## 1 Introduction  
With the current state of economic development and increasing transportation demands, more and more tunnels are being put into construction [1]. Tunnels can be referred to as a special type of semi-enclosed passageway. When vehicles pass through the tunnels, the exhaust gas will accumulate and therefore increase the concentration of harmful substances such as carbon monoxide, nitrogen oxides, and particulates, which will deteriorate the operating environment in the tunnel [2, 3] and even cause a series of health problems such as dyspnea and cardiovascular diseases [4, 5]. Usually, ventilation equipment is adopted to dilute the pollutant concentration in the tunnels [6, 7, 8]. In recent years, due to being restricted by landforms and architectural patterns, more and more twin tunnels with small spacing are being constructed [9, 10]. During the operation of this kind of tunnel, not only do the pollutants of the outlet tunnel influence the local environment of the tunnel exit [11], but the pollutants also mix with new incoming air at the adjacent inlet to form circulating air, consequently intensifying the pollution level at the internal environment of the inlet tunnel and resulting in increasing costs of ventilation system construction and operation. At present, many scholars have studied the circulating air at the portals of twin tunnels with small spacing. Through three-dimensional numerical simulations and field tests, [PERSON] et al. [12] presented how the mixed volume of pollutants correlated with the tunnel spacing and airflow velocity at the inlet/outlet. They concluded that the mixed proportion of the polluted airflow could be reduced by increasing the outlet air velocity and lateral distance of the tunnel portals as well as decreasing the inlet air velocity. Through a model test, [PERSON] et al. [13] established a testing ventilation system of a close tunnel model and conducted a series of experiments. The results showed that when inlet and outlet airflow velocity were equal, the mixed volume of pollutants was barely influenced by the airflow velocity but would decrease with the increase of the tunnel spacing. Based on the experimental data, they also obtained the quantitative relationship between the mixed volume of pollutants and inlet/outlet airflow velocity and tunnel spacing. Through numerical simulation, [PERSON] et al. [14] further analyzed the influence of ambient wind on the pollutant diffusion at the tunnel portals. They concluded that under adverse ambient wind conditions, the degree of influence of the exhaust pollutant at outlet air tunnel portal on the inlet air tunnel lessened as the environmental air slowed down. According to previous findings [12; 15], the circulating air can be effectively weakened by establishing midboards, green belts, or building horizontal ventilation tunnels. It should be noted that many closely spaced twin tunnels with staggered inlet and outlet exist due to landform restrictions. For instance, for the Baziling Tunnel in the Hurong highway, the inlet is about 5 m behind the outlet. At the north opening of the #1 tunnel of the Zizhi Tunnel in Hangzhou, the inlet is about 19 m before the outlet; at the south opening of the #3 tunnel, the inlet is about 44 m before the outlet. The circulating air can be effectively influenced by staggering the inlet and outlet air tunnel portals. For example, by increasing the length of outlet air tunnel portal and repositioning it at a point 5 m away from the inlet, the proportion of polluted airflow mixed with fresh airflow will be lower than 5% [13]. However, previous research mostly focused on closely spaced twin tunnels with aligned inlet and outlet air tunnel portals, and studies relating to the circulating air of the staggered inlet and outlet are still inadequate.