# Numerical Simulation of Air–Water Two-Phase Flow on Stepped Spillways behind X-Shaped Flaring Gate Piers under Very High Unit Discharge

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}·s

^{−1}[2]. Their hydraulic characteristics have been widely and deeply investigated in the past several decades [3,4]. Sánchez-Juny et al. [5] measured the pressures on the step surfaces for skimming flow and found that pressures on the horizontal faces of the steps are always positive, while on the vertical faces, maximum pressure occurs in the vicinity of the internal edges, and the minimum values are often negative, appearing slightly below the step tips. Ohtsu and Yasuda [6], Li [7], and Zhang et al. [8] also determined a similar pressure distribution in their experiments. Frizell and Renna [9] used a specialized low-ambient pressure chamber (LAPC) to produce cavitation, and conducted a detailed investigation on the critical cavitation index and the shear strain rate close to the steps. They discovered that the cavitation inception was due to negative pressures, and was located at the highly intense shear layer slightly above the step tips. In terms of the absolute threshold value for the cavitation index, [9] proposed a value about four times the friction factor, i.e., about 0.5 for typical stepped spillways [10], while the critical inception index was acknowledged to be 0.20 to 0.25 for smooth spillways [11].

_{max}, to be approximately 25 m

^{2}·s

^{−1}, whereas Pfister et al. [13] reported q

_{max}= 30 m

^{2}·s

^{−1}, and Liang et al. [14] reported q

_{max}= 40 m

^{2}·s

^{−1}. Boes [10] provided a graph and a table to determine the threshold unit discharge for several typical chute slopes and step heights based on a critical cavitation index of 0.5. For higher unit discharge, bottom aerators must be arranged upstream to supply additional air [15]. As an alternative to unit discharge, critical velocities are provided in [2,16,17], which are around 15 to 20 m/s.

^{2}·s

^{−1}, i.e., 5 to 8 times larger than the critical unit discharge for common stepped spillways. The FGP concept was put forward in China in 1978 to allow for an energy dissipator at Ankang dam, where bedrock and river dynamics conditions would have required the construction of a large-sized stilling basin. Fortunately, the laboratory model test showed that the FGP concept allowed for a reduction of the stilling basin length at Ankang dam by 1/3 to 1/2 [18]. Subsequently, the FGP concept became popular at many Chinese dams. For example, FGP was combined with a flip bucket in 1980 at Panjiakou dam, enabling a reduction of the depth of the downstream plunge pool by roughly 14% and relocation of the scouring region approximately 20 to 30 m downstream. Then, in 1993, the FGP concept was used combