Abstract

Piston action describes the phenomenon that air at the train nose is pushed forward by the increased pressure and air at the train rear is drawn forward by the decreased pressure when a train passes through a tunnel. The variation of pressure changes the thermal environment inside the tunnel and promotes the frost damage generation in cold season. In this paper, a fluid-thermal-solid involving piston action is proposed. A high-speed railway tunnel that is located in the northeast of China is set in the background, and the influence of outside air temperature and train velocity on the thermal environment inside the tunnel are simulated and analyzed. The piston action can significantly change the temperature inside the tunnel, especially at the ends of tunnel. The temperature distribution is characterized by three zones, including disturbed zones at two sides of tunnel and undisturbed zone at tunnel middle. The freezing length is closely related to air temperature and train velocity. And also, the lengths are different at vault and rail of tunnel portal, which indicates that the anti-freezing measure should be different at these positions considering the cost. This paper can help us understand the temperature evolution process of tunnels under the action of piston wind and provide some guidelines for structural design of cold regional tunnels.

1 Introduction

Cold regions account for 43.5% of total terrestrial area in China [1], which stores a lot of mineral resources. With the development of the modern transportation system, the high-speed railways have been rapidly expanded to cold regions. By the end of 2020, the total length of high-speed railway reaches to 37,900 km in China. Due to the extremely cold environment, the subgrade and tunnel are subjected to serious frost damage [25]. In the northeast and northwest of China, some tunnels cannot be used for 8–9 months in a year based on the survey [6]. Cracks and water leakage caused by frost damage frequently occur on tunnels in winter, such as in Norway, Russia, Alaska, Sweden, Japan, and so on [711]. These potential adverse factors will increase the risk of train operation and reduce the tunnel life. Therefore, how to protect the tunnels from frost damage is a key point in tunnel design and construction.

Until now, some scholars have proposed the active heating method and passive insulation method to minimize the frost damage, such as installing electric heating pipe and steam pipe [12,13], laying insulation materials [14], and establishing insulation doors [15]. Although these measures are helpful to protect the tunnel from frost damage, frost damage still occurs. The air temperature and wind characteristics have a significantly effect on the thermal environment inside the tunnel, which have been validated and acknowledged in previous literatures [16,17]. But the effect of piston action on the thermal environment inside the tunnel was not given enough attention in most previous literatures. The researches considering the piston action induced by train running mainly focused on air pressure and velocity instead of air temperature [18,19], and most of the studies are carried out on the subway tunnel [20,21]. The effect of piston action on the thermal environment inside the tunnel can be found in Ref. [22], but unfortunately, which mainly studies the temperature distribution in horizontal section but vertical section. On the other hand, the frost damage is different on tunnel vault and railway ballast because the temperature distribution is non-uniform in vertical section [23]. As the heat transfer process is related to flow characteristics and finally decides the thermal environment inside the tunnel, the investigation of temperature distribution in vertical section caused by piston action would help us to determine the location of frost damage and provide some guidelines for cold regional tunnels in design and construction process.

The Jilin–Tumen–Hunchun Passenger Dedicated Line is one of the main transportation routes in northeast of China. The frost damage appeared on some tunnels along the line in the winter of 2016. Although the tunnels have been maintained for many times, frost damage still exists, such as the sidewall and track icing, the secondary lining, and drainage ditch freezing. Therefore, the railway workers must clear and carry out the accumulated ice at night to ensure the normal operation of the trains, which increases the operation cost of railway. Besides, the anti-freezing design of these tunnels refers to the engineering experience of tunnels that service for conventional train in the same area, and the 5 cm thick thermal insulation material is laid between the secondary lining and the primary support near tunnel entrance and exit, but this method is not obviously suitable for high-speed railway tunnels. The piston action maybe a key factor for frost damage in high-speed railway tunnel because the air flow is drastic.

Based on this, the aims of this paper are to (1) analyze the influence of piston action on thermal environment inside the tunnel, (2) reveal the influence of air temperature outside the tunnel and train velocity on the heat transfer process between air and tunnel structure, and (3) determine the freezing length along the tunnel longitudinal direction. This study can provide a reference for cold regional tunnels in the design and construction process.

