## 3 Modeling  
The problems are solved in the nonstationary formulation, with a dynamically structured network in the ANSYS Fluent 14.5 environment, using resources of the Siberian Supercomputer Sharing Center of the Siberian Branch of the Russian Academy of Sciences, on the G6-cluster in the multisequence mode (48 cores). The k-epsilon Realizable turbulence model, near-wall treatment non-equilibrium and PISO algorithm recommended for unsteady flows were used in the simulation. To simulate the movement of the train, the dynamic mesh layering method was used with interfaces between the movable and stationary parts. The shape of the grid elements is mainly quadrangular; in the areas of the movable grid on the boundary of the calculated area, the cells have the shape of regular quadrangles. The dimensions of the elements of the calculated grid are accepted with the condition that the dimensionless parameter y+, which characterizes the quality of the grid for modeling the wall layer on the wall of the train and tunnel, does not exceed 100. This corresponds to the recommendations for the accepted turbulence model. The material is air with the properties of an ideal gas, taking into account the compressibility. For the 3 km long tunnel, the total physical time in the model was 442 s at a time step of 0.0002 s. In the calculation, 9822 saves were made every 225 time steps (which totaled 0.045 s). One save containing the current state of the dynamic network and the calculated results had a volume of 65 Mbytes. The total number of time steps was 2,209,950. The finite element mesh contained 291,947 cells and 635,038 points. For the 6 km long tunnel, the total physical time in the model was 703 s at a time step of 0.0002 s. During the calculation, 6390 saves were made every 550 time steps (in 0.11 s). The volume of a single save containing the current state of the dynamic network and the calculated results was 89 Mbytes. The total time steps were 3,515,000 in number. The finite element mesh contained 454,975 cells and 990,888 points. Figure 3 demonstrates the geometry of the plane axially symmetric calculation model of a tunnel with a train. The total length of the model domain is 5152 and 8152 for the 3000 and 6000 m long tunnels, respectively. At the initial time, the train 1000 m in length is immobile at a distance of 70 m from the entry to the tunnel portal. The modeling assumes the mode of train motion as follows:  
1. the train uniformly accelerates in an open space section, 50 long, from 0 to 11.1 m/s (40 km/h);
2. the train skirts the remaining 20 m to the tunnel entrance at the above-specified velocity, enters the tunnel, moves in it and leaves the tunnel within 364 s for a 3 km tunnel, and within 625 s for a 6 km tunnel;
3. at the moment when the distance from the exit portal to the end of the last car of the train is 20 m, the train begins uniformly decelerating in a span of 50 m from 11.1 to 0 m/s;
4. having stopped, the train stands for 60 s in the open space outward of the tunnel. The estimated skin friction drag per 1 km in the tunnel is 0.000347 k\(\upmu\) (1 k\(\upmu\) \(\approx\) 9.81 N\(\cdot\)s\({}^{2}\cdot\)m\({}^{-8}\)). Figures 4 and 5 present the numerical modeling results for a 3000 m long tunnel: the air pressure and flow rates per cross sections of the tunnel, as well as the air pressure at the front and back surfaces of the train. Figure 4a shows the longitudinal plan of the tunnel with marked sections 1-1, 2-2 and 3-3. Sections 1-1 and 3-3 lay at the distance of 10 tunnel diameters from the inlet and outlet portals, respectively, and section 2-2 is in the middle of the tunnel.