Numerical study on hydraulic characteristics of rotating stepped dropshafts in deep urban tunnels

dropshaft, flow patterns, pressure distribution, rotating steps, terminal velocity

As an important component of the deep tunnel drainage system for dealing with urban waterlogging, the rotating stepped dropshaft has been proposed due to its small air entrainment. However, the hydraulic characteristics inside the shaft still need to be fully studied. In this study, the flow patterns, water velocity, and pressure in the rotating stepped dropshaft under different flow rates and geometric parameters were studied using a three-dimensional numerical model. The results show that increasing the central angle of the step and reducing the step height can both reduce the terminal velocity. A theoretical formula for predicting the terminal velocity was established and well validated. The connection between the shaft and the outlet pipe poses a severe threat to the structural safety due to alternating positive and negative pressures. Wall-attached swirling flow generates a circular high-pressure zone at the bottom of the dropshaft and the larger the flow rate, the greater the pressure gradient at the center of the bottom. By using the momentum theorem and considering the impact pressure range of the swirling flow, the shaft bottom pressure can be predicted reasonably well.

HIGHLIGHTS

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The hydraulic characteristics of rotating stepped dropshafts are analyzed and the terminal velocity is evaluated theoretically and well validated.
The pressure distribution inside rotating stepped dropshafts and the impact effect of the wall-attached swirling flow on the shaft bottom are determined.
A simplified theoretical model is established to predict the impact pressure of the shaft.

INTRODUCTION

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In recent years, due to climate change and urban sprawl, short-duration heavy rainfall has been increasing. In order to address issues such as urban waterlogging during extreme weather events like heavy rainstorms, many cities have proposed the use of deep tunnel drainage systems as an important engineering solution for flood control and drainage (Wang et al. 2016). Dropshafts as critical structure in the deep tunnel drainage system play a key role in guiding the long-distance downward flow of shallow water. Commonly used dropshaft structures include plunging dropshaft (Rajaratnam 1997), baffle dropshaft (Odgaard et al. 2013), vortex-flow dropshaft (João & Ricardo 2019), and helicoidal-ramp dropshaft (Zhang et al. 2022). The rotating stepped dropshaft, as a new type of vortex-flow dropshaft structure, the rotating step structure provides the support to make the water flow form a swirling flow, and when the water falls, it creates the vortex rolling, vortex aeration and self-aeration on the surface of the water flow in the triangular area between the upper and lower levels of the step, resulting in a high energy dissipation efficiency (Ren et al. 2021). Furthermore, the stepped structures are less prone to cavitation damage (Frizell et al. 2013; Shen et al. 2019), which is beneficial in reducing water–air disasters in deep tunnels, making it highly promising for a wide range of applications.

Under different flow conditions, the main flow patterns in the rotating stepped dropshaft are nappe flow, transition flow and skimming flow (Ren et al. 2021). The napped flow and transition flow generate obvious standing waves on the outer wall of the dropshaft. The blade-shaped steps effectively reduce the height of the standing waves and optimize the radial flow velocity distribution (Sun et al. 2021). The maximum transport capacity of the dropshaft is controlled by the geometric parameters of the steps and dropshaft curvature (Sun et al. 2023). Adding a sill at the end of the horizontal step can enhance energy dissipation, but it also reduces the transport capacity of the dropshaft (Shen et al. 2019). To prevent the falling jet from damaging the bottom, a certain thickness of water cushion is usually set at the bottom of the dropshaft (Chanson 2010). When the falling jet impacts the water cushion in the plunging dropshaft, it forms intense turbulence and consumes most of the energy (Puertas & Dolz 2005), but the influence of the specific jet form of the rotating stepped dropshaft on the bottom pressure is still unclear. Although existing studies have discussed the energy dissipation methods for each flow pattern (Ren et al. 2021) and the maximum water depth on the stepped ramp initially decreases and then increases with the increase of the value of the center angle of the step (Qi et al. 2019), they have not predicted the velocity changes inside the dropshaft. The terminal velocity is related to the impact pressure of the water flow on the wall and the frequency of the pressure pulsation cannot cause step resonance to ensure the safety of the shaft structure (Liao et al. 2019).

