Numerical Simulation of Transient Two-Phase Flow in the Filling Process of the Vertical Shaft Section of a Water Conveyance Tunnel

Abstract

Long-distance water conveyance systems require controlled filling after initial operation or maintenance. This process is complex and challenging to manage accurately. It involves transient two-phase flow with rapid velocity and pressure changes, which can risk pipeline damage. Studying the filling process is thus essential to ensure the safe and efficient operation of the system. Combining a specific engineering case, this work investigates gas–liquid two-phase flow in tunnel sections during filling. We employ a coupled Volume of Fluid (VOF) multiphase model and a Realizable k-ε turbulence model for our simulations. Hydraulic parameters (flow patterns, pressure, velocity) are analyzed using the results. Key findings indicate that higher filling flow rates destabilize the process. Gas retention behavior in low-pressure caverns varies, and gas–liquid eruptions occur at shaft water surfaces. Increased flow rates also intensify phase–pattern transitions, elevate peak pressure and velocity values, and amplify pressure pulsations and velocity fluctuations. Furthermore, faster gas transport in low-pressure caverns triggers flow instability, compromising exhaust efficiency.

1. Introduction

Tunnel water conveyance is a common method in long-distance water transfer systems, generally categorized into pressurized and free-surface flow. When the tunnel itself is long, the flow may alternate between free-surface and pressurized conditions during conveyance, resulting in significant changes in pressure and velocity. This can subject the tunnel structure to unfavorable stress states, potentially causing damage and jeopardizing project safety. The initial operation of a long-distance water transfer system involves complex filling procedures, which are critical for the system’s safety and stability and represent one of the most essential operational conditions. The filling process involves intricate, transient air-water two-phase flow, characterized by highly variable flow patterns and significant fluctuations in pressure and velocity. The hydraulic characteristics vary depending on the tunnel shape, gate opening combinations, and other factors. Therefore, determining key hydraulic parameters such as the water surface profile, pressure, and velocity changes during filling is of great significance for ensuring the safe and stable operation of the tunnel.

Currently, most domestic and international research on tunnel segments primarily relies on one-dimensional numerical simulations, while three-dimensional simulations are mainly applied to spillways, inverted siphons, and similar structures. Yang et al. [1] studied the Yellow River Diversion Project to Shanxi and proposed the ‘virtual discharge method’ to simulate the filling process of free-surface tunnels. Their work demonstrated that this method could accurately predict pressure surges and flow instabilities during the filling process. Yang et al. [2] combined the four-point implicit difference method with the virtual discharge method to simulate the Tianjin section of the South-to-North Water Diversion Project. They validated the model’s ability to capture liquid column separation and rejoining during transient flow. Guo et al. [3] analyzed transient air-water flow during slow pipeline filling and established governing equations for pressure waves, highlighting the risks of gas entrapment in V-shaped pipelines. Jiang et al. [4] derived two-phase transient flow equations to analyze the transient process of liquid column separation in pipelines and validated the model’s accuracy through experiments. Olsen et al. [5] applied the k-ε turbulence model to 3D spillway simulations, demonstrating the importance of turbulence resolution for predicting discharge coefficients. Bhajantri et al. [6] further validated 2D turbulence models for hydropower spillways, showing their effectiveness in capturing flow separation and pressure variations. Zhou et al. [7] conducted numerical simulations of the filling process in pressurized pipelines containing air pockets, tracking the air-water interface using the VOF model. The results showed irregular movement of the air-water interface during filling, with the water column near the bottom wall passing through the pipe end first, accompanied by hydraulic jumps. Wang [8] focused on transient flow during pipeline filling, employing an interface-tracking method to simulate the transient two-phase flow interface in pipelines. She conducted simulations for different pipeline configurations and established a relatively comprehensive filling prediction model. Wang et al. [9] discussed a complex transient flow calculation model for large-scale pipeline filling, analyzed potential liquid column separation and cavitation during filling, and summarized computational models and numerical methods for air valves in water hammer protection. They also outlined future research directions for transient flow in pipeline filling processes. Wang et al. [10] used Fluent software to numerically simulate the filling stage of a long-distance free-surface diversion bifurcation tunnel, analyzing changes in the water surface profile and flow field. Their experimental validation confirmed the accuracy of the numerical simulations. He [11] and others used Realizable k-ε turbulence model to numerically simulate a high head and large discharge volume of a vertical shaft cyclone floodway, obtained the distribution law of pressure, flow pattern, cavity diameter and other hydraulic elements, and compared with the experimental results to verify that the results are in good agreement. Chen et al. [12] used Fluent software to simulate the shaft floodway and obtain hydraulic parameters, such as the water surface line and flow velocity, which matched well with the experimental results. Coronado et al. [13] proposed a quasi-static model to predict the extreme value of the gas pressure in the process of pipeline filling. Zhou [14] and others, after accounting for the elasticity of the water body, the compressibility of the gas, and other characteristics, deduced the establishment of a mathematical model for the pipeline filling process with stagnant gas clouds, which accurately simulates the transient pressure of the gas clouds in the pipeline. Chen [15] and others proposed a mathematical model for the change in the airbag on top of the pipe during pipeline filling, analyzing the influencing factors of airbag elimination, which are primarily the flow rate and topography. They summarized the results of removing the airbag. Lu et al. [16] analyzed the accuracy of Standard k-ε, RNG k-ε, and Realizable k-ε turbulence models in simulating the water-air interaction in pipelines employing a CFD method, and the results showed that the Standard k-ε turbulence model can simulate the pressure peaks and fluctuation cycles better. Fang et al. [17] analyzed the water-air interaction in the water filling process of a T-type pipeline based on the VOF model, and the result was summarized. Chen [18] and others established a pipeline filling and venting simulation test loop system with adjustable pipeline inclination (0–30°) to address the issue of air plugging in undulating pipelines. The results showed that the flow patterns and hydraulic characteristics of the downturned pipelines varied under different pipeline inclinations.

