## Abstract

With regard to high water head and large flow velocity in the spillway tunnels of hydraulic projects in China, the aerator device has been introduced and is widely used to prevent cavitation damage. The bottom rollers in the nappe cavity below the aerator device are a serious concern in designing suitable cavity regimes; however, observation of roller size may be inaccurate due to high flow turbulence and de-aeration in the jet impact region. In this study, a novel approach is proposed to predict roller sizes using pressure distribution of the bottom rollers. Pressure distribution characteristics are experimentally investigated under different geometrical parameters of aerator device and hydraulic conditions. The results specify the influence of the relative step height and working gate opening on pressure distribution. The simplified estimating formula of pressure distribution is derived within relative errors of 15%. The evaluation of the applicability of the proposed equation shows test data are in good agreement with the calculated value. Research results provide a reference for estimating bottom rollers of similar engineering.

## NOTATION

*e*Arch-gate opening

*h*_{o}Pressure flow depth

*h*_{e}Flow depth at the outlet of arch-gate

*H*_{P}Normalized pressure

*v*_{o}Approach flow velocity

*p*Pressure of bottom rollers

*X*_{P}Normalized location

*t*_{s}Drop-step height of the aerator device

*T*_{s}Relative drop-step height

*φ*The bottom slope to the horizontal plane

*ρ*Fluid density

## INTRODUCTION

Aeration is widely applied in sewage sludge treatment in ships (Zhu *et al.* 2016) and deep-shaft systems (Xiong *et al.* 2016), contaminant removal for constructed wetlands (Auvinen *et al.* 2017; Masi *et al.* 2017; Sun *et al.* 2017), hydraulic efficiency improvement in hydraulic turbines (Bunea *et al.* 2017) and cavitation erosion prevention in spillway tunnels of large-scale hydropower projects with high heads (Liu *et al.* 2011; Li *et al.* 2016a, 2016b; Wang *et al.* 2018a, 2018b). It has been evident that cavitation erosion could be substantially reduced when an air concentration of 1.5–2.5% is provided in the flow on the surface of the materials (Russell & Sheehan 1974; Wu *et al.* 2017). In the applications of air-entrainment technology, many kinds of aerators have been developed (Ruan *et al.* 2007). For aeration cavities below aerator devices in a tunnel, previous studies (Chanson 1989a, 1989b; Rutschmann & Hager 1990; Pfister 2011) declared that air entrainment discharge is an important parameter influencing air concentration; therefore, favorable cavity regimes were the main considerations in aeration situations (Guenther *et al.* 2013; Felder & Chanson 2014). However, severe bottom rollers in the ventilated cavity may worsen cavity regimes, cease to protect the chute surface and even act potentially as a source of cavitation erosion (Chanson 1990; Qian *et al.* 2016). Accordingly, large numbers of relevant literatures have been conducted to improve cavity regimes, including restraining bottom rollers (Wang *et al.* 2005; Su *et al.* 2009; Wu *et al.* 2011), increasing cavity length (Qi *et al.* 2007; Wu & Ruan 2008; Ma *et al.* 2010) and optimizing the jet impact angle (Pfister 2012) on the chute downstream of the aerator device.

It is universally known that the appearance of the bottom rollers will greatly decrease the efficiency of air entrainment, and the net cavity length becomes the key factor dominating the air discharge in the flow. Accordingly, it is essential to predict and observe the bottom roller sizes. To date, Chanson (1995), Yang *et al.* (2000) and Wu *et al.* (2013) selected different control volumes to examine the calculation of bottom rollers based on the jet trajectory equation and the momentum equation. In highly turbulent water flows, air is entrained by the high-intensity turbulent eddies' proximity to the air-water interfaces (Wei *et al.* 2016; Valero & Bung 2018), and nappe entrainment develops on both the lower and upper jet free surfaces (Valero & Bung 2016). Moreover, in the vicinity of the cavity the rollers entrain additional quantity of air plunging jet entrainment, and the cavity boundaries are indeterminable visually as high flow turbulence and rapid redistributions of air concentration occur in the impact region. Hence, in prototype observation or physical model tests, it is anticipated potentially that the direct observation of the bottom rollers perhaps remains inaccurate due to visual error. Here, the studies on dynamic pressure head distribution of the bottom rollers may be a better approach to estimate roller characteristics and cavity regimes.

