Aerodynamic Characteristics When Trains Pass Each Other in High-Speed Railway Shield Tunnel

Abstract

The characteristics of the aerodynamic effects of high-speed trains passing in a shield tunnel were studied using the three-dimensional, compressible, unsteady Reynolds-averaged Navier-Stokes (RANS) equations for the simulation analysis. Numerical calculations were compared with dynamic model tests to verify the reliability of the numerical simulations. The results showed that the compression wave characteristics of high-speed trains in shield tunnels were consistent with those in molded concrete tunnels. When high-speed trains met in the middle of the shield tunnel, the positive and negative peak attenuation rates of shield tunnels were higher than the positive and negative peak attenuation rates of molded lining tunnels, and the maximum pressure attenuation rate could reach 57.8%. At the same time, the micro-pressure wave of the former was reduced by 10.78%, compared with those of the latter. When meeting cars at different locations, the maximum pressure at the intersection in the center of the tunnel was significantly higher than those at other intersection points in the tunnel. Different intersection positions and different tunnel lining structures had relatively little influence on the aerodynamic drag and lateral force, while train speed had a significant influence.

1. Introduction

In the actual operation of high-speed trains, train intersections are inevitable. Especially with the rapid development of modern society and the increasing number of people traveling, the pressure on passenger transportation is increasing. For existing lines, increasing the operating frequency of high-speed trains has become the most effective measure to meet transportation demand, and the increase in the operating frequency will inevitably increase the possibility of high-speed trains meeting in tunnels. When trains traveling in opposite directions enter a tunnel, their heads and tails will form pressure waves upon entry. At the same time, the generated pressure waves will partially flow out of the tunnel in the form of micro-pressure waves at the entrance and exit of the tunnel, and the other part forms a reflected wave at the entrance and exit of the tunnel. When trains travelling in opposite directions pass a certain position in the tunnel, a sudden change in the volume will generate a new compression wave when the two trains meet. The original compression wave and the expansion wave propagating in the tunnel combine and superimpose. The aerodynamic effect is not only more complicated than the aerodynamic effect of a single train passing through the tunnel, but also, a more intense pressure will result when the two trains meet in the tunnel. This will cause more intense pressure fluctuations and may cause the train to undergo serpentine motion. With the development of high-speed trains in many countries, such aerodynamic problems have become an important issue for railway technical designers [1]. Therefore, under the current line conditions in China, it is of great significance to study the aerodynamic characteristics of different tunnel lining structures and different intersection positions in the tunnel when two trains meet.

In recent years, scholars around the world have carried out research using theoretical analysis, actual vehicle tests, dynamic model tests, and numerical simulations. Baker [2] provided an overview of train aerodynamics in two parts. This included the exploration of the nature of the airflow around a train under different working conditions—open air, with or without cross winds—and in limited geometric conditions and tunnels. Based on the description of the entire flow field, the issues involved in the design and operation of modern trains were considered. Based on theoretical analysis, some scholars have studied the influence of the tunnel length on the amplitude of the train surface pressure change during an intersection [3], as well as the relationship between the maximum negative pressure peak possible in the tunnel and the speed of the passing trains [4]. Ogawa and Fujii [5] solved the three-dimensional, compressible Euler equations through an enhanced solution algorithm to handle the moving boundary problem. It was concluded that when two trains meet, the high-pressure area around the nose tip of the train first pushes the train away laterally, but, when the lateral forces are arranged side by side, the lateral force changes direction and pushes the trains away from each other.

An actual vehicle test can directly and effectively measure the pressure and aerodynamic parameters of the tunnel wall in a real environment [6,7,8,9]. For example, Liu et al. [6] found that as the train speed increased, the maximum peak-to-peak pressure change position moved toward the entrance of the tunnel, which was mainly driven by the change of the negative pressure peak. Furthermore, the length of the train led to a significant difference in the peak pressure of the tunnel wall in the middle of the tunnel. Ko et al. [7] studied the propagation characteristics of the pressure wave and discussed the spatial variations of the induced pressure peak of the train in the tunnel. Zhou [8] evaluated the rationality of the tunnel section parameters and the design parameters of the auxiliary tunnel, based on a field test. Liu et al. [9] analyzed the formation mechanism, amplitude distribution, and attenuation of the pressure wave through a series of on-site measurements.

Compared with field tests, model test applications have the advantages of low costs and simple operations. However, dynamic model tests began relatively late [10], and their advantage is that they can simulate the relative motion between the train and the ground. For example, the dynamic model test bench of the Institute of Mechanics of the Chinese Academy of Sciences can simulate the relative movement of a train and the ground and verify the reliability of numerical simulations [11,12]. Baker used dynamic model tests to reveal that the peak winds of trains under all closed conditions were greater than those of open-air conditions. In a tunnel, the “piston effect” seems to be the main reason for the increase in the peak intensity of the train wind [13]. Miyachi et al. [14] used model experiments to study the pressure wave generated by a train passing through a branch and the pulse wave that radiated from the main tunnel through the exit of the branch. At the same time, simple acoustic theory was used to clarify the correlation between the pressure wave in the tunnel and the pulse wave radiating from the exit.

At present, with the rapid development of computer technology, numerical simulations are not only convenient to perform but also yield accurate results. Thus, numerical simulations have been widely used [15,16,17,18,19,20,21,22,23,24,25,26,27,28]. They can also greatly reduce the costs of actual vehicle tests and dynamic model tests. Some scholars have used numerical simulations to study the impact of aerodynamic effects when two trains meet, mainly focusing on the flow field distribution [15], traffic safety issues [16], changes in the pressure wave amplitude on the surface of the train [17,18,19], and the influence of different train head lengths on the average slipstream velocity and tunnel wall pressure [20,21]. Sun et al. [22] also studied the influence of four streamlined structures on aerodynamic performance and flow mechanism based on numerical simulations. Other scholars [23,24,25,26,27,28] studied the propagation of aerodynamic loads and pressure waves when trains intersected in a tunnel through numerical simulations and compared the aerodynamic performances of open lines and tunnels.