The above numerical simulation of the water filling process is primarily designed for a specific working condition, and there is limited research on the influence of water filling flow rate, pipe diameter, and other factors on the non-constant flow characteristics of the water filling process in pipelines and tunnels. Most existing studies rely on 1D simulations or simplified 2D models, which cannot resolve 3D flow instabilities in vertical shaft sections and therefore cannot observe the changes in the flow state within the pipeline. Currently, the study of water filling flow rate is mainly used for small flow rates. However, for water transfer tunnels and other projects, it cannot play a better role in guiding the actual water transfer project’s filling flow rate. To ensure the safe and stable operation of the water transfer system filling process, it is necessary to investigate the non-constant flow characteristics of the water transfer system charging process.

Therefore, this paper combines specific engineering examples, establishes the geometric model of the tunnel section, meshes the computational domain [19], and conducts a three-dimensional transient two-phase flow numerical simulation of the water filling process based on the multiphase flow method and the turbulence model, coupled with Fluent numerical simulation software. The flow conditions and flow field changes in the water filling process were finally determined, providing a reference basis for engineering operations.

2. Mathematical Modeling

2.1. Mathematical Modeling

In this paper, a mathematical model of two-phase water-air flow in a tunnel section is established by coupling the Volume of Fluid (VOF) method with the Reynolds-averaged turbulence model, proposed by Hirt and Nichols [20]. Through the improvement of the Marker and Cell (MAC) method, this approach is an effective method for dealing with free surfaces, which is primarily applicable to simulating the interface of incompressible fluid movement between two or more immiscible fluids. Its basic governing equations are as follows:

Continuity equation:

$${\displaystyle \frac{\partial \rho }{\partial \mathrm{t}}}+\nabla \left(\rho \overrightarrow{u}\right)=0$$

(1)

where $\mathrm{t}$ is the time, s; $\overrightarrow{u}$ is the mixture velocity vector, m/s; $\rho$ is the mixture density, defined in terms of the volume fraction $\alpha$:

$$\rho ={\alpha }_w{\rho }_w+\left(1-{\alpha }_w\right){\rho }_{\alpha }$$

(2)

where ${\rho }_w$ is the liquid-phase density; ${\rho }_{\alpha }$ is the gas-phase density; ${\alpha }_w$ is the liquid-phase volume fraction.

VOF Volume Fraction Equation:

$${\displaystyle \frac{\partial {\alpha }_w}{\partial \mathrm{t}}}+\nabla \left(\overrightarrow{u}{\alpha }_w\right)=0$$

(3)

Momentum equation:

$${\displaystyle \frac{\partial \rho \overrightarrow{u}}{\partial \mathrm{t}}}+\nabla \left(\rho \overrightarrow{u}\overrightarrow{u}\right)=-\nabla p+\nabla \left[\left(\mu +{\mu }_t\right)\left(\nabla \overrightarrow{u}+{\left(\nabla \overrightarrow{u}\right)}^T\right)\right]$$

(4)

where the turbulent kinetic energy viscosity factor ${\mu }_{\mathrm{t}}=\rho {\mathrm{c}}_{\mu }{\displaystyle \frac{{\mathrm{k}}^2}{\epsilon }},c_{\mu }$ is an empirical constant; $\mu$ is the mixture dynamic viscosity:

$$\mu ={\alpha }_w{\mu }_w+\left(1-{\alpha }_w\right){\mu }_{\alpha }$$

(5)

where ${\mu }_w$ is the liquid-phase dynamic viscosity; ${\mu }_{\alpha }$ is the gas-phase dynamic viscosity.

The turbulent kinetic energy $k$ equation:

$$\rho {\displaystyle \frac{\partial k}{\partial t}}+\nabla \left(\rho \overrightarrow{u}k\right)=\nabla \left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right)\nabla k\right]+P_k-\rho \epsilon$$

(6)

Dissipation rate $\epsilon$ equation:

$$\rho {\displaystyle \frac{\partial \epsilon }{\partial t}}+\nabla \left(\rho \overrightarrow{u}\epsilon \right)=\nabla \left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right)\nabla \epsilon \right]+c_1\rho {\displaystyle \frac{\epsilon }{k}}{P}_k-c_2\rho {\displaystyle \frac{{\epsilon }^2}{k+\sqrt{\nu \epsilon }}}$$

(7)

where $P_k$ is the Turbulent Kinetic Energy Production Term:

$$P_k={\mu }_t{S}^2$$

(8)

where $S$ is the Magnitude of the Strain Rate Tensor:

$$S=\sqrt{2S_{ij}\cdot S_{ij}}$$

(9)

The coefficients in the equation:

$${\sigma }_k=1.0,{\sigma }_{\epsilon }=1.2,c_2=1.9,c_1=\max \left\{0.43,{\displaystyle \frac{\eta }{\eta +5}}\right\}$$

