## Abstract

In view of high water head and large discharge in the release structures of hydraulic projects, the inverted arch plunge pool has been put forward due to higher overload capability and stability. Impact pressure on the bottom is a serious concern in design safety precautions, however, the quantitative impact pressure distribution in the inverted arch plunge pool is not yet elucidated. In this study, a novel approach is presented to estimate the impact pressure of an inverted arch plunge pool. Impact pressure characteristics are experimentally investigated under different hydraulic conditions. The results detailed the effect of relative discharge coefficient and the deflection angle relative to the vertical central axis of the plunge pool bottom. The predicting formulas of impact pressure distribution are derived within small relative errors, and the proposed approaches have good applicability in three case studies. The achievements of this investigation are used to define issuance parameters relevant for engineering practice.

## NOTATION

*B*width of inverted arch plunge pool

*h*_{P}impact pressure value

*h*_{Pa}time-average pressure value

*h*_{PM}maximum impact pressure value

*h*_{b}height of inverted arch plunge pool

*H*_{d}water depth of plunge pool

*H*_{P}normalized impact pressure value

*H*_{PM}normalized maximum impact pressure value

*q*unit flow discharge

*q*/(*H*_{d}(*g*· Δ*H*)^{0.5})relative discharge coefficient

*Q*approach flow discharge

*R*radius of inverted arch plunge pool

*s*height of the sill

*v*inflow velocity of jet flow

*x*_{P}distance from the measuring point to the dam body

*x*_{PM}position of maximum impact pressure value

*X*_{P}normalized position of impact pressure value

*X*_{PM}normalized position of maximum impact pressure value

*α*central angle of inverted arch plunge pool

*ρ*flow density

*θ*deflection angle relative to the vertical central axis of plunge pool bottom

- Δ
*H*distance between the water level of dam and the flow surface

## INTRODUCTION

Many hydraulic projects with large discharge and high head are under construction worldwide, and energy dissipation remains a major item in hydraulic engineering (Wu *et al.* 2018; Wang *et al.* 2019). High-velocity jets issued from high-head dams have the potential to generate scour and erosion when impacting rocky riverbeds, which may run the risk of influencing the stability of the dam itself (Castillo & Carrillo 2017; Movahedi *et al.* 2018). The plunge pool is installed downstream of the dam to cushion the jet impingement through the atmosphere and dissipate the energy of the falling water, especially for large flow discharge and power (Ervine *et al.* 1997). The impact pressure of jet impingement acting on the bottom of the plunge pool and inside rock fissures transmits to the rock joints, which contributes to block uplift and instability of the plunge pool (Sun & Chen 2003; Melo *et al.* 2006). Therefore, the stability of a protective structure for the plunge pool is not only a hot topic in the field of water conservancy engineering safety, but also a key technical problem to be solved in hydropower engineering construction (Xu *et al.* 2002a, 2002b, 2002c; Li *et al.* 2016).

Plunge pool geometry should be optimized considering the characteristics of bedrock and local geology to increase anti-instability capability (Yue *et al.* 2015). Investigations relating to the geometry of the plunge pool bottom have been conducted including flat bottom, inverted arch bottom and lining slope without bottom protection (Manso *et al.* 2007; Deng *et al.* 2008), among which the inverted arch plunge pool is of interest due to its larger overload capability, higher stability and lower excavation quantity (Lian *et al.* 2009). So far, the inverted arch plunge pool has been operated in Inguri Dam in Georgia, Susqueda Dam in Spain, Laixiwa Dam in China, and others, and hydraulic engineers and researchers have paid attention to the characteristics of the inverted arch plunge pool.

Zhang *et al.* (2015) qualitatively performed velocity distribution and characteristics of maximum impact pressure for a submerged impact jet in an inverted arch plunge pool. The rather similar case of velocity and pressure distribution caused by jets falling on a plunge pool was studied by Sun *et al.* (2009), while Xu *et al.* (2002a, 2002b, 2002c) implied uplift force characteristics on the bottom of an inverted arch plunge pool subjected to jet impingement and Yang *et al.* (2013) quantitatively proposed the maximum uplift forces accounting for flow discharge, pool depth and block thickness, etc. Liu *et al.* (2002) suggested the concentration of air entrainment distribution by means of numerical simulation.

