Airflow characteristics of the stepped dropshaft in the deep tunnel storage system

As a result of the global climate change, cities are experiencing more intense rainfall events, causing a series of problems associated with urban flooding. As an example, Zhengzhou was hit by a once-in-a-millennium rainstorm disaster on 20 July 2021: cumulative rainfall reached 217.8 mm in 24 h, causing severe traffic disruption, casualties, and property damage (Peng et al. 2023). With increasing urbanization, the increase in impervious surface changes the urban water cycle, increasing the frequency and damage of urban flooding year on year (Vasconcelos & Wright 2011). Deep tunnel storage systems have been implemented or are being considered in many countries to solve urban flooding (Yang et al. 2019). The system, which typically consists of main tunnels of tens of meters underground and several dropshafts, is capable of collecting, transporting, and storing large volumes of stormwater. These dropshafts can safely and efficiently convey the runoff to the underground tunnels, achieving high energy dissipation and avoiding the occurrence of water choking (Guo & Song 1991). During heavy rainfall, the water flow in the dropshafts carries a large amount of air into the underground tunnels, which is difficult to vent and can lead to trapped air pockets and subsequent pressure surges and geysers (Leon et al. 2019; Zhang et al. 2022). Therefore, it is necessary to study the airflow characteristics of the dropshafts.

Typical dropshafts include plunging dropshaft, vortex dropshaft, baffle dropshaft, and helicoidal-ramp dropshaft (Ren et al. 2021). For the plunging dropshaft, a free-falling jet impacts on the cushion at the dropshaft bottom or the opposite wall of the dropshaft (Chanson 2007; Liu et al. 2022). For the vortex dropshaft, water flows tangentially into the dropshaft and along the wall, creating an air core in the dropshaft center (Zhao et al. 2006). For the baffle dropshaft, a vertical wall divides the dropshaft into dry and wet sides with a series of baffles positioned on the wet side to drain water and dissipate its energy; the air carried by water on the wet side can be released from the dry side through the air holes in the vertical wall (Odgaard et al. 2013). The helicoidal-ramp dropshaft has a continuous helical ramp on the dropshaft wall for water discharge and a central exhaust pipe surrounded by ramps for air discharge through air holes in the pipe wall (Ansar & Jain 1997). The plunging and vortex dropshafts have no additional exhaust structures; hence, their entrained air is more easily carried into the underground tunnels (Zhang et al. 2016).

For the plunging dropshaft, air can be carried into the dropshaft in several ways (Chanson 2007): (1) entrained by the inflow impinging the dropshaft wall; (2) dragged by the falling water currents; (3) carried by the falling water plunging on the bottom cushion; and (4) driven by the turbulent outflow. Ma et al. (2016) found that falling water can break into many drops, transferring the momentum of the drops to the surrounding air and dragging the air downward, thus causing significant air entrainment. The effects of water flow discharge Q, dropshaft height H, and dropshaft diameter D on air inflow discharge Q_in have been investigated in previous studies. Rajaratnam et al. (1997) indicated that the air inflow discharge Q_in increases with Q, but the relative air inflow discharge Q_in/Q decreases with Q/(gD⁵)^0.5. They reported that Q_in/Q = 1.40 for Q/(gD⁵)^0.5 = 0.10 in a plunging dropshaft with H/D = 6.6. Camino et al. (2015) and Ma et al. (2016) reported that Q_in/Q of a plunging dropshaft with H/D = 20.0 can be as large as 40 for Q/(gD⁵)^0.5 < 0.004. The air inflow discharge can be influenced by the underground tunnel conditions as well. Granata et al. (2011) found that when the underground tunnel ventilation is blocked, the air inflow discharge shows a sudden drop. Ma et al. (2016) suggested that the weir installed on the bottom of the underground tunnel can increase the water depth and decrease the airflow space over the water surface, thus limiting the air inflow discharge and the airflow into the underground tunnel.