Typically, at the end of the nappe, flow is subject to a rapid change of pressure distribution from a minimum pressure gradient to a maximum pressure gradient at the impact point (Chanson 1989a, 1989b). Bottom roller sizes could be reflected and calculated by dynamic pressure head in the cavity submerged region upstream of the impact point. Nevertheless, for spillway tunnels, the existing literature has followed with interest in pressure distribution on aerator devices (Steiner *et al.* 2008; Yamini *et al.* 2015) and sidewalls (Zhang *et al.* 2011; Li *et al.* 2016a, 2016b). Little literature has addressed pressure head characteristics of the bottom rollers in a single investigation except for Shi *et al.* (1983), who issued the pressure distribution function in close proximity to the nappe impact point to analyze maximum pressure. In fact, the pressure distribution of the bottom rollers is affected by many factors, to which careful attention should be paid, according to the performance of the flow. For the aerator on the bottom of the different release structures, different hydraulic characteristics are shown. To the authors' knowledge, so far no study has ever systematically assessed the influence of the geometrical parameters of the aerator device and hydraulic conditions on pressure distribution.

The object of the present research is a flood-spillway tunnel in the Nuozhadu Hydropower Project (Yunnan Province, China). This paper is to experimentally observe and discuss the dynamic pressure distribution of the bottom rollers, as well as the effect of geometric and hydraulic parameters on pressure distribution. Meanwhile, the estimating formula of pressure distribution is presented, and its application is checked by the comparison of the test values in the previous literatures with the calculated values.

## THEORETICAL CONSIDERATION

According to the experimental observations, the definition sketch of the aerator device can be described by Figure 1, in which the geometric parameters contain *t*_{s} as the drop-step height of the aerator device and *φ* as the bottom slope to the horizontal plane. Flow upstream of the arch-gate is pressure flow. The hydraulic parameters are *h*_{o}, *h*_{e} and *v*_{o} as pressure flow depth, flow depth at the outlet of arch-gate, and approach flow velocity, respectively.

*φ*is a constant value. Select

*h*

_{o},

*ρ*and

*v*

_{o}representing

*L*,

*M,*and

*T*as the independent basic dimensionless quantities, and Equation (1) could be rewritten in a dimensionless variable as

Note the opening of arch-gate *e* = *h*_{e}/*h*_{o}; moreover, *T*_{s} = *t*_{s}/*h*_{o} can be defined as relative drop-step height. Furthermore, *ρ* is unchanged as the test fluid is water, and the energy equation of flow shows the degrees of the arch-gate opening can dominate approach flow velocity *v*_{o}. Consequently, dimensionless analysis addressing bottom pressure is mainly affected by the relative drop-step height and arch-gate opening.

## EXPERIMENTAL SETUP AND METHODOLOGY

Nuozhadu Dam's discharge tunnel model was designed at a scale of 1/40 based upon the Froude similitude in the High-speed Flow Laboratory of Hohai University (Nanjing, China). The experimental setup consists of a pump, an approach conduit, a large feeding basin with a maximum discharge *Q**=* 400 l/s, a model of the spillway tunnel with an aerator device and a flow return system with a discharge measured weir instrument. The physical model of the spillway tunnel, made of Perspex, is shown in Figure 2. The size of the chute with pressure flow before the gate chamber is 13.59 × 0.125 × 0.3 m (length × width × height). The length and slope of the pressure tunnel are 15 m and 2.68^{o} respectively. Spillway arch-gates are effectively used on the spillway tunnels of various projects due to favorable operating and flow characteristics (Kazemzadeh-Parsi 2014). Furthermore, the degrees of sluice gate opening could control hydraulic and air-entrainment characteristics of the approach flow (Abdolahpour & Roshan 2014; Wu *et al.* 2015). Therefore, the arch working gate is set at the outlet of the gate chamber and the following drop-step aerator is installed as an aeration facility. The tested spillway tunnel downstream of the drop-step aerator has a slope of 11.31^{o} and width of 0.205 m. Thirty-five time-average pressure gauging points are placed on the chute bottom with a 5 cm spacing along the flow direction for measurement of pressure. Water level, measured using pointer gauges of 0.1 mm minimum count, is 2.50 m, corresponding to the range of the experimental Froude numbers of 1.89–3.33.