With regard to energy dissipation for large discharge, multi-layer jets and jet collision in air have been widely applied in release structures thanks to higher air entrainment and jet energy dissipation (Jiao *et al.* 2012), which results in jet break-up (Diao & Yang 2002). It has been evident that the impact pressure is significantly correlated to the aeration effect of jets and the plunge pool (Pinheiro & Melo 2008; Deng *et al.* 2015), as well as the degree of the jet break-up (Castillo *et al.* 2015). Generally, high-velocity multi-layer jets issued from the dam body will progressively incorporate air by the high-intensity turbulent eddies' proximity to the air–water interfaces (Valero & Bung 2018), and a large additional quantity of air is entrained between jet boundaries and the plunge pool surface. Pool depth could dissipate a part of a jet's energy and the remaining energy will be converted to impact pressure impinging on the bottom of the plunge pool (Duarte *et al.* 2016).

The existing literature has followed with interest the impact pressure distribution in a plunge pool with a flat bottom, and the pressure head is self-similar and follows the Gaussian distribution (Tian *et al.* 2005; Manso *et al.* 2008; Duarte *et al.* 2015). Borghei & Zarnania (2008) focused on the distribution of mean and extreme pressure fluctuations on the sidewalls for different regimes of plunging jets, and the effects of pool dimension on pressure distribution. Jiang *et al.* (2007) stated that a dentated sill could weaken the impact pressure in the pool. Nevertheless, the geometry of a plunge pool is a key element in the definition of the impact pressures on the bottom (Manso *et al.* 2009), and little literature has addressed the impact pressure distribution in the inverted arch plunge pool except for Xu *et al.* (2002a, 2002b, 2002c), who numerically and qualitatively simulated the pressure distribution of the inverted arch plunge pool. Additionally, Sun *et al.* (2018) quantitatively proposed an approach to estimate maximum impact pressure of the inverted arch plunge pool as a function of the flow discharge ratio of crest spillways and seven outlets for jet collision in air. In fact, impact pressure distribution in an inverted arch plunge pool is affected by many parameters, particularly the remarkable difference of flow depth along the perpendicular of the flow direction due to the special structure of the inverted arch plunge pool. To the authors' knowledge, no detailed investigation has ever systematically and quantitatively assessed the influence of the hydraulic conditions and geometrical parameters of the inverted arch plunge pool on pressure distribution.

The object of this study is the inverted arch plunge pool of Baihetan Hydropower Project (289-m-high double-curvature concrete arch dam), which is located downstream of the Jinsha River, between the regions of Ningnan County in Sichuan Province and Qiaojia County in Yunnan Province, China (Espada *et al.* 2018). This paper aims at discussing the impact pressure distribution of an inverted arch plunge pool quantitatively for the jet collision in air, as well as the effect of hydraulic parameters on impact pressure distribution. Meanwhile, an estimating approach is proposed and three case studies adopting the proposed method are presented to demonstrate and to verify the applicability of the comprehensive approach.

## DIMENSIONAL ANALYSIS

*H*as the distance between the water level of the dam and the flow surface of the plunge pool and

*H*

_{d}as the water depth of the plunge pool. Additionally, the sill (height:

*s*) is employed downstream of the inverted arch plunge pool to generate enough cushion. Figure 1(b) depicts the front view of the inverted arch plunge pool, in which the geometric parameters are

*B*,

*α*,

*R*and

*θ*as the width, central angle and radius of the inverted arch plunge pool and the deflection angle relative to the vertical central axis of the plunge pool bottom, respectively. Typically, the main parameters influencing the impact pressure are as follows: (1) hydraulic parameters: Δ

*H*,

*H*

_{d}, the discharge of approach flow

*Q*and the inflow velocity of jet flow

*v*; (2) geometric parameters:

*s*,

*B*,

*R*,

*α*and

*θ*; (3) flow physical properties: flow density

*ρ*etc. Therefore, the impact pressure

*h*

_{P}can be expressed as:

