## Abstract

The pressure surges for water filling in deep stormwater storage tunnels (DSSTs) with entrapped air-pockets seriously threaten the safety of pipeline structures and even lead to the destruction of urban water infrastructure. Hence, this paper develops a volume of fluid (VOF) model to study pressure surges in a two shafts and one tunnel system. Research works under different initial air-pocket parameters are carried out, while the VOF model is verified by the empty tunnel water filling experiment in the same system. The results show that the maximum pressure increases with an initial length/diameter ratio decrease or air volume fraction increase. Also, the extreme pressure with entrapped air-pocket at a certain scale can reach 1.6 times the static pressure (30 m). With the increase of initial air volume fraction, the frequency of pressure surges slows down, while the maximum pressure gradually increases and approaches a specific value within a fixed air volume fraction range (0.2–5.0%). The maximum pressure of air-pocket at different initial positions shows a downward trend from the middle of the tunnel to two sides, while the pressure of the air-pocket near a higher shaft will be slightly higher. The proposed model can be extended to systems with multiple shafts or air-pockets, and the conclusions have reference significance for structure design and water filling control in the DSST.

## HIGHLIGHTS

The pressure surges and the water–air interface change law with two-sided inflow impacting entrapped air-pocket in deep stormwater storage tunnels are studied.

The maximum pressure of entrapped air-pocket can reach 1.6 times the static pressure (30 m).

The pressure surges are closely related to the initial length/diameter ratio, air volume fraction, and center position of the air-pocket.

## INTRODUCTION

The deep stormwater storage tunnel (DSST) system is a new type of municipal engineering dealing with large-scale urban water circulation problems. It relieves pressures of underground pipe networks through planned and targeted water storage and drainage, in order to reduce urban local waterlogging and sewage confluence pollution. DSST systems have been practically applied in large cities such as Chicago (Cataño-Lopera *et al.* 2014) and Shanghai (Wu *et al.* 2016), and have achieved relatively good drainage and storage effects. However, during the storage process with rapid water filling from shafts, it is unrealistic to completely discharge air in the DSST and thus air-pocket will be entrapped. With the flow pattern of the tunnel changing from an open channel flow to a pressurized flow, the pressurized rainwater impacting on the entrapped air-pocket will cause abnormal pressure surges, and even endanger the safety of the DSST system (Vasconcelos & Wright 2005).

The pressure surges with the entrapped air-pocket are subordinate to typical air–water two-phase flow issues in pressurized pipelines. For pipelines with entrapped air-pocket, most of the research works mainly focus on constructing a mathematical model, including the elastic water model (EWM) and rigid column model (RCM). Among them, the EWM is widely used in research works of pressure surges in closed pipelines (Guarga *et al.* 1996) or air released in a single hole (Zhou *et al.* 2002). For a closed pipe containing air-pocket in the end or air releasing in an exhaust hole, the air-pocket impacted by pressurized water with valve opening will produce several times direct water hammer pressure (Zhou *et al.* 2011a, 2011b, 2018), and the elasticity of water cannot be ignored in this event. In contrast, the wave velocity of the air phase is much smaller than that of the liquid phase, and the elasticity of air can be ignored (Lee 2006). Except for the system dominated by elastic deformation of water, in the pipeline with entrapped air-pocket dominated by inertial force, the RCM is more convenient and the calculation accuracy is guaranteed (Guarga *et al.* 1996). In addition, the applicability of the RCM is also affected by air-pocket parameters and inlet pressure (Lee & Martin 1999). In fact, for rapid water filling impacting entrapped air-pocket at the end of a pipe, the pressure simulation result of the RCM is consistent with that of the experiment (Huang & Zhu 2021). However, the assumption that the water–air interface is perpendicular to the conduit axis in the RCM is difficult to apply in the complex alternating process of open and full flow (Hou *et al.* 2014), especially under the condition of multiple inflows of the DSST system.

