## 5 Conclusions  
This study employs an improved delayed detached eddy simulation (IDDES) turbulence model to establish a three-dimensional numerical model of the lining-debris-air-tunnel-train. The aerodynamics of falling debris under extreme wind conditions induced by single-train and trains-crossing scenarios are analyzed. Key findings are summarized as follows:  
1. Falling debris exhibits three-dimensional motion under both single-train and trains-crossing conditions. In the \(X\)Z plane, the debris trajectory opposes the train's direction of travel; in the \(Y\)Z plane, motion is downward (-\(Z\) direction). Debris flight distance is greater under single-train conditions than during trains crossing. At train speeds of 250-350 km/h, the average flight distances in the \(X\) and \(Z\) directions under single train conditions are 10.3 times and 5.5 times greater, respectively, compared with trains crossing. Under single-train conditions, debris accelerates in the negative \(X\) and \(Z\) directions. During trains crossing, debris velocity in the \(X\) and \(Z\) directions fluctuates around zero;
2. The rotation and angular velocity of falling debris exhibit two distinct stages: an initial phase of regular, limited-range changes followed by a second phase of intensified fluctuations. In both single-train and trains-crossing scenarios, debris rotation primarily occurs about the \(X\) and \(Z\) axes, with negligible rotation about the \(Y\) axis. At 300 km/h, the maximum rotation angles about the \(X\) and \(Z\) axes reach 783\({}^{\circ}\) and 535\({}^{\circ}\) under single-train conditions, compared with 685\({}^{\circ}\) and 254\({}^{\circ}\) under trains-crossing conditions;
3. In both single-train and trains-crossing scenarios, the \(C_{Fx}\) and \(C_{Fz}\) forces acting on the falling debris show less fluctuation in the first half of the fall but become more turbulent in the second half. Under trains-crossing conditions, the fluctuations of \(C_{Fx}\) and \(C_{Fz}\) are more intense, although the impulses of \(C_{Fx}\) and \(C_{Fz}\) are significantly lower compared with the single-train conditions. At a train speed of 300 km/h, the impulses of \(C_{Fx}\) and \(C_{Fz}\) under the single-train conditions are 8.6 and 4.5 times higher than those under the trains-crossing conditions, respectively, which explains the shorter flight distance observed under the single-train conditions;
4. The \(C_{Mx}\) of the falling debris is the most prominent, with the \(C_{My}\) value substantially smaller than \(C_{Mx}\) and \(C_{Mz}\). At a train speed of 300 km/h, the positive peaks of \(C_{Mx}\) and \(C_{Mz}\) under the single train condition reach 2.76 and 1.86, respectively, while the corresponding values under the trains-crossing conditions are 2.54 and 1.01. Notably, the positive peaks of \(C_{My}\) under both single-train and trains-crossing conditions are significantly lower, measuring only 0.039 and 0.079, respectively;
5. Under single-train conditions, the surrounding airflow in the \(X\)Z plane moves front-to-rear, causing debris trajectories to oppose the train's direction. In the \(Y\)Z plane, airflow near the train body is predominantly downward (-\(Z\) direction), aligning with debris fall from the vault. During trains crossing, debris positioned between trains experiences lower wind speeds with unstable direction. This instability causes shorter flight distances, increased trajectory deviations, and greater velocity fluctuations. The debris' relative surface velocity decreases, generating smaller, more numerous vortex structures. The aerodynamic coefficient exhibits reduced magnitude but an increased fluctuation range. It is important to note that this study focuses on understanding the aerodynamic behavior variations of irregularly shaped falling debris from tunnel linings under the influence of train slipstream. Future research will delve into the intrinsic mechanisms of how aerodynamic pressure causes debris to detach from the secondary lining. **Author Contributions:** W.-C.Y.