Pressure distribution of the bottom rollers below the aerator device was investigated under seven cases (Table 1) consisting of four drop-step heights and four degrees of sluice gate opening. Four drop-step heights of aerator devices varied from dimensionless value *T*_{s} = 0.067, 0.083, 0.100 and 0.125, corresponding to *t*_{s} = 2.00 cm, 2.50 cm, 3.00 cm and 3.75 cm (*h*_{o} = 30.00 cm is the pressure flow depth), respectively. Second, the effect of sluice gate opening was studied, as the effect of hydraulic parameters upon pressure distribution of bottom rollers at *e* = 1.00, 0.75, 0.50 and 0.25 with the same *T*_{s} = 0.125.

Cases name . | T_{s}
. | e
. | Symbol . |
---|---|---|---|

M1 | 0.067 | 1.00 | ◆ |

M2 | 0.083 | 1.00 | ● |

M3 | 0.100 | 1.00 | □ |

M4 | 0.125 | 1.00 | △ |

M5 | 0.125 | 0.75 | ▪ |

M6 | 0.125 | 0.50 | × |

M7 | 0.125 | 0.25 | – |

Cases name . | T_{s}
. | e
. | Symbol . |
---|---|---|---|

M1 | 0.067 | 1.00 | ◆ |

M2 | 0.083 | 1.00 | ● |

M3 | 0.100 | 1.00 | □ |

M4 | 0.125 | 1.00 | △ |

M5 | 0.125 | 0.75 | ▪ |

M6 | 0.125 | 0.50 | × |

M7 | 0.125 | 0.25 | – |

## RESULTS AND DISCUSSION

### Normalization

Figure 3 depicts the sketch of a streamwise section along the chute with pressure distribution (rotating the model presented in Figure 2 by 11.31 degrees counterclockwise), and plots the pressure head line (*h*_{P=}*p*/*ρg*, where *g* is an acceleration of gravity). The origin of the coordinate system (*x*, *y*) is at the chute bottom below the drop-step. The nappe, flowing into the chute bottom downstream through the aeration step, results in the maximum pressure head value (*h*_{PM}) at the corresponding position (*x*_{PM}) in the vicinity of the nappe impact point and finally to the hydrostatic pressure gradually far downstream. In a cavity formed below the nappe, the bottom pressure rises from the inception point of the bottom rollers to the maximum value. In fact, a sub-pressure beneath the nappe is produced by which air is entrained into the flow; here, we only discuss pressure distribution in the submerged region rather than the air-exposed region in the cavity.

*x*

_{P}) of the bottom inside the cavity, the normalized pressure

*H*

_{P}can be defined as:

Consequently, the normalized pressure starts at *X*_{P} = *H*_{P} = 0; that is, at the origin of the coordinate system. The maximum value *x*_{PM} and *h*_{PM} are located at *X*_{P} = *H*_{P} = 1.

### Pressure distribution

Figure 4 displays the variation of normalized pressure *H*_{P} against normalized location *X*_{P}. It could be noticed that the data present an approximate inception point of *X*_{P} = 0.55 for all seven cases; that is to say, the bottom rollers start close to the center of the cavities. Subsequently, the pressure rises rapidly in the range of *X*_{P} = 0.55–1.0 due to the increasing depth of the bottom rollers and the superposition of nappe and rollers.

*H*

_{P}and

*X*

_{P}in Figure 4, let where

*n*is an index and depends on the geometric parameters of the aerator device and the hydraulic parameters of the flow. On the basis of Equation (3), we have

To separately maintain the analysis of *T*_{s} and *e* effect upon *n*, we present the variations of *H*_{P} against *X*_{P} for different *T*_{s} and *e* at *X*_{P} ≥ 0.55, as shown in Figure 5(a) and 5(b), respectively. Based on Figure 5, the values of *n* are 6.036, 5.267, 4.451, 3.250, 3.851, 4.415 and 5.052, corresponding to M1-M7, respectively.

Clearly, *n* increases with the decreasing *T*_{s} or *e* in the range of the present investigation. It is presumed that this phenomenon depends on bottom roller characteristics in response to jet trajectories described with based point-mass parabola versus the take-off conditions. It is well-known that bottom rollers are associated with the impact angle of the lower trajectory of the jet to the bottom, and the roller intensifies with the increase in impact angle (Pfister & Hager 2010a, 2010b). Theoretical analysis of free projectile indicates that the step height increment produces a larger impact angle, while small *e* implies a large Froude number, which subsequently reduces the impact angle (Qian *et al.* 2014). Hence from the bottom roller analysis standpoint, the decrease of *T*_{s} or *e* contributes to larger *n* by reducing the jet impact angle in the cavity, and results in relatively small values of the bottom pressure at identical *X*_{P}.

*R*

^{2}= 0.989):

The empirical relationship of Equation (10), valid for 0.067 ≤ *T*_{s} ≤ 0.125, 0.25 ≤ *e* ≤ 1.00, shows that normalized pressure head *H*_{P} negatively correlates with the relative step height and arch-gate opening for identical *X*_{P}. In fact, the higher pressure means larger submerged depth in the cavity; therefore, Equation (10) could predict the cavity length and inception point of the bottom rollers on the basis of pressure distribution regardless of visual error, and thus better estimate cavity regimes.

*H*

_{Pcal}is obtained based on Equation (10); both

*T*

_{s}and

*e*could be obtained by means of Table 1; the dotted lines indicate the range of the relative error (

*Err*) of 15% on the basis of

Note that no data scatter more than 15% on basis of Equation (10) and the maximum value of the relative error is 14.2%. The accuracy of Equation (10) considering geometrical and hydraulic characteristics of the pressure distribution of the bottom rollers is believed to be acceptable in conjunction with the current experimental uncertainty.

### Comparison with previous results

To evaluate the applicability of the proposed equation, three representative examples are chosen, and their experimental or numerical results available in the literature are compared with the calculated results using Equation (10). On the basis of the experimental time-average pressure value along the center of the chute bottom, Li *et al.* (2016a) discussed pressure distribution for three bottom aerators without and with the lateral aerator, according to a tunnel with free flow in a practical project with the high dam. However, we have only compared the 2# aerator depicted in Figure 8(a), as *T*_{s} and *e* of this aerator agree with the scope of Equation (7). Shilpakar *et al.* (2017) numerically simulated pressure distribution on the spillway tunnel of the Jinping-I Hydropower Project for four aerators with a fully arch-gate opening, and the comparison between the simulated value and the calculated value is shown in Figure 8(b) for 1#, 2# and 4# drop-step aerator which are in the scope of Equation (10). Ma *et al.* (2014) experimentally and numerically investigated pressure distribution of the chute bottom for a certain drop-step aerator in the arch-gate closing process; herein, we selected *e* = 3/4, 2/3 and 1/2 to calculate the normalized pressure head, as shown in Figure 8(c). From Figure 8(a)–8(c), it can be seen that the calculated results obtained on the basis of Equation (10) are in good accord with those obtained from the previous model tests and numerical simulations.

## CONCLUSIONS

With respect to the potential visual error of observing the bottom rollers below the aerator device, a new methodology of predicting bottom roller sizes using pressure distribution is introduced.

The performance of pressure distribution under different geometrical parameters of aerator device and hydraulic parameters shows that the bottom rollers start closely at the center of the cavity, according to the inception point of the dynamic pressure head at the relative position *X*_{P} = 0.55, and the decrease of relative step heights or arch-gate opening generates small values of normalized pressure at an identical normalized position in the scope of current issues.

The new prediction of pressure distribution on the basis of relative step heights and the arch-gate opening is provided. Comparison with our experimental data demonstrates that the maximum relative error of the present estimating equation is 14.2%. Moreover, comparisons of test results presented in the previous literatures with values calculated by Equation (10) demonstrate the good applicability of the new equation for estimating pressure distribution.

The limitation of this methodology mainly depends on the specific geometry of the physical model and the proposed prediction is valid for 0.067 ≤ *T*_{s} ≤ 0.125, 0.25 ≤ *e* ≤ 1.00.

## ACKNOWLEDGEMENT

The research presented herein is financially supported by the National Natural Science Foundation of China (51579076, 51779081), the Fundamental Research Funds for the Central Universities (2017B618X14), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0425).