(10)

$$\eta ={\displaystyle \frac{Sk}{\epsilon }}$$

(11)

For the Realizable model, the empirical coefficients are no longer constant and are determined by the following equation [21]:

$$c_{\mu }={\displaystyle \frac{1}{A_0+A_S{U}^{*}\frac{k}{\epsilon }}}$$

(12)

The coefficients in the above equation are:

$$U^{*}=\sqrt{S_{i,j}{S}_{i,j}+{\tilde{\mathsf{\Omega }}}_{i,j}{\tilde{\mathsf{\Omega }}}_{i,j}}$$

(13)

$$A_0=4.04,A_s=\sqrt{6}\cos \phi ,\phi ={\displaystyle \frac{1}{3}}\arccos \left(\sqrt{6}W\right),W={\displaystyle \frac{S_{i,j}{S}_{j,k}{S}_{k,i}}{{\left(S_{i,j}{S}_{i,j}\right)}^{3/2}}}$$

(14)

The numerical simulation of the two-phase flow in the tunnel section can be performed by combining the equations from (1) to (14).

2.2. Physical Models

According to the specific design parameters and shape of the project, the original tunnel section was scaled and simplified. The calculation model was constructed in equal proportions using Unigraphics NX (UG) modeling software, which is divided into five sections, with the specific dimensional parameters listed in

Table 1. Calculating basic parameters of the model.

Serial Number	Project	Length (m)	Longitudinal Slope	Cross-Sectional Form
1	Box Culvert	1	1/3000	Rectangle, 97.5 × 120 mm
2	Low-pressure tunnel inlet shaft			Inner diameter 150 mm, Height 805 mm
3	Low-pressure tunnel section	6.742	1/5000	Round shape, Inner diameter 117.5 mm
4	Low-pressure cave outlet shaft			Inner diameter 150 mm, Height 103.45 mm
5	Pressure-free outlet section	1	1/5000	Round shape, Inner diameter 110 mm

. The calculation model is shown in Figure 1.

3. Mesh Generation and Numerical Method Settings

3.1. Mesh Generation

After the geometric model is constructed, meshing is required for numerical calculations. ANSYS ICEM CFD 2019 can repair geometric model defects and generate high-quality structured meshes with high computational accuracy and superior convergence characteristics. Structured meshes feature orderly node distribution where each node shares identical neighboring cells, whereas unstructured meshes exhibit irregular node patterns, involve more complex generation processes, and require longer computation times—though they demonstrate stronger adaptability for complex boundary problems [19]. For the computational model in this study, structured meshing was employed. In this study, ICEM CFD 2019 is used to generate meshes for the 3D model. The research focuses on regular and simple rectangular box culverts and cylindrical pipes, so structured meshes are employed. To ensure the height of the first boundary layer grid, the boundary layers are refined, resulting in three sets of meshes with grid counts ranging from 1.61 million (161 w) to 3.24 million (324 w) in the fluid domain. When the grid count increases from 1.61 million to 2.42 million (242 w), noticeable changes occur in the velocity and pressure values, with a maximum deviation of approximately 6.92%. However, when the grid count further increases from 2.42 million to 3.24 million, the changes in velocity and pressure values become minimal, remaining essentially consistent. The overall variation trend of turbulent kinetic energy (TKE) is consistent with the velocity trend. This indicates that the differences in parameters between the 2.42 million and 3.24 million grids are negligible, confirming that the grid count meets the independence requirement. Additionally, when the grid count exceeds 2.4 million, the computational time increases significantly, with the time step decreasing from 0.0001 s to 0.00001 s, resulting in a substantial rise in computation time. Therefore, after comprehensive consideration, the 2.42 million structured mesh is selected for simulation. The final mesh division is shown in Figure 2, and the grid independence verification is illustrated in Figure 3.

3.2. Boundary Condition Settings

Proper specification of boundary conditions is critical for accurately simulating the transient two-phase flow behavior during the tunnel filling process. The boundary conditions must realistically represent the physical scenario while maintaining numerical stability throughout the simulation. For our water conveyance tunnel model, we carefully selected boundary conditions that account for the distinct gas and liquid phases, the transient nature of the filling process, and the specific geometry of the tunnel system.

(1)

Inlet Boundary. The inlet boundary is divided into upper and lower sections. The upper section serves as the gas-phase inlet, while the lower section is the liquid-phase inlet. The gas-phase inlet boundary is set as a pressure inlet, and the liquid-phase inlet boundary is set as a velocity inlet. The velocity magnitude is determined by the inflow rate and the cross-sectional area of the inlet water level, with the velocity direction set perpendicular to the boundary condition.

(2)

Outlet Boundary. The outlet is in direct contact with the air and is set as a pressure outlet, with the pressure fixed at atmospheric pressure.

(3)

Wall Boundary Condition. The entire water conveyance tunnel wall is treated as a solid wall boundary, with all solid walls set as no-slip boundaries. The standard wall function method is applied for treatment [22].

