Three-dimensional numerical simulation of stepped dropshaft with different step shapes

The deep tunnel system is increasingly used worldwide for stormwater conveyance and storage, providing a robust and effective means of preventing urban waterlogging. In the system, the dropshaft, with the function of conveying stormwater to the deep tunnels underground, often runs under conditions of high falling head and large discharge. Based on the standard stepped dropshaft, a blade-shaped stepped dropshaft was proposed in order to control the potential standing wave and improve discharge capacity. Its hydraulic characteristics in respect of flow pattern, flow rate distribution, time-averaged pressure and energy dissipation were investigated by numerical simulation. Compared with the standard stepped dropshaft, the blade-shaped stepped dropshaft generated a more uniform flow rate distribution in the radial direction, therefore effectively decreasing the height of the standing wave near the external wall. The negative pressure areas that easily existed on the vertical wall of steps were well controlled. The energy dissipation of the bladeshaped stepped dropshaft was as high as that of the standard stepped dropshaft. Therefore, the blade-shaped stepped dropshaft could be a preferable design for the deep tunnel system.


INTRODUCTION
With the rapid development of urbanization, the increase of population and the decrease of water area have led to great pressure on the sewer and storage tunnels of the city (Vasconcelos & Wright ). Due to global climate change, the pressure may further increase with frequent intense rain events leading to urban water problems such as waterlogging (Zhang et al. ). Deep storage tunnel systems, generally including several dropshafts and underground tunnel, can be effective in preventing the urban water problems, and have been implemented or is under consideration in many cities (Guo & Song ; Changnon ). In this system, the stormwater runoff can be diverted to the deep tunnels through dropshafts, and then diverted to the river or sewage plant, or temporarily stored in the tunnels (Guo & Song ). The dropshaft often runs with the conditions of high falling head and large discharge, which may lead to operational issues related to energy dissipation and cavitation, and even cause disasters like geysers (Rajaratnam et al. ; Vasconcelos & Wright ). Therefore, it is important to take into account the hydraulic characteristics in the dropshaft.
Dropshafts can be divided into four typical types: the vortex dropshaft, baffle dropshaft, plunging dropshaft and helicoidal-ramp dropshaft (He et al. ). Previous research indicated that the helicoidal-ramp dropshaft shows a better performance for high falling head and large discharge with good energy dissipation and exhaust effect (Kawasaki et al. ). It includes a vertical circular dropshaft and a continuous helicoidal ramp attached to the dropshaft wall.
The water can skim down along the ramp, and the air can be released from the center hollow column without any extra ventilation device (Ansar & Jain ). The energy dissipation occurs as distributed friction loss along the helicoidal ramp without any extra energy dissipators (Kennedy et al. ).
Inspired by the helicoidal-ramp dropshaft, Wu et al.
() proposed a stepped dropshaft in order to provide well aeration to decrease the risk of cavitation erosion and to further increase the energy dissipation. Figure 1(a) is a definition sketch of the standard stepped dropshaft. By changing the ramps to successive steps, the flow skimming over the ramps could be converted to successive nappes impacting steps with vortices beneath them. On the one hand, the energy dissipation can increase due to nappe breakup in the air and impact on the steps as well as full or partial hydraulic jumps. On the other hand, the intense turbulence resulting from the vortice increases the self-aeration from the free surface, which can decrease the risk of cavitation. The results showed that the end sills on the steps can increase the energy dissipation but decrease the discharge capacity of the stepped dropshaft.
The standing wave, as an important flow phenomenon, is associated with the discharge capacity. When the height of the standing wave is too large to touch the bottom of the upper step, the discharge capacity of the stepped dropshaft will be limited. Therefore, it is necessary to control The density ρ and the viscosity μ can be expressed as: where ρ w and ρ a are the density of water and gas, respectively; μ w and μ a are the viscosity of water and gas, respectively.
The variables and their attributes represent air or water, or a mixture of them at any control volume. The tracking of the interface between air and water is accomplished by the continuity equation as follows: where x i and u i are the coordinates and velocity components, respectively (i ¼ 1, 2, 3), t is the time.

