## Abstract

The hydraulic characteristics of the lock chamber section in a bottom outlet are complex. Particularly, cavitation is a prominent problem when the flow in the tunnel transitions from pressurized to unpressurized conditions. As a countermeasure, sidewall contractions are often installed at the end of the pressurized section. In this paper, with numerical simulations and vacuum tank experiments, we analyzed the design parameters of several prototype projects and studied the flow properties and cavitation characteristics of the transition section. The four main factors that influence cavitation risks are the contraction ratio of the transition section, the top press slope, the length of the transition section, and the shape of the orifice. The study of a real-world project demonstrates that by implementing a severe constriction degree, the top press slope can be slowed down, and the pressurized transition section can be extended, thereby reducing cavitation risks and improving flow patterns. These research results provide a reference for the optimal design of hydraulic structures.

## HIGHLIGHTS

This paper analyses four factors that affect the cavitation of the bottom discharge tunnel.

This paper collects the design parameters of 23 lock chambers and summarizes the value range and design law of these design parameters.

The cavitation and flow pattern of the LW hydropower station are verified through the vacuum tank experiment and numerical simulation.

## INTRODUCTION

Some literature studied the cavitation and flow characteristics of outlets using numerical simulations and model experiments. Aminoroayaie Yamini *et al.* (2021) used Flow-3D to study the hydrodynamic performance and cavitation of the bottom outlet at Sardab Dam. They analyzed flow patterns at the different gate openings and the working status of the branch. Kyriakopoulos *et al.* (2022) used numerical simulation and model experiments to study the flow pattern and vorticity of different distances between the butterfly valves. Through 3D flow simulation, Sadat Helbar *et al.* (2021) investigated the effect of the area, number and shape of the bottom outlet gates on the velocity, concentration, and volume of the removed sediments and the dimensions of the flushing cone. Through three-dimensional numerical simulation, Nikseresht *et al.* (2012) studied pressure, and cavitation index under different conditions of the gate openings and water levels of the reservoir. Dargahi (2010) used numerical simulation and model experiments to study the flow characteristics of Aswan Dam bottom outlet gates under different operating modes. Li *et al.* (2016) observed that the degree of gate opening has a great effect on the pressure and flow cavitation index of side walls. The lateral expansion and partial gate opening increased the risk of cavitation erosion on side walls downstream. Based on the results of numerical simulations, Jamali Rovesht & Manafpour (2019) observed that cavitation indices are lower than the critical amount (0.2) on the tunnel bottom for 10–80% of service gate openings.

There some literature conducted model tests under normal atmospheric pressure to study the flow pattern of the spillway tunnel and prevent cavitation risk. In order to ensure the safe operation of the gate under high water levels, the arc-shaped working gate was usually used, and the sudden expansion and sudden fall structure type was set at the end of the lock chamber to increase the aeration of the downstream channel (Feng *et al.* 2001). Abdolahpour & Roshan (2014) carried out model tests at normal atmospheric pressure and studied a bottom outlet for high-velocity discharges past from the control gate. According to the measured pressure and velocity data, they calculated the distribution of cavitation numbers along the bottom and between the two gates. Mohaghegh & Wu (2010) studied various geometry parameters for the discharge tunnel in the atmospheric model and discriminate different flow regimes under different conditions. Mohaghegh & Wu (2009) studied the influence of different angles of top press slope on the downstream cavity length. They observed that steep top press slope results in shorter bottom cavity length downstream in big discharges. In addition, for the flow with high *Fr* number (*Fr* > 3.4), the length of the bottom cavity increases monotonously with the increase in the step height. Zhou *et al.* (2018) proposed an expression for calculating the geometrical morphology and the length of bottom and lateral cavities. Based on experimental results, Li *et al.* (2021) developed an improved model for calculating airflows in the cavity downstream of a bottom offset. Wang *et al.* (2019) studied the influence of the relative step height and working gate opening on pressure distribution. Wenhai *et al.* (2007) combined with the right bank flood relief tunnel of the Jiudian Gorge water conservancy pivot project to carry out experimental research on the sudden expansion and sudden fall section of the flood relief tunnel, the top press slope of the lock chamber is 1:6 and the area contraction ratio is 0.88. Jamali *et al.* (2023) studied the pressure and cavitation number in the transition zone of the discharge tunnel with different Fr number. The change of expansion angles in the transition zone will bring great changes to the pressure and cavitation number in the transition zone The pressure and cavitation number in the transition zone were obtained by atmospheric pressure experiment and numerical simulation.

