### 4.1 Temperature distribution

To observe the temperature distribution in the tube, the initial ambient temperature of 293 K and the air pressure of 0.1 atm was taken as an example. Figure 9 shows the temperature distribution on the train surface. Line A and line B represent the intersection lines between the upper and lower surface of the train and the *xy* plane, respectively. Line C represents the intersection line between the left surface of the train and the *xz* plane. As it can be seen from Fig. 9, the temperature of the train surface is between 308 K and 340 K, which is higher than the initial ambient temperature. The highest temperature is located at the stagnation point of the nose tip of the head car, near 340 K, and the lowest temperature is located at the nose tip of the tail car, about 315 K. In addition, the temperature distribution of the nose tip of the head and tail car shows a large gradient. The temperature distribution on the middle part of the train body is relatively flat, and the temperature gradient is small.

Figure 10 shows the temperature distribution on the tube wall. Along the tube wall, temperature data on six lines were collected, and the arrangement of these six lines is shown in the figure. With the operation of the train, under the condition that the flow field is stable, it can be found that the temperature difference on the cross section of the same tube wall in front of the wake is small. Because of the complex wake area, the temperature distribution on the tube wall which is located behind the train fluctuates with different amplitude. And there is a low temperature region in the wake area, which makes the temperature of the tube wall also very low. Behind the low temperature region, some locations on the tube wall reach higher temperatures. Subsequently, the temperature distribution on the tube wall tends to be consistent.

Figure 11 shows the change of Mach number along the gap between the train and the tube. The air flowing into the gap between the train and the tube is similar to the air flowing into the Laval nozzle. The nose tip of the head car together with the tube wall forms a convergent section, and the nose tip of the tail car together with the tube wall forms a divergent section. As shown in Fig. 11, the air flows into the gap from the nose tip of the head car at subsonic speed and accelerates continuously in the gap, then the velocity exceeds the local sound velocity at the shoulder of the tail car. Because the transonic fluid no longer follows the principle of “The smaller the cross section, the faster the airflow velocity; the larger the cross section, the slower the airflow velocity”. On the contrary, the larger the cross section, the faster the velocity. Therefore, the airflow continuously accelerates and expands, creating expansion waves and compression waves behind the train. As a result, a supersonic flow and low-pressure region are formed in the wake, shown in Fig. 12. In addition, because the tube wall and the train surface are adiabatic, the flow field in the tube can be regarded as isentropic flow. Eventually, the temperature in the supersonic flow region becomes very low, forming a low temperature region.

When the airflow velocity in the flow field reaches the local sound velocity, it is possible to generate shock waves. Figure 12 shows the contours of pressure, temperature and Mach number. As analyzed earlier, the Mach number of the train’s tail exceeds 1 and the maximum Mach number reaches 2. Therefore, shock waves are generated at the rear of the train. Figure 12(b) shows the horizontal cross-section of the contour of pressure, temperature and Mach number in the wake area, respectively. It can be seen from the figure that the expansion wave has an obvious influence on the structure of the flow field in the wake area. Due to the limitation of the tube wall, the continuous reflection and interaction of the compression wave significantly change the structure of the flow field in the tube and complicate the wake.

### 4.2 The influence of initial ambient temperature (*T*
_{0}) on flow field

In the future, several high-speed trains will continuously run in the tube, so the ambient temperature inside the tube will be constantly rising. For the trains running in the tube at different times, the initial ambient temperature (*T*_{0}) of the flow field is different. The earlier the train runs, the lower the ambient temperature, and the later the train runs, the higher the ambient temperature may be. Accordingly, this section mainly discusses the influence of *T*_{0} on the structure of flow field. *T*_{0} assumed in the calculation cases in this section are 243 K, 293 K, 343 K and 393 K, respectively. The assumptions for some of *T*_{0} may not be in line with the actual situation, which is only for exploratory discussion here.

Figure 13 shows the pressure distribution on the intersection line between the train upper surface and the *xy* plane. It can be seen from Fig. 13 that pressure distribution under different *T*_{0} is roughly approximate as a whole. As *T*_{0} rises, the pressure on the train surface decreases gradually. At the nose tip of the tail car, the pressure distribution at *T*_{0} = 243 K is similar to that at *T*_{0} = 293 K. When the *T*_{0} is 343 K and 393 K, the pressure distribution at the nose tip is quite different from that of *T*_{0} = 243 K and 293 K. In addition, the maximum pressure on the train surface gradually decreases, and the minimum pressure gradually rises, as shown in Fig. 14. The pressure change on the train surface further affects the aerodynamic drag. Figure 15 shows the aerodynamic drag of the train at different *T*_{0}. As shown in the figure, the aerodynamic drag decreases with the increase of *T*_{0}. And the aerodynamic drag of the train at *T*_{0} = 393 K is about 50 kN lower than that of *T*_{0} = 243 K. The aerodynamic drag consists of pressure drag and shear force. The pressure drag accounts for 90% of the aerodynamic drag, so the pressure drag plays a leading role and is the main reason for the decrease of the aerodynamic drag. Therefore, the decrease of pressure drag is the main factor leading to the decrease of aerodynamic drag.

To explore the influence of *T*_{0} on the Mach number, the Mach number on the straight line near the top of the tube was collected, as shown in Fig. 16. The *x*-coordinate of the straight line ranges from − 10 to 110 m, and the *x*-coordinate of the nose tip of the tail car is about 80 m, so this is also the position at the beginning of the wake area. It can be seen that the Mach number distribution in front of the wake area is basically consistent at different *T*_{0}. However, the distribution of Mach number in the wake area is quite different. Combined with the contour of the Mach number (as shown in Fig. 17) in the wake area, it can be found that *T*_{0} has a great influence on the supersonic flow region. When *T*_{0} is low, there is a longer supersonic flow region appearing in the wake area, and when *T*_{0} increases, the supersonic flow region becomes shorter. Moreover, when *T*_{0} rises from 243 K to 343 K, the change of supersonic flow region is more obvious, while when *T*_{0} rises from 343 K to 393 K, the change of supersonic flow region is relatively small. Meanwhile, the maximum Mach number on this straight line also decreases with the increase of *T*_{0}. Especially, when *T*_{0} rises from 293 K to 393 K, the maximum Mach number changes greatly.

Subsequently, the pressure data was collected on the straight line which used in Fig. 16, and the pressure curves at different *T*_{0} were plotted in Fig. 18. It can be found from Fig. 18 and Fig. 19 that *T*_{0} also has a great influence on the air pressure. And with the rise of *T*_{0}, the air pressure gradually decreases. The influence of *T*_{0} on the air pressure in the wake is similar to the influence of *T*_{0} on the Mach number in the wake. The reason for this phenomenon is probably due to the acceleration of the expansion of the airflow, resulting in faster airflow velocity and lower pressure, as analyzed in Section 4.1. Therefore, the air pressure will change with the change of the Mach number. Finally, according to the isentropic flow, the structure of the temperature field in the wake will also change with the Mach number and pressure.

As shown in Fig. 20, it can be seen from the contour of the temperature field in the wake area that *T*_{0} has a great influence on the structure of temperature field. When the *T*_{0} rises from 243 K to 293 K, the length of the low temperature region in the wake area is shortened from about 98 m to about 21 m, but the contours of their low temperature region have a similar profile. When *T*_{0} continues to rise to 343 K and 393 K, the length of its low temperature region is not only shortened, but also the profile of its temperature contour is changed.