2 Tunnel Site Description

Yushuchuan tunnel is located in Antu county, Jinlin province. The total length is 2190 m. The maximum buried depth is 165 m, as shown in Fig. 1. The wholly weathered granite distributes on the mountain surface and the thickness is 4.5 m.

Fig. 1
Outline of the studied tunnel
Fig. 1
Outline of the studied tunnel
Close modal
The underground water belongs to bedrock fissure water. The normal water inflow and maximum water inflow of the tunnel are 600 m3/d and 1200 m3/d, respectively. The average annual precipitation is 814 mm. The climate type of the tunnel site belongs to humid and semi-humid continental monsoon climate in North sub-temperate zone. The maximum snow depth is 0.5 m and the maximum frozen depth is approximately 1.75 m based on the records of local meteorological stations [24]. The extreme temperature recorded in history is −40.2 °C. The mean annual air temperature is 5.72 °C based on the in situ monitoring, as shown in Figs. 2(a) and 2(b). The changes of air temperature can be simplified as a sine function, which can be found in Eq. (1).
T=5.72+17.11sin(2πt365182π365)
(1)
Fig. 2
Meteorological station at tunnel entrance and the variation of air temperature: (a) meteorological station at tunnel inlet and (b) air temperature changing process outside the tunnel
Fig. 2
Meteorological station at tunnel entrance and the variation of air temperature: (a) meteorological station at tunnel inlet and (b) air temperature changing process outside the tunnel
Close modal

3 Governing Equation and Modeling Procedures

3.1 Governing Equation of Heat Transfer.

Assume that the effect of air and water migration on heat transfer of surrounding rock can be neglected during the freeze–thaw process and the properties of surrounding rock are isotropous [25]. The governing equation considering the ice-water phase change for heat transfer in surrounding rock can be described as follows:
InthawedzonechρTht=x(λhThx)+y(λhThy)
(2)
InfrozenzonecfρTft=x(λfTfx)+y(λfTfy)
(3)
The maximum unfrozen water content can be expressed as
θu=a1|T|b
(4)
a1 = 0.14 and b1 = 0.45 in this study.
The latent heat of surrounding rock can be expressed as
L=ρQ(θθu)
(5)
where Q = 334.5 kJ/kg and θ = 0.15 in this study.
Assuming that the phase change occurs in the temperature region denoted by (Tm + ΔT), then the equivalent heat capacity Cp and the equivalent thermal conductivity λp can be expressed as follows [26]:
Cp={Cf,L2ΔTCh,+Cf+Ch2,T<TmΔTTmΔTTTmΔTT<TmΔT
(6)
λp={λfλf+λhλf2ΔT[T(TmΔT)]λhT<TmΔTTmΔTTTmΔTT<TmΔT
(7)
The governing equation should meet continuity condition and energy conservation equation in the freezing front s(t):
Tf(s(t),t)=Th(s(t),t)=Tm
(8)
λfTfn1λhThn1=Lds(t)dt
(9)
Initial condition is as follows:
T|t=0=T0(x,y)
(10)

The temperature boundary conditions are as follows:

  1. The temperature boundary of the mountain surface is known, that is:
    T=T¯(x,y,t)
    (11)
    where T¯(x,y,z) is a known function of space and time. In this paper, a sine function is employed in order to be consistent with the actual ground temperature changes.
  2. The heat flux on the mountain bottom is fixed and can be written as
    λTn=qT(x,y,t)
    (12)
    The energy equation can be expressed as
    ρcaTt=x(λaTx)+y(λaTy)ρca((uiT)x+(ujT)y)
    (13)

3.2 Governing Equation of Fluid–Solid Coupled.

The continuity equation, momentum equation, and standard kε equation model are adopted in order to calculate the airflow.

