## Abstract

Cavity length has been proven to have a significant effect on the air entrainment of an aerator in flood discharge engineering; however, the estimation of both the bottom and lateral cavity lengths downstream of a sudden fall-expansion aerator remain unclear. This research conducts a series of experiments involving various approach-flow conditions and geometric parameters of an aerator. An improved solution of the cavity length for the bottom and lateral cavities is established. The proposed equation was validated through the data of experiments and predecessor formulas, and exhibited a higher precision than other methods. Both the transverse turbulence and the axial dynamic pressure are found to be related to the formation of a lateral cavity. The present method involving the lateral cavity length was developed based on dimensional analysis and experimental test. The geometric morphology of a lateral cavity exhibits a parabolic shape, which is similar to that of the bottom cavity.

## SYMBOL LIST

- ,,
the acceleration along the

*x*-direction,*y*-direction and*z*-direction the width of the sudden fall-expansion aerator

the air concentration

air concentration of 2% and 90%, respectively

the coefficient of air resistance

- , , ,
the width, thickness, and length of control volume

the transverse force

*F*_{a}buoyancy of the air on a nappe

*F*_{r}Froude number

*F*_{τ}air resistance

*g*gravitational acceleration

*G*gravity of control volume

the depth of flow

the flow depth

the coefficient

the length of the lateral cavity length

the length of the bottom cavity length

*m*the weight of control volume

*N*_{1,2}surface tensions

*p*_{1,2}pressures in the

*x*-direction- ,
the pressures close to the free surface and internal cavity, respectively

sub-pressure beneath the cavity

the radius of curvature

the Reynolds number (−)

the drop height of the aerator

*T*time

the lateral diffusion velocity

the cross-sectional mean velocity

the fluctuating velocity

the cross-sectional mean velocity at the pressure outlet

- ,
*y*,the horizontal, transverse and vertical coordinates of the nappe profile

the slopes upstream of the aerator

the slope downstream of the aerator

the dynamic viscosity coefficient

the coefficient of surface tension

the emergence angle of the flow jet

- , ,
density of the air, water, and air–water mixtures

## INTRODUCTION

Flow aeration by aerators is an inexpensive and effective measure to prevent cavitation erosion in hydraulic engineering (Pfister *et al.* 2011; Hager & Boes 2014). Sudden fall-expansion (vertical drop and lateral enlargement) aerators are frequently adopted in flood discharge tunnels of high-head dams. The cavity length downstream of a sudden fall-expansion aerator includes the bottom cavity length and the lateral cavity length. Cavity length is one of the most important parameters that affect the air entrainment of an aerator (Wu & Ruan 2008; Wu *et al.* 2011; Wu & Ma 2013).

Dimensional concepts complemented with semi-empirical assumptions and simplified theoretical approximations have usually been adopted to estimate the bottom cavity length downstream of a bottom aerator. Those can be divided into four categories: (i) dimensional analysis (Shi *et al.* 1983; Pan & Shao 1986; Xia & Zhang 1999; Mohaghegh & Wu 2009; Pfister & Hager 2010); (ii) projectile theory (Rutschmann & Hager 1990; Ni 1993; Chanson 1995; Yang *et al.* 1996); (iii) the infinitesimal method (Wu & Ruan 2008); and (iv) numerical simulations (Zhang *et al.* 2011; Aghebatie & Hosseini 2016). Most of the methods do not or only partially consider the effect of the exit angle, flow depth, fluctuating velocity and aeration on the bottom cavity length. In terms of the scope of application, most formulas are poor in generality and low in accuracy.

Compared to the bottom cavity, there is little research on the lateral cavity length; this may be because of the unknown mechanism of the formation of the lateral cavity. A lateral deflector built on the sidewall can increase the cavity length in sudden fall-expansion aerators (Nie *et al.* 2003). These results indicate that the slope of the outlet and the shape of the lateral deflector should be considered in the estimation of the cavity length. Liu *et al.* (2008) proposed that the lateral cavity length was influenced by both the lateral aerator and the bottom aerator. The effects of aerator conformation, the flow velocity and the fluctuating velocity on the lateral cavity were analyzed by Yue *et al.* (2009), Wang *et al.* (2013) and Liu *et al.* (2015). Additionally, some empirical equations have also been established for calculating the lateral cavity length (Yue *et al.* 2009; Liu *et al.* 2015).

Although there are several methods to estimate the cavity length of a bottom aerator or a lateral aerator, few of these methods are suitable for calculating the cavity length downstream of a sudden fall-expansion aerator. A theoretical analysis based on the projectile law and micro-element method was adopted in this study to further the knowledge of cavity length and geometric morphology along the nappe downstream of a sudden fall-expansion aerator on a spillway. The improved expressions were verified through the extensive experimental data from this study and other investigators.