*h*

_{p}] = [

*ρ*][

*v*]

^{2}and [

*Q*/

*B*] = [

*H*

_{d}][

*g*]

^{−1/2}[Δ

*H*]

^{−1/2}, according to dimensional theory, and Equation (1) can be expressed in dimensionless quantities as: with

*q*as unit discharge (

*q*=

*Q*/

*B*), and

*q*/(

*H*

_{d}(

*g*· Δ

*H*)

^{0.5}) can be defined as relative discharge coefficient. Furthermore,

*ρ*is constant as the experimental fluid is tap water, and the jet trajectory equation addresses that Δ

*H*could dominate the inflow velocity of jet flow

*v*. Hence, dimensional analysis states that the pressure distribution of the inverted arch plunge pool mainly depends on the relative discharge coefficient and the deflection angle relative to the vertical central axis of the plunge pool bottom.

## EXPERIMENTAL SETUP AND METHODOLOGY

The integrated hydraulic model of Baihetan Hydroelectric Project was designed at a scale of 1:50 at the State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Nanjing Hydraulic Research Institute (Nanjing, China). The experimental setup consists of a large-scale underground reservoir, an approach conduit, a pump with a maximum discharge of 2,400 l/s, an integrated model including the dam body, inverted arch plunge pool, three spillway tunnels, power generation system and a flow return system with a rectangular discharge measured weir instrument. The models of the dam body and plunge pool are made of concrete and Perspex, respectively. The dam body, with a height of 5.48 m, is installed with six crest spillways and seven outlets. With a view to the actual situation of the terrain, the side wall links to the plunge pool with a slope of 1:1 (left bank) and 1:0.4 (right bank), moreover, the central angle (*α*), radius (*R*) and height (*h*_{b}, see Figure 1(b)) of this inverted arch plunge pool are 1.305 rad, 2.14 m and 4.40 m, respectively. The height of the sill is 0.84 m and the length of this inverted arch plunge pool is 7.60 m.

The model in operation is photographed in Figure 2 (flow discharge = 1.379 m^{3}/s). Pressure readings start at the position of 1.80 m downstream of the dam body using conventional pressure taps, and 270 impact pressure gauging points are placed on the bottom of the inverted arch plunge pool with a 20 cm spacing along the flow and deflection angle of 0.122 rad spacing perpendicular to the flow direction. Therefore, the positions of pressure measurement points are *θ**=* 0, −0.070 rad, −0.192 rad, −0.314 rad and −0.436 rad on the left of the vertical central axis of the plunge pool bottom (set as negative values in order to distinguish them from the right of the vertical central axis of the plunge pool bottom), and *θ**=* 0.122 rad, 0.244 rad and 0.366 rad on the right of the vertical central axis of the plunge pool bottom. The results reported are the average of results obtained from three repeated tests, and the measuring error of impact pressure is ±5%.

The impact pressure distribution of the inverted arch plunge pool was investigated under three large discharges *Q* = 1.703 m^{3}/s, 1.379 m^{3}/s and 1.194 m^{3}/s, corresponding to relative discharge coefficient *q*/(*H*_{d}(*g* · Δ*H*)^{0.5}) = 0.0533, 0.0462 and 0.0422, respectively.

## EXPERIMENTAL RESULTS AND DISCUSSION

### Time-average pressure distribution

Impact pressure is calculated by time-average pressure minus the hydrostatic pressure of the inverted arch plunge pool, and hydrostatic pressure is invariable for a certain *q*/(*H*_{d}(*g* · Δ*H*)^{0.5}). Figure 3 presents the time-average pressure (*h*_{Pa}) along the streamwise section of flow direction for different *θ* where *x*_{p} means the distance from the measuring point to the dam body, and plots the pressure head line. The origin of the coordinate system (*x*, *y*) is at the position of 1.80 m downstream from the dam body, that is to say, the initiation point of impact pressure *x*_{P0} = 1.80 m. To get uniform pressure in the plunge pool, the jet trajectory is adjusted to approach the horizontal central line of the plunge pool bottom.