The hydraulic diameter D can be expressed as
D=4AP
(14)
The continuity equation can be described as
ρt(ρui)xi=0
(15)
The momentum equation can be written as
(ρui)t+(ρuiuj)xi=pxi+xj[μ+μt(uixj+ujxi)]
(16)
The μt can be calculated as
μt=ρCμk2ε
(17)
The transport equation of turbulent kinetic energy coefficient and dissipation rate of turbulent kinetic energy for the incompressive fluid can be expressed as
(ρk)t+(ρkui)xi=xj[(μi+μiσk)kxj]+Gkρε
(18)
(ρε)t+(ρεui)xi=C1εεkGkC2cρε2k+xj[(μi+μiσε)εxj]
(19)
where C1ɛ, C2c, σɛ, and σk are model coeficient, which are 1.47, 1.92, 1.33, and 1.0, respectively.
The Gk can be written as
Gk=μt(uixj+ujxj)uixi
(20)
The convective heat transfer coeffieient between fluid and solid inside the tunnel can be obtained on the basis of standard temperature wall function, which is given by
hw=ρcpcu1/4k1/2T*=qTwTp
(21)
T*={Pryp++12ρPrcμ1/4kp1/2qμp2(yp+<yTp+)Prt[1κln(Eyp+)+B]+12ρPrcμ1/4kp1/2q{Prtμp2+(PrPrt)uc2}(yp+>yTp+)
(22)
The term B can be computed by
B=9.24[(PrPrt)3/41][1+0.28e0.007Pr/Prt]
(23)

3.3 Governing Equation of Dynamic Mesh.

The dynamic mesh model is very good at simulating the change of boundary with time. There are three conventional methods to solve the dynamic mesh model: they are dynamic layering, spring smoothing, and local remeshing. The dynamic layer can merge and segment the mesh automatically together with avoiding the negative volume generation in the calculation process, which is adopted in this paper [27].

The conservation equation for a general scalar on a random control volume can be expressed as
ddVρϕdV+Vρ(uug)dA=VρΓϕdA+VSϕdV
(24)
The time derivative in Eq. (24) is obtained by first-order backward difference formlua.
ddtVρϕdV=(ρϕV)n+1(ρϕV)nΔt
(25)
The (n + 1)th time-step volume, Vn+ 1, can be determined as follows:
Vn+1=Vn+dVdtΔt
(26)
To meet the mesh conservation rate, the derivative of control volume with respect to time can be described as
dVdt=VugdA=jnfug,jAj
(27)
The dot product ug,jAj on each control volume face can be calculated as
ug,jAj=δVjΔt
(28)

3.4 Numerical Model.

In order to obtain the initial temperature distribution of tunnel for analyzing the effect of piston action, a full-scale 2d model consists of the mountain and the tunnel is established at first. The distance between the bottom of geometric model and tunnel inverted arch is 30 m, and the height of tunnel is 10 m. The thermal parameters of surrounding rock and linings are listed in Table 1. And also, the heat flux at the bottom of mountain is 0.07 W m−2 [28].

Table 1

Thermal parameters of the surrounding rock and the tunnel structure [29]

λf (W m−1 °C−1)λh (W m−1 °C−1)Cf (J kg−1 °C−1)Ch (J kg−1 °C−1)L (J m−3)ρ (kg m−3)
Rock3.493.392110211.05 × 1072200
Preliminary lining2.231.002300
Secondary lining1.740.932500
λf (W m−1 °C−1)λh (W m−1 °C−1)Cf (J kg−1 °C−1)Ch (J kg−1 °C−1)L (J m−3)ρ (kg m−3)
Rock3.493.392110211.05 × 1072200
Preliminary lining2.231.002300
Secondary lining1.740.932500

The 2d model is also employed to simulate piston action by replacing 3d model and the accuracy was confirmed [20]. The geometric model includes the air domain at tunnel portal with the dimension of 500 m × 100 m (width × height). The thickness of secondary lining and preliminary lining are 0.40 m and 0.25 m, respectively. The geometric model of train is based on the CRH 380A (the full name is Harmony CRH380A Electric Multiple Unit, Manufactured by CRRC Corporation Limited, the maximum velocity is 486.10 km/h), as shown in Fig. 3.

Fig. 3
Geometric model for piston action
Fig. 3
Geometric model for piston action
Close modal

The geometric model is based on the following assumptions:

  1. The slope is assumed to be 0 deg.

  2. Considering the effect of the train bogie, pantograph, and traction motor on air flow and heat transfer is limited [30], these parts can be neglected in the model.