3.3. Numerical Calculation Methods

The water filling process is a transient flow problem, so it is solved using a pressure-based transient solver. To ensure computational accuracy, three-dimensional double precision is employed. The 3D geometric model was meshed using ICEM CFD, and transient simulations were performed with ANSYS Fluent 2019. The VOF method is coupled with the Realizable turbulence model for the solution. The Realizable k-ε turbulence model dynamically calculates turbulence intensity through coupled equations for turbulent kinetic energy (k) and dissipation rate (ε). Inlet turbulence intensity was initialized using I = 0.16·Re^−1/8, while near-wall effects were modeled via standard wall functions. The PISO algorithm is adopted for pressure-velocity coupling. Gradient terms are evaluated using the Green-Gauss node-based gradient method. For the momentum equations, turbulent kinetic energy equation, and dissipation rate equation, the convective terms are discretized using a second-order upwind scheme, while the diffusive terms employ central differencing. In this computational model, all residual factors are set to a specific value, and the time step is fixed at 0.001 s. Figure 4 summarizes the complete numerical simulation workflow, from geometric modeling to post-processing.

3.4. Test Validation

We adopted the experimental results of pressure variations at monitoring points during the water filling process from the work of Fan J.R. [23] to validate the accuracy of the numerical model established in this study. Using the same settings as the water conveyance tunnel simulation, a pressurized pipeline model was constructed to monitor pressure changes at designated points during the filling process, with results compared against the experimental measurements reported by Fan J.R. [23].

Figure 5 illustrates the geometric configuration of the pressurized pipeline model used for validation and the locations of monitoring points. The pipeline consists of a front horizontal section (L₁ = 2.2 m), a vertically ascending section (L₂ = 0.2 m), an upper horizontal section (L₃ = 1.5 m), a vertically descending section (L₄ = 0.2 m), and a rear horizontal section (L₅ = 2.2 m), with a diameter of D = 0.2 m. The water filling velocity was simulated at 0.6 m/s. Monitoring points were positioned as shown in Figure 5b, consistent with the experimental setup, to record pressure variations during the filling process.

As shown in Figure 6, the pressure variations obtained from the numerical model align well with the overall trend of the experimental results. A comparison of peak pressure values at different monitoring points between numerical simulations and experimental measurements reveals the following: For P₁, the simulated peak pressure is 8.61 kPa versus the measured value of 8.48 kPa, yielding an error of 1.53%. For P₂, the simulated peak pressure is 2.61 kPa versus the measured value of 2.51 kPa, yielding an error of 3.98%.

The overall error of the calculations remains within an acceptable range. Therefore, the numerical model employed in this study demonstrates sufficient reliability and is suitable for subsequent research.

4. Numerical Calculation Results and Analysis

Based on the established three-dimensional geometric model, water filling operations were conducted on the water conveyance tunnel section at five different flow velocities. Figure 5 and Figure 6 present the transient flow characteristics of air-water two-phase flow at representative filling velocities of 0.3 m/s and 0.6 m/s, respectively.

4.1. Flow State During the Water Filling Process

Based on the calculation results, contour plots of the water volume fraction at different times along the two-dimensional central cross-section of the tunnel segment are exported, as shown in Figure 7. In the plots, red represents the water phase, while blue represents the gas phase. Figure 7 illustrates the two-phase flow state of water and air during the filling process of the tunnel segment under the 0.3 m/s condition. Observations reveal the following: Initially, the tunnel is filled with air. After filling begins, water flows into the box culvert segment at a constant velocity, moving gradually to the right in an open-channel flow pattern. By 20.3 s, the water levels in both shafts rise further, and the leading edge of the backflow in the low-pressure tunnel reaches the middle of the tunnel. The flow state near the water surface in the right shaft becomes complex, with gas exiting the tunnel roof at high speed and entraining some water near the surface. At 26.6 s, water begins to exit the outlet. Meanwhile, in the left shaft, gas migrates upward in the form of bubbles. By this time, the water phase occupies a significant portion of the low-pressure tunnel. Around 28.9 s, the backflow in the low-pressure tunnel reaches the inlet. The water level in the left half of the tunnel is relatively flat, with wavy flow observed in the middle, while the right half is nearly complete. The water level in the right shaft drops, and no water exits the outlet. By 31.2 s, water flows from right to left, raising the water level. Some sections of the tunnel are filled. Most of the gas in the tunnel accumulates near the top of the middle section, with higher water levels on both sides. Gas escapes from both ends, with a higher exit velocity from the right shaft, causing turbulent flow at its water surface. Around 37 s, most of the gas in the low-pressure tunnel has been expelled, with only residual gas trapped near the roof. The remaining gas gradually migrates to the right and exits in the form of bubbles with the water flow, while almost no bubbles escape from the left shaft. By 57.6 s, more gas is expelled, but a continuous strip of gas remains trapped near the roof of the right section of the low-pressure tunnel, still exiting from the right shaft. Subsequently, gas is slowly expelled, and by approximately 75.4 s, most of the gas in the low-pressure tunnel has been removed, except for a small air pocket trapped near the inlet of the left low-pressure tunnel. The rest of the tunnel is nearly filled with liquid. By 91.8 s, the flow state in the entire tunnel segment stabilizes, with minimal changes in hydraulic parameters, indicating the completion of filling. However, a small air pocket remains trapped near the inlet of the low-pressure tunnel and is not expelled.