In addition to the mathematical models, the volume of fluid (VOF) method is a computational fluid dynamics (CFD) two-phase flow model solving the stratified flow of incompatible liquids (Hirt & Nichols 1981). It can describe the transient deformation of the water–air interface, which is also applicable to pipelines with entrapped air-pocket (Hargreaves *et al.* 2007), especially for alternating open channel and pressurized flow. Some experiments show that, for geysers caused by air discharge through different orifices to the shafts, the VOF model simulation results were very close to the experiments, especially for internal flow characteristics and transient movement of the water–air interface (Zhou *et al.* 2011a, 2011b; Cong *et al.* 2017; Qian *et al.* 2020). The other experiments reflect that there is an obvious heat exchange process in the air–water interaction in the elbow pipe of the horizontal–vertical pipeline, and obvious white fog and local temperature rise can be observed in the experiment and the VOF simulation separately (Zhou *et al.* 2018). For a similar pipeline with water filling simulated by VOF, the maximum transient pressure in the release process can reach 70 m (Martins *et al.* 2017). In addition, the VOF model can also be used to calculate the downstream orifice wall deformation with the water column impacting air-pocket, coupled with solid analysis (Maddahian *et al.* 2020). Although the VOF model can accurately locate the air-pocket position and describe the process of air-pocket release, it lacks applicability for the multi-inlet and multi-branch DSST system for a long time of simulation.

The existing experimental research of the DSST system shows that air-pocket entrapped is unavoidable due to the complicated situation of open channel flow reflected waves in the water filling process with multi-shaft inflow (Vasconcelos & Wright 2005, 2009). However, due to the unclear law of air-pocket retention, most studies still tend to predetermine specific air-pocket positions, especially to study pressure surges or geysers of the manhole with unilateral inflow or tunnel diameter changes (Wright *et al.* 2011; Vasconcelos & Leite 2012). For the multiple inflows model, inflow conditions have been evaluated for the statistical size and number of entrapped air-pockets in a four-shaft system (Vasconcelos & Wright 2017). In order to study pressure surges of multi-inflow water impacting entrapped air-pocket, parameter conditions of air-pocket must be given, and the multi-inflow DSST system models must be constructed, of which the simplest one is the two shafts and one tunnel system. For this simple model, the CFD three-dimensional (3D) numerical simulation based on the VOF model can undoubtedly monitor the pressure surges and describe the water–air interface transient changes. The present work aims to study the mechanism for pressure surges of DSST systems with entrapped air-pocket during the water filling process. Specifically, this study focuses on (1) establishing a 3D numerical model of the two shafts and one tunnel system and comparing simulation results of the water filling process with test; (2) predicting pressure surges of different initial air-pocket length/diameter ratios; (3) simulating pressure surges of different initial air-pocket volume fractions; and (4) comparing pressure surges with different initial air-pocket center positions.

## METHODOLOGY

### A 3D numerical model

^{3}/s at the bottom of each shaft, and the water beyond the top of the shaft overflows as the water level rises. In order to ensure that the grid quality meets the requirements of calculation accuracy and efficiency at the same time, the hexahedral structured grid of fluid domain of the two shafts and one tunnel model is divided based on the ANSYS ICEM software, the quality of the whole simulation grid element is more than 0.6 as shown in Figure 1 (right).

### Governing equations

*ρ*is the density of the mixture,

*τ*is the stress tensor,

*g*is the gravity vector,

*U*and

*p*denote the velocity and pressure shared by the phases, respectively.

*k–ε*model based on the renormalization group statistics technique. The model introduces the relevant terms of rotation and curvature into the turbulent viscosity, and corrects the consumption of the standard model according to the mean square vorticity pulsation dynamic equation. The dissipation rate equation, the turbulent kinetic energy

*k*equation, and the turbulent dissipation rate

*ε*equation are:where , , , , , . For other parameters, they are either fixed values or can be solved according to known parameters.

### Boundary and initial conditions

In order to reduce the impact of the water filling drop of shafts on the energy equations, the bottom of two shafts is defined as flow inlet, and an extension section is added at the bottom to ensure the full development of inflow. The top of two shafts is set as pressure outlet, the wall of shafts and tunnel is treated as non-slip velocity condition. The initial water–air interface position is defined by the method of selected area to patch, while the gas phase (air) model is defined as an ideal gas and the liquid phase (water) model is defined as a compressible liquid. It is assumed that there is no heat exchange in the whole process. The simulation of water filling process needs to consider the influence of gravity, and the acceleration is defined as 9.81 m/s^{2}, pointing to *Z*-axis direction. The solver adopts the pressure benchmark PISO algorithm for transient numerical simulation, with a time step of 0.02 s and a convergence standard of 10^{−4}. In order to obtain the maximum transient pressure value in the rising process of the shaft water level, the time is set as 300 s.