  3. Since four carriages are enough to reflect the characteristics of airflow [31], then the train dimension is determined with length of 103 m, height and width of 3.7 m and 3.38 m, respectively.

  4. The initial position of train is 150 m away from the tunnel entrance to ensure the full development of airflow around train body.

Moreover, assembly mesh is constructed with ANYSIS ICEM 19.0, including the quadrilateral mesh of 17,523 and the triangular mesh of 216,307. The boundary conditions are shown in Table 2; the tunnel entrance and exit are pressure inlet and pressure outlet, respectively. The train, tunnel wall, and ground are set as the wall boundary conditions. The adjacent cells of fluid zone and solid zone are set as the interface boundary conditions.

Table 2

Boundary conditions

PositionBoundary type
A1B1, A1E1, B1C1Pressure inlet
A2B2, B2E2, A2C2Pressure outlet
E1E2, C1C2No-slip wall
D1D2Interface
PositionBoundary type
A1B1, A1E1, B1C1Pressure inlet
A2B2, B2E2, A2C2Pressure outlet
E1E2, C1C2No-slip wall
D1D2Interface

4 Results and Discussions

4.1 Temperature Distribution of Tunnel Without Considering the Piston Action.

The initial thermal conditions of the mountain play a decisive role in the occurrence and degree of frost damage in the tunnel. In some high geothermal regions, the heat in surrounding rock can be used to eliminate frost damage. Meanwhile, to accurately assess the temperature distribution inside the tunnel, the analysis of initial thermal condition of the mountain must be carried out.

First of all, the theoretical temperature is calculated based on the geothermal theory. For the temperature distribution under the ground surface, it consists of three regions along the vertical direction, including variable temperature region, constant temperature region, and increasing temperature region [32]. The relationship can be expressed as the function of buried depth, as shown in Eq. (29):
TH=Z+GH
(29)
where Z is 7.5 °C and G is 0.02 °C/m in this study [28]. The temperature of the surrounding rock at the maximum buried depth ranges from 11.00 °C to 11.20 °C.

Then the initial temperature distribution of mountain can be calculated by the proposed model involving ice-water phase. The simulated temperature at the bottom of the mountain is 11.10 °C (Fig. 4(a)). Compared with the theoretical temperature (the ranges from 11.00 °C to 11.20 °C), the simulated results are more consistent with theoretical results, indicating numerical results with a well reliability.

Fig. 4
Temperature distribution of the mountain before and after excavation: (a) before excavation and (b) after excavation: (i) global view of the mountain, (ii) tunnel entrance, and (iii) tunnel exit
Fig. 4
Temperature distribution of the mountain before and after excavation: (a) before excavation and (b) after excavation: (i) global view of the mountain, (ii) tunnel entrance, and (iii) tunnel exit
Close modal

The effect of solar radiation on temperature distribution is also neglected [33]. The temperature distribution of tunnel after construction is calculated only considering the natural convection heat transfer. Figures 4(b-i) show the temperature distribution of tunnel after excavation. The air temperature decreases at tunnel entrance and exit after the excavation, because the natural convection heat transfer is between air inside and outside the tunnel. Moreover, the frost depth at entrance and exit is approximately 4.8 m (Figs. 4(b-ii) and 4(b-iii)). However, except that temperature at entrance and exit of tunnel decreases slightly, the temperature at secondary lining and preliminary lining as well as the surrounding rock are higher than 0 °C, which suggests that the frost damage only appeared at two sides of the tunnel without considering piston action.

4.2 Influence of Air Temperature.

The air temperature inside the tunnel is directly influenced by the air temperature outside the tunnel due to piston action. The frost damage in the range of 0–50 m away from tunnel entrance is more serious based on the survey. To show the changes of air temperature and temperature of tunnel structures, some representative positions that are located 50 m away from the entrance were recorded in the simulation process, which are located at secondary lining surface and secondary lining interior. Meanwhile, considering that the train always runs at the velocity lower than the designed velocity (250 km/h) and the air temperature outside the tunnel ranges from −30 °C to 0 °C in cold season, five air temperatures (0, −5, −10, −20, and −30 °C) are used to analyze the effect of air temperature on temperature distribution inside the tunnel. Here, the train velocity is 200 km/h.

