# Aerodynamic Characteristics When Trains Pass Each Other in High-Speed Railway Shield Tunnel
## 2 Numerical Model
### Computing Area and Boundary Conditions  
The initial conditions of the flow field have a very important effect on the convergence of the calculation, especially for unsteady flows. To ensure that the flow field around the train was fully developed before the train entered the tunnel, since the two trains were facing each other, when the intersection position in the tunnel was varied, the minimum distance between the nose of the train and the entrance of the tunnel was taken as 100 m. According to EN14067:6-2018, a distance of 100 m produces a realistic simulation of the whole process of a train passing through the tunnel. The ratio of the cross section of the train to the calculated cross-sectional area of the entrance and exit of the tunnel should be less than 0.01, the height H should not be less than eight times the characteristic height, and the length L should not be less than 16 times the characteristic height. When two trains entered the tunnel at the same speed, the final calculation domain dimensions of the tunnel entrance and exit were length \(\times\) width \(\times\) height = 121.6H \(\times\) 32.43H \(\times\) 16.2H. The calculation domain and boundary conditions of the model are shown in Figure 1, where the x-axis is along the direction of the train, and the y-axis and z-axis are along the transverse and vertical directions of the tunnel, respectively. For the boundary conditions, firstly, the normal velocity of the train wall surface, tunnel wall surface, ground surface and tunnel opening boundary surface was zero, which is the non-slip wall boundary condition, that is, the normal component of the pressure gradient is 0, which is simplified to the mathematical formula \(\partial p/\partial n=0\). Secondly, during the actual operation of the train, when the pressure wave propagated and rushed out of the tunnel, it would dissipate into the surrounding atmosphere, and would not be constantly reflected like in the tunnel. In the process of simulation calculation, it was impossible to simulate the unlimited range of atmospheric flow field. At this time, the pressure far field condition needed to be set, see Figure 2b for details. The non-reflection boundary condition was based on the Riemann invariant, whereby the compression wave is transmitted without reflection after reaching the surface of the calculation area, and the boundary set at this time was the standard atmospheric pressure, that was, \(p=101,325\)\(Pa\). Thirdly, since the \(k-\varepsilon\) model was used to simulate turbulence, turbulence model boundaries were required. When calculatinng the flow in the boundary layer near the train surface, the wall function method, commonly used in engineering turbulence calculation, was used [32]. We assumed  
Figure 1: Schematic diagram of computational region and boundary condition: (a) side view and (b) top view. that the distance between the node A near the surface and the train surface was \((\delta h)_{A}\), which is defined as the dimensionless distance \(y_{A}^{+}\):  
\[y_{A}^{+}=\frac{\rho c_{\mu}^{1/4}\overline{\mathrm{k}}_{A}^{1/2}(\delta h)_{A}}{\mu} \tag{8}\]  
where \(\mu\) is the dynamic viscosity coefficient of the fluid; \(\overline{\mathrm{k}}_{A}\) is the eddy kinetic energy at point \(A\); \(\rho\) is the fluid density. The shear stress \(\tau_{w}\) on the train surface is:  
\[\tau_{w}=\lambda_{A}(\frac{\delta\overset{\rightarrow}{\partial}}{\delta h}) _{A} \tag{9}\]  
In the formula, \(\lambda_{A}\) is the equivalent viscosity coefficient, and its value depends on the position of node \(A\). That is, it changes with the change of \(y_{A}^{+}\). \[\lambda_{A}=\left\{ \begin{array}{cc}\mu&y_{A}^{+}\leq 11.6\\ \frac{\rho c_{\mu}^{1/4}\mathrm{k}_{A}^{1/2}(\delta h)_{A}}{\ln(E_{x}y_{A}^{ +})/\kappa_{0}}&y_{A}^{+}>11.6\end{array} \right. \tag{10}\]  
In the formula, \(\kappa_{0}\) is the Karman constant, taking 0.41; \(E_{x}\) is the empirical constant, taking 0.143.