In this study, a three-dimensional numerical model was developed based on experiments (Wu et al. 2018) and verification methods (Qi & Zhang 2019; Qi et al. 2019). The model and verification method have detailed, valid and reliable experimental data, which can be used to verify the accuracy of the numerical model. Based on the validated numerical model, the flow inside a rotating stepped droplet shaft is investigated and the effects of different step structures on the flow field inside the shaft are examined. The main objectives are as follows: (1) to investigate the flow patterns inside the shaft with and without internal wall structures, analyze the influence of changes in the height, width, and center angle of the steps on the terminal velocity, and derive a formula for the terminal velocity and (2) to determine the pressure distribution on the wall boundary and the impact of wall-attached swirling flow on the bottom of the dropshaft and to establish a simplified model for predicting the bottom pressure.

METHODOLOGY

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Numerical model

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Numerical models of the rotating stepped dropshaft are shown in Figure 1, with an internal wall (Figure 1(a)) and without an internal wall (Figure 1(b)). The model mainly consists of an inlet pipe, dropshaft, outlet pipe, and outlet tank. The diameter of the inlet pipe is D_in = 0.3 m with a length of L_in = 3.0 m. The connecting section between the inlet pipe and the dropshaft adopts a quarter elliptical curve (Zhao et al. 2006). The diameter of the dropshaft is D = 0.4 m with a height of H = 4.0 m. The depth of plunge pool is H_p = 0.1 m. The diameter of the outlet pipe is D_out = 0.4 m, with a length of L_out = 2.5 m. The geometric structure of the steps is controlled by the vertical step height h, central angle θ and step width w (Figure 1). To maintain the system's ventilation and accurately simulate the external atmospheric pressure, ventilation pipes and air columns are set at the top of the dropshaft and outlet tank with the air column walls exposed to atmospheric pressure. For the dropshaft with an internal wall, its internal wall diameter is d = D – 2w and circular ventilation holes with a diameter of 4 cm are set on the internal wall below each horizontal steps.

Figure 1

Numerical model: (a) dropshaft with internal wall and (b) dropshaft without internal wall.

The Ansys Fluent was used to simulate the two-phase flow of air and liquid in the model with air defined as ideal air and water defined as incompressible fluid. Based on the volume of fluid (VOF) method, the air–liquid interface and volume fraction can be tracked, effectively handling complex-free surface problems (Hirt & Nichols 1981). The volume fractions of air and water can be represented by α_a and α_w, respectively, obtaining expressions for density ρ and viscosity μ:

(1)

(2)

where ρ_w and ρ_a are the density of water and air, respectively; μ_w and μ_a are the dynamic viscosities of water and air, respectively.

The tracking of the air–liquid interface can be achieved by solving the following continuity equation:

(3)

where t represents time; u_i and x_i are the components of velocity and coordinate, respectively (i = 1, 2, 3).

The κ–ω turbulent model based on the shear stress transfer (SST) model was used to simulate the coupled flow of water and air in the dropshaft (Devolder et al. 2017). The inlet is set as a mass flow rate inlet boundary, while the outlet and the upper part of the vent pipe are set as atmospheric pressure boundaries to simulate the atmospheric environment.

Model verification and simulating cases

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This study conducted a simulation analysis on mesh numbers of 1.32 million, 1.59 million, 1.91 million and 2.29 million, as shown in Figure 2(a). The entire computation domain was partitioned using a polyhedron–hexahedron composite mesh, with local mesh refinement applied to the inlet pipe and dropshaft to meet the accuracy requirements. When the mesh number is 1.91 million, the maximum pressure at the bottom of the shaft is 6.71 kPa, which differed by −0.66% compared to 2.29 million. Further increasing the mesh numbers does not significantly change the simulation results. Therefore, the mesh number of 1.91 million is used for the simulation analysis. Simulations were performed for three time-steps of 0.001, 0.005, and 0.01 s, as shown in Figure 2(b). Decreasing the time-steps resulted in minimal pressure changes, while the relative error of water velocity under time-steps of 0.005 and 0.01 s is 0.64%. Therefore, a time-step of 0.005 s is uniformly adopted for all the simulation cases. This study compares the simulated values with the experimental values of the pressure at various points near the ninth-level step in the experimental model of Wu et al. (2018) to verify the accuracy of the numerical model, as shown in Figure 2(d). When the flow rate are Q = 3.55 L/s and Q = 11.46 L/s, the maximum relative errors of the P/h between the simulated data and experimental results are −6.86 and −5.40%, respectively. This indicates that the numerical model under study has good reliability and can be used for the analysis of flow patterns and pressures inside the dropshaft.

