \\ \hline Coarse mesh & 15 & 10 mm & 29.82 million \\ Fine mesh & 20 & 1.5 mm & 69.22 million \\ \hline \end{tabular}
\end{table} 
Table 1: Mesh parameters. Figure 4: Wind tunnel models: (**a**) the computational train model; (**b**) the train model in the wind tunnel. Table 2 lists the coefficients obtained by two mesh configurations as well as the experiment. It can be reported that total drag coefficients (W-Cd) obtained by both kinds of mesh are nearly the same, but the drag coefficient for each carriage of the fine mesh configuration agrees better with that of the experiment. The errors of H-Cd, M-Cd, and T-Cd with fine mesh are 1.6%, 0.1%, and 9.4%, respectively. Therefore, although the pressure coefficients along the longitudinal section of the streamlined shape for both mesh configurations agree well with the experiment, the fine mesh configuration was adopted for current study due to its excellent performance in predicting the drag coefficients for each carriage of the train model.