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
- Fa
buoyancy of the air on a nappe
- Fr
Froude number
- Fτ
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
- N1,2
surface tensions
- p1,2
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
, thickness: 
): the density of the air–water flow is 
, and the weight is 
. The forces acting on the control volume include the following: 
Sketch of the flow over an aerator.
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).
is the sub-pressure beneath the cavity; 
and 
are the pressures close to the free surface and internal cavity, respectively; and 
, 
and 
are the density of the air, water and air–water mixture flow, respectively.
and 
. The flow jet was assumed to be formed by a solid particle, and its flight time 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.
for a control volume of unit height (width 
and length 
). The forces acting on the control volume include the following: 
The definitions of the symbols are the same as those noted above.
Flow pattern of the aerator with lateral enlargement.
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.
and downstream 
of the aerator, the lateral width 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) × 106, 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 
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 
.
, the Froude number 
, the bottom slopes 
and the height of the aerator 
. Therefore, those parameters are summarized in the dimensionless parameter 
, which can be written as follows: 
Comparison of the experimental data and calculation results: (a) 
; (b) 
.
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).
Experimental flow conditions
| Series | (ts, b) /(m,m) |  |  |  (×106) |
|---|---|---|---|---|
| 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 | (ts, b) /(m,m) | | | (×106) |
|---|---|---|---|---|
| 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 |
Schematic of the experimental test equipment.
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.
Comparison of the experimental data with the calculated values of the trajectory of the bottom cavity: (a) (ts, b) = (4.5 cm, 4.5 cm), 
= 0%; (b) (ts, b) = (2.5 cm, 2.5 cm), 
= 25%; (c) (ts, b) = (2.5 cm, 2.5 cm), 
= 10%; (d) (ts, b) = (4.5 cm, 4.5 cm), 
= 10%.
Comparison of the experimental data with the calculated values of the trajectory of the bottom cavity: (a) (ts, b) = (4.5 cm, 4.5 cm), = 0%; (b) (ts, b) = (2.5 cm, 2.5 cm), = 25%; (c) (ts, b) = (2.5 cm, 2.5 cm), = 10%; (d) (ts, b) = (4.5 cm, 4.5 cm), = 10%.
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 relative error of various methods.
The cumulative error of various methods.
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.
Comparison of the experimental data with the calculated values of trajectory of the lateral cavity: (a) (ts, b) = (4.5 cm, 4.5 cm), 
= 0%; (b) (ts, b) = (4.5 cm, 4.5 cm), 
= 25%; (c) (ts, b) = (2.5 cm, 2.5 cm), 
= 10%; (d) (ts, b) = (6.5 cm, 6.5 cm), 
= 10%.
Comparison of the experimental data with the calculated values of trajectory of the lateral cavity: (a) (ts, b) = (4.5 cm, 4.5 cm), = 0%; (b) (ts, b) = (4.5 cm, 4.5 cm), = 25%; (c) (ts, b) = (2.5 cm, 2.5 cm), = 10%; (d) (ts, b) = (6.5 cm, 6.5 cm), = 10%.
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.
Comparison of the calculated and experimental data.
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).
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