Figure 8 shows the two-phase flow pattern of water and air during the tunnel filling process under the condition of 0.6 m/s. Observations reveal the following: At the initial stage of filling, the tunnel is filled with air. Once filling begins, water flows into the box culvert section at a constant velocity and moves to the right in a non-full flow state. At approximately 6.5 s, the water reaches the outlet of the pressurized tunnel and flows into the right shaft, causing the water level in the right shaft to begin rising. By 8 s, the water level in the right shaft has surpassed the top of the low-pressure tunnel. As the liquid flows upward along the shaft wall, its velocity decreases due to gravity, eventually losing momentum and collapsing downward, resulting in a leftward backflow. However, the water levels in both the left and right shafts continue to rise. By approximately 10.7 s, the water level in the left shaft exceeds that of the first box culvert segment, causing the water to flow back to the left along the culvert. The box culvert segment gradually becomes filled, while the water level in the right shaft also rises, with water flowing out of the outlet. At this stage, the low-pressure tunnel contains more air and less water. As filling progresses, the water level in the low-pressure tunnel gradually rises from right to left, and the air is steadily expelled through the left shaft. The flow regime in the left shaft becomes complex, with large air bubbles escaping upward.

At 12.4 s, an air-water eruption phenomenon occurs in the left shaft, characterized by a high proportion of air in the water. The water phase approaches the inlet of the shaft, and the first segment becomes full, transitioning into a pressurized state. Subsequently, the water level gradually recedes, and the air phase is slowly expelled. At approximately 15 s, a gas–liquid ejection phenomenon also occurs in the right shaft. The high velocity of the gas phase drives the liquid phase upward, causing a change in flow regime. Currently, the gas phase in the low-pressure tunnel is discharged from both shafts. By 19.8 s, the flow regime in the left shaft stabilizes, with small bubbles rising and being discharged, while the water level in the low-pressure tunnel remains lower on the left side and higher on the right side. The backflow of water in the low-pressure tunnel moves relatively slowly, whereas the gas phase flows faster. As the gas phase reaches the exit of the low-pressure tunnel, its velocity increases, allowing it to rapidly enter the shaft and flow upward along the left wall. The liquid phase remains closer to the right wall, resulting in a chaotic flow regime within the shaft, with the gas phase being discharged in large gas pockets. Subsequently, water in the low-pressure tunnel continuously replenishes the gas phase, which is gradually discharged. Meanwhile, the flow regime in the left shaft shows no significant changes, and no bubbles rise to the surface. During this period, the gas phase in the low-pressure tunnel is discharged through the right shaft, with large gas pockets gradually breaking down into smaller bubbles due to the interaction of the two phases, which are then carried and discharged within the liquid phase. By approximately 34.3 s, the low-pressure tunnel is mainly filled with the liquid phase, with only a small amount of gas remaining near the roof. At approximately 50.5 s, the trapped gas in the low-pressure tunnel migrates from left to right and is discharged through the right shaft. By 65 s, the water filling reaches a relatively stable state, and the trapped gas phase in the low-pressure tunnel is completely discharged. At this point, the low-pressure tunnel is filled with the liquid phase, marking the completion of water filling in the entire tunnel section. All hydraulic parameters within the tunnel section cease to change.

4.2. Flow Pattern Variation in the Pipeline Section

An analysis was conducted on the gas–liquid two-phase flow patterns during the tunnel filling process at different time intervals. The flow patterns throughout the entire process were summarized under the typical operating condition of a 0.6 m/s water filling velocity, with radial cross-sections established in the left vertical shaft. As shown in Figure 9, at the initial stage of filling, the water phase enters the pipe section at a constant velocity and flows to the right. During this stage, the two phases exhibit clear stratification. When the water flow exits the box culvert section and collides with the shaft wall, it accelerates downward. A portion of the water enters the low-pressure tunnel, while most of it falls to the bottom of the shaft. This process involves water flow carrying bubbles and gas clusters, as illustrated in Figure 9a. As filling progresses, the water levels in the left and right shafts rise rapidly, while the low-pressure tunnel contains relatively little water. The upper half of the tunnel remains entirely in the gas phase. When the water level in the left shaft exceeds the elevation of the box culvert section, the water begins to flow back into the left side. The right half of the box culvert section fills with water, while the left half develops a gas pocket in the upper region, resulting in slug flow. At this stage, the flow regime in the left shaft becomes highly dynamic, with bubbles and gas clusters rising through the water, as shown in Figure 9b. By 13.5 s into the filling process, the water level in the left shaft is significantly higher, and the number of bubbles in the water decreases. Small bubbles gradually coalesce into larger gas clusters, transitioning into plug flow. Near the inlet of the low-pressure tunnel, the flowing water contains numerous bubbles that continuously rise to the surface, replenishing the upper gas phase. In the right half of the low-pressure tunnel, the two phases remain distinctly stratified, with a small amount of gas compressed and fragmented into bubble flow along the left wall of the shaft, as depicted in Figure 9c. After approximately 20.6 s, the flow regime in the left shaft stabilizes, with almost no bubbles remaining in the water. In the low-pressure tunnel, the water flows back to the left in a slug flow pattern, and the water level gradually increases. Meanwhile, the flow regime in the right shaft transitions into plug flow and bubble flow, with gas–liquid ejection phenomena diminishing progressively, as shown in Figure 9d. The gas phase in the low-pressure tunnel is gradually expelled, forming multiple liquid slugs Figure 9e. By this stage, the low-pressure tunnel is nearly filled with water, with only residual gas trapped near the ceiling of the tunnel. The remaining gas is expelled in the form of bubbles through the right shaft until the filling process concludes. At this point, the water phase volume fraction stabilizes, and the low-pressure tunnel is filled with water, with no remaining gas phase.