For the intersection of high-speed trains in a tunnel, different tunnel lining structures may cause different aerodynamic effects when the trains meet. At the same time, the intersection position of the train in the tunnel may not be at the midpoint of the tunnel, which could affect the pressure wave. When two high-speed trains travel at different speeds, the aerodynamic load acting on the surface of the train may also change. Although many scholars have studied the aerodynamic loads on the surfaces of trains and the wall pressures of tunnels, there have been few studies on the wall pressure and the aerodynamic load on the surface of a train based on the shield tunnel lining structure. Therefore, it is necessary to establish a simulation model with tunnel–train intersection coupling based on the three-dimensional, compressible, and unsteady Reynolds-averaged Navier-Stokes equations to compare and study the aerodynamic characteristics of the train surface and different tunnel lining structures when two trains pass each other in a tunnel.

2. Numerical Model

2.1. Governing Equations

To numerically simulate the problem of high-speed trains meeting in a tunnel, the conservation of mass, momentum, and energy must be satisfied. Since the values of the Reynolds number in the aerodynamics of high-speed railways exceed 10⁶, the air moves in a turbulent state in the tunnel. Therefore, the $k-\epsilon$ turbulence model and the Navier-Stokes equation were used in this study, combined with the ideal gas equation of state [29].

The formulae of the Reynolds-averaged Navier-Stokes equations can be expressed generally as follows:

$$\frac{\partial \rho \psi }{\partial t}+\mathrm{div}(\rho \overrightarrow{V}\psi -{\mathsf{\Gamma }}_{\psi }\times \mathrm{grad}\psi )=q_{\psi },$$

(1)

where $\rho ,\psi ,\overrightarrow{V},{\mathsf{\Gamma }}_{\psi }$ and $q_{\psi }$ represent the density, a general variable, the velocity vector, the generalized diffusion coefficient, and the generalized source term, respectively.

For the continuity equation, the terms in Equation (1) take the following values:

$$\psi =1,{\mathsf{\Gamma }}_{\psi }=0,q_{\psi }=0.$$

(2)

For the momentum equation in the $x$-direction, the terms in Equation (1) are as follows:

$$\psi =u,{\mathsf{\Gamma }}_{\psi }={\mu }_{eff},q_{\psi }=-\frac{\partial p}{\partial x}+\mathrm{div}({\mu }_{eff}\frac{\partial \overrightarrow{V}}{\partial x}),$$

(3)

where $u$ is the velocity in the $x$-direction, ${\mu }_{eff}$ is the effective viscosity coefficient, and $P$ is the pressure. The momentum equation in the $y$- and $z$-directions had similar expressions.

For the energy equation, the terms used in Equation (4) are as follows:

$$\psi =T,{\mathsf{\Gamma }}_{\psi }=\frac{\lambda }{C_P},q_{\psi }=\frac{S_h}{C_p},$$

(4)

where $T$ is the temperature, $C_p$ is the specific heat capacity, $\lambda$ is the thermal conductivity, and $S_h$ is the internal heat source term.

For the turbulent kinetic energy equation,

$$\psi =k,{\mathsf{\Gamma }}_{\psi }=\mu +\frac{{\mu }_t}{{\sigma }_k},q_{\psi }=G_k-\rho \epsilon ,$$

(5)

where $k$ is the turbulent kinetic energy, $\mu$ is the laminar flow viscosity coefficient, ${\mu }_t$ is the turbulent viscosity coefficient, ${\sigma }_k$ is the pulsating kinetic energy Prandtl number, $\epsilon$ is typically the turbulent dissipation rate, and $G_k$ is the generation term.

For the turbulent dissipation rate equation,

$$\psi =\epsilon ,{\mathsf{\Gamma }}_{\psi }=\frac{{\mu }_{eff}}{{\sigma }_{\epsilon }},q_{\psi }=\frac{\epsilon }{k}(C_1{G}_k-C_2\rho \epsilon \frac{k}{k+\sqrt{\upsilon \epsilon }}),$$

(6)

where ${\sigma }_{\epsilon }$ is the Prandtl number of the dissipation rate, $C_1$ and $C_2$ are empirical constants, and $\upsilon$ is the viscosity coefficient of the fluid’s molecular motion.

The fluid is assumed to be an ideal gas, whose equation of state is given as follows:

$$P=\rho RT,$$

(7)

where $R$ is the molar gas constant.

The above equations form a closed equation system that can be solved numerically.

2.2. Computational Model

The high-speed train model used in this paper had the same head and tail trains, and it was an eight-carriage train. The width and height of the train were 3.38 and 3.7 m, respectively. The head and tail streamline length was 12 m, and the cross-sectional area was 11.2 m². The detailed structures, such as the pantograph, windshield structure, and bogie, were neglected, and the model was a smooth train body. The total length of the train was 203 m. The parameter details are shown in

Table 1. Parameters for simulation.

Category	Tunnel length, Line Spacing (m)	Cross-Sectional Area of the Tunnel (m2)	Type of Train	Cross-Sectional Area of Train (m2)	Train Length, Width, Height (m)	Train Speed (km/h)
Parameters	1000, 5	100	CRH380A	11.42	203, 3.38, 3.7	300~400

.

The high-speed railway tunnel was a single-hole double-track tunnel with a clear area of 100 m², and the height and width of the double-track tunnel were 8.78 and 13.2 m, respectively. Tunnels with two lining structures were considered: a molded lining tunnel and a shield tunnel. Considering the actual assembly method of the shield tunnel segments, 2 m was taken as a unit body. Since the bolt holes in the tube slices were small, the requirements for the graphics processing unit (GPU) of the computer were extremely high. Thus, finally, the bolt hole on each tube slice was equivalent to a large bolt hole.