Figure 2

Model validation: (a) mesh independence analysis; (b) time-steps sensitivity analysis; (c) location of pressure monitoring points; (d) comparison between simulated and experimental values.

All cases of this study are shown in Table 1. The case A0 is used to observe the flow characteristics inside the shaft without an internal wall. The cases A series is set with θ = 60° to generate skimming flow and by adjusting h and w, the effect of the water velocity inside the shaft is explored to predict the terminal velocity. The cases B series is set with θ = 90° and by adjusting h and w, it generates napped flow and transition flow. They are mainly used to study the pressure distribution on the shaft wall under different flow rates and the impact of wall-attached swirling flow on the bottom of the shaft. The case C1 is set with θ = 120° to generate napped flow and combined with the cases A and cases B series to explore the combined effect of θ on water velocity and bottom pressure of the shaft.

Table 1

List of simulated cases

Cases	Center angle θ (°)	Step height h (m)	Step width w (m)	Flow rate Q (L/s)	Plunge pool depth H_p (m)	Drop height H (m)	Shaft diameter D (m)	Notes
A0	60	0.06	0.10	5–40	/	/	/	No internal wall
A1	60	0.06	0.10	5–25	0.1	4.0	0.4	Internal wall diameter: d=D–2w
A2			0.12
A3			0.14
A4		0.09	0.10
A5			0.12
A6			0.14
A7		0.12	0.14
B1	90	0.09	0.10
B2			0.12
B3			0.14
B4		0.12	0.14
C1	120	0.12	0.14

Cases	Center angle θ (°)	Step height h (m)	Step width w (m)	Flow rate Q (L/s)	Plunge pool depth H_p (m)	Drop height H (m)	Shaft diameter D (m)	Notes
A0	60	0.06	0.10	5–40	/	/	/	No internal wall
A1	60	0.06	0.10	5–25	0.1	4.0	0.4	Internal wall diameter: d=D–2w
A2			0.12
A3			0.14
A4		0.09	0.10
A5			0.12
A6			0.14
A7		0.12	0.14
B1	90	0.09	0.10
B2			0.12
B3			0.14
B4		0.12	0.14
C1	120	0.12	0.14

RESULTS AND DISCUSSION

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Flow patterns

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The flow patterns inside the dropshaft with an internal wall structure are shown in Figure 3. The flow patterns can be divided into three types: napped flow, transition flow, and skimming flow (Ren et al. 2021). In the napped flow, the water tongue produces a hydraulic jump when it impacts the downstream step. A significant standing wave phenomenon is formed on the outer wall surface and an air cavity underneath the water tongue, as shown in Figure 3(a). For the transition flow, the hydraulic jump and the standing wave on the outer wall surface are weakened. The wall-attached swirling flow gradually becomes prominent, while the air cavity on the inside is gradually filled with water, as shown in Figure 3(b). The standing wave of the skimming flow almost disappear, smooth water surface lines are formed on the outer wall, and the air cavity at the step angle is replaced by circulating vortex, as shown in Figure 3(c). If it is unfolded in two dimensions, it will approximate to the skimming flow on a stepped spillway where the flow section slides downwards along the pseudo-bottom (Rajaratnam 1990), as shown in Figure 3(d).

Figure 3

Flow patterns diagram: (a) napped flow; (b) transition flow; (c) skimming flow; and (d) unfolded skimming flow.