Figure 5 illustrates the temperature distribution inside the tunnel under different air temperatures. It can be found that the temperature gradually increases from two sides of tunnel to tunnel middle. The reason for this phenomenon is that the negative pressure generated at train rear leads to the cold air outside the tunnel to be carried towards tunnel exit. According to the distribution of 0 deg isotherm, the freezing length can be determined. Along with air temperature decreasing from 0 °C to −30 °C, the freezing lengths at vault are 0, 505.03, 539.60, 585.77, and 616.33 m, respectively. The lengths at rail are 0, 603.22, 639.75, 682.62, and 713.31 m, respectively. It shows that the lengths at vault are longer than that at rail and the difference between two of them are 0, 98.20, 100.10, 96.85, and 96.96 m, respectively. The difference in turbulence intensity is the major factor that causes the difference in the lengths between vault and rail. Because the distance between train bottom and rail is less than that between train top and vault when the train is running in the tunnel, the air turbulent is relatively stronger near the rail. Therefore, the negative pressure near rail is greater than that near vault and the cold air near rail is more convenient to be sucked into the tunnel. Under this circumstance, the anti-freezing measures at the rail should be strengthened appropriately in the design and construction of the tunnels, such as extending the insulation layer near the rail and digging the deeper insulation gutter.

Fig. 5
Temperature distribution of the tunnel with different air temperatures after train passes through the tunnel: (a) 0 °C, (b) −5 °C, (c) −10 °C, (d) −20 °C, and (e) −30 °C
Fig. 5
Temperature distribution of the tunnel with different air temperatures after train passes through the tunnel: (a) 0 °C, (b) −5 °C, (c) −10 °C, (d) −20 °C, and (e) −30 °C
Close modal

Besides, temperature variation at these characteristic positions with different air temperatures is shown in Fig. 6. From Fig. 6(a), it can be seen that the temperature at the secondary lining surface had no dramatic changes before 5.5 s (the train rear arrives at the characteristic positions at 5.5 s). After 5.5 s, the temperature at the secondary lining surface decreases sharply because the cold air is sucked into the tunnel and heat exchange occurred between warm air and cold air. Then, when the train runs away from the characteristic positions, the temperature at the secondary lining surface gradually comes to stability because the piston action becomes weaker.

Fig. 6
Temperature at the secondary lining surface and secondary lining interior under different air temperatures outside the tunnel
Fig. 6
Temperature at the secondary lining surface and secondary lining interior under different air temperatures outside the tunnel
Close modal

Additionally, the rules of temperature change at secondary lining interior are similar to that at the secondary lining surface during the train passing. However, the abrupt time of temperature of secondary lining interior is lagged compared with the secondary lining surface, and the temperature decrease is also minor (Fig. 6(b)). The reason is that the temperature of the secondary lining surface decreases due to the forced convention heat transfer, but for the temperature of secondary lining interior, it decreases due to the heat conduction. When the train passes through the tunnel with a high velocity, the rate of forced convection heat transfer is faster than the heat conduction.

It can also be seen from Figs. 6(a) and 6(b) that the temperature of the characteristic points is related to the temperature outside the tunnel. When the air temperature is 0, −5, −10, −20, and −30 °C, respectively, the corresponding temperature decreases at the secondary lining surface and at secondary lining interior are 4.9, 8.3, 11.6, 18.3, 25.1 °C and 0.7, 1.2, 1.7, 2.7, 3.7 °C, respectively. Under the action of piston action, the heat transfer efficiency between air and tunnel structure is proportional to the absolute value of the temperature outside the tunnel.

4.3 Influence of Train Velocity.

Train velocity has a significant influence on the intensity of piston action [34]. At this point, taking the designed speed of the high-speed railway (HSR) as the upper bound and the speed of the conventional train as the lower bound (80 km/h), five different train velocities (80, 130, 160, 200, and 250 km/h) are used to analyze the effects of train velocity on temperature distribution. Here, the air temperature of the tunnel is −30 °C.