For the vortex dropshaft, the flow clings to the wall and spirals down while an air core forms in the center. Jain & Kennedy (1983) found that due to the pressure gradient in the center air core, the entrained air converges toward the dropshaft center and rises against the moving water. It is believed that the air inflow discharge in the vortex dropshaft is less than that in the plunge-flow dropshaft. They showed that in vortex dropshafts, Q_in/Q ≤ 0.40 for H/D = 25.0 and Q_in/Q ≤ 0.60 for H/D = 11.0, when Q/(gD⁵)^0.5 < 1.00. Ogihara & Kudou (1997) found that in vortex dropshafts, Q_in/Q ≤ 1.10 for H/D = 25.0 and Q_in/Q ≤ 1.40 for H/D = 37.5, when Q/(gD⁵)^0.5 < 1.50. It indicated that Q_in/Q decreases with H/D at large water discharges while increasing with H/D at small water discharges. Zhao et al. (2006) found that Q_in/Q in a vortex dropshaft of H/D = 67.0 is comparable to that in a plunging dropshaft.

Some literatures have studied the flow regimes, energy dissipation, and air concentration of the stepped dropshaft. Referring to the research on the stepped spillway, the flow regimes of the stepped dropshaft were divided into the nappe flow, the transition flow, and the skimming flow, according to whether the cavity occurs between the nappe and the step surface (Wu et al. 2018; Qian et al. 2022; Peng et al. 2024). Influenced by the centrifugal force, water is concentrated on the external wall and forms standing waves (Wu et al. 2017; Qi et al. 2018). Ren et al. (2021) found that the standing wave peak increases with the water flow discharge Q, which can be up to three times the water depth in the internal wall. Sun et al. (2023) defined the maximum water flow discharge of the stepped dropshaft when the peak of the standing wave just reaches the above steps, and provided the empirical formulas to predict the standing wave characteristics and the maximum allowable water flow discharge. For the energy dissipation and air concentration, the stepped dropshaft can achieve a high energy dissipation rate of 96% and an air concentration on the step horizontal plane of up to 3.5%, indicating good energy dissipation performance and high air entrainment to prevent cavitation risk (Liao et al. 2019; Shen et al. 2019).

Previous research on the stepped dropshaft has focused mainly on its basic hydraulic characteristics, while its airflow characteristics require further study. In the present study, the airflow characteristics of the stepped dropshaft were investigated by an experiment. The relationship between the flow regimes and the airflow characteristics was analyzed. The influences of water flow discharge, step height, step rotation angle, air vent pipe diameter, underground tunnel blockage on the air inflow discharge, air vent discharge, air discharge into underground tunnel, and the exhaust capacity (the ratio of the air vent discharge to the air inflow discharge) were analyzed. Empirical formulas of these airflow characteristics of the stepped dropshaft were established for prediction and as a guide for practical design and operation.

The airflow in the stepped dropshaft consisted of three parts: the air inflow discharge Q_in carried by water flow into the dropshaft, the air vent discharge Q_out released through the central exhaust pipe, and the air discharge into the underground tunnel Q_outd, where Q_in=Q_out + Q_outd. For an easy and accurate measurement of Q_in, the entrances of water and air for the inlet conduit should be separated. In the plunging dropshaft experiments of Ma et al. (2016) and Zheng et al. (2017), the inlet conduit entrance was designed as a right-angled downward elbow with a separate air inflow pipe on top. As the water surface rises in the chamber, air can be blocked by water from entering the inlet conduit through the water entrance. Therefore, air can only enter the dropshaft through the air inflow pipe, i.e. Q_in = v_a1πd₁²/4, where v_a1 is the measured air flow velocity in the air inflow pipe, and d₁ is the diameter of the air inflow pipe. The results of Ma et al. (2016) and Zheng et al. (2017) showed that for 0 ≤ Q* ≤ 1.40 and d₁ ≥ 0.05 m, installation of the air inflow pipe as an independent air entrance should not change the actual pattern and discharge of the air and water flow in the dropshaft. Therefore, this study adopted the inlet conduit entrance design with reference to Ma et al. (2016) and Zheng et al. (2017), as shown in Figure 2(b), with the diameter of the air inflow pipe d₁ as 0.05 m. To obtain the air vent discharge Q_out, an air vent pipe was arranged at the top plane of the dropshaft. Diameter d₂ of the air vent pipe ranged from 0.01 to 0.05 m for studying its influence. After measuring the air flow velocity v_a2 in the air vent pipe, there is Q_out = v_a2πd₂²/4.