2.3. Solver Description

Studies have shown that when $M^2{L}_{tu}/L_{tr}\ll 1$, the compressibility of the fluid should be considered, where M is the Mach number, which is defined as the ratio of the speed of the high-speed train to the speed of sound in air, $L_{tu}$ is the length of the tunnel, and $L_{tr}$ is the length of the train. In this work, the tunnel length was 1000 m, the speed of sound was 340 m/s, the high-speed train length was 203 m, and the speed was 350 km/h. The value calculated using the Mach number formula above was about $0.4\ll 1$. Thus, the compressibility of the fluid should be considered. In this paper, the unsteady, viscous, compressible Navier-Stokes (N-S) equations and the renormalization group (RNG) two-equation turbulence model were selected for the simulation. This model has been widely proven to be effectively applied for the study of aerodynamic effects in high-speed railway tunnels and is sufficient to verify the reliability of the calculation results [30,31]. The governing equations were discretized by the finite volume method, the fluid pressure and velocity were coupled by the Pressure-Implicit with Splitting of Operators (PISO) algorithm, and the pressure was corrected by an iterative method. The second-order upwind style was used to discretize the convection and diffusion terms, and the second-order implicit format was used to discretize the time derivative. The calculation time step was 0.002 s, and the number of iterations per time step was 20. In the simulation, the ambient temperature was 15 °C, the atmospheric pressure was 101,325 Pa, and the standard air density was 1.225 kg/m³.

2.4. Computing Area and Boundary Conditions

The initial conditions of the flow field have a very important effect on the convergence of the calculation, especially for unsteady flows. To ensure that the flow field around the train was fully developed before the train entered the tunnel, since the two trains were facing each other, when the intersection position in the tunnel was varied, the minimum distance between the nose of the train and the entrance of the tunnel was taken as 100 m. According to EN14067:6-2018, a distance of 100 m produces a realistic simulation of the whole process of a train passing through the tunnel. The ratio of the cross section of the train to the calculated cross-sectional area of the entrance and exit of the tunnel should be less than 0.01, the height H should not be less than eight times the characteristic height, and the length L should not be less than 16 times the characteristic height. When two trains entered the tunnel at the same speed, the final calculation domain dimensions of the tunnel entrance and exit were length × width × height = 121.6H × 32.43H × 16.2H. The calculation domain and boundary conditions of the model are shown in Figure 1, where the x-axis is along the direction of the train, and the y-axis and z-axis are along the transverse and vertical directions of the tunnel, respectively.

For the boundary conditions, firstly, the normal velocity of the train wall surface, tunnel wall surface, ground surface and tunnel opening boundary surface was zero, which is the non-slip wall boundary condition, that is, the normal component of the pressure gradient is 0, which is simplified to the mathematical formula $\partial p/\partial n=0$. Secondly, during the actual operation of the train, when the pressure wave propagated and rushed out of the tunnel, it would dissipate into the surrounding atmosphere, and would not be constantly reflected like in the tunnel. In the process of simulation calculation, it was impossible to simulate the unlimited range of atmospheric flow field. At this time, the pressure far field condition needed to be set, see Figure 2b for details. The non-reflection boundary condition was based on the Riemann invariant, whereby the compression wave is transmitted without reflection after reaching the surface of the calculation area, and the boundary set at this time was the standard atmospheric pressure, that was, $p=\mathrm{101,325}Pa$. Thirdly, since the $k-\epsilon$ model was used to simulate turbulence, turbulence model boundaries were required. When calculatimng the flow in the boundary layer near the train surface, the wall function method, commonly used in engineering turbulence calculation, was used [32]. We assumed that the distance between the node A near the surface and the train surface was ${(\delta h)}_A$, which is defined as the dimensionless distance $y_A^{+}$:

$$y_A^{+}=\frac{\rho c_{\mu }^{1/4}{\overline{\mathrm{k}}}_A^{1/2}{(\delta h)}_A}{\mu }$$

(8)

where $\mu$ is the dynamic viscosity coefficient of the fluid; ${\overline{\mathrm{k}}}_A$ is the eddy kinetic energy at point A; $\rho$ is the fluid density.

The shear stress ${\tau }_w$ on the train surface is:

$${\tau }_w={\lambda }_A{(\frac{\delta \overrightarrow{v}}{\delta h})}_A$$

(9)

In the formula, ${\lambda }_A$ is the equivalent viscosity coefficient, and its value depends on the position of node A. That is, it changes with the change of $y_A^{+}$.

$${\lambda }_A=\left\{\begin{array}{c}\mu \\ \frac{\rho c_{\mu }^{1/4}{\overline{\mathrm{k}}}_A^{1/2}{(\delta h)}_A}{\ln (E_x{y}_A^{+})/{\kappa }_0}\end{array}\right.\begin{array}{c}{y}_A^{+}\le 11.6\\ y_A^{+}>11.6\end{array}$$

(10)

In the formula, ${\kappa }_0$ is the Karman constant, taking 0.41; $E_x$ is the empirical constant, taking 0.143.

