## 1 Introduction
tunnels under single-train and trains-crossing conditions. [PERSON] et al. [27] conducted full-scale tests to examine the influence of various factors such as train length, type, speed, tunnel length, and station geometry on the slipstream effect. They also simulated the slipstream effect when one train was stationary in the tunnel while another passed at normal speed. [PERSON] et al. [28] employed numerical simulation methods to investigate the impact of train head length and shape on the slipstream effect in tunnels under trains-crossing conditions. These studies collectively demonstrate that the slipstream effect in tunnels under trains-crossing conditions is more pronounced than under single-train conditions. Hence, conducting a comparative study on the influence of slipstream on the flight behavior of falling debris under both single-train and two-train intersection conditions is necessary. Such analysis plays a role in ensuring the operational safety of high-speed railways. Several scholars have conducted research on the impact of train slipstream on the flight trajectory of falling debris in the presence of a single train. [PERSON] et al. [29] employed the improved delayed detached eddy simulation (IDDES) method to investigate the falling track and force characteristics of debris within the train slipstream. They analyzed the variations in the trajectory of falling debris under different falling times. Additionally, [PERSON] et al. [30] developed a comprehensive three-dimensional refined gas-solid coupling model of air-train-tunnel-falling debris. Their study delved into key factors including the shape (block and sheet), size, density, circumferential position of falling debris, and train speed, to assess their influence on the trajectory of falling debris. The researchers also delineated safety and danger zones based on whether the position of the fall posed a threat to the high-speed railway's operation [6]. However, it is noteworthy that the aforementioned studies concentrate only on the flight trajectory of falling debris in the context of a single train. There is a scarcity of research comparing the distinctions in the flight trajectories of falling debris under conditions involving a single train and trains crossing. In this research, we employ the IDDES turbulence model to develop a three-dimensional refined numerical model of falling debris-tunnel-air-train. The primary focus is on simulating the flight trajectory of the falling debris under the influence of the train slipstream. We conducted a moving model test to validate the accuracy of our numerical simulations. The impact of the train slipstream on the flight trajectory of the falling debris is thoroughly analyzed, considering scenarios involving both single train and trains crossing conditions. Section 2 provides a comprehensive description of the numerical simulation method. In Section 3, we meticulously verify the correctness of our simulation results, ensuring that the flight trajectory of the falling debris aligns with the outcomes of the moving model test. Section 4 is a detailed analysis of the flight characteristics, the variation in aerodynamic load, and the structural characteristics of the flow field surrounding the lining falling debris, all in response to the influence of the train slipstream. Section 5 includes the main conclusions.