Table 1 concludes the study ranges of the geometry parameters of the stepped dropshaft. The step rotation angle θ ranged from 75° ≤ θ ≤ 120°, the step height b ranged from 0.098 m ≤ b ≤ 0.147 m, the diameter of the air vent pipe d₂ ranged from 0.01 m ≤ d₂ ≤ 0.05 m, and the airflow height over the water surface in the underground tunnel h_a ranged from 0.01 m ≤ h_a ≤ 0.15 m.

The water flow discharge was measured by an electromagnetic flowmeter (MTF-S200-7-APMetran China) with an accuracy of 0.1 m³/h. The flow regimes of the stepped dropshaft were captured by a digital camera (SONY FDR-AX60, Japan). The air velocity of inflow and vent pipes was measured by an anemometer (TSI VelociCalc Model 9545, USA) with an accuracy of 0.01 m/s. To exclude the scale effects of the stepped dropshaft flow, the minimum Reynolds number Re must be roughly 10⁵, and the minimum Weber number We should be about 10⁴ (Boes & Hager 2003), which were satisfied for the stepped dropshaft in this study.

For the nappe flow, a succession of nappes impacts on each step, followed by partial or full hydraulic jumps. An air cavity is formed beneath each nappe, which is partially filled by a backflow cushion, as shown in Figure 3(a). Due to the nappe impact and the hydraulic jump, weak air entrainment occurs, and air is carried by water flow in the form of bubbles. The bubbles in the nappe flow are large in size but small in number, and a part of the nappe flow remains transparent and unaerated.

For the transition flow, the air cavity near the internal wall is completely filled by the backflow, while the air cavity near the external wall remains unfilled, as shown in Figure 3(b). With the increase of water flow discharge, there is significantly stronger flow turbulence and hydraulic jump, leading to the enhancement of air entrainment and the water flow carrying more and smaller bubbles. Due to the intense water–air intersection, some fluctuations and splashes are observed on the free surface of the water flow, which favor the air being dragged downstream by the transition flow.

For the skimming flow, the air cavity completely disappears and is replaced by the recirculating vortex, and the water flow skims over the steps as a coherent stream, as shown in Figure 3(c). The free surface air entrainment is further enhanced, and a large number of tiny bubbles are carried and delivered downstream by the skimming flow. Due to the violent turbulence at the water–air interface, the free surface of skimming flow has stronger fluctuations and splashes, causing the air to move downstream, dragged by the high-speed skimming water stream.

In this study, the thresholds of Q*_N–T and Q*_T–S among the nappe flow, transition flow, and skimming flow in the stepped dropshaft are listed in Table 1. Q*_N–T is defined as the dimensionless water flow discharge when the air cavity beneath the nappe remains unfilled near the internal wall, while it begins to be filled with the backflow near the external wall, i.e., the threshold from the nappe flow to the transition flow. Q*_T–S is defined as the dimensionless water flow discharge when the air cavity begins to completely disappear and be replaced by the recirculating vortex, i.e., the threshold from the transition flow to the skimming flow. Therefore, Q*_N–T and Q*_T–S can be considered as fixed values for a given stepped dropshaft geometry, as shown in Table 1. The experimental observation shows that from the nappe flow to the transition flow, and finally, to the skimming flow, the bubble-carried air increases due to stronger water–air interaction and free surface air entrainment, and the surface-dragged air increases as well due to enhanced surface fluctuations and splashes.