The value of k on node A can be solved according to the k equation, and can be determined by the following formula:

$${\epsilon }_A=\frac{c_{\mu }^{3/4}{k}_A^{2/3}}{{\kappa }_0{(\delta h)}_A}$$

(11)

Finally, a sliding mesh method can reduce the computing cost, increase the computational efficiency, and accurately simulate mutual movement between tunnels and high-speed trains. Thus, a sliding grid was used in the simulations. The interface boundary was used for the contact surface between the moving area and the static area. The slip area (Region-A and Region-B) and the static area (Region-C) are shown in Figure 1b. Since two trains meet in the tunnel, in any control volume, when simulating the interface information transfer of the moving boundary, the integral conservation equation of the generalized scalar $\phi$ of the slip grid is:

$$\frac{d}{dt}{\displaystyle \underset{V}{\int }\rho \phi dV}+{\displaystyle {\int }_{\partial V}\rho \phi (u-u_g)dA}={\displaystyle {\int }_{\partial V}\mathsf{\Gamma }\nabla \phi •dA}+{\displaystyle {\int }_V{S}_{\phi }dV}$$

(12)

where $\partial V$ is the boundary of the control volume; $u_g$ is the slip velocity of the slip grid; $V$ is the velocity vector; $A$ is the control volume; $\mathsf{\Gamma }$ is the control area; gg is the diffusion coefficient; $S_{\phi }$ is the source term.

The computational domain was discretized as a hexahedral grid. The grid of the first layer of the boundary layer in this paper was 0.00012 m, and the grid size outside the train body was 0.005 m, and the minimum grid size of the tunnel wall was 0.25 m. The grid of the locomotive, the cross-sectional hexahedral grid of the tunnel entrance, and the partially enlarged grid of the bolt holes are shown in Figure 2. The cross-sectional dimensions of the fluid domain were width × height = 4.4 m × 4.7 m, and the distance between the two fluid domains was 0.6 m. Since there were bolt holes in the lining of the shield tunnel, when the grid was generated, the grid density was increased in the vicinity of the bolts. The grid is shown in Figure 2.

2.5. Monitoring Point Layout

A total of 13 monitoring sections were established along the longitudinal direction of the tunnel, and the distances from the tunnel entrance were 20, 50, 100, 200, 400, 500, 600, 800, 900, 950, 980, 1020, and 1050 m. Five measurement points were arranged on each monitoring section, and the heights of the measurement points from the track surface were 1.5, 3.0, and 8.78 m, as shown in Figure 3b. Figure 3a shows the distribution of measurement points in the longitudinal direction of the tunnel. The two rows represent the left track and the right track of the high-speed train running in the tunnel. It needs to be emphasized that to study some measurement points in detail, only some of the monitoring points were selected in the analysis process presented below.

3. Validation

3.1. Grid Validation

In order to verify that the grid scheme adopted in this paper was dense enough and reduce the calculation error, grid independence research was carried out by refining the grid. Two different numbers of mesh models were used, in which the mesh number of the coarse mesh model was about 47 million, and the mesh model of the fine mesh model was 62 million. Figure 4 is the pneumatic pressure curve of the central measuring point of the 1000 m tunnel (1.5 m from the rail surface). By comparing the aerodynamic pressure curves of the calculation models with different numbers of meshes, it could be seen that the aerodynamic pressures of the two mesh models with different numbers had a small difference and good consistency. Under the premise of maintaining the calculation accuracy and saving resources, this paper finally adopted the calculation model of coarse grid for numerical analysis.

3.2. Model Verification

A 1:8 dynamic model experiment consisting of a tunnel model and a train model was used to verify the results of the numerical simulation. The dynamic model adopted a three-car marshalling of the CRH train series. The total length of the train model was 9.9 m, as shown in Figure 5. The data collection equipment was an INV3010 model data collector, a 24-bit high-precision data collector, and a model 0815C-15 Endevco air pressure sensor. The locations of the measurement points, the tunnel length, the tunnel section type, and the train length in the numerical simulation were consistent with the dynamic model test. The speed of the high-speed train was 299 km/h, and its initial position was 100 m from the tunnel entrance. Pneumatic pressure sensor measurement points were arranged at a distance of 20 m from the tunnel entrance near the tunnel side, and the height of the measurement points from the track surface was 0.4 m.

As shown in Figure 6, at t = 0 s, the nose tip of the train just arrived at the tunnel entrance, and at t = 0.839 s, the train just left the tunnel completely. The analysis of the measurement point data in Figure 6 revealed that the fluctuations and peak size of the field measured data and the numerical simulations were basically consistent. The pressure at the measurement points changed correspondingly by the alternating action of the compression wave, expansion wave, and their reflected waves at the entrance of the train. The compression wave caused the pressure at the measurement point to rise, and the expansion wave caused the pressure at the measurement point to drop. After the train entered the tunnel, the two were in good agreement. After t = 0.5 s, the maximum positive peak value and the minimum negative peak value at the measurement point had large deviations. The reason may have been that the speed of the train decreased slightly after entering the tunnel. Based on comprehensive analysis, the numerical model established in this paper and the calculation results were reliable.

4. Results and Discussion

4.1. Pressure Wave Characteristics in Tunnels

Figure 7 shows the pressure–time history curve of the point of measurement on the wall at the center of the shield tunnel. The point of measurement was located on the tunnel wall of the left track at a height of 1.5 m from the track surface. The train travel speed was 350 km/h. Figure 7 shows that when a train passed through the shield tunnel, the positive and negative peak pressure loads at the central point of measurement of the lining structure were 1878.5 and −3057.2 Pa, respectively. After exiting the tunnel, they were 2445.9 and −2106.2 Pa, respectively. The pressure load ranges when the train passed through the tunnel and after it exited were 4935.7 and 4552.1 Pa, respectively. When the two trains met at the center of the tunnel, the positive and negative peak pressure loads at the central measuring point of the lining structure were 4149.2 and −6319.9 Pa, respectively. After exiting the tunnel, it was 4519.3 and −3794.3 Pa, respectively. The pressure load ranges of the train passing through the tunnel and after exiting the tunnel were 10469.1 and 8313.6 Pa, respectively. Thus, the trains met at the center of the tunnel at a constant speed, and the range of the aerodynamic pressure load changes during the passing phase of the train and after the train exited the tunnel were 2.12 times and 1.83 times that of a single vehicle, respectively. After the intersecting trains exited the tunnel, the pressure load range in the tunnel was still 1.68 times the pressure load range when a train was traveling in the tunnel. This fully showed that the intersection of the two trains had a greater impact on the aerodynamic load of the tunnel, which further demonstrated the need to study the aerodynamic loads of the intersecting pressure waves in a high-speed railway tunnel.

