## 2 Methodology
### Numerical Method  
Two numerical simulation methods were used to capture complex small-scale vortex structures around a train, i.e., the LES [26; 27] and hybrid RANS/LES [12; 19; 22] methods. However, the former has relatively high requirements for grid quality and computing resources. In this study, we employed the IDDES method proposed by [PERSON] et al. [28] as a hybrid RANS/LES approach to simulate the unsteady aerodynamic performance of two MTs passing each other. In the hybrid method, the RANS method is used to model attached flows in the near-wall regions, whereas the LES method is used to resolve largely separated flows in the remaining computational domain regions [28; 29]. Furthermore, this method can effectively solve the existing problems of classical detached eddy simulations or delayed detached eddy simulation models that contain model stress dissipation, grid-induced separation, and logarithmic layer mismatch [28; 30]. Consequently, IDDES has been widely used to simulate external flows around a train, particularly small-scale vortex structures [12; 18; 19; 22; 23]. To combine the delayed detached eddy simulation and wall modelling LES, the IDDES length scale is redefined as follows:  
\[l_{\text{hyb}}=\widetilde{f}_{\text{d}}(1+f_{\text{e}})l_{\text{RANS}}+\left( 1-\widetilde{f}_{\text{d}}\right)l_{\text{LES}}, \tag{1}\]  
where \(l_{\text{RANS}}\), \(l_{\text{LES}}\), and \(l_{\text{hyb}}\) are the lengths of the hybrid, RANS, and LES models, respectively; \(\widetilde{f}_{d}=\max\{(1-f_{dt}),f_{\text{B}}\}\), where \(f_{\text{dt}}\) and \(f_{\text{B}}\) are the delay and empirical blending functions, respectively, and \(f_{\text{e}}\) is an elevating function. We selected the IDDES method based on the SA and SST k-\(\omega\) models to compare the difference in capturing flow fields based on the one-equation model (SA) and the two-equation model (SST k-\(\omega\)). In the SA model, \(l_{\text{RANS}}=d_{\text{w}}\), where \(d_{\text{w}}\) is the distance to the wall, and \(l_{\text{LES}}=C_{\text{DES}}\psi\Delta\), where \(C_{\text{DES}}\) is the empirical constant and \(\psi\) is a low-Reynolds number correction. In the SST k-\(\omega\) model, \(l_{\text{RANS}}=\sqrt{k}/\left(C_{\text{tr}}\omega\right)\), where \(k\) is the turbulent kinetic energy and \(\omega\) is the turbulent dissipation rate. The sub-grid length scale \(\Delta\) is defined as follows:  
\[\Delta=\min\{\max[C_{\text{w}}d_{\text{w}},C_{\text{w}}h_{\text{max}},h_{ \text{wn}}],h_{\text{max}}\}, \tag{2}\]  
where the empirical constant \(C_{\text{w}}\) is equal to 0.15. \(h_{\text{max}}\) is the maximum size of the cell in three directions, or wall distance, and \(h_{\text{wn}}\) is the mesh step in the direction normal to the wall.  
To facilitate the comparison, we also chose the unsteady RANS method to calculate the aerodynamic performance of two MTs passing each other. In this study, the two-equation model SST k-\(\omega\) was used to solve the RANS equations, with the turbulent eddy viscosity \(\mu_{t}\) calculated as follows:  
\[\mu_{t}=\frac{k}{\omega}. \tag{3}\]  
Figure 4: Schematic of computational domain and boundary conditions.  
Unsteady compressible Navier-Stokes equations were solved using a finite-volume solver STAR\(-\)CCM+. The time step \(\Delta t\) was fixed at 5 \(\times\) 10\({}^{-5}\) s, which was smaller than that reported previously [1; 31]. After the initial development of the flow, the two MTs were operated with a total time step of 2070. For each time step, the maximum number of iterations was set as 30, and the residuals of all physical equations were below 1 \(\times\) 10\({}^{-4}\). Less than 1% of cells in the computational domain had a Courant-Friedrichs-Lewy (CFL) number greater than one. According to [PERSON] [32], a minor violation of CFL conditions has a minimal effect on the flow field. In this study, five cases (including three turbulence models, as well as the coarse and fine mesh of the SST-IDDES model) were performed at the National Supercomputing Centre in Wuxi, China.