### Experimental test

*We*and Reynolds number

*Re*are ∼5.28 × 10

^{6}and ∼6.2 × 10

^{4}, respectively. Considering the entrapped air-pockets existing in the tunnel, the air–water scale effects are insignificant (Pfister & Chanson 2014).

^{3}/s) through valves (5) in front of the pumps inlet. The pumps (4) and valves (5) are closed after 205 s (converted to prototype resemble time is 1,127.5 s) water inflow process taking the initial inflow time of two shafts as 0 s. The pressure surges of water–air interaction are monitored by six pressure sensors (7) at the bottom of tunnel. The sensors’ signal sampling frequency is 128 Hz, the measurement range is from −50 to 50 kPa, and the measurement uncertainty is 0.1%. The pressure signal is connected to a PC computer (9) through an analog-digital card (8) for experiment data collection. The transient pressure data of the monitoring point near the low-level shaft are obtained for filtering, and the relative head at the bottom of lower tunnel entrance

*H*under the original model is obtained by resemble conversion, as shown in Figure 3(a).

_{s}### CFD model validation and grid sensitivity analysis

Using CFD models to simulate pressure surges under different initial air-pocket parameters can be realizable, but for experiment, it is extremely difficult to control the shape and position of air-pocket especially for multiple or suspended bubbles. For the same CFD physical model, describing the pressure surges of the water filling process in empty tunnel or pressurized pipe with entrapped air-pocket, the control equations and multiphase model settings are consistent. Therefore, if the simulation of empty tunnel water filling process using the CFD model is credible, it can also be reliable for the process of water impacting entrapped air-pocket. The comparison of relative head *H _{s}* at the bottom of lower tunnel entrance in the original model between experimental data and CFD simulation results is shown in Figure 3(a). It can be clearly seen that the transient pressures estimated by CFD compare very well with those obtained experimentally, and the relative difference in maximum pressure head is less than 1%. The good agreement indicates that the CFD model can be used to characterize the maximum pressure of empty tunnel water filling process in the simplified system. On this basis, taking the maximum relative head at the bottom of the tunnel entrance

*H*as the parameter variable, the grid dispersion error is evaluated according to the GCI convergence index of the Richardson extrapolation method recommended by the American Society of Mechanical Engineers. According to the requirements of GCI grid sensitivity analysis (Celik

_{m}*et al.*2008), four grid schemes with different average unit grid volume parameters are set. The refinement coefficients of the two adjacent schemes are greater than 1.3. Based on the simulation results of the grid convergence index of the parameter variable, as shown in Figure 3(b), grid 3 is selected and the total number is about 3.89 × 10

^{5}.

## RESULTS AND DISCUSSION

### Effects of initial air-pocket length/diameter ratios

*L*and the diameter

*D*of the air-pocket. At the extreme working condition of air-pocket blocking water flow, define the initial parameters

*D*=

*D*(diameter of tunnel) = 10 m,

_{t}*L*/

*D*= 7.57,

*φ*(air volume fraction) = 3.8% as the control group of

_{a}*L*/

*D*= 100,

*φ*= 3.8%, and the transient pressures are shown in Figure 4(a). It can be seen that there is little difference in frequencies of pressures under different initial length/diameter ratios air-pockets. Also, the maximum pressures of the extreme working condition (

_{a}*L*/

*D*= 7.57) reaches 48.97 m, which is 4.5% larger than

*L*/

*D*= 100. The further analysis of the maximum pressures is shown in Figure 4(b), where the maximum pressures decrease with the increase of length/diameter ratios. Under the premise that the air-pocket length does not exceed 1/4 of tunnel total length, the absolute head

*H*is reduced by 10.5% at most when

_{a}*φ*= 3.8%, and by 8.0% when

_{a}*φ*= 0.7%. The maximum pressure when

_{a}*φ*= 0.7% reaches 48.25 m, which is 60.8% higher than the absolute static pressure (30 m) with no air-pocket produced by relative control height of shafts. In summary, it can be concluded that the existence of air-pocket with a certain scale (

_{a}*φ*≥ 0.7%) will produce pressure surges that are 60% higher than the static pressure of water, which is potentially harmful to the tunnel structure.