When the two high-speed trains entered the tunnel at the same time, the compression wave generated by the front of the car and the expansion wave generated by the rear of the car propagated and superimposed in the tunnel, after which they were reflected at the tunnel port. This caused the pressure distribution in the tunnel at the time of the intersection to become more complicated. Figure 8 shows the propagation of the pressure wave and the corresponding change when the trains met at the midpoint of the tunnel, in which Figure 8a is a schematic diagram of wave system propagation; Figure 8b is a pressure time history diagram. The horizontal line in Figure 8a represents the position of the sensor on the tunnel lining. The bold black and blue lines represent the positions of the head and tail of train A at different times, respectively, and the bold red and pink lines indicate those of train B. The relatively thin black solid line, blue solid line, red solid line, and pink solid line represent the compression waves generated by trains A and B in the tunnel, which are labeled with the letter C, e.g., C_AH, C_BH, C_AH1, C_BH1, C_BT, and C_AT. Similarly, the relatively thin black dotted line, blue dotted line, red dotted line, and pink dotted line represent the expansion waves generated by the head of train A, the head of train B, the tail of train A, and the tail of train B in the tunnel, respectively, which are labelled with the letter E, e.g., E_AT, E_BT, E_AT1, E_BT1, E_AH, and E_BH. The compression and expansion waves generated in the tunnel are represented by green solid lines and dashed lines, respectively, when trains A and B exited the tunnel. Compression or expansion waves propagate along the tunnel. When a pressure wave reaches the tunnel exit, part of the wave is released to the outside in the form of micro-pressure waves. The resulting expansion/compression wave is always weaker than the previous expansion/compression wave. Furthermore, since the propagation speed of the wave was 340 m/s greater than the speed of the high-speed train (97.22 m/s), the wave propagation line was steeper than the displacement line between the head and tail of the train.

Figure 8b shows the pressure changes with time at a height of 1.5 m from the track surface at the center of the tunnel. As shown in Figure 8b, under the three different tunnel lining structures, the pressure change trends of the tunnel wall were similar. Due to the differences of the lining structures, the peak pressures at the tunnel wall were different. The peak pressure of the molded lining structure overall was greater than the wall pressure of the shield tunnel. The reason was that there were a large number of bolt holes in the shield tunnel. At this time, the pressure wave was reflected more, and the non-linear effect caused a greater reduction in the peak value. A more specific analysis is presented in Section 4.2.

Figure 8b also shows that since the two trains entered the tunnel at the same time, the compression wave generated by the nose of the train propagated into the tunnel, and the compression waves C_AH and C_BH generated by trains A and B were superimposed in the center of the tunnel, as shown in Figure 8b-(1). At t = 2.76 s, the pressure was slightly reduced due to the low-pressure area of the train shoulder. Between t = 2.95 and 3.88 s, as the train moved further into the tunnel, the frictional force between the train and the wall of the tunnel on the air in the annular space gradually increased, causing the pressure to rise slowly. When the two trains moved by each other, the space between the two trains gradually decreased, and finally, the first pressure peak reached 4017 Pa. As the expansion waves E_AT and E_BT generated when the train’s rear entered the tunnel were also superimposed at the central measurement point of the tunnel, the pressure at the measurement point decreased, as shown in Figure 8b-(2). The compression waves (C_AH and C_BH) propagated to the exit of the tunnel and were reflected back. When expansion waves (E_AH and E_BH) propagated to the central measurement point of the tunnel again, the pressure at the measurement point at this time dropped again, as shown in Figure 8b-(3). When the nose tip of the train passed the measurement point, the pressure of the wall measurement point dropped sharply, as shown in Figure 8b-(4). When the two trains met, the compression wave reflected by the expansion wave was superimposed at the center of the tunnel. At this time, the pressure at the point of measurement in the center of the tunnel first decreased and then increased. For example, when the tails of the two trains met, the aerodynamic pressure on the tunnel wall increased sharply, as shown in Figure 8b-(5). After this, the pressure at the point of measurement in the center of the tunnel fluctuated with the propagation and reflection of the compression wave and the expansion wave. Figure 8b-(6,7) show that the compression or expansion wave caused by the head or tail of the train exiting the tunnel met at the central measurement point of the tunnel, causing the wall pressure to rise or fall, respectively.

To more clearly display the change in pressure in the tunnel and on the surface of the train as the train passed through, Figure 9 shows the pressure distributions at different times for high-speed trains meeting at constant speed. The tunnel ground pressure changed drastically with the progress of the train crossing. Before the high-speed trains rendezvoused (see t = 5.0 and 6.0 s), the pressure near the train was negative, and that at the front of the train was positive. As the high-speed trains approached each other, the negative pressure increased sharply, and the area of the negative pressure region increased with the increase in the intersection area (see t = 6.5 and 7.0 s) in Figure 9. After the train head of one train met the tail of the opposite train (see Figure 9 at t = 7.0 sand 7.5 s), the negative pressure near the train was reduced. At t = 9.0 s, after the trains had passed each other, the pressure near the train gradually returned to its state before the intersection (t = 5.0 s), but the negative pressure was lower than that before the intersection. The pressure cloud map shown in Figure 9 was consistent with the changes at the tunnel central measurement point shown in Figure 8, and the cloud map can clearly explain the changes of pressures acting on the trains and the ground pressure as the trains passed.