Another important way was to conduct a model test in the vacuum tank to reduce environmental pressure and used the hydrophone to monitor the sound generated by cavitation bubble collapse to judge the cavitation risk of the discharge tunnel. Wang *et al.* (2022) used the vacuum tank to observe the cavity length and entrained air of chute aerators under various atmospheric pressures. Liu *et al.* (2020) used the vacuum tank to reduce the ambient pressure environment to observe the atomized rain of a high-velocity waterjet.

Some literature works have made prototype observations on actual engineering projects. Zhenlian *et al.* (2006) carried out the prototype observation of the gas doping facility of the sudden expansion and sudden fall step of the lock chamber in the bottom tunnel.

To assess cavitation occurrence at the dam outlet, three primary methods are commonly employed: evaluating the critical cavitation number based on experience, conducting vacuum tank tests, and prototype observations. But the prototype observations tend to be costly, the vacuum tank experiment offers an efficient approach for studying cavitation risks.

There are many papers that studied bottom outlet structures. Apart from these studies, to the best of our knowledge, there is little research on the cavitation characteristic of the different top press slopes of the tunnel when drastic reduction in the outlet flow area.

How to design the shape of the transition section to achieve better cavitation characteristics under severe flow area contractions? This issue is not clear. Therefore, we studied the impact of different top press slopes on the cavitation characteristics of water flows under severe flow area contraction.

Although many papers have used numerical simulation and model experiments to study the cavitation and flow characteristics at the outlet of dams, relatively little literature combines numerical simulation and vacuum tank to study cavitation characteristics of the outlet of a dam. The vacuum tank experiment can effectively determine whether the structure occurs cavitation, but it is difficult to measure flow field data in the vacuum tank model experiment. Therefore, we used numerical simulation to obtain flow field data and calculate the water flow cavitation number. In this paper, we combine numerical simulation and a vacuum tank to study the cavitation characteristics of the transition section.

In order to understand the effects of different transition sections on the characteristics of cavitation, we first collected the design parameters of multiple projects. Also, then based on the experience of these projects we have provided suggestions for contraction ratio *η*, orifice shape *h/B*, the top press slope *i*, and the transition section length *L*. (in section 3.1). Then we proposed the following hypothesis: ‘*extending the length of the transition section may make the cavitation characteristics better, under severe flow area contraction*’. According to this hypothesis, we carried out a numerical simulation to obtain pressure and velocity field (in Section 3.2), and then implemented a vacuum tank experiment of different transition sections to verify (in Section 3.3). With significant contraction in the flow section, extending the length of the transition section appropriately will achieve better cavitation performance.

## RESEARCH METHOD

As shown in Figure 2, there are some key design parameters of the lock chamber. These parameters would affect the flow characteristics and cavitation prevention. They are tunnel diameter *D*, transition section *L*, the top press slope of the tunnel, contraction ratio , orifice height *h*, orifice width *B*. In this paper, the ratio of orifice height *h* and width *B* is used to express the change in the orifice shape. When = 1, the orifice is a square orifice; when *h/B* < 1, the orifice is wide-flat orifice; when >1, the orifice is narrow-high orifice. The main influencing factors of the hydraulic and cavitation characteristics of the lock chamber include the transition section length *L*, contraction ratio , and top press slope *i*. In this paper, , , , , , , , . Where *D* = 12 m is the diameter of the discharge tunnel. is the inlet area of the section and is the outlet area of the section, then *η* is the contraction ratio of the lock chamber outlet orifice. is the flow velocity at the orifice, *h* is the height of the orifice, *L* is the length of the transition section, Δ*b* is the width of lateral expansion, *L _{lateral}* is the length of the lateral cavity, and

*θ*is the angle of incidence after lateral diffusion of flow.

We think that the top press slope, the length of the transition section, the contraction ratio, and the orifice shape *h/B*, will have an obvious impact on the cavitation characteristics of the transition section. The reasons for our selection of these four parameters are as follows.