_{a}Under the constant initial volume fraction of entrapped air-pocket, *D* = *D _{t}* = 10 m is the most dangerous working condition. Compared with the others, the frequency is faster and the transient energy is larger. This is due to the radial deformation of air-pocket is affected by buoyancy, and the deformation rate and inertial force increases. Through experimental observation and daily experience, it can be found that the air-pocket in the tunnel always tends to appear in the form of a larger length/diameter ratio after a period of standing. Compared with shafts full flow time (177 s), the air-pocket radial deformation time is much shorter. Therefore, under a long time of water filling process, it is difficult for the air-pocket to stay in the form of blocking water flow. Since the air-pocket occupying the tunnel cross-section is the most dangerous working condition, the structural design and water filling control obtained by the risk assessment are the most conservative. Therefore,

*D*=

*D*= 10 m is used in the following research to define the initial diameter of air-pocket.

_{t}### Effect of initial air-pocket volume fractions

*D*remains 10 m unchanged, the air volume fraction is adjusted by changing the length

*L*to obtain the groups of different air-pocket volume fractions, the transient pressures of air volume fraction

*φ*= 0.2% and

_{a}*φ*= 2.0% are compared as shown in Figure 5(a). It can be seen that the amplitude and frequency of the pressure fluctuation are quite different with different air volume fractions. The maximum pressure of air-pocket with

_{a}*φ*= 2.0% is 47.66 m, which is 26.1% higher than that of

_{a}*φ*= 0.2%. Since the frequency of the entire pressure surges changes unequally during the water filling process, the specific cycle time after the occurrence of maximum pressure is defined as

_{a}*T*, and the cycle time of

_{s}*T*with

_{s}*φ*= 2.0% is obtained is 26.44 s, which is 164.4% more than that of

_{a}*φ*= 0.2%. In addition, it can be observed that the fluctuation range gradually slowed down and approached a constant value in a limited time (300 s) after the transient pressure with

_{a}*φ*= 0.2% peaked, while this process may take longer time with

_{a}*φ*= 2.0%. To explain this phenomenon, more air volume fractions need to be analyzed.

_{a}Taking the maximum pressure and cycle time *T _{s}* as the references, the comparison of 0.2–5.0% air-pocket with different initial air volume fractions is obtained as shown in Figure 5(b). It can be seen that with the increase of

*φ*, the maximum pressure and cycle time

_{a}*T*all show a gradual increasing trend. Among them, the maximum pressure increased by 29.0% in total, and its growth trend gradually slowed down and finally approached about 49 m. Also, the total increase of cycle time is 282.0%, except for

_{s}*φ*= 0.5%, the growth rate under other working conditions of

_{a}*T*is relatively close. In general, the influence of air volume fraction on the amplitude and frequency of pressure surges is obvious, especially the frequency, which is basically linear with the air volume fraction. However, in the range of

_{s}*φ*= 2.0–5.0%, the influence of air volume fraction on the maximum pressure is small, while the pressures may be more affected by the static pressure of shaft water level. It can be speculated that with the further increase of air volume fraction and gradually occupying most of the tunnel, the increase of

_{a}*φ*may lead to further reduce of the maximum pressure. But due to the shape of tunnel structure and the form of water filling process, entrapped air-pocket with large volume fraction could not exist with air discharging from shafts in the DSST system.

_{a}### Effect of initial air-pocket center positions

To explain the phenomenon, the maximum pressures at different air-pocket centers and the pipeline center (P1000 m) are compared as shown in Figure 6(b), where Pothers is defined as the monitoring point located at the air-pocket center. It can be seen that, except for P250 m and P1000 m, the maximum pressures at the center of other air-pockets is higher than that of the tunnel center. The maximum pressure of the air-pocket center at different initial positions shows a downward trend from the middle to two sides, while the pressure of air-pocket near higher shaft will be slightly higher. This is because that the air-pockets close to both sides of tunnel are subjected to different pressures on both sides, and the deformation and compression of it with the movement of water flow are less. At the same time, there will be some air overflow in the shafts. Also, the maximum pressure point appears at 1,250 m instead of the center of the tunnel. Due to the influence of tunnel slope, although the initial inflow of two shafts is equal, the final outflow is different, and a large amount of water flows from the high-level shaft to the low-level shaft. At the point slightly higher than the center of the tunnel, the sum of static pressure and dynamic pressure on both sides of the entrapped air-pocket is relatively close. At this time, the compression deformation of the air-pocket is the largest, and the maximum transient pressure is also the largest.