4.2. Aerodynamic Effects of Different Tunnel Lining Structures

This section mainly considers the aerodynamic effects of different tunnel lining structures when high-speed trains entered the tunnel simultaneously at constant speeds of 350 km/h. Figure 10 shows the comparison of the aerodynamic pressure at the midpoints of the tunnels with different lining structures. For both the integral lining tunnel and a shield tunnel, the overall trends of the curves were the same. Since the shield tunnel lining was formed by splicing segments and there were many bolt holes in the segments, when a high-speed train passed through the tunnel, the pressure wave was more disturbed. Due to the air viscosity, tunnel wall roughness, and nonlinear effects, more energy of the pressure wave could be consumed, so that the pressure amplitude of the shield tunnel decayed faster. When the train left the tunnel, as the pressure wave in the tunnel propagated and reflected, due to frictional energy consumption and wave system superposition, the amplitude of the pressure wave on the tunnel lining structure gradually decreased.

Figure 11 shows the time history curves of the pressure at various measurement points when two trains of constant speed met in the middle of the tunnel. The pressure amplitude changed with the longitudinal position of the monitoring point. The pressure amplitudes of the measurement points near the entrance and exit of the tunnel were relatively small, while the peak positive and negative pressures at the central measurement point of the tunnel were the largest. For both the shield tunnel and the integrally molded lining tunnel, the pressure curves of the measurement points at 200 and 800 m were almost completely coincident. That is, the pressure value of the measurement point 200 m from the tunnel entrance was the same as the pressure value of the measurement point at a distance of 200 m from the tunnel exit. At this time, the pressures at the measurement points at both ends were symmetrically distributed. The maximum positive pressure of 4519.3 Pa at the central monitoring point in the tunnel appeared at t = 14.66 s. According to Figure 7, the maximum positive pressure peak was caused by the superposition of the compression waves C_BT1, C_AT1, C_AH2, and C_BH2. The first positive pressure peak of 4149.2 Pa appeared at t = 4.227 s. At this time, the positive pressure peak was caused by compression waves C_AH and C_BH and the air viscosity. The maximum negative pressure peak (−6319.9 Pa) occurred at t = 7.15 s when the negative pressure around the train was the largest. The maximum positive and negative peak pressures at the measurement points of the lining structure 200 and 800 m from the tunnel entrance both appeared in the fluctuation period after the train completely exited the tunnel. Therefore, it is necessary to study the aerodynamic load changes in two different stages: when the train passes through the tunnel and when the train completely exits the tunnel.

When the left and right lines in a tunnel both have high-speed trains running and intersections occur, the micro-pressure wave outside the tunnel entrance will inevitably be very different from when a train travels on a single line. The micro-pressure wave is induced by the compression wave generated by the train entering the tunnel and radiating to the surroundings after reaching the exit of the tunnel. The right-track train will also pass through the measurement point of the micro-pressure wave when it is not entering the tunnel, which will cause the pressure value at the measurement point to increase sharply. Therefore, for the pressure curve at each measurement point, only the micro-pressure part formed by the divergence of the compression wave pulse should be analyzed. Figure 12 shows the micro-pressure wave curve when the constant-velocity trains crossed in the middle of the tunnel and the single train was 20 m outside the tunnel exit. From the moment when the peak appeared in Figure 12, the micro-pressure waves were all formed by the initial compression wave generated by the train entering the tunnel and the radiation diverging from the tunnel exit. Figure 12 and

Table 2. Comparison of the peak values of the micro-pressure wave 20 m from the exit of the tunnel.

Lining Structure Type	Integral Lining Tunnel—100 m2	Shield Tunnel—100 m2		MPW Variation Rate
Situation	Intersection at the midpoint	Single train	Intersection at the midpoint	Comparison of different lining tunnels	Single train and intersection
MPW maximum (Pa)	45.54	39.6	40.63	10.78%	2.6%

show that when the high-speed trains crossed in the center of the molded lining tunnel, the micro-pressure waves (positive and negative peaks) generated were the largest. Compared with the micro-pressure waves generated by the trains passing through the integrally molded lining tunnel, the micro-pressure waves of the shield tunnel were reduced by 10.78%. In contrast, when the high-speed trains traveled in one-way and two-way directions in the shield tunnel, the micro-pressure wave when the two trains met in the center of the tunnel increased by 2.6%.

Figure 13 shows the pressure decay rate at the central measurement point of the tunnel at a height of 1.5 m from the track surface after the train exited the tunnel. The black solid line and the dashed line in Figure 13 represent the pressure attenuation rates of the positive and negative peaks of the same attenuation period under different tunnel lining structures, respectively, which were calculated by the formulae $(p_{I_i-\max }-p_{S_i-\max })/p_{I_i-\max }$ and $(p_{I_i-\min }-p_{S_i-\min })/p_{I_i-\min }$.

The positive and negative peak pressures of the shield tunnel at different cycle numbers are represented by $p_{S_i-\max }$ and $p_{S_i-\min }$, respectively, and those of the molded lining tunnel are represented by $p_{I_i-\max }$ and $p_{I_i-\min }$, where $i$ is the cycle number (1, 2, 3,…, 10), “S” represents the shield tunnel, and “I” stands for molded lining tunnel. Similarly, the blue solid line and the dashed line represent the positive and negative peak pressure decay rates of different cycle periods under the shield tunnel, respectively, which were calculated by the formulae $(p_{S_i-\max }-p_{S_{i+1}-\max })/p_{S_i-\max }$ and $(p_{S_i-\min }-p_{S_{i+1}-\min })/p_{S_i-\min }$. The red solid line and the dashed line represent the positive and negative peak pressure attenuation rates of different cycle periods under the molded lining tunnel, respectively, which were calculated by the formulae $(p_{I_i-\max }-p_{I_{i+1}-\max })/p_{I_i-\max }$ and $(p_{I_i-\min }-p_{I_{i+1}-\min })/p_{I_i-\min }$.