Turbulence model
The shear stress transport rotation-curvature correction where ω ¼ specific dissipation rate, k ¼ turbulent kinetic energy, μ T ¼ turbulent eddy viscosity, d ¼ the distance to the nearest wall, and P k and P ω ¼ production terms. β, β*, σ k , σ ω and σ ω2 are the model coefficients.
The curvature correction term ( f r1 ) can increase the sensitivity to the curvature and rotation of the streamline, which is more in line with the spiral bending flow of the water flow inside the dropshaft (Spalart & Shur ). The empirical formula ( f rotation ) for curvature correction is: where the constant parameters are C r1 ¼ 1.0, C r2 ¼ 2.0, C r3 ¼ 1.0. rÃ andr are related to the strain rate tensor (S) and the magnitude of vorticity (Ω): Numerical algorithm The Geo-Reconstruct can obtain the face fluxes whenever a cell is filled with water or air and be used when the cell is The dimensionless flow rate is Q* ¼ Q/(g(R À r) 5 ) 1/2 , where Q is the flow rate, g is the acceleration of gravity.
Numerical predictions were carried out for the range of the flow rates was Q ¼ 0.012-0.048 m 3 /s, which corresponded to Q* ¼ 0.078-0.311. The minimum time step was 0.001s and the max iterations per time step were set as 20.

Model validation
The experiment results (     Flow rate distribution  dropshaft is more uniform than that of the standard stepped dropshaft. The reason will be described below via the velocity distribution on the step horizontal surface.      There is an obvious negative pressure zone on each step vertical wall. Figure 8 presents the dimensionless pressure distribution p n /ρgb on the negative pressure zone with y/L Y for the eighth to eleventh steps at Q* ¼ 0.155, where p n is the vertical average pressure along the water depth.

Velocity distribution
With increasing y/L Y , p n /ρgb first decreases under the influence of the centrifugal force, and then increases near the external wall due to the sidewall resistance. Corresponding to the time-averaged pressure distribution on the step horizontal surface, p n /ρgb shows periodic for the standard stepped dropshaft: p n /ρgb of the eighth and tenth steps are similar with the minimum p n /ρgb at y/L Y ¼ 0.67; p n /ρgb of the ninth and eleventh steps are similar with the minimum p n /ρgb at y/L Y ¼ 0.93 (Figure 8(a)). For the blade-shaped stepped dropshaft, p n /ρgb from the ninth to eleventh steps are similar, with the minimum p n /ρgb at y/L Y ¼ 0.68. Figure 9 shows the minimum p n /ρgb and the dimensionless negative pressure area s/lb, where s is the average negative pressure area for the eighth to eleventh steps vertical walls. It indicates that both p n /ρgb and s/lb linearly decrease with the increase of Q*. With the blade-shaped stepped dropshaft, the minimum p n /ρgb are greater and the s/lb are less than that of the standard stepped dropshaft.

Energy dissipation
The energy dissipation η of the dropshaft is expressed as follows: where E 1 and E 2 are the total energy at the inflow and outflow sections, respectively; Z 1 and Z 2 are the elevation heads of the inlet and outlet channels, respectively; p 1 / ρg ¼ 0 and p 2 /ρg ¼ 0 are the pressure heads of the inlet A comparison with the results of a plunging dropshaft (Chanson ) indicates that η of the stepped dropshaft is similar to that of the plunging dropshaft in some cases. It is believed that η is related to the multiple parameters of the dropshaft, which is needed further studies.

CONCLUSIONS
In order to control the standing wave and improve the discharge capacity, the blade-shaped stepped dropshaft was proposed in this study to generate a more uniform  Based on these findings, it can be found that compared with the standard stepped dropshaft, the blade-shaped stepped dropshaft can improve the discharge capacity, improve the pressure distribution, and keep a good energy dissipation. Therefore, it is a potentially safe and efficient dropshaft design that can be applied to deep tunnel systems.