Figure 7 shows the temperature distribution under different train velocities. Based on the temperature distribution of air inside the tunnel, the temperature distribution can be divided into three zones based on the temperature distribution of air inside the tunnel, including the disturbed zones at tunnel entrance and exit, the undisturbed zone at the tunnel middle. The lengths of the disturbed zone at entrance and exit of the tunnel are within 1000 m and 10 m, respectively, while the undisturbed zone is within 1180 m. At the disturbed zones, the effect of piston action is drastic. So, the forced convection heat transfer between cold air and warm air is efficient, resulting in the air temperature at these zones decreased significantly. But, at the undisturbed zone, the air temperature slightly changes during the train passing through the tunnel because the forced convection heat transfer is weak. It can also be found that the freezing length is closely related to train velocity. When the train velocities are 80, 130, 160, 200, and 250 km/h, the lengths at vault of tunnel entrance are 638.83, 624.68, 622.23, 617.45, and 600.56 m, respectively, while the lengths at tunnel exit are 4.35, 12.43, 17.89, 22.94, and 29.36 m, respectively. Overall, the lengths decrease with the increase of train velocity at tunnel entrance, while it is opposite to that at tunnel exit. The freezing length of tunnel entrance and exit is different, which is mainly related to two ways of cold air entering the tunnel. One is that the cold air to be sucked into tunnel because of the annular negative pressure region generated at the top of train body. Another is that the cold air is sucked into the tunnel by negative pressure generated at train rear. The heat exchange between cold air and warm air inside the tunnel is accompanied by the running train.

Fig. 7
Temperature distributions after the train passes through the tunnel with different train velocities: (a) 80 km/h, (b) 130 km/h, (c) 160 km/h, (d) 200 km/h, and (e) 250 km/h
Fig. 7
Temperature distributions after the train passes through the tunnel with different train velocities: (a) 80 km/h, (b) 130 km/h, (c) 160 km/h, (d) 200 km/h, and (e) 250 km/h
Close modal

For tunnel entrance, the negative pressure at the rear of train plays a key role in the freezing length. When train runs at a slow velocity, the duration time of train from start to stop increased and the duration of heat exchange becomes longer. Therefore, the cold air can move further under the action of residual piston action and buoyancy, resulting in the temperature inside the tunnel decreasing greatly. For tunnel exit, the annular negative pressure at the top of train body plays a decisive role in the freezing lengths. The reason for this is that the turbulence intensity of air in the annular negative pressure region increases with the increase of train velocity. The increased negative pressure can cause more cold air to be sucked into the tunnel.

Figure 8 shows temperature variation at these two characteristic positions during the train passing through the tunnel with different velocities. The temperatures at the surface and interior of secondary lining are almost invariant before the train arrives. However, once the train passes through the characteristic positions, the temperature decreased abruptly. When the train velocity is 80, 130, 160, 200, and 250 km/h, respectively, the temperatures at the secondary lining surface and at secondary lining interior decrease to 19.54, 22.06, 19.54, 23.88, 25.06, 26.12 °C and 0.61, 0.76, 0.82, 0.88, 0.93 °C, respectively. The reason is that the faster the train runs, much cold air is sucked into the tunnel by piston action, which has a stronger cooling effect on the tunnel structure.

Fig. 8
Temperature at the secondary lining surface and secondary lining interior under different train velocities: (a) secondary lining surface and (b) secondary lining interior
Fig. 8
Temperature at the secondary lining surface and secondary lining interior under different train velocities: (a) secondary lining surface and (b) secondary lining interior
Close modal

5 Conclusions

Based on the proposed fluid–solid coupled numerical model, the effect of piston action on temperature distribution of the tunnel in vertical section is analyzed. The effects of air temperature and train velocity on the thermal environment inside the tunnel are studied. Some valuable conclusions are drawn as follows:

  1. The secondary lining surface at the entrance and exit of the tunnel is subjected to a freeze–thaw cycle after the train passes through the tunnel. The temperature distribution inside the tunnel is significantly affected by train velocity and air temperature.