## THEORETICAL CONSIDERATIONS

### Bottom cavity length

where *v* is the cross-sectional mean velocity, approximately equal to the mean velocity at the pressure-outlet; is the coefficient of surface tension ; *R* is the radius of curvature; and is the coefficient of air resistance, (Yang *et al.* 1996).

*T*was defined as and can be calculated by modifying the mean velocity at the pressure outlet as , where is the mean velocity. Based on the results of Pope (2000), the fluctuating velocity can be calculated as , where is the kinematic viscosity coefficient,

*C*is the mean air concentration, and is the emergence angle of the jet. According to the present experiments and the results of Wu & Ruan (2008), an improved expression for the emergence angle is given as ; is the angle with respect to the horizontal;

*g*is the acceleration of gravity.

### Lateral cavity length

As is generally known, hydrodynamic pressure is always generated when water flows in a pressure-tunnel. The cross-sectional distribution of hydrodynamic pressure is that it is higher near the center and lower near the circumference (Ai & Shen 1995). The water cannot diffuse because of the restriction of the surrounding wall. When the water flow arrives at the sudden fall-expansion aerator, the pressure around the water surface immediately changes to atmospheric pressure whereas the hydrodynamic pressure inside the water remains high. The internal high pressure must inevitably spread to the surrounding water-free surfaces, resulting in the water spreading. Additionally, the high-speed water within the intense turbulence and the Reynolds stress can affect the transverse diffusion flow too (Ghasemi *et al.* 2015). The two factors mentioned can be integrated and expressed as a transverse force *F* that results in the formation of the side cavity.

The definitions of the symbols are the same as those noted above.

*x*–

*y*coordinates) can be written as where is the acceleration along the transverse direction, is the lateral diffusion velocity, and

*b*represents the width of the lateral expansion.

Equation (7) is derived from Equation (5), including two main parameters ( and ), which are crucial to determining the cavity length. A dimensional analysis was performed to determine those two parameters in this research.

*b*and the drop height ; (2) the flow velocity and the depth

*h*; (3) the pressure in the lateral cavity and the gravitational acceleration

*g*; and (4) the density of air–water , the dynamic viscosity coefficient , and the surface tension . In this study, the Reynolds number is approximately (0.89–1.24) × 10

^{6}, which is large enough to ignore the viscous force. There is no negative pressure in the lateral cavity because of good ventilation conditions. The surface tension has less effect on a high velocity flow. Then, the dimensional analysis of the cavity length yields The independent dimensions

*h*, and

*g*, which include , and , were employed in the present dimensional analysis, namely, , and . The remaining variables can be represented by the three basic dimensions as follows: ; ; ; ; ; ; and .

According to the principle of dimensional harmony, the exponents are determined as follows: , and for , namely, . Similarly, , and .

## VERIFICATION OF THE PRESENT METHOD

The present equations of cavity length were validated through comprehensive experimental tests. Figure 4 shows a schematic of the rectangular flume setup. The experimental parameters are summarized in Table 1. The chute size downstream of the aerator is 3.5 m × 0.3 m × 0.5 m (length × width × height).