The flow regime in the left vertical shaft exhibited greater complexity and variability, with pronounced turbulence intensity. The flow regime reached its maximum disorder between 6.5 s and 10.7 s, during which severe two-phase mixing occurred. Large gas bubbles with complex morphologies were entrained in the water column as it moved upward. This phase was characterized by significant gas venting from the low-pressure tunnel to the left side, with most gases discharged through the left vertical shaft. Subsequently, the flow regime gradually stabilized, though striped gas phases continued to ascend in the water column with decreasing frequency. By approximately 20.6 s, the flow regime in the left vertical shaft attained full stabilization, with essentially no gas phase discharge observed.

Overall, during the water-filling process, four flow patterns emerge: stratified flow (gas and liquid phases separated by a distinct interface), slug flow (intermittent liquid slugs with entrained gas bubbles), bubbly flow (dispersed gas bubbles in liquid), and plug flow (elongated gas pockets in liquid). As the filling velocity increases, the flow patterns become increasingly complex and transition more frequently, leading to changes in the two-phase flow characteristics. In the low-pressure tunnel, the flow pattern initially transitions from a stratified two-phase flow of gas and water to slug flow. By the middle and late stages of filling, the water level in the tunnel approaches full-pipe conditions, with minimal gas. At this point, the flow pattern gradually shifts from slug flow to bubbly flow, as trapped gas pockets are compressed, broken, and escape in the form of bubbles. In the two vertical shafts, the flow patterns primarily alternate between bubbly flow and plug flow. Among these, bubbly flow persists the longest, as the trapped gas pockets in the low-pressure tunnel migrate toward both sides in the form of small bubbles before eventually escaping from the water surface.

To further investigate the changes in two-phase flow patterns during the water-filling process, the slug flow pattern, which exhibits significant variations in the low-pressure tunnel, was selected for examination. A segment of the slug flow variation along the length was captured, as shown in Figure 10.

The figure reveals that as the water flows from left to right, the position of the liquid slug continuously changes, and the shape of the slug gradually elongates. A distinct stratification exists between the water and gas on both sides of the slug, but the gas–liquid interface is not stable. As the water flows, bubbles gradually appear within the slug, and the gas phase penetrates the slug body, with the gas phases on both sides being interconnected. Near the slug, the gas above the water surface moves rightward with the flow, and the gas phase velocity here exceeds that of the liquid phase. During the rightward flow, the faster-moving gas phase traverses the liquid phase, causing bubbles to migrate with the water flow within the slug body. Below the slug, the water flow velocity is relatively low, while it increases near the left and right surfaces of the slug. The presence of slug flow obstructs gas migration, leading to rapid changes in flow velocity, which in turn causes pressure fluctuations at this location. The intermittent nature of slug flow affects the flow regime within the tunnel, and the resulting pressure pulsations may induce fatigue damage to the tunnel structure.

4.3. Pressure Analysis

To study the pressure field in the tunnel section during the water-filling process, this paper analyzes the pressure at 44.6 s under the condition of 0.3 m/s, when the flow is relatively stable. The corresponding pressure field is shown in Figure 11.

Analyzing the static pressure distribution in the tunnel section shown in Figure 11 reveals the following: The liquid-phase static pressure in the low-pressure tunnel is primarily determined by the height of the water level. In contrast, the gas-phase static pressure is caused by the compression resulting from the interaction between the gas and liquid phases. Consequently, the static pressure at the trapped air pockets remains the same. The maximum static pressure occurs at the bottom of the left and right vertical shafts, reaching 5.29 kPa. At the final steady state of water filling, the pressure value is 5.397 kPa. Due to the free-surface flow in the inlet and outlet sections, their static pressures are relatively small. The location of the maximum pressure within the tunnel section remains unchanged under different water-filling conditions.

To observe the temporal variation in pressure in the tunnel section during the water-filling process, nine monitoring points (P1–P9) were set up in the left and right vertical shafts and the low-pressure tunnel, as shown in Figure 12. The coordinates of the pressure monitoring points (P1–P9) are as follows: P1: x = 0.06 m, y = 1.15 m, z = 0 m; P2: x = 0.06 m, y = 1.15 m, z = −0.35 m; P3: x = 0.06 m, y = 1.15 m, z = −0.47 m; P4: x = 0.06 m, y = 1.4 m, z = −0.41 m; P5: x = 0.06 m, y = 3.5 m, z = −0.41 m; P6: x = 0.06 m, y = 6 m, z = −0.41 m; P7: x = 0.06 m, y = 8 m, z = −0.44 m; P8: x = 0.06 m, y = 8 m, z = −0.355 m; P9: x = 0.06 m, y = 8 m, z = −0.46 m. Additionally, five cross-sectional planes along the radial direction of the tunnel N1 to N5 were selected to monitor average pressure changes. Time-dependent pressure data were obtained through calculations and plotted as curves, as illustrated in Figure 13. A comparative analysis of pressure variations under two working conditions—0.3 m/s and 0.6 m/s—is presented.

Figure 13 shows the pressure variation over time at the monitoring points under the condition of 0.3 m/s. As can be seen from the pressure curves of the nine monitoring points in Figure 13, the pressure at these points exhibits significant amplitude fluctuations during the early stage of water filling, specifically before 40 s. In particular, the pressure at monitoring points P4–P6 in the low-pressure tunnel fluctuates intensely, with nearly identical trends. In terms of pressure increase, from the start of water filling until 40 s, the water phase occupies most of the tunnel space, while the gas phase is almost entirely expelled. Consequently, the pressure rises rapidly during this water-filling and gas-venting process. However, due to the interaction between the two phases and changes in flow patterns, pressure fluctuations occur. After 40 s, the water level in the entire low-pressure tunnel stabilizes, with only a small amount of gas remaining unvented near the tunnel crown. As a result, the pressure at the monitoring points increases slowly, and once the water filling stabilizes, the pressure remains essentially constant.

