## 2 Methodology
### Mesh Sensitivity  
The pressure coefficient \(C_{P}\) is defined by Equation (4), where \(\rho\) is the air density (1.225 kg/m\({}^{3}\)), \(P\) is the static pressure, \(P_{\infty}\) is the reference pressure (equal to zero), and \(u_{tr}\) is the train speed (55.56 m/s). The results of grids with different densities were computed using the IDDES method based on the SST k-\(\omega\) model to validate the grid sensitivity. The time-history pressure coefficients of monitoring points No. 3 and No. 8 on the train surface for different density grids are presented in Figure 7. The peak-to-peak pressure coefficient (\(\Delta C_{P}\)) is shown in Figure 7a. For monitoring point No. 3, the difference in \(\Delta C_{P}\) between the coarse grid and fine grid was found to be 4.1%, and that between the medium grid and fine grid was 0.5%. For monitoring point No. 8, the difference in \(\Delta C_{P}\) between the coarse grid and fine grid was 3.1%, and that between the medium grid and fine grid was 0.6%. The differences in \(\Delta C_{P}\) between the medium and fine grids were found to be rather small. \[C_{P}=\frac{P-P_{\infty}}{0.5\rho u_{tr}^{2}} \tag{4}\]  
Figure 8 compares time-history curves of resultant velocity computed by the coarse, medium and fine grids at y = 0.25 m and z = 0.125 m (namely point g in Figure 3). The non-dimensional resultant slipstream velocity \(U\) was defined as follows:  
\[U=\frac{\sqrt{u^{2}+v^{2}+w^{2}}}{u_{tr}}, \tag{5}\]  
where \(u\), \(v\), and \(w\) are the streamwise, spanwise, and vertical velocity, respectively, and \(u_{tr}\) is the train velocity. The resultant velocity predicted by three meshes is consistent each other before the tail nose passing, while it shows a large deviation between the coarse and medium as well as fine meshes when wake flow passing. The iso-surfaces of the second invariant of the velocity gradient, Q, is an effective approach to describe the three-dimensional instantaneous flow field structures around a train. In this study, we used it to investigate the basic law of the interaction of the wake vortex on the intersection side when two MTs pass each other. According to [PERSON], et al. (2009), the Q can be expressed as follows:  
\[\text{Q}=\frac{1}{2}\left(\left\|\mathbf{\Omega}\right\|^{2}-\left\|\mathbf{S }\right\|^{2}\right)=\frac{1}{2}\left[\frac{1}{2}\left(\frac{\partial\overline {u_{i}}}{\partial\overline{x_{j}}}+\frac{\partial\overline{u_{j}}}{\partial \overline{x_{i}}}\right)^{2}-\frac{1}{2}\left(\frac{\partial\overline{u_{i}} }{\partial\overline{x_{j}}}-\frac{\partial\overline{u_{j}}}{\partial\overline {x_{i}}}\right)^{2}\right]=\frac{1}{2}\frac{\partial\overline{u_{i}}\partial \overline{u_{j}}}{\partial\overline{x_{i}}\partial\overline{x_{j}}}, \tag{6}\]  
where \(\mathbf{\Omega}\) is the rate-of-rotation tensor, and \(\mathbf{S}\) is the rate-strain-tensor. Figure 9 shows the instantaneous iso-surfaces of Q = 20,000 s\({}^{-2}\) of three sets of meshes at the \(t\) = 0.08 s position,  
Figure 7: Transient pressure of monitoring points for different density mesh: **(a)** No. 3 and **(b)** No. 8.  
coloured according to the resultant velocity \(U\). The \(V_{2}-V_{2}{{}^{\prime}}\) and \(V_{1}-V_{1}{{}^{\prime}}\) regions in the figure indicate that the fine grid can capture more small-scale vortices than the medium and coarse grids with the same Q value. In the near-wake region, the number of small-scale vortices captured by the coarse grid is significantly smaller than that captured by the fine grid, whereas the difference in the flow structures between the medium and fine grid was found to be rather small. Combing with the mesh independence study on the pressure, the resultant velocity and the Q criterion, the medium grid was found to have sufficient resolution for the purpose of the present study. Figure 8: Time-history curves of resultant velocity with different mesh at y = 0.25 m and z = 0.125 m (namely point \(g\) in Figure 3).