## DISCUSSION

*φ*= 5.0%,

_{a}*D*= 10 m and

*P*= 1,000 m is shown in Figure 7.

From the unmarked air–water two-phase distribution diagram in the figure and subsequent, the change of air-pocket during water filling process can be divided into the following three stages. (1) Deformation stage on both sides (0–5.0 s): due to the upward deformation of air-pocket under the influence of buoyancy, the water–air interface on both sides gradually changes from perpendicular to the tunnel axis to the water flow incoming direction. Also, the water cross-section changes along the gradient of water–air boundary line, the water inflows on both sides near the center of air-pocket continue to concentrate, the velocity increases, and finally converges to one point. (2) Air-pocket separating stage (5.0–10.65 s): The water inflows on both sides converge to a point, which converts the kinetic energy of water into potential energy, resulting in surge at the center point. The water level rises rapidly, and finally reaches the top of tunnel, separating the air-pocket into two parts. (3) Axial deformation stage: The two separated air-pockets gradually move away from the center. With the subsequent pressure surges, the air-pocket deformation is mainly through the axial direction (due to that the length diameter ratio of the tunnel is too large 200:1, the subsequent air–water two-phase distribution is not given in the paper).

It can be seen from Figure 4(b) that the air-pocket length/diameter ratio has an influence on maximum pressure, and the largest maximum pressure occurs when *D* = *D _{t}* = 10 m. However, with the radial deformation of air-pocket in the early pressure rise stage of the water filling process, the difference of air-pocket height is gradually eliminated, but it has an impact on the maximum pressure in the subsequent pressure fluctuation stage. In fact, it can be seen from Figure 4(a) that when air-pocket height is smaller, the pressure rises faster in the early stage of water filling process. In this paper, it is assumed that the axial deformation inertia force caused by different air-pocket height causing this phenomenon, but there is no corresponding evidence. In addition, the air-pocket separates into two parts in the early stage of water filling process, the inevitable condition triggering this phenomenon is difficult to measure, and the influence of multi air-pockets on subsequent pressure surges is also unknown.

## CONCLUSIONS

The DSST system, a new type of urban waterlogging drainage facility, is prone to geysers or pressure surges under the conditions of heavy rainfall and rapid inflows. This work aimed to provide a general view on the relationship between the maximum pressures of water filling process and the entrapped air-pocket parameters. Although the two shafts and one tunnel model studied cannot represent the actual geometry of the DSST system, it can fully reflect the characteristics of air-pocket deformation and pressure surges. The relevant conclusions are expected to be extended to the study of multiple shafts and multiple air-pockets.

The CFD model has been widely used in the study of pipeline transient pressures with specific entrapped air-pocket position, but it is much more difficult to control the state and position of air in the experiment. Hence, this paper verifies the accuracy of the CFD model through a water filling experiment of the two shafts and one tunnel model. The pressure surges affected by initial length/diameter ratio, air volume fraction, and the center position of air-pocket are listed as follows:

The maximum pressures with fixed air volume fraction increase with the decrease of initial length/diameter ratios, and the extremum one is achieved with the initial air-pocket blocking water flow. The existence of air-pocket with a certain scale (

*φ*≥ 0.7%) will produce pressures that are 60% higher than the static pressure._{a}The influence of air volume fraction on the amplitude and frequency of pressure surges is obvious, especially the frequency, which is basically linear with the air volume fraction. In the range of

*φ*= 0.2–5.0%, the maximum pressure increased by 29.0% and the cycle time_{a}*T*increased by 282.0% in total._{s}When the initial position of the air-pocket is not in the center of tunnel, the transient pressure at the corresponding position will fluctuate violently in the initial stage and tend to be stable before 50 s. The maximum pressure of air-pocket at different initial positions shows a downward trend from the middle to two sides, while the pressure of air-pocket near higher shaft will be slightly higher.

In this paper, only the influence of initial air-pocket parameters is considered, and there is a lack of detailed and verifiable explanation the air-pocket deformation and pressure transient change in the process of water filling. In the follow-up study, the effects of multi air-pocket parameters and inflow difference between two shafts on the pressure surges of entrapped air-pocket should be studied.

## ACKNOWLEDGEMENT

The authors acknowledge the financial support given by the National Natural Science Foundation of China (Nos 52179062 and 51879087) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No KYCX22_0652).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Effect of Pressurization and Expulsion of Entrapped air in Pipelines*

*PhD thesis*