  2. When air temperature decreases from 0 °C to −30 °C, the freezing lengths at vault are 0, 505.03, 539.60, 585.77, and 616.33 m, respectively, and the lengths at rail are 0, 603.22, 639.75, 682.62, and 713.31 m, respectively. Therefore, the anti-freezing fortificated length at rail should be longer than that at vault.

  3. When the train velocity ranges from 80 to 250 km/h, for the vault at the tunnel entrance, the freezing lengths are 638.83, 624.68, 622.23, 617.45, and 600.56 m, respectively, while for the vault at the tunnel exit, the lengths are 4.35, 12.43, 17.89, 22.94, and 29.36 m, respectively. Therefore, the anti-freezing fortified length at the tunnel entrance should be longer than that at exit.

  4. Because the piston action can enhance the heat exchange between cold air outside the tunnel and the tunnel structure, piston action must be considered in the design and construction of tunnels in cold regions. The temperature distribution is characterized by three zones, including disturbed zones at two sides of the tunnel and undisturbed zone at tunnel middle. The lengths of the disturbed zone at entrance and exit of the tunnel are within 1000 m and 10 m, respectively, while the undisturbed zone is within 1180 m.

Acknowledgment

This study was financially supported by the Second Tibetan Plateau Scientific Expedition and Research (STEP) Program (Grant No. 2019QZKK0905) and the Guangdong Provincial Key Laboratory of Modern Civil Engineering Technology (2021B1212040003). All the sources of support are gratefully acknowledged. The authors would like to thank the anonymous reviewers of this paper for their constructive comments.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The authors attest that all data for this study are included in the paper.

Nomenclature

     
  • f =

    frozen state

  •  
  • h =

    thawed state

  •  
  • i =

    coordinate component in x

  •  
  • k =

    turbulent kinetic energy coefficient

  •  
  • n =

    forward direction of the freezing front

  •  
  • p =

    mean pressure of air

  •  
  • q =

    heat flux

  •  
  • t =

    time

  •  
  • u =

    mean velocity of air

  •  
  • u =

    velocity vector

  •  
  • A =

    cross-sectional area

  •  
  • C =

    heat capacity

  •  
  • E =

    empirical constant

  •  
  • G =

    geothermal gradient

  •  
  • H =

    depth below the constant temperature region

  •  
  • L =

    latent heat of ice-water phase transition

  •  
  • P =

    wetted perimeter of cross section

  •  
  • T =

    temperature

  •  
  • V =

    control volume

  •  
  • Z =

    temperature in the constant temperature region

  •  
  • A =

    area of the dynamic mesh

  •  
  • ca =

    specific heat capacity of air

  •  
  • hw =

    convective heat teansfer coefficient

  •  
  • kp =

    total turbulent kinetic energy at the first layer

  •  
  • n1 =

    forward direction of the freezing front

  •  
  • uc =

    average velocity under the condition of yp+=ytp+

  •  
  • ug =

    velocity of the dynamic mesh

  •  
  • Gk =

    turbulent kinetic energy

  •  
  • SΦ =

    source term

  •  
  • Tm =

    freezing temperature

  •  
  • T0 =

    initial temperature

  •  
  • Tp =

    node temperature of the wall at the first layer

  •  
  • TH =

    temperature at a specified depth

  •  
  • Tw =

    wall temperature

  •  
  • T* =

    dimension of temperature

  •  
  • yp+ =

    normalized distance

  •  
  • Prt =

    turbulent Prandtl number

  •  
  • Pr =

    Prandtl number

  •  
  • T¯(x,y,z) =

    a known function of space and time

  •  
  • Γ =

    diffusion coefficient

  •  
  • ɛ =

    dissipation rate of turbulent kinetic energy

  •  
  • κ =

    Kalman constant

  •  
  • λ =

    thermal conductivity

  •  
  • λa =

    thermal conductivity of air

  •  
  • μ =

    dynamic viscosity of air

  •  
  • μi =

    laminar flow viscosity coefficient

  •  
  • μt =

    turbulent viscosity of air

  •  
  • μp =

    average velocity of the fluid near the wall

  •  
  • ρ =

    medium density

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