From the pressure variation curves of the five monitoring surfaces shown in the figure, it can be observed that the pressure values of monitoring surfaces N1 and N5 are relatively small. The pressure fluctuation of monitoring surface N1 is minor, while that of N5 is more pronounced. At approximately 20 s, the pressure on surface N5 begins to change, and around 32 s, water flows out from the outlet section, causing a rapid increase in pressure. Subsequently, the pressure fluctuations become smaller but remain unstable. For the three monitoring surfaces N2–N4 in the low-pressure tunnel, their pressure fluctuations resemble those of the monitoring points, with similar trends and largely overlapping pressure curves.

Summarizing the pressure variations across these monitoring surfaces, the pressure fluctuations in the low-pressure tunnel are more significant. At the same time, those in the two vertical shafts are smaller and exhibit a stratified distribution. The amount of trapped gas in the low-pressure tunnel affects the pressure changes—the more gas present, the greater the pressure variation. The pressures at the inlet and outlet monitoring surfaces are relatively low, with a maximum of about 62 Pa, whereas the maximum pressure in the low-pressure tunnel reaches approximately 4.5 kPa. Due to the minimal positional difference between the monitoring points in the left and right vertical shafts, their pressure magnitudes are close, and their fluctuation trends during the early and middle stages of water filling are also similar. The pressure in the left vertical shaft appears first, followed by changes in the right vertical shaft. The pressure fluctuations in the right vertical shaft persist longer, which can be attributed to the gas discharge process in the low-pressure tunnel during the later stages of water filling, resulting in sustained pressure fluctuations.

Figure 14 shows the curve of face pressure variation over time at monitoring points in the tunnel under the 0.6 m/s condition. From the pressure variation graphs of the nine monitoring points, it can be observed that the pressure change process can be divided into two stages. The first stage is a rapid pressure increase phase, where the pressure exhibits a fluctuating rise from the start of water filling until 30 s. In terms of magnitude, the pressure increases from 0 Pa to approximately 6 kPa. This stage belongs to the rapid water-filling phase, characterized by a significant inflow rate and rapid air expulsion, resulting in a pulsating pressure increase. After 30 s, the pressure gradually increases and stabilizes. Among the monitoring points, P3 exhibits the highest pressure after stabilization, while P1 shows the lowest. Regarding the pressure on the monitoring faces, the pressure on face N1 initially increases, then decreases, and finally tends to rise slowly. The analysis suggests that this is due to the significant inflow rate in this condition. After water filling begins, the water levels in the two vertical shafts rise rapidly, while the low-pressure tunnel takes longer to fill. This process causes backflow in the inlet culvert section, transforming it from free-surface flow to pressurized flow, resulting in a gradual increase in pressure. Subsequently, air flows leftward along the tunnel roof, causing a pressure drop. Eventually, as water fills the pipeline, the pressure stabilizes. On monitoring face N5, the pressure slightly decreases in the later stage of water filling, attributed to the increased flow velocity reducing the pressure. Meanwhile, the pressure in the outlet section begins to fluctuate and gradually increases after 10 s before stabilizing. The pressure variation trends on the monitoring faces in the low-pressure tunnel are similar, with the most pronounced fluctuations, and their pressure values are nearly identical.

A comparison of the pressures under the two conditions reveals that when the filling velocity increases from 0.3 m/s to 0.6 m/s, the pressure inside the tunnel rises. The primary reason is the excessive inflow rate relative to the small computational model size, resulting in slower water filling in the low-pressure tunnel. The water level in the left vertical shaft quickly rises above the tunnel crown, hindering the expulsion of air. When the water level in the right vertical shaft also exceeds the low-pressure tunnel, air expulsion is further obstructed, causing the first culvert section to transition from free-surface flow to pressurized flow, thereby increasing the internal pressure. However, the limitations in model scale primarily affect quantitative thresholds rather than qualitative patterns. The fundamental flow mechanisms, flow regime transitions, and relative relationships among parameters retain broad applicability.

4.4. Velocity Analysis

Regarding the flow characteristics of the water-gas two-phase flow in the computational model, this paper selects the two-phase velocity field at 17.3 s during the water-filling process under typical conditions of 0.6 m/s for analysis. To facilitate observation, local regions of the flow field are magnified. Figure 15 shows a schematic diagram of the selected flow field locations in the tunnel section, with four typical positions chosen for flow field analysis. Figure 16 presents the local velocity vector diagram and velocity contour map of the two-dimensional planar flow field along the length of the tunnel section. In the velocity vector diagram, red represents water, blue represents the gas phase, the size of the arrows indicates the magnitude of the velocity—larger arrows denote higher velocities at that location—and the direction of the arrows represents the velocity direction.