For the top press slope, a slower slope will bring a longer transition section, which is conducive to water flow adjustment, but the protection ability of the pipeline top will be weakened. For the contraction ratio, a more severe contraction ratio can also cause drastic changes in the flow pattern. When it is used together with the lateral aeration facilities, after the flow out of the orifice, the pressure on the side walls of the open channel may undergo adverse drastic changes due to the rapid diffusion of the flow toward the sides wall. For the orifice shape , when >1, the flow is compressed into a thin and tail shape, and the pressure distribution at the top may be bad at the outlet. When <1, the flow is compressed into a flat shape, and when used in conjunction with the lateral aeration facilities, the flow may have a stronger impact on the side wall after being out from the orifice.

The LW dam is an earth dam with a crest elevation of a bottom outlet is 2,617.5 m (m.a.s.l.), a design water level is 2,702 m (m.a.s.l.) and a dam height of 239 m. The flow rate at the design water level is 2,530 m^{3}/s, and the velocity of the lock chamber outlet is 32.8 m/s. Therefore, the Reynolds number is calculated by orifice width *B*, , and the Froude number = 3.7. The original design of the transition section parameters is shown in Table 1.

Tunnel diameter D (m)
. | Orifice width B (m)
. | Orifice height h (m)
. | Top press slope (i)
. | Transition section length (L/D)
. | Contraction ratio (η)
. | Orifice shape (h/B)
. |
---|---|---|---|---|---|---|

12 | 10 | 7.7 | 1:6 | 2.87 | 0.681 | 0.77 |

Tunnel diameter D (m)
. | Orifice width B (m)
. | Orifice height h (m)
. | Top press slope (i)
. | Transition section length (L/D)
. | Contraction ratio (η)
. | Orifice shape (h/B)
. |
---|---|---|---|---|---|---|

12 | 10 | 7.7 | 1:6 | 2.87 | 0.681 | 0.77 |

### Numerical simulation method

Numerical simulation using Flow-3D to calculate various cases with a grid size of and the total mesh cell counts was 7.8 million, as shown in Figure 2(a) and 2(b). The numerical simulation size adopts the LW dam prototype size, with a tunnel diameter of . The inlet used the pressure inlet and set up the fluid elevation as 5.8D. The outlet was set to free outflow, the vent was set to pressure outlet, 0 Pa, and the other region was set to the wall. The following equations are used to control the fluid motion. The continuity equation. ; The momentum equation:, where = 1,000 m^{3}/s, the water density. Turbulent model is used standard *k-ε* model. *k* Equation: ; the turbulent dissipation rate *ε* is given by the equation:, turbulent viscosity coefficient, , and are constants. We used Volume-Of-Fluid (VOF) method to deal with a free surface. Fractional area-volume obstacle representation (FAVOR): to simulate the rigid surfaces and some components, such as geometric boundaries. The VOF method was utilized to deal with water air interface. The flow mode is an incompressible fluid.

Item . | Definition . |
---|---|

Model input | Specified pressure, fluid elevation H = 70.06 m (P, color red) |

Lateral air vent | Specified pressure, 0 Pa (P, color green) |

Model output | Outflow (O, color red) |

Walls | Wall, no-slip (W, color blue) |

Total number of computational mesh | 7,884,066 |

Item . | Definition . |
---|---|

Model input | Specified pressure, fluid elevation H = 70.06 m (P, color red) |

Lateral air vent | Specified pressure, 0 Pa (P, color green) |

Model output | Outflow (O, color red) |

Walls | Wall, no-slip (W, color blue) |

Total number of computational mesh | 7,884,066 |

The mesh block size is: *X*_{min} = −60 m, *X*_{max} = 105 m, *Y*_{min} = −7 m, *Y*_{max} = 7 m, *Z*_{min} = −18 m, *Z*_{max} = 7 m.

The velocity and pressure fluctuations of high-speed water flow are significant. In order to obtain statistically significant results, we used the time average method to extract pressure and velocity data, which time average within 5 s. We pay more attention to the cavitation number in the near wall region, so we extract data at a distance of 0.2 m from the wall.