For the ratio of the air vent discharge to the air inflow discharge Q_out/Q_in, it fluctuates between 0.75 and 0.90 with the variation of b/r₂ when 0.10 <Q* ≤ 0.31. When larger water flow discharges of 0.31 <Q* ≤ 0.56 occur, the increase of b/r₂ clearly reduces Q_out/Q_in, which can even be less than 0.50 when b/r₂ = 0.59. This is related to the fact that the increase of b/r₂ leads to significant fluctuations of water surface in the transition flow and the skimming flow, blocking the air holes intermittently and, therefore, decreasing Q_out/Q_in, i.e., the exhaust capacity of the stepped dropshaft.

The reason why Figures 7 and 8 show the nappe flow, transition flow, and skimming flow and Figures 9 and 10 show only the nappe flow and the transition flow is that the thresholds of Q*_N–T and Q*_T–S among the nappe flow, transition flow, and skimming flow are also influenced by the geometrical parameters of the stepped dropshaft. In Figures 9 and 10, the effects of d* and (h_t − h_a)/h_t are obtained by studying cases M5–M13 with b/r₂ = 0.42 and 0.59 and θ = 120°. Q*_N–T and Q*_T–S increase significantly with b/r₂ and θ, making it difficult to generate the skimming flow in the stepped dropshafts with b/r₂ ≥ 0.42 and θ ≥ 120° in the range of 0 ≤ Q* ≤ 0.56 in this study. Noting that the effects of d* and (h_t − h_a)/h_t on Q_in/Q, Q_out/Q remain similar under different water flow discharges and regimes, it can be supposed that the obtained effects of d* and (h_t − h_a)/h_t in the nappe flow and the transition flow as shown in Figures 9 and 10 should also be applied to the skimming flow.

The correlation coefficients for Equations (1) and (2) are both R² = 0.84. Equations (2) and (3) are applicable for the stepped dropshafts under the nappe flow, transition flow, and skimming flow regimes, with 0 ≤ Q* ≤ 0.56, 0.39 ≤ b/r₂ ≤ 0.59, 1.31 ≤ θπ/180 ≤ 2.09, 0 ≤ d* ≤ 0.10, and 0.30 ≤ (h_t − h_a)/h_t ≤ 1.00.

The correlation coefficient for Equation (4) is R² = 0.84.

Equations (2) and (3) show that Q_in/Q and Q_out/Q decrease with Q*, b/r₂, θπ/180, and (h_t − h_a)/h_t. Q_out/Q also increases with d*, while Q_in/Q is independent of d*. Equation (4) shows that Q_out/Q_in increases with Q*, θπ/180, and d* while decreases with b/r₂ and (h_t − h_a)/h_t. On the other hand, Q*, b/r₂, θπ/180, d*, and (h_t − h_a)/h_t have similar magnitudes in Equations (2)–(4), indicating that their coefficients can reflect their effects and importance. It is demonstrated that Q*, b/r₂, and θπ/180 have similar importance on Q_in/Q, Q_out/Q, and Q_out/Q_in, (h_t − h_a)/h_t has a much greater effect on Q_out/Q and Q_out/Q_in than Q_in/Q, and d* has the most significant effect on Q_out/Q and Q_out/Q_in.

The reason for (h_t − h_a)/h_t having different effects on Q_in/Q and Q_out/Q is that the pressure difference between the inside dropshaft and the outside atmosphere increases with (h_t − h_a)/h_t, which enhances the water–air interaction in the dropshaft, resulting in more air being carried into the underground tunnel by the dropshaft outflow and a significant increase in Q_outd/Q and a decrease in Q_out/Q (Zheng et al. 2017). In addition, the experimental study of the plunging dropshaft by Ma et al. (2016) indicated that Q_in/Q is mainly related to the drag force of the water flow but has a relatively limited relation with (h_t − h_a)/h_t. The reason for d* having the most significant effect on Q_out/Q and Q_out/Q_in is that it represents the dimension of the central exhaust pipe of the stepped dropshaft, which directly influences the cross section and resistance of the airflow and, therefore, has a significant effect on the air vent discharge and the exhaust capacity.

