## 1 Introduction  
In the past three years, coronavirus disease 2019 (COVID-19) has led to over 769 million infections and about 7 million deaths, as reported by the World Health Organization's Coronavirus Dashboard ([[https://covid19.who.int/](https://covid19.who.int/)]([https://covid19.who.int/](https://covid19.who.int/)) (accessed on 30 August 2023)). Although many countries or regions have successfully contained the spread of the epidemic using measures such as isolation, closures, and vaccination, some countries are still experiencing the impact of COVID-19-related variants. The outbreak has drawn significant attention from scholars worldwide regarding virus transmission [1, 2, 3, 4, 5, 6]. Subway transportation networks are of high convenience to residents commuting in various areas of metropolitan cities and have become a major mode of transportation for urban dwellers [7, 8]. However, the high passenger density and relatively enclosed environment in subway carriages also provide suitable conditions for spreading the virus [9, 10], allowing the virus to spread through the urban subway network. Understanding the patterns of droplet transmission during subway operations is of great significance for future epidemic prevention policies and improvements to subway ventilation systems. Airborne transmission through droplet nuclei (aerosols) is a major transmission mode for the COVID-19 virus [11]. Virus carriers can release pathogen-bearing droplets by talking,coughing, sneezing, or breathing [12; 13; 14]. These droplets can either settle on surfaces due to gravity or remain suspended in the air for an extended period. Small droplets (\(<\)10 \(\upmu\)m) rapidly transform into smaller droplet nuclei due to evaporation within one second of being released into the air [15]. At this point, the effect of gravity on the droplets can be ignored, and the dispersion of droplets is primarily governed by the airflow in the environment [16]. In the confined environment of a subway carriage, the high passenger density makes it difficult for passengers to maintain a safe social distance [17] from each other. Passengers may make contact with droplets settled on surfaces or directly inhale suspended droplets in the air, significantly increasing their risk of cross-infection. However, the short distances between subway stations, frequent train stops, and high passenger flow within the carriages present significant challenges in conducting experiments to study the movement characteristics of droplets inside subway carriages. Theoretical multiphase flow and droplet physics models [18; 19] have provided convenient conditions for studying droplet transmission in various scenarios. In relevant studies, the large eddy simulation (LES) method [20; 21] and the renormalization group (RNG) \(k\)-\(\varepsilon\) model [22; 23; 24; 25] are widely used to predict indoor airflow. The Lagrangian method is commonly used to track the movement of droplets. Within it, the droplets are assumed to be spherical particles whose volume is influenced by the evaporation rate, mainly determined by environmental parameters [26; 27; 28]. [PERSON] and [PERSON] [29] modeled the coughing process as a turbulent jet to investigate the droplet diffusion and obtained travel distances with different droplet sizes. [PERSON] et al. [30] used computational fluid dynamics (CFD) simulation to predict the cough jet dispersion in an airline cabin. To date, studies have evaluated SARS-CoV-2 spread via heating, ventilation, and air conditioning (HVAC) systems in various practical settings, such as underground car parks [31], laboratories [32], hospital wards [33; 34; 35; 36], restaurants [37], restrooms [38; 39], elevators [40], public transportation [41; 42; 43; 44], classrooms [45], cafeterias [46], airplanes [47], dental clinics [48], factories [49], etc. These research efforts provide essential references for understanding droplet diffusion in different environments and formulating strategies to prevent cross-infections.