## 1 Introduction  
The Russian economy has been expanding exports to Asia-Pacific countries in recent years. As a consequence, the volume of rail traffic to the ports of the Russian Far East has continued to grow. By the end of 2017, the rail transportation volume exceeded 180 Mt and persisted rising [1]. For the avoidance of deficient transportation, high investments are made in the infrastructure of the Baikal-Amur Mainline. One of the vulnerable spots in the planned setup toward the enhancement of the rail transportation capacity is the length of the rail tunnels. The Baikal-Amur Mainline has three tunnels longer than 3 km: the Kuznetsovsky Tunnel (3890 m), the Baikal Tunnel (6686 m) and the Severomuysky Tunnel (15,343 m). The most troublesome operating period of the tunnels is winter. In this period, owing to extremely low air temperatures, water permeates through the tunnel lining turns into ice mount, which endangers the movement of trains. The available ventilation installations at the tunnel faces fail to prevent ice coating on the lining because of two reasons. First, the absorption of heat in tunnels is an intermittent process. A cold train, when running in a tunnel, actively assimilates thermal energy from the tunnel air and spends this energy for self-heating. For instance, the heat-absorption rate of a freight train passing the Severomuysky Tunnel is 10-12 MW at the air temperature of \(-\)38--\(-\)44 \({}^{\circ}\)C [2]. Second, the heat generated by the heat-and-fan units installed at the tunnel faces is used inefficiently. Air has an excessively high temperature at the tunnel faces, while its temperature in the middle of the tunnel is insufficient to prevent ice coating [3]. Efficiently providing warm air along a tunnel requires combining the heat-and-fan units, the tunnel ventilation network and the piston effect. In order to achieve this, it is necessary to understand the dynamics of air distribution in the tunnel during train movements. A study of the piston effect in the long tunnels of the Baikal-Amur Mainline has been carried out in this study. The research on the piston effect of moving trains and vehicles in tunnels [4; 5; 6; 7; 8; 9; 10; 11; 12] was mostly studied in tunnels up to 1000 m long with vehicles and trains not exceeding 150-200 m in length. Throughout this article, we discuss freight trains that are 1200-1500 m long. It takes a long time for a train to pass an extended tunnel; for instance, travel through the Severoumysky Tunnel takes 25 min at the velocity of 40 km/h. Passage through the inlet and outlet portals of the tunnel takes 90.1 s. An excess of air pressure at the front face of the loco initiates air flow in the tunnel and in-between the train and the tunnel walls. After the whole train is inside the tunnel, air flow is additionally influenced by the rarefaction at the back face of the tail car. As the train continues moving, the air drag between the front face of the locomotive and the outlet tunnel portal decreases while the resistance to air flow between the tail car and inlet portal increases. Then, the locomotive exits the tunnel, and the high pressure at its front face has no longer influences the air flow in the tunnel. After the whole train leaves the tunnel, air flows remain there for some time because of the momentum. It is obvious that the air distribution in extra-long railway tunnels has complex dynamics. It takes into account the airflow around the moving train and influences the dynamics of heat-absorption in the tunnel. The mechanisms of these processes are necessary to develop efficient ventilation and heating circuits in tunnels. Thus, it is of current concern to analyze air distribution under the piston effect in extra-long traffic tunnels. [PERSON] and [PERSON] [13] proposed a modeling method for air distribution caused by the piston effect in subway tunnels. The method was trialed, and the calculated results agreed well with the in-situ measurements of the air flow rates in the subways of Novosibirsk and Yekaterinburg.