### Numerical model validation

### Experiment method

In high-altitude areas, the decrease in air density can reduce the ambient pressure. To assess the low ambient pressure effect, it is helpful to conduct a relevant experiment in an artificial low-pressure environment. The vacuum tank has been applied to mechanism research and practical applications in many fields. The experiment was conducted on a lock chamber model test and low ambient pressure at the State Key Laboratory of Hydraulics and Mountain River Engineering of Sichuan University.

The modeled using a model scale 35, the velocity scale is , and the volume of flow scale is . Therefore, the pipeline diameter of the experiment model is . The flow rate of the engineering prototype is 2,530 m/s, so the maximum discharge of model is = 349 L/s:

*g*= 9.81 m/s

^{2}is gravitational acceleration,

*u*is velocity. The smaller the cavitation number is, the easier it is to occur cavitation. For the vacuum tank experiment, it used an air extraction pump to reduce the ambient air pressure in the tank.

The cavitation number of the actual engineering project is , the cavitation number of vacuum tank model is .

In order to achieve the cavitation number similarity criterion, we reduced the ambient air pressure in the tank, the cavitation number of the actual engineering project is equal to the cavitation number of model, i.e., .

All experiment data are obtained under the 0.38 m ambient pressure inside the vacuum tank. In addition, we would gradually adjust the ambient pressure inside the vacuum tank to obtain the incipient cavitation number of the body type.

*E*is the cavitation sound energy, is the background sound energy. According to the sound difference and relative sound energy ratio , four types of cavitation intensities are shown in Table 3.

Cavitation type . | No cavitation . | Weak cavitation . | Significant cavitation . | Intense airing . |
---|---|---|---|---|

Sound difference | 0–5 dB | 5–7 dB | 7–10 dB | 10 to +∞dB |

E/E_{0} relative energy ratio | 0–2 | 2–4 | 4–6 | >6 |

Cavitation type . | No cavitation . | Weak cavitation . | Significant cavitation . | Intense airing . |
---|---|---|---|---|

Sound difference | 0–5 dB | 5–7 dB | 7–10 dB | 10 to +∞dB |

E/E_{0} relative energy ratio | 0–2 | 2–4 | 4–6 | >6 |

## RESULTS AND DISCUSSION

### Lock chamber transition section and orifice shape

*P*to

*x*derivative, then . In the outlet section of the tunnel (100–120 m), the mainstream downward deviation, the upper part of the pipe flow velocity decreases, so the pressure increases gradually, so . Because the pressure at the top of this section of the pipe gradually increases to zero, the pressure at the top of this section of the pipe (60–120 m) must be negative. In order to eliminate or mitigate the negative pressure at the top of the outlet section of the pressurized tunnel, the top press slope is used at the outlet of the tunnel to increase the pressure at the top of the tunnel. As shown in Figure 7(c) , the transition section of the discharge tunnel often adopts the contraction on three sides, i.e., contraction on both sides and top press slope on the top plate, which makes the overflow area decrease along the pipe and achieves the purpose of eliminating negative pressure. In Figure 7(c), after adopting the form of three-sided contraction, the pressure inside the pipe is greater than 0 Pa.

In addition, the shape of the orifice also has an effect on the pressure distribution at the top of the tunnel. The study shows that when , the pressure distribution and cavitation number of a wide-flat orifice (*h/B**<* 1) is better than other orifice shapes; when , with low water level working conditions, narrow- tall orifice shape (*h/B**>* 1) performance is better, and the square orifice (*h/B**=* 1) is suitable for use in applications with large variations in water level (Zeyu 1982, 1988).

*η*, top press slope

*i*, transition section length and orifice shape

*h/B*to prevent negative pressure at the top of the transition section, negative pressure zone and water wing near the downstream side wall. As shown in Figure 8, the relationship between the four factors is to be considered comprehensively to select a reasonable design.

*η*, the top press slope, the length of the transition section

*L*/

*D*, and orifice shape

*h/B*. The geometric parameters of each project are plotted in Figure 9. In Figure 9(a) and 9(d), for the orifice shapes, more projects use wide-flat orifices and square orifices with

*h/B*ranging from 0.8 to 1.0, and few projects with

*h/B*> 1.2, because these projects use narrow, tall orifices in combination with their layout considerations. The top press slope for most of the projects was between 1:6 and 1:10, the contraction ratio was mostly concentrated between 0.7 and 0.9, and the length of the transition section is mostly between 1.5 and 2.5. For the contraction ratio , they are more likely to adopt a steep top press slope

*i*of about 1:6, because the contraction degree is small. This indicates that when the degree of contraction is small, the projects tend to use steeper top press slopes to prevent low pressure at the top of the tunnel.