From Figure 14, the flow state and velocity distribution of the air-water two-phase flow in each tunnel section can be observed under the 0.6 m/s condition at 17.3 s. At Section (1), the water level in the shaft aligns with that in the box culvert section. The water flow in the box culvert moves from left to right, while the water level in the low-pressure tunnel is approximately half the diameter of the tunnel. The velocity vectors at the top of the low-pressure tunnel are larger and directed to the left, indicating that the air inside the tunnel flows to the left at a relatively high speed. The water flow in the low-pressure tunnel also exhibits a high velocity, with dense and numerous velocity vectors. The velocity contour reveals that the maximum speed occurs near the tunnel crown, reaching about 1 m/s, while the water flow velocity near the tunnel bottom is approximately 0.6 m/s. At Section (2), the water level is lower on the left and higher on the right. The left segment shows a higher water flow velocity, whereas the right segment, due to the elevated water level, has a very low velocity. The velocity vectors are dense and large on the left but decrease in size on the right, with the largest arrows located near the tunnel crown. The velocity contour indicates that the water flow speed on the left is about 1.0 m/s, while the maximum air velocity near the crown reaches approximately 1.4 m/s. The air velocity decreases near the water surface but increases again closer to the tunnel crown. At Section (3), the water level is high, and the velocity vectors in the water phase are small, whereas those above the water phase become significantly larger. Thus, the water flow velocity is low, while the air velocity is high. The velocity contour indicates that the maximum air velocity is 2.3 m/s, while the maximum water velocity is approximately 0.7 m/s. Additionally, the water flow speed on the right segment of this section is very low, nearly stagnant. At Section (4), the two-phase flow pattern is complex, with air pockets migrating through the water. The velocity vectors and contours indicate that the water phase velocity is significantly smaller than the air phase velocity, with the latter reaching a maximum of approximately 12 m/s, while the water phase peaks at 1.2 m/s. Due to the high-speed air discharge, splashing occurs near the water surface, resulting in variations in the flow pattern.

In summary, at this moment, the air phase velocity in the tunnel sections is generally higher than the water phase velocity, and the magnitude of the air velocity significantly influences the changes in flow pattern.

Due to the complex two-phase flow characteristics exhibited in region (4), we selected this region for turbulent kinetic energy (TKE) analysis. Figure 17 illustrates the temporal variation in turbulent kinetic energy (TKE) in region (4). As shown, during the initial phase, TKE remains relatively low and primarily concentrated in the vertical direction, with weaker turbulence in corner regions due to geometric constraints. In the intermediate phase, TKE intensifies dramatically and expands spatially, driven by flow separation and vortex formation. Although the total TKE continues to rise in the late phase, the core diffusion zone contracts, indicating ongoing energy dissipation. This evolution reveals the generation and decay mechanisms of bend-induced turbulence, demonstrating the need for structural optimization to mitigate erosion risks from high-TKE zones on tunnel walls, thereby enhancing operational stability in long-distance tunnels.

5. Conclusions and Future Work

For the water filling process in the vertical shaft section of the tunnel, numerical simulation methods were employed to determine the flow characteristics and variations in hydraulic parameters during the filling process:

(1)

When filling at a flow velocity of 0.3 m/s, the flow regime in the tunnel remains stable, with minor fluctuations in pressure and velocity. The gas discharge in the low-pressure tunnel is relatively smooth, primarily occurring as small gas clusters or bubbles. Some gas retention is observed at the inlet of the low-pressure tunnel. In contrast, at a flow velocity of 0.6 m/s, the water-filling and gas-discharge processes proceed more rapidly. Gas migrates in the form of large clusters within the low-pressure tunnel and the two shafts. This discharge process alters the two-phase flow dynamics, resulting in increased pressure and velocity fluctuations in the low-pressure section of the tunnel. Additionally, gas–liquid eruptions occur near the water surface in the shafts, leading to intense mixing of water and gas. However, no gas retention is observed in the low-pressure tunnel.

(2)

The flow patterns observed in the tunnel primarily include stratified flow, slug flow, bubbly flow, plug flow, and wavy flow. As the filling velocity increases, transitions between these two-phase flow patterns become more frequent. In the third section of the low-pressure tunnel, the flow pattern mainly transitions from slug flow to bubbly flow, with slug flow causing significant pressure fluctuations.

(3)

Pressure variations in the low-pressure tunnel are strongly influenced by gas presence. Greater gas content leads to more intense pressure fluctuations. The pressure fluctuations are demarcated at approximately 40 s: fluctuations are more pronounced before 40 s and gradually stabilize afterward. The filling velocity also affects pressure changes—higher velocities result in higher maximum pressures in the tunnel. As the filling velocity increases, velocity fluctuations in the tunnel intensify, and the gas discharge rate in the low-pressure tunnel rises. Notably, velocity fluctuations in the low-pressure tunnel are the most significant due to gas migration, while minor fluctuations are observed in the left and right shafts. The turbulent kinetic energy at the exit of a long-distance tunnel exhibits phased evolution over time.

This study has several limitations. First, the numerical simulations were not validated against independent experimental data due to the current unavailability of physical prototypes for the specific tunnel configuration analyzed. The absence of dedicated experiments for this tunnel geometry introduces uncertainties in flow pattern predictions, particularly for complex gas–liquid interactions near the shaft walls. Second, the computational grid resolution (2.42 million elements) and time step (0.001 s) were optimized for computational efficiency, which may limit the resolution of small-scale interfacial phenomena, such as bubble breakup and coalescence.

Future experimental campaigns are recommended to validate the numerical results. A scaled laboratory setup with high-speed imaging and pressure sensors could provide detailed insights into transient flow patterns and gas retention mechanisms. This would enhance the reliability of numerical models for real-world tunnel designs.