The average relative errors of Q_in/Q, Q_out/Q, and Q_out/Q_in are 7.8, 14.3, and 9.0%, respectively, and their maximum relative errors are all less than 20%, indicating that Equations (2)–(4) can well predict the airflow characteristics in the stepped dropshaft.

Previous research has studied the effect of the airflow of dropshafts on their discharge capacity and energy dissipation. Jain & Kennedy (1983) showed that the inadequate air inflow into the dropshaft can create the phenomenon of flow choking and result in a reduction in discharge capacity, and the effect of airflow on the water flow regime and discharge capacity becomes weak when Q_in/Q ≥ 0.15. Zheng et al. (2017) found that for a dropshaft with free water outlets, the airflow has a limited effect on the energy dissipation of the water flow. For a dropshaft with pressurized water outlets, it is difficult for the air carried by the water flow to flow smoothly out of the dropshaft, which enhances the water–air interaction and the energy dissipation in the dropshaft. In this study, the stepped dropshaft has an independent central exhaust pipe to provide a free air outlet and maintains Q_in/Q ≥ 0.30 in the nappe flow, transition flow, and skimming flow regimes. Therefore, according to Jain & Kennedy (1983) and Zheng et al. (2017), the airflow characteristics of the stepped dropshaft should have a limited effect on the flow regime, discharge capacity, and energy dissipation of the water flow.

In the present study, the airflow characteristics in the stepped dropshaft were experimentally investigated. The influences of inflow condition, geometry parameters, and outlet condition of the stepped dropshaft were considered, including the water flow discharge Q, step height b, step rotation angle θ, diameter of air vent pipe d₂, and airflow height over water surface in the underground tunnel h_a. The conclusions were summarized as follows:

With the increase in water flow discharge, the flow regimes in the stepped dropshaft were classified as the nappe flow, the transition flow, and skimming flow. Due to the different characteristics of water–air interaction from the nappe flow to the transition flow, and finally, to the skimming flow, both the bubble-carried air and surface-dragged air increase. Correspondingly, both the air inflow and air vent discharges increase with the water flow discharge, whereas the relative air inflow and air vent discharges decrease with the water flow discharge, significantly less than the vortex and plunging dropshafts. The exhaust capacity in the stepped dropshaft, represented by the ratio of the air vent discharge to the air inflow discharge, reaches its maximum at the threshold between the nappe flow and the transition flow regimes.

It was found that both the relative air inflow and air vent discharges decrease with the increase of the water flow discharge, step height, step rotation angle, and underground tunnel blockage, while they are independent and increase with the outlet diameter of the central exhaust pipe, respectively. According to the variations of air inflow and air vent discharges, the exhaust capacity of the stepped dropshaft overall increases with the water flow discharge, step rotation angle, and the outlet diameter of the central exhaust pipe and decreases with the step height and the underground tunnel blockage.

For predicting the airflow characteristics in the stepped dropshaft, the formulas of the relative air inflow and air vent discharges as well as the exhaust capacity of the stepped dropshaft were established, which were expressed by the parameters of water flow discharge, step height, step rotation angle, outlet diameter of the central exhaust pipe, and underground tunnel blockage. These formulas were proven to have good prediction performances.

The airflow characteristics must be carefully considered when designing dropshafts for deep tunnel storage systems because they directly affect the water flow discharge and storage capacity of the system and may lead to risks related to the generation, transport, and ejection of air pockets. This study can be helpful in understanding and predicting the airflow characteristics of stepped dropshafts, and is significant for designing stepped dropshafts with good exhaust capacity. Considering the potential scale effect on the water–air interaction and airflow, it is suggested that a series of physical models at different scales should be used to study in the future.

The authors gratefully appreciate the financial support from the National Natural Science Foundation of China (No. 52279063), the Natural Science Foundation of Jiangsu Province (Nos. BK20231462, BK20241520), Jiangsu Innovation Support Programme for International Science and Technology Cooperation (No. BZ2023047), and the Fundamental Research Funds for the Basic Research of Public Welfare Scientific Research Institutes (No. Y124016).

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

The authors declare there is no conflict.