As shown in Figure 9(b), when the Froude number is greater than 2, the wide-flat orifice and square orifice is used in many projects, and the narrow-high orifice is used in a few projects. Figure 9(c) reflects the relationship between the top press slope and the length of the transition section. When the top press slope *i* is steep, the corresponding transition section is short; when the top press slope *i* is slow, the corresponding transition section length is large.

In Figure 9, some projects had been damaged before. Longyang Gorge (Baoqin & Taishan 2000) The orifice shape *h/B* = 1.4, and a top press slope is 1:6. It is a narrow-height type orifice shape, and high number of the orifice. The pressure distribution at the top section of the narrow-height orifice is undesirable at high numbers (Zhongyi & Xia 1986; Meilan 2009). Although no damage to the top of the tunnel of Longyang Gorge dam was seen, damage to the sidewall occurs, and the cavitation prevention performance at this location is weak. Liujiaxia hydropower station (Linsheng 1999) has *h/B* = 1.3 and an number of 7 and damage due to sediment had occurred.

According to Figure 9, the analysis can be summarized as follows: high water level discharge tunnel mostly uses wide-flat and square orifices, and the orifice ranges from 0.8 to 1.0. The contraction ratio *η* of the orifice can range from 0.7 to 0.9, the top press slope *i* can range from 1:6 to 1:10, and the length of transition section *L*/*D* in the range of 1.5–2.5D. These experiences can be referred to in the actual engineering design.

### Numerical simulation of practical engineering

We would optimize the body design of the LW hydropower station according to the rules summarized above. This contraction ratio is relatively small compared with other projects *η*(0.7,0.9). Compared with other projects in Section 3.1, the contraction ratio of the LW hydropower station is small. The top press slope *i* is also steeper, resulting in a short length of the transition section and a drastic change of flow section in the transition section, which had a negative impact on the hydraulics characteristics. Referring to the design law obtained above, the press slope *i* of the LW hydropower station was changed from 1:6 to 1:8, thus being able to extend the length of the transition section with 2.87D and making the water flow more smoothly. Because of the arrangement of the LW hydropower station lock chamber, this optimization is only for the top press slope and the length of the transition section. The numerical simulation working condition parameters are shown in Table 4.

Top press slope () . | Transition section length () . | Orifice shape () . | Contraction ratio () . |
---|---|---|---|

1:4 | 1.43 | 0.77 | 0.681 |

1:6 | 2.15 | ||

1:8 | 2.87 | ||

1:10 | 3.58 | ||

1:12 | 4.30 |

Top press slope () . | Transition section length () . | Orifice shape () . | Contraction ratio () . |
---|---|---|---|

1:4 | 1.43 | 0.77 | 0.681 |

1:6 | 2.15 | ||

1:8 | 2.87 | ||

1:10 | 3.58 | ||

1:12 | 4.30 |

Projects . | i = 1:6 top press slope
. | i = 1:8 top press slope
. |
---|---|---|

Transition section length, L/D | 2.15 | 2.87 |

Contraction ratio, | 0.68 | |

Orifice height to width ratio, | 0.77 |

Projects . | i = 1:6 top press slope
. | i = 1:8 top press slope
. |
---|---|---|

Transition section length, L/D | 2.15 | 2.87 |

Contraction ratio, | 0.68 | |

Orifice height to width ratio, | 0.77 |

*i*for numerical simulation. The numerical simulation is used to make a comparison, as shown in Figure 10. The top press slope

*i*change is 1:4 to 1:12, and the water level difference is set to with a corresponding discharge is 2,603 m

^{3}/s

*.*In Figure 11, the top press and water cavitation number of the five slopes are shown, and the pressure and velocity data are time-averaged over 5 s and dimensionless processed as . For the transition section of the lock chamber, the concern is that the top press slope

*i*is too slow leading to the appearance of negative pressure. In Figure 11, no negative pressure appears at the top of the tunnel whether it is a steep slope or a gentle slope, so it is safe to change the slope within a reasonable range. Therefore, the top of the tunnels is safe.

