## 3 Results and Discussion
### Aerodynamic Forces  
The drag-, side-, and lift-force coefficients were calculated as follows:  
\[C_{d}=\frac{F_{d}}{0.5\rho u_{tr}^{2}S^{\prime}},\ C_{s}=\frac{F_{s}}{0.5\rho u _{tr}^{2}S^{\prime}},\ C_{l}=\frac{F_{l}}{0.5\rho u_{tr}^{2}S^{\prime}}, \tag{7}\]  
Figure 17: Streamlines projected onto the vertical planes at \(x=0\) at (**a**) \(t_{2}\) and (**b**) \(t_{3}\). Figure 18: Parameters \(h_{0}\), \(h_{1}\), and \(w\) obtained with different turbulence models at (**a**) \(t_{2}\) and (**b**) \(t_{3}\). where \(S\) is the cross-section area in the middle of the train; \(C_{d}\), \(C_{s}\), and \(C_{l}\) are the drag, side, and lift force coefficients, respectively; \(\rho\) is air density; and \(u_{tr}\) is train speed. \(F_{d}\), \(F_{s}\), and \(F_{l}\) are the drag, side, and lift forces, respectively. A MT running in the positive direction of \(x\) coordinates was used to analyse the aerodynamic force. The results of the drag, side, and lift force coefficients obtained with different turbulence models are shown in Figure 19. Figure 19a,b show the time-history of the drag force coefficients of the head and tail cars, respectively. It can be observed from the figure that the maximum absolute value of the drag force coefficient of the head car is around the position where two head noses meet, and the maximum absolute value of the drag force coefficient of the tail car is around the position where two tail noses meet. The drag force coefficient obtained with the SA\(-\)IDDES turbulence model is remarkably different from that obtained with the SST k\(-\omega\) and SST\(-\)IDDES turbulence models. For the head and tail cars, the SA\(-\)IDDES and SST\(-\)IDDES turbulence models exhibit the maximum differences in the peak values of the drag force coefficient (as depicted in the green ellipse), which are 9.3% and 4.5%, respectively. Figure 19: Aerodynamic force coefficient: (**a**) resistance of the head car; (**b**) resistance of the tail car; (**c**) side force of the head car; (**d**) side force of the tail car; (**e**) lift force of the head car; (**f**) lift force of the tail car. (NN, NT, and TT denote the intersection of the train noses (\(t\) = 3.37 \(\times\)\(10^{-2}\) s), intersection of the train nose and tail of other train (\(t\) = 5.18 \(\times\)\(10^{-2}\) s), and intersection of the train tails (\(t\) = 6.99 \(\times\)\(10^{-2}\) s), respectively.). Figure 19c,d show the time histories of the side force coefficients of the head and tail car, respectively. The differences in the variation trend and the amplitude of the side force of the head and tail cars obtained by different turbulence models are considerably small, and the differences in the peak values are all lower than 6%. For the head car, a negative peak amplitude of approximately \(-0.64\) is generated near a position where the two head noses meet. A positive peak amplitude of approximately 0.69 is generated near the position where the head and tail noses meet. For the tail car, a negative peak amplitude of approximately \(-0.42\) is generated near a position where the two tail noses meet, and a positive peak amplitude of approximately 0.7 is generated near a position where the head and tail noses meet. Figure 19e,f show the time histories of the lift force coefficients of the head and tail cars, respectively. As can be observed from Figure 19e, the turbulence model has minimal effect on the lift force of the head car before the head nose of a train arrives at the middle of the other train, while it has a strong influence after that position. The positive and negative peak amplitudes obtained with the different turbulence models were approximately 0.14 and \(-0.14\), respectively. As shown in Figure 19f, the lift force of the tail car is significantly different for the different turbulence models. For the maximum value of the lift force (green ellipse in Figure 19f), the value computed with the SST\(-\)IDDES model is 11% larger than that computed with the SA\(-\)IDDES model.