As shown in Figure 13, after the train left the tunnel, as the fluctuation period increased, the peak pressure of the tunnel lining structure at the point of measurement gradually attenuated. The positive and negative peak attenuation rates of the shield tunnel were higher than those of the molded lining tunnel. In the first two fluctuation cycles, the pressure decay rates of the positive and negative peaks were relatively large, ranging from 16.1% to 28.6%. In the third pressure fluctuation period, the pressure decay rate increased significantly, and the maximum pressure decay rate was the positive peak pressure decay rate of the shield tunnel, which could reach 57.8%. During the subsequent pressure fluctuation period, the pressure decay rate in the molded lining tunnel was maintained at 4.1–9.3%. The pressure decay rate in the shield tunnel was maintained at 7.8–15.3%. The common trend was that the attenuation rate of the same measurement point showed a downward trend as the number of cycles increased. Figure 13 shows that for the aerodynamic pressure decay rates on different tunnel linings, as the fluctuation period increased, the pressure decay rate gradually increased, while the growth rate gradually slowed.

4.3. Aerodynamic Effects at Different Intersection Positions

The aerodynamic effects of the high-speed trains when they met in the shield tunnel were analyzed. The intersection points were selected as 100, 200, 300, 400, and 500 m from the tunnel entrance. The nose of the left train was 900, 700, 500, 300, and 100 m from the tunnel entrance, and the nose of the right train was 100 m away from the tunnel exit. The two trains traveled into the tunnel at a speed of 350 km/h. The pressure changes on the shield tunnel lining structure and the micro-pressure waves 20 and 50 m outside the tunnel exit under five working conditions were compared and calculated. Figure 13 shows the comparison of the pressure peaks in the tunnel at different intersections during the whole process of the train traveling in the tunnel. Figure 14 shows the pressure peaks of the lining structure of the shield tunnel under the two-stage conditions as the train traveled through (Figure 14a) and after completely exiting the tunnel (Figure 14b).

Figure 14a shows that the pressure amplitude varied with the longitudinal position of the tunnel monitoring point. When the high-speed trains met in the middle of the tunnel, the pressure amplitude near the tunnel entrance was smaller, but it was larger in the middle of the tunnel. When the high-speed trains met in the center of the tunnel, maximum positive and negative peak pressures appeared. The peak pressure appeared at the central measurement point of the tunnel, and the measurement point of the peak pressure was symmetric. In these five intersection conditions, when the trains met 100 and 200 m from the tunnel entrance, the pressure peak generated was the lowest. The maximum positive peak pressure was much smaller than that when the trains met at the center of the tunnel. When the train intersected at 100, 400, and 500 m from the tunnel entrance, the positive and negative peak pressures at the intersection point were maximal. When trains met at 200 and 300 m, the maximum positive and negative peak pressures on the shield tunnel lining structure deviated from the intersection position. There may have been three reasons for the deviation [3]: (a) the initial compression wave from the head of the train entering the tunnel and the expansion wave formed by reflection at the tunnel exit, (b) the initial expansion wave generated when the rear of the train entered the tunnel and the compression wave formed by its reflection, and (c) the mutual pressure change caused by two trains meeting in the tunnel.

Figure 14b shows that after the trains completely exited the tunnel, when the trains had intersected at distances of 200 and 500 m from the tunnel entrance, the maximum positive peak pressure was still greater than the positive peak pressure when the train was traveling in the tunnel. The overall value was less than the peak pressure when the train was traveling in the tunnel. This further verified the necessity of studying the two different stages of the train passing through the tunnel.

Figure 15 shows the pressure curve of the micro-pressure wave outside the tunnel opening at different intersection positions. The micro-pressure waves generated when two trains intersected in the center of the tunnel were analyzed in Section 4.2. In this section, only the micro-pressure waves in the four working conditions are discussed. For the pressure curve at each measurement point, only the micro-pressure part that formed by the divergence of the compression wave pulse is analyzed. Figure 15 shows that a low- pressure wave formed 50 m from the tunnel exit caused by trains meeting at a distance of 200 m from the tunnel entrance, with a pressure of 23.6 Pa. On the whole, the micro-pressure wave at 20 m and 50 m outside the shield tunnel exit was less than 50 Pa and 20 Pa, respectively. It basically met the requirements of the tunnel exit micro-pressure wave in 7.0.2 of the “Technical Specifications for Dynamic Acceptance of High-speed Railway Engineering.” The moments when the micro-pressure wave outside the tunnel exit reached peak values were t = 12.32 s (12.41 s), 10.29 s (10.37 s), 8.25 s (8.35 s), and 6.17 s (6.26 s). These were caused by radiation divergence, due to the initial compression wave generated by the left-line train entering the tunnel and being transmitted to a distance of 20 m (50 m) outside the tunnel exit.

4.4. Aerodynamic Load on Surface of Train

When the high-speed train entered the shield tunnel, the pneumatic load acting on the train carriage expanded in the X-, Y-, and Z-directions, corresponding to the aerodynamic resistance of the train, the aerodynamic side force, and the aerodynamic lift, respectively. Each train carriage was divided into several segments along the direction of travel, the averages of the static pressure data over time were computed at the bottom surface, top surface, leeward side, and windward side for each segment in each operating condition. When the two trains intersected in the tunnel, the impact on the pneumatic lift was small, and the torque was calculated based on the pneumatic side force. The pneumatic resistance and pneumatic side force mainly depend on the speed, lining structure, and intersection position. The formulae used to calculate the aerodynamic resistance F_x and the aerodynamic lateral force F_y on a single carriage of the train are as follows:

$$F_x=0.5\rho v_t{}^2{C}_d{A}_{tr1},$$

(13)

$$F_y=0.5\rho v_t{}^2{C}_{s}A{’}_{tr},$$

(14)

where $\rho$ is the air density (kg/m³), $v_t$ is the train speed, $A{’}_{tr}$ is the area of the side of the train and $A_{tr1}$ is the cross-sectional area of the train.

When high-speed trains meet in a tunnel, due to the space limitations in the tunnel, the transient amplitude of the aerodynamic load will significantly affect the safety of the train operation. Figure 16 shows a time history curve of aerodynamic drag at different train speeds.

Table 3. Maximum aerodynamic resistance of the train surface under different working conditions.

Different Conditions	Train Speed (km/h)			Intersection Position (m)
	300	350	400	100	200	300	400	500
$F_{\mathrm{x}-\max }(\mathrm{kN})$	21.9	30.116	39.48	25.63	22.13	19.81	28.44	30.22

shows the maximum aerodynamic resistance of the train surface under different working conditions. Figure 16 and Table 3 show that as the train speed increased, the aerodynamic drag on the train surface gradually increased. The aerodynamic drag generated at a train speed of 400 km/h in the shield tunnel was 80.3% higher than that at a train speed of 300 km/h. At the same speed (350 km/h), the maximum aerodynamic drag and the minimum aero-dynamic drag generated by trains in the lining/shield tunnels were 29.7/30.1 kN and −32.8/−32.3 kN, respectively. Thus, the maximum aerodynamic drag difference of the lining and shield tunnels was about 1.35%, and that of the minimum aerodynamic drag was 1.52%. The aerodynamic drag was greatly affected by the train speed. Under the same blocking ratio, the influences of the two different tunnel lining structures on the aerodynamic drag were almost negligible.

Figure 17 shows the aerodynamic lateral force of the train surface. As shown in Figure 17a, there was a slight difference between the maximum positive and negative lateral forces when two trains met in the shield tunnel and the molded lining tunnel. The maximum positive and negative side pressures of the train head in the lining/shield tunnels were 25.2/25.3 kN and −20/−21.4 kN, respectively. Thus, the maximum positive side pressure difference of the train head in the lining and shield tunnels was about 1.35%, and that of the minimum positive side pressure was 1.52%. The maximum positive and maximum negative side pressures of the middle train in the lining/shield tunnels were 13.1/13.2 kN and −13.6/−13.7 kN, respectively. Thus, the maximum positive side pressure difference of the middle train in the lining and shield tunnels was about 0.8%, and that of the minimum positive side pressure was 0.7%. The maximum positive and maximum negative side pressures of the tail train in the lining/shield tunnels were 10.14/11 kN and −28.1/−28.3 kN, respectively. Thus, the maximum positive side pressure difference of the tail train in the lining and shield tunnels was about 8.5%, and that of the minimum positive side pressure was 0.7%.

As shown in Figure 17b, as the train speed increased, the maximum lateral force on the head of the train increased more than it did for the middle and trailing trains. This showed that, compared with different tunnel lining structures, the influence of the train speed on the aerodynamic lateral force on the surface of the train was slightly greater.

Figure 18 shows the maximum aerodynamic lateral forces on the surface of the train for different intersection positions in the tunnel. The data in Figure 18 and Table 3 show that when the trains met at different positions in the shield tunnel, as the train gradually passed the central measurement point of the tunnel, the maximum aerodynamic resistance showed a trend of first decreasing, and then increasing. At the central measurement point of the tunnel, the aerodynamic resistance reached a maximum value of 30.22 kN. The overall differences in the maximum aerodynamic lateral forces, whether it was at the head, middle, and rear of the train, were not large. This indicated that when the trains met at different locations in the tunnel, the impact of the intersection position on the aerodynamic lateral force was not significant.

5. Conclusions

The three-dimensional, compressible, and unsteady RANS method was used to study the aerodynamic characteristics of high-speed trains when they met in a shield tunnel. By comparing the numerical calculation results with dynamic model tests, the reliability of the numerical simulation was verified. The conclusions were as follows:

(1)

Through detailed analysis of wave diagrams and pressure cloud diagrams of high-speed trains meeting in a shield tunnel, the mechanism of the pressure changes on the train surface and lining structure during the intersection of the two trains was explained intuitively and clearly.

(2)

Based on the analysis of the aerodynamic effects of different tunnel lining structures, it was revealed that the shield tunnels were affected by bolt holes, the tunnel wall roughness, and nonlinear effects, and the pressure amplitude decayed faster. When the high-speed trains met in the middle of the tunnel, the positive and negative peaks of the micro-pressure wave of the shield tunnel were reduced by 10.78% and 4.39%, respectively, compared with the micro-pressure wave of the molded lining tunnel. Therefore, the use of a shield tunnel lining can reduce the micro-pressure wave at the exit of the tunnel.

(3)

After the train left the tunnel, the positive and negative peak attenuation rates of the shield tunnel were higher than the positive and negative peak attenuation rates of the molded lining tunnel. In the first three fluctuation cycles, the pressure decay rates of the positive and negative peaks became gradually larger, and the maximum pressure decay rate in the shield tunnel could reach 57.8%. In the subsequent period of pressure fluctuations, the pressure decay rate in the molded lining tunnel was maintained at 4.1–9.3%, while that in the shield tunnel was maintained at 7.8–15.3%.

(4)

The maximum positive and negative pressure values on the tunnel lining structure when high-speed trains intersected in the middle of the tunnel were both located at the central measurement point of the tunnel, and the maximum pressure at this measurement point was significantly greater than that at other intersections in the tunnel. The values of the micro-pressure waves 20 and 50 m outside the tunnel exit, caused by different intersection positions, could meet the specification requirements.

(5)

The train speed had a greater influence on the aerodynamic drag and aerodynamic lateral force on the surface of the train, while different intersection positions and different tunnel lining structures had relatively little influence on the aerodynamic drag and aerodynamic lateral force.