Figure 12(c) shows the flow cavitation number of open-channel sections under different top press slopes. Through the experiment in the vacuum tank, we detected cavitation in the open-channel section on the NO.4 hydrophone (Figure 5), so we calculated the incipient cavitation number of the two types. For a top press slope of 1:8, . The incipient cavitation number of the body shape of the top press slope 1:6, . As shown in Figure 12(c), the cavitation number of flows for each case is basically greater than 0.05. The cavitation number in the downstream part of the top press slope of 1:6 is smaller than the incipient cavitation number = 0.122. Therefore, there is a risk of weak cavitation. In contrast, for the top press slope = 1:8 body type, cavitation number *K* are greater than the incipient cavitation number Ki = 0.024. There is no risk of cavitation for the top press slope .

We finally optimize the original design top press slope 1:6 to top press slope 1:8, the length of the transition section is changed from 2.15*D* to 2.87*D*, other parameters keep the original design orifice height to width ratio *h/B* = 0.77, contraction ratio *η* = 0.68.

### Vacuum tank experiment of practical engineering

Based on the analysis above, the two body types are now compared in a vacuum tank experiment. The vacuum experiment results are obtained for similar vacuum conditions. The specific design parameters are shown in Table 5.

Under the condition that the gate was fully open, according to the vacuum tank test results, it can be calculated that at the gate opening of 100%, the incipient cavitation number of the body shape of the top press slope 1:8 was = 0.024, the incipient cavitation number of body shape for top press slope 1:6 was = 0.122. In Figure 14, weak cavitation occurs downstream during high water level conditions when the top press slope was 1:6. Combined with the Figure 9 numerical simulation results, we can see that the cavitation number *K* of the open channel is smaller than the incipient cavitation number = 0.122. Therefore, the weak cavitation phenomenon occurs. Considering that the discharge tunnel has the need of gate partially opening in actual application because the top press slope 1:8 has better performance under different gate openings. Finally, top press slope 1:8 was selected as the optimized body type.

## CONCLUSION

For the bottom discharge tunnel, the top press slope *i*, transition section *L*, orifice shape *h/B*, and contraction ratio *η* are important design parameters. The design should ensure that the low pressure at the top of the transition section is within a reasonable range, but also consider reducing the height of the water wing and the risk of cavitation of the side wall. This paper analyzes the flow and cavitation characteristics of the pressurized to unpressurized lock chamber through data statistics, numerical simulations, and vacuum tank experiment results to obtain the following conclusions.

- (1)
It is recommended that the contraction ratio of the lock chamber section is between 0.7 and 0.9; the top press slope

*i*is 1:6–1:10, among which the top press slope of 1:8 is better for the downstream the water wing control. The orifice shape mostly uses the square orifice and the wide-flat orifice, and the orifice height to width ratio*h/B*is better around 0.8–1.0. The range of the transition section length*L*is between 1.5*D*–2.5*D*, and the body shape with a longer transition section*L*has better hydrodynamic characteristics. - (2)
For practical engineering with the high number and the small contraction ratio

*η*, we can use a longer transition section*L*/*D*, a slower top press slope*i*, and a wide-flat orifice shape. This design can effectively solve the water wing and cavitation problems for the bottom tunnel with a high Froude number. - (3)
The water flow in the pressurized to unpressurized lock chamber is unusually complex, especially when the lateral sudden expansion facility is used in the lock chamber of the high-velocity flow discharge tunnel. In order to effectively control the hydraulics and cavitation problems in the chamber section. we need to comprehensively consider the relationship between the contraction ratio , orifice shape , the top press slope, and the transition section length

*L*. - (4)
For cavitation characteristics, the four factors top press slope

*i*, transition section length*L*, contraction ratio , and orifice height to width ratio*h/B*are interdependent, and in specific engineering, selecting appropriate design parameters should comprehensively consider the mutual influence of these four factors.

## ACKNOWLEDGEMENTS

This research was funded by the National Natural Science Foundation of China (No. 52130904).

## 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

**2000**(

**2001**(

**1999**(

**2009**(

**2007**(

**1982**(

**1988**(

**2006**(

**1986**(