## 3 Numerical Algorithms
### Boundary Conditions  
The geometry of TP1 is shown in Figure 6. TP1 is an eight-grouped model, with the simplified bogies and windshields included. Meanwhile, components such as pantographs, doors, and windows are all neglected. For TP2, TP3, and TP4, the difference from TP1 only exists in the streamlined shapes. The cross section area S is 11.2 m, while the train body height H is 3.5 m. For the convenience of analyzing the aerodynamic performance of each component, the orders of each carriage and each bogie are prescribed as shown in Figure 6. From small to large the numbers represent the parts from the head to the tail, while the streamlined part of the train is also named independently for further analysis.  
The x axis is set along the train axis, with the origin set on the middle of the train. The z axis is set to the vertical direction, as Figure 5 shows. The y axis could be decided in accordance with the right-hand rule. The computational domain is shown in Figure 5. Taking the height of the train as the characteristic length H, the inlet boundary is 55H away from the head nose and the outlet boundary is 85H away from the tail nose. The height and width of the computational domain are 28 H and 56H, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **H-Cd** & **M-Cd** & **T-Cd** & **W-Cd** \\ \hline Wind tunnel & 0.1346 & 0.0808 & 0.1323 & 0.3477 \\ Coarse mesh & 0.1276 & 0.0770 & 0.1565 & 0.3611 \\ Fine mesh & 0.1325 & 0.0809 & 0.1447 & 0.3581 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of Cd among numerical simulations and wind tunnel experiment.  
Figure 5: Plots of pressure coefficients (Cp) of the head and the tail profile: (**a**) head; (**b**) tail.  
Figure 6: Geometry of TP1.  
The velocity of the train is 350 km/h, so that the Mach number is 0.2858. As a result, the incoming flow could be assumed as incompressible. The turbulence model used with DDES is the SST k-w two-equation model. The temporal terms for all of the DDES simulations were discretized by using a second-order implicit scheme. The diffusive and subgrid fluxes were discretized with a second-order central difference scheme. The convective term was discretized using a second-order upwind scheme. Studies from [PERSON] [2] reveal that the Strouhal number of the shedding vortex in the wake ranges from 0.11 to 0.14. As a result, the time step used in the current study is 5 \(\times\) 10\({}^{-4}\) s [6]. The number of inner iterations in each time step is 10. The total computational time is 2 s.  
The velocity inlet boundary with a velocity of 350 km/h and the pressure outlet boundary with the relative pressure of 0 Pa were adopted for the INLET and OUTLET boundary in Figure 7. The lateral sides and top of the computational domain were set as slip walls, while the ground was set as the moving wall with a speed of 350 km/h in the same direction as the inlet stream. For the simulation in a crosswind condition the velocity of the incoming flow is the resultant velocity of the train and the wind. The INLET boundary and the windward side boundary were taken as the inlet boundaries. Meanwhile, the OUTLET boundary and the leeward side boundary were set as the pressure outlet boundaries. The rest of the boundary conditions remain the same.