Series . | (t_{s}, b) /(m,m)
. | . | . | (×10^{6})
. |
---|---|---|---|---|

1 | (0.025,0.000) | 10% | 4.95–7.42 | 0.89–1.34 |

2 | (0.045,0.000) | 10% | 4.95–7.42 | 0.89–1.34 |

3 | (0.065,0.000) | 10% | 4.95–7.42 | 0.89–1.34 |

4 | (0.000,0.025) | 10% | 4.95–7.42 | 0.89–1.34 |

5 | (0.000,0.045) | 10% | 4.95–7.42 | 0.89–1.34 |

6 | (0.000,0.065) | 10% | 4.95–7.42 | 0.89–1.34 |

7 | (0.025,0.025) | 10% | 4.95–7.42 | 0.89–1.34 |

8 | (0.045,0.045) | 10% | 4.95–7.42 | 0.89–1.34 |

9 | (0.065,0.065) | 10% | 4.95–7.42 | 0.89–1.34 |

10 | (0.045,0.045) | 0% | 4.95–7.42 | 0.89–1.34 |

11 | (0.000,0.045) | 0% | 4.95–7.42 | 0.89–1.34 |

12 | (0.045,0.000) | 0% | 4.95–7.42 | 0.89–1.34 |

13 | (0.045,0.045) | 25% | 4.95–7.42 | 0.89–1.34 |

14 | (0.000,0.045) | 25% | 4.95–7.42 | 0.89–1.34 |

15 | (0.045,0.000) | 25% | 4.95–7.42 | 0.89–1.34 |

Series . | (t_{s}, b) /(m,m)
. | . | . | (×10^{6})
. |
---|---|---|---|---|

1 | (0.025,0.000) | 10% | 4.95–7.42 | 0.89–1.34 |

2 | (0.045,0.000) | 10% | 4.95–7.42 | 0.89–1.34 |

3 | (0.065,0.000) | 10% | 4.95–7.42 | 0.89–1.34 |

4 | (0.000,0.025) | 10% | 4.95–7.42 | 0.89–1.34 |

5 | (0.000,0.045) | 10% | 4.95–7.42 | 0.89–1.34 |

6 | (0.000,0.065) | 10% | 4.95–7.42 | 0.89–1.34 |

7 | (0.025,0.025) | 10% | 4.95–7.42 | 0.89–1.34 |

8 | (0.045,0.045) | 10% | 4.95–7.42 | 0.89–1.34 |

9 | (0.065,0.065) | 10% | 4.95–7.42 | 0.89–1.34 |

10 | (0.045,0.045) | 0% | 4.95–7.42 | 0.89–1.34 |

11 | (0.000,0.045) | 0% | 4.95–7.42 | 0.89–1.34 |

12 | (0.045,0.000) | 0% | 4.95–7.42 | 0.89–1.34 |

13 | (0.045,0.045) | 25% | 4.95–7.42 | 0.89–1.34 |

14 | (0.000,0.045) | 25% | 4.95–7.42 | 0.89–1.34 |

15 | (0.045,0.000) | 25% | 4.95–7.42 | 0.89–1.34 |

The trajectories of the lower nappe (Equation (4)) are used to fit the experimental data. The result proves the goodness of fit illustrated in Figure 5. The geometric shape of the bottom cavity presents a parabolic distribution. The results also indicated that the present expression has an error of 12%, reflecting the fluctuation of bottom cavity length, caused by the air–water mixture flow.

*et al.*1982), Dongjiang, Lubuge, Fengjiashan, Nuozhadu, Shitouhe, Wujiangdu and Xizhijiang (Ni 1993). Here, the relative error between the calculated results and the measured values , and the root mean square error were adopted (see Equation (12)) to verify the precision of different formulas: The root mean square error, the relative error and cumulative error of the formulas are shown in Figures 6 and 7, respectively. Generally, the results demonstrate that the present method has a relative higher accuracy than other methods. The reason may be that this research accounts for more factors which affect the cavity length. It is noted that the accuracy of the method of Yang

*et al.*(1996) is slightly lower than the present method. The reason may be that it does not account for the influence of aeration on the cavity.

Equation (7) was also tested against the experimental tests (Figure 8), indicating a good agreement between the predictions and observations. The result shows that the morphology of the lateral cavity obeys a quadratic parabolic curve, caused by the gradually decay of both the internal pressure and the Reynolds stress, which is consistent with the theoretical analysis.

To validate the precision of the equation of lateral cavity length, the data of these experiments and Wang *et al.* (2013) and Liu *et al.* (2015) are plotted in Figure 9. It can be seen that the present calculation results are consistent with the test data. Again, it is proved that Equation (7) can be adopted for calculating the lateral cavity length. Additionally, some discrete data still exist, caused by errors of the measuring instruments or the different definitions of the cavity length.

## CONCLUSIONS

In this study, the factors affecting the cavity length downstream of a sudden fall-expansion aerator are investigated for various aerator sizes and approach-flow velocities. The following conclusions can be drawn:

- (1)
An improved expression for calculating the geometrical morphology and the length of bottom and lateral cavities is established. When compared with other empirical equations, the proposed equation has a higher precision.

- (2)
Preliminary analysis of the formation mechanism of the lateral cavity revealed that the lateral cavity is mainly caused by dynamic pressure and transverse turbulent stress. The morphology of the lateral cavity exhibited a quadratic parabolic distribution, which is similar to the curve of the bottom cavity.

## FUNDING

This work was partly supported by China Postdoctoral Science Foundation (grant 2016M602716), the CAS ‘Light of West China’ Program (grant Y6R2220220), the National Science Fund for Distinguished Young Scholars (grant 51625901), the Natural Science Foundation of China (grant 51579165) and the Young Scientists Research Fund of the Institute of Mountain Hazards and Environment CAS (grant Y6K2080080).

## REFERENCES

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