The water flow inside the shaft without internal wall detaches from the stepped ramp and the splashing water scatters throughout the central cavity of the shaft, as shown in Figure 4(a). The broken water flow will affect the air demand of the shaft. The greater the degree of water fragmentation, the larger the contact area between the water and air, enhancing the drag effect of the water flow on the air and resulting in increased air flow into the shaft through the top vent pipe. Figure 4(b) shows the variation of air demand inside the shaft. The degree of water fragmentation is indicated by M_c/M_s, where M_c and M_s represent the mass of water in the central cavity of the shaft and on the stepped ramp, respectively. It can be seen that the flow rate increases, the air demand of the shaft first decreases and then increases with the degree of water fragmentation.

Figure 4

Flow in the dropshaft without the internal wall structure: (a) internal flow pattern of case A0 and (b) variation of air demand in case A0.

Water velocity distribution

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The variation of water velocity with different drop heights in the rotating stepped dropshaft is shown in Figure 5(a). Under the influence of gravity, the water velocity continuously increases. If above a given drop height the average amplitude of the water velocity on the rotating steps is within 2%, it can be considered that the water velocity no longer changes with the falling height, which indicates the appearance of the terminal velocity. Therefore, the height at which this occurs can be considered as the starting position of the terminal velocity. It can be observed that the greater the flow rate, the higher the falling height before reaching the terminal velocity. When reaching the terminal velocity, the variation of sectional flow velocity on each horizontal step is the same along the flow direction, as shown in Figure 5(b). When the flow rate is Q = 5 L/s, the water velocity increases toward the end of the horizontal steps. When the flow rate increases to Q = 15 L/s, the water velocity first increases and then remains relatively stable due to the reduction in backflow and cavity region. Additionally, the impact jet at high flow rates forms a pressure gradient greater than the centrifugal force in the rear section of the horizontal steps and the secondary flow generated by this contributes to the relative stability of the flow in this region (Qi et al. 2019). From Figure 5(c), at low flow rates (Q ≤ 10 L/s), the flow pattern is mainly in the state of napped flow and the influence of the step structure on the terminal velocity is minimal. At high flow rates (Q ≥ 15 L/s), the flow of water is gradually continuous, increasing the central angle θ and reducing the step height h can both decrease the terminal velocity. The variation in step width w does not have a significant impact on the terminal velocity at different flow rates.

Figure 5

Water velocity in dropshafts under case A1: (a) water velocity with different flow rate; (b) velocity of section on the horizontal steps; and (c) terminal under different step structures.

The energy loss in the napped flow is mainly caused by the fragmentation and aeration of the plunging jet, the impact of the plunging jet on the downstream step and the mixing of the plunging jet with the backflow (Ren et al. 2021). The energy dissipation in the skimming flow mainly comes from the wall shear stress τ₁ (Zhao et al. 2006) and the average Reynolds shear stress τ₂ (Rajaratnam 1990) generated by the circulating vortex below pseudo-bottom. The transition flow combines the characteristics of the napped flow and the skimming flow and therefore possesses both of the above-mentioned energy dissipation mechanisms.

The control volume was selected to analyze the hydraulic characteristics of the skimming flow. A cross-section of the flow is chosen as the control volume and it slides down along the pseudo-bottom, as shown in Figure 6(a). The forces acting on the control volume include gravity, frictional forces, and dynamic water pressure (Zhang et al. 2022). When the terminal velocity is reached, the acceleration of the control volume is zero and therefore the following equation can be derived:

(4)

where G = ρgA, G represents the gravity acting on the control volume, A is the cross-sectional area of the control volume; α is the characteristic inclination angle of the stepped ramp. Due to the higher free liquid level on the outer wall than on the inner wall, the influence of the inner wall can be neglected. Therefore, F_f = yτ₁ + wτ₂, where

and

⁠, f₁ and f₂ are Darcy–Weisbach friction coefficients, c_f is the fluid friction coefficient; ρ is the density of the water; v is the water velocity; F₁ and F₂ are the dynamic water pressure acting on the front and back of the control volume, and when the terminal velocity is reached, assume F₁ = F₂.

Figure 6

Hydraulic characterization analysis: (a) schematic diagram of the control volume and (b) comparison of the theoretical and simulated terminal velocity of the skimmed flow.

