## 4 Results and Discussion
### Aerodynamic Characteristics of Falling Debris  
Figure 13 compares the aerodynamic force coefficients in the \(X\) and \(Z\) directions for falling debris under single-train conditions and trains-crossing conditions at speeds ranging from 250 to 350 km/h. The impulse (\(I\)), representing the integral value of the aerodynamic force coefficient over time, is also provided in the figure to describe the cumulative effect of aerodynamic force on the motion of the falling debris. Figure 12: Rotational angular velocity of falling debris: (**a1–a3**) single train; (**b1–b3**) trains crossing. Under single-train conditions, the \(C_{Fx}\) and \(C_{Fz}\) values of falling debris exhibit less fluctuation in the first half of the fall but become highly erratic in the second half. Most of the time, the \(C_{Fx}\) of falling debris is negative, indicating that it experiences thrust in the negative direction, leading to a flight trajectory opposite to the direction of train operation. The \(C_{Fz}\) value of falling debris is primarily negative in the first half of the fall with minimal fluctuation, while in the second half, if fluctuates violently around zero. The impulse values of \(C_{Fz}\) are consistently negative, causing the falling debris to move away from the train. As the train speed increases, the impulse of the falling debris gradually increases. When train speed increases from 250 km/h to 350 km/h, the impulse of \(C_{Fx}\) increases by 167%, and the impulse of \(C_{Fz}\) increases by 65%. Under the conditions of trains crossing, the \(C_{Fx}\) and \(C_{Fz}\) values of falling debris fluctuate near zero, with slight fluctuations in the first half and more intense fluctuations in the second half compared with the single-train conditions. However, the fluctuation of \(C_{Fx}\) and \(C_{Fz}\) values of falling debris under the conditions of trains crossing is more pronounced, indicating a more complex flow field in the tunnel during intersection. The impulses of \(C_{Fx}\) and \(C_{Fz}\) are significantly lower than the corresponding values under single train conditions. For instance, when the velocity reaches 300 km/h, the impulses of \(C_{Fx}\) and \(C_{Fz}\) in the scenario of a single train are 8.6 and 4.5 times greater, respectively, than those in the case of intersecting trains. This discrepancy significantly contributes to the notably shorter flight distance observed in the trains crossing scenario. Figure 14 illustrates a comparison of the aerodynamic moment coefficient of falling debris in three directions, considering both a single train and trains crossing at speeds ranging from 250 to 350 km/h. As depicted in Figure 14, irrespective of the operating conditions, the \(C_{Mx}\) of falling debris is consistently the largest, with \(C_{My}\) significantly smaller than both \(C_{Mx}\) and \(C_{Mz}\). At a train speed of 300 km/h, the positive peaks of \(C_{Mx}\) and \(C_{Mz}\) in the single-train scenario reach 2.76 and 1.86, respectively, while the corresponding values in the trains crossing scenario are 2.54 and 1.01. Notably, the positive peak values of \(C_{My}\) in the single-train and  
Figure 13: Force coefficient of falling debris: (**a1,a2**) single train; (**b1,b2**) trains crossing. trains-crossing scenarios are only 0.039 and 0.079, respectively. This discrepancy elucidates why the rotation of falling debris around the \(Y\) axis is less pronounced. The aerodynamic moment coefficient of falling debris exhibits a two-stage pattern, with more pronounced characteristics under the trains-crossing conditions. In comparison to the first stage, the moment coefficients during the second stage display sharper fluctuations but the law is not significant. Under trains-crossing conditions, the aerodynamic moment coefficient of falling debris primarily oscillates around zero during the first stage, with an intensified fluctuation amplitude towards the end of this stage.