It can be noticed that the trend lines of time-average pressure are similar for different *θ*, and larger *θ* results in large pressure head value for identical *x*_{P} in the scope of 1.80 ≤ *x*_{P} (m) ≤ 4.80. The multiple jets, impinging on the plunge pool, produce the maximum pressure value (*h*_{PM}) at the corresponding position (*x*_{PM}) in the vicinity of the impact point (*x*_{P} ≈ 3.60 m) and then the time-average pressure decreases rapidly to nearly zero at *x*_{P} ≈ 4.80 m. Theoretically, the maximum impact pressure value exists at the intersection of the jet impingement point with the pool bottom (stagnation point) when the velocity is zero, and the slowed velocity contributes to the pressure increment in the vicinity of the impingement region (Beltaos & Rajaratnam 1977; Ervine *et al.* 1997). The jet is deflected outwards from the impacting region and the velocity parallel to the bottom increases with radial distance from the stagnation point, which results in the pressure decreasing (Duarte *et al.* 2015).

Subsequently, the pressure line increases to a constant value and *θ* has a minor effect on *h*_{Pa} due to the rise of water level in the vicinity of the sill. Here, this paper only discusses impact pressure distribution rather than pressure changes ascribed to other reasons, consequently, the scope of the impact pressure is valid for 1.80 ≤ *x*_{P} (m) ≤ 4.80.

Generally, impact pressure distribution characteristics are expressed as the impact pressure value versus corresponding positions. However, the comparisons of impact pressure distribution under several hydraulic conditions are more complex as variations of the position of the jet impact point occur. Here, a reasonable approach is employed using the maximum impact pressure head characteristics (*x*_{PM}; *h*_{PM}). For impact pressure value *h*_{P} at the position *x*_{P}, the normalized impact pressure *H*_{P} can be represented by *H*_{P} = *h*_{P}/*h*_{PM} at *X*_{P} = (*x*_{P} − *x*_{PM})/(*x*_{PM} − *x*_{P0}). Thus, the scope of normalized impact pressure upstream and downstream of the impact point varies from −1 ≤ *X*_{P} < 0 and *X*_{P} > 0, respectively. The maximum elevations *H*_{PM} = 1 are located at *X*_{PM} = 0.

### Normalized impact pressure distribution

Figure 4 shows the variation of *H*_{P} against *X*_{p} for different relative discharge coefficients. The analysis of the data suggests two types of distributions, depending on the position of maximum impact point. For −1 ≤ *X*_{P} ≤ 0, i.e., for the position upstream of the maximum impact point, the pressure line is slowly peaked, whereas it is sharp if *X*_{P} > 0 since impact pressure dissipates fast downstream of the impact point. Secondly, the rising slopes of lines for different relative discharge coefficients demonstrate a remarkable difference, and a larger relative discharge coefficient produces a larger normalized impact pressure value.

Figure 5(a)–5(c) display the effect of *θ* on the normalized impact pressure distribution for three relative discharge coefficients. Note that the trend lines in Figure 5 resemble those in Figure 4, and *H*_{P} presents an approximately symmetric distribution on two flanks of the vertical central axis of the plunge pool bottom. Moreover, the increase of absolute value of *θ* contributes to a larger *H*_{P} at identical *X*_{P}, and it is anticipated potentially that this phenomenon is associated with the decrease of flow depth with the absolute value of *θ* increasing on account of the special construction of the inverted arch plunge pool.

*X*

_{P}= 0, and the data of Figures 4 and 5 scatter about the trend lines: with where

*m*is a composite function considering

*q*/(

*H*

_{d}(

*g*· Δ

*H*)

^{0.5}) and

*θ*on the basis of Equation (3), and

*α*

_{1},

*β*

_{1},

*γ*

_{1},

*α*

_{2},

*β*

_{2},

*γ*

_{2}are undetermined coefficients. According to Figures 4 and 5 and the characteristics of the exponential function,

*m*increases with the decrease of

*q*/(

*H*

_{d}(

*g*· Δ

*H*)

^{0.5}) or

*θ*in the range of this investigation.