Due to the varying slope of the stepped ramp along the radial direction and is subject to the centrifugal force, the majority of the flow is concentrated near the outer wall (Sun et al. 2021). Therefore, this study is based on the radial position r of the center of gravity of the control volume and according to Figure 3(d), the characteristic slope sinα of the step ramp can be obtained (θ in radian):

(5)

By combining Equations () and (), we obtain the expression for the terminal velocity of the skimming flow:

(6)

The terminal velocity and relative error for different operating conditions are shown in Figure 6(b), where v_T and v_S are theoretical and simulated values, respectively. According to the Plexiglas material used in the experimental model, f₁ = 0.01. Based on the experimentally validated simulation data, the reasonable range of f₂ based on the inverse derivation of Equation (6) is about 0.06–0.08 (Wu et al. 2018). The maximum error between v_T and v_S is 7.14% and as the flow rate increases from 20 to 25 L/s, the average error further decreases from 4.07 to 3.44%. This is because flow rate increases the development of the skimming flow becomes more complete and closer to the ideal state described by Equation (6). In conclusion, the theoretical analysis has achieved good results and provides a reference for estimating the terminal velocity of the skimming flow.

Internal pressure distribution

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As shown in Figure 7(a), the wall pressure gradually increases with the increase of the drop distance and the pressure remains relatively stable after reaching a certain height. Two distinct areas of pressure formed on the surface of the horizontal step (Figure 7(b)) due to jet impact on the horizontal step back section and the presence of air cavity and backflow in the front section. Furthermore, at the connection between the shaft and the outlet pipe (dashed area in Figure 7(b)), there is a drastic pressure fluctuation due to the sudden change in wall structure. For example, when Q = 20 L/s, the high pressure at this location has exceeded 6 kPa, while the low pressure can reach −2 kPa. This intense fluctuation of positive and negative pressures may cause safety issues for the shaft structure (Liu et al. 2022).

Figure 7

Pressure in dropshafts: (a) pressure on the outer wall of the shaft changes with the drop distance; (b) pressure distribution on the shaft wall; and (c) pressure distribution at shaft bottom.

Figure 7(c) depicts the pressure distribution at the bottom of the dropshaft, where the white line represents the location of the step end and arrows indicate the flow direction. When Q = 5 L/s, the tangential velocity of the wall-attached swirling flow is small. After passing through a turning angle of approximately 130°, it impacts on the water cushion surface, resulting in a pressure peak and a small high-pressure region near the outer edge of the shaft bottom. The larger the flow rate, the greater the tangential velocity and the larger the angle at which the water flow impacts the water cushion. This causes the location of the bottom pressure peak to move counterclockwise along the swirling direction and the range of the high-pressure region continuously extends along the bottom edge. When Q = 25 L/s, a complete circular high-pressure region forms along the shaft bottom edge.

The time average pressure at different dropshaft bottom monitoring points under different flow rates is shown in Figure 8. As the flow rate increases, the pressure increment at each point near the shaft bottom edge becomes larger, resulting in a larger pressure gradient toward the center of the shaft bottom. From Q = 5 L/s to Q = 25 L/s, the pressure gradient difference between point P9 and P3 increases by approximately 4.5 times. This is because the higher flow rate carries more energy and can generate larger impact pressure. Furthermore, the presence of annular hydraulic jump can also cause certain pressure differences.

Figure 8

Pressure at monitoring points (Case B1): (a) time average pressure at P1–P5 and (b) time average pressure at P6–P9.

Figure 9 illustrates the variation of dropshaft bottom pressure under different stepped structures when the flow rate is Q = 20 L/s. Equation () reveals that increasing the central angle θ and reducing the step height h both can decrease the slope of the step, resulting in a slower water velocity reaching the water cushion and reducing the impact pressure on the shaft bottom. Compared to θ and h, the variation of step width w has a smaller effect on the stepped ramp and the velocity of the water reaching the cushion layer is similar, leading to insignificant differences in bottom pressure.

Figure 9

Average and maximum pressure at shaft bottom under different step structures.

