## 1 Introduction  
The number of underwater tunnels has been increasing in recent years, and a large number of underwater tunnels with special V-shaped structures are under construction [1, 2]. The single-ended closure of tunnels during construction often makes it difficult for pollutants to be exhausted and fresh air to enter. This leads to poor air quality and endangers the lives of construction workers [3, 4]. As shown in Figure 1, the V-shaped tunnel is bored single-ended and sequentially through a downhill section, a horizontal section, and an uphill section. Due to the different structures and the corresponding differences in the flow characteristics during tunnel construction, the previous experience of ventilation in normal flat tunnels is no longer applicable. Therefore, it is necessary to carry out a systematic study of the ventilation characteristics during the construction of V-shaped tunnels. In the past few years, many scholars have studied the flow field and pollutant transport law in tunnels [5, 6, 7, 8, 9, 10]. Previous studies have shown that the structure of the ventilation flow field during tunnel construction can be divided into the jet zone, vortex zone, and return zone. The characteristics of the return zone are often the focus of research, as it relates to the ability of pollutants to be discharged smoothly during tunnel construction. Further, the air return velocity \(u_{\text{return}}\) has been widely studied as an important parameter for measuring the return zone, with the advantages of being easy to measure and having a direct effect on the emission of pollutants. [PERSON] et al. [11] studied the flow form of backflow in ventilated tunnels and derived a semi-empirical equation for the distribution law of the mean air backflow velocity. The equations demonstrate that the mean air backflow velocity is related to the tunnel width, height, duct diameter, and jet velocity. Then, [PERSON] et al. [12] investigated the applicability of the backflow velocity requirement (0.15 m/s) under conventional size tunnels in the construction of super-large cross-section underground works by constructing a numerical model. The results indicated that this wind speed was not sufficient to keep CO concentrations in the tunnel below an acceptable level. [PERSON] et al. [13] studied the distribution of air velocity values in the tunnel ventilation area by varying the ventilation parameter settings. The study showed that increasing the cross-sectional area of the tunnel reduced the velocity of the airflow in the return zone. Furthermore, other scholars have evaluated the optimal level of \(u_{\text{return}}\) in tunnels by utilizing concepts like dead zones and the average age of air [14; 15]. Additionally, some scholars have also conducted a lot of research on the distribution of wind speed in tunnel construction ventilation under high-temperature environmental hazards [16; 17]. In the contaminant transport study, [PERSON] and [PERSON] [18] used parameters such as the TBM (Tunnel Boring Machine) advance rate to predict pollutant influx into the tunnel based on coal mine experience. However, more studies have focused on the distribution patterns of pollutants in tunnels [19; 20; 21]. The effects of ventilation parameter settings have been frequently emphasized. [PERSON] et al. [22] considered the effects of the air supply volume and the distance of the duct outlet from the working face on the distribution of dust concentration. The study revealed that for the optimal ventilation effect, each air supply volume corresponds to a specific installation location of the air duct outlet. [PERSON] et al. [23] investigated the gas distribution characteristics in tunnels under full-section excavation and step excavation methods under the influence of the distance between the duct outlet and the working face. [PERSON] et al.