*q*/(

*H*

_{d}(

*g*· Δ

*H*)

^{0.5}) ≤ 0.0533 and 0 ≤ |

*θ*| ≤ 0.436 rad, indicate the normalized impact pressure

*H*

_{P}is inversely proportional to

*q*/(

*H*

_{d}(

*g*· Δ

*H*)

^{0.5}) and

*θ*. The statistical information of an empirical relation is evaluated by determination of the correlation coefficient (

*R*

^{2}), as an equation with higher

*R*

^{2}value may exhibit high accuracy. Here, the accuracy is acceptable since

*R*

^{2}of Equations (7) and (8) is 0.983 and 0.976, respectively.

*H*

_{Pcal}can be calculated by Equations (7) and (8), and the two dotted lines demonstrate the range of the relative error (

*Err*) of ±25% as: It is noticed that only few data scatter more than ±25% from Equations (7) and (8), and the average relative error value is 11.45%. Accordingly, the fitting and prediction precision can be guaranteed with respect to a high degree of flow turbulence and uncertainty.

### Applicability in other hydraulic projects

To evaluate the usability of the proposed equations, we compare the available numerical or experimental data of three representative hydraulic projects with the results calculated by Equations (7) and (8). Wang *et al.* (2009) numerically and experimentally studied the impact pressure distribution along the horizontal central axis of the inverted arch plunge pool bottom of Changtangang Hydroelectric Project in Hunan Province, China. In prototype, *α* = 1.288 rad, *R* = 30 m and *B* = 50 m, and the calculated data are plotted in Figure 8(a) for *q*/(*H*_{d}(*g* · Δ*H*)^{0.5}) = 0.0517 with *Q* = 913 m^{3}/s. Sun (2008) investigated the impact pressure distribution on the inverted arch plunge pool of Laxiwa Hydroelectric Project located on the Yellow River by means of a comparison between experimental and numerical results, and the prototype parameters of this inverted arch plunge pool are *α* = 1.287 rad, *h*_{b} = 12 m, *R* = 60 m and *B* = 72 m. At the maximum flow discharge *Q* = 6,310 m^{3}/s, the experimental data and the calculated results using Equations (7) and (8) are shown in Figure 8(b). Sun *et al.* (2009) experimentally analyzed impact pressure distribution along the flow direction on the basis of the inverted arch plunge pool of Xiluodu Hydroelectric Project located in the upper main reach of the Yangtze River with *α* = 1.357 rad, *R* = 81.25 m and *B* = 108 m; here, we select *θ* = 0, 0.276 rad, 0.354 rad and 0.506 rad to calculate the normalized impact pressure value due to the restriction of the present data. Figure 8(c) shows the comparison results for *Q* = 31,496 m^{3}/s. In Figure 8(a)–8(c), note that the calculated lines using Equations (7) and (8) are a good tally with the experimental and numerical results in other hydraulic projects, therefore, the proposed predictive models in this paper are fairly reliable to forecast the impact pressure distribution of the inverted arch plunge pool.

## CONCLUSIONS

In view of the bottom stability in the inverted arch plunge pool for large discharges, a new methodology of predicting the impact pressure distribution under various parameters is proposed. The novel predictive equations of the impact pressure distribution upstream and downstream of the jet impact point are proposed as Equations (7) and (8).

The investigation of the impact pressure distribution shows that the increase of relative discharge coefficient and deflection angle relative to the vertical central axis of the plunge pool bottom generates a larger normalized impact pressure at the same normalized position in the scope of the current research.

The analysis of relative error demonstrates the majority of calculated data scatters less than 25% compared with experimental results. Furthermore, the new proposed predictive models are of great reliability since the calculated data by means of those equations are in good accord with the experimental and numerical data of another three hydraulic projects. The proposed approaches thus offer a comprehensive method of impact pressure distribution, which further provides an important theory for engineering design of the inverted arch plunge pool.

## ACKNOWLEDGEMENT

This research has been financially supported by the National Key Research and Development Program of China (2016YFC0401705).