Similar to the plunging dropshaft (Liu et al. 2022) the bottom pressure of the rotating stepped dropshaft is mainly determined by the velocity at which the wall-attached swirling flow impacts the water cushion. A control volume is established to analyze the forces acting on the bottom of the shaft (as shown in Figure 10(a)), resulting in the following momentum equation:

(7)

where h_b is the average water depth at the bottom of the shaft, which can be determined by the water level inside the downstream outlet pipe; A_b is the range of influence of the shaft bottom impact pressure, which is the area occupied by half of the pressure at the stagnation point (Castillo et al. 2014); F is the impact force exerted on the bottom of the shaft. Compared to v_y1, v_up is so small that it can be ignored (Puertas & Dolz 2005), where v_y1 is the vertical velocity at which the swirling flow reaches the water cushion and v_up is the velocity at which water escapes upwards from the shaft bottom.

Figure 10

Simulation of overflow nappe impingement jets

Swirling flow impact analysis: (a) wall-attached swirling flow impact water cushion and (b) comparison of theoretical and simulated shaft bottom pressure.

The distance between the end of the stepped ramp and the bottom of the shaft is small and the swirling flow quickly impinges on the bottom, if the influence of wall friction is ignored, it can be obtained:

(8)

where ΔH is the distance from the end of the stepped ramp to the water cushion; v_y0 = vsinβ represents the vertical velocity of the water flow when it leaves the stepped ramp, where v is the average velocity of section of the water flow. The shaft bottom pressure can be obtained by combining Equations () and ():

(9)

The theoretical bottom pressure of the dropshaft was calculated based on Equation (9) and compared with the simulated values, as shown in Figure 10(b). The results show that the average error between theoretical values and simulated values is 5.70%, Equation (9) can well predict the bottom pressure of rotating stepped dropshaft. This can be used to determine whether it is necessary to add thicker water cushion or other structural safety protection measures. The calculations were also compared with the test results of the plunging dropshaft (Liu et al. 2022), as also shown in Figure 10(b). Different from the rotating stepped dropshaft, the impact pressure in the plunging dropshaft is generated by the annular flow and bounced jet, range can cover the whole bottom of the shaft. Although the bounced jet in the plunging dropshaft undergoes fragmentation during the falling process, the water flow reaches a high velocity when it reaches the water cushion due to the lack of channel support. Therefore, even though the height of the plunging dropshaft is lower than that of the rotating stepped dropshaft, a significant impact pressure at the bottom can still be generated.

CONCLUSIONS

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This study employed the SST κ–ω turbulence model combined with the VOF method to conduct three-dimensional numerical simulation of the rotating stepped dropshaft. By discussing and analyzing the flow patterns, the water velocity distribution and pressure distribution inside the shaft, the following conclusions can be drawn:

The flow patterns within the shaft with internal wall can be classified into napped flow, transition flow and skimming flow. However, the shaft without internal wall is filled with fragmented water flow. The resistance of the skimming flow comes from the shear stress on the wall and the average Reynolds shear stress generated by the circulating vortex beneath the pseudo-bottom. A theoretical formula based on this can provide a good estimation of the terminal velocity magnitude.
The water velocity is related to the flow rate and the stepped structure. Increasing the central angle of the step and reducing the step height can both decrease the terminal velocity and alleviate the pressure at the bottom of the shaft.
When water flows through the connection between the shaft and the outlet pipe, it poses a severe alternating threat to the structural safety due to extreme positive and negative pressures. The wall-attached swirling flow create a circular high-pressure zone at the edge of the bottom. The shaft bottom pressure consists of hydrostatic pressure and swirling flow impact pressure. By utilizing the momentum theorem and considering the impact pressure range of the swirling flow, the shaft bottom pressure can be reasonably predicted.

ACKNOWLEDGEMENTS

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The authors gratefully appreciate the financial support from the Natural Science Foundation of Zhejiang Province (No. LQ22E090002) and the Natural Science Foundation of Ningbo (No. 2023J092).

DATA AVAILABILITY STATEMENT

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

CONFLICT OF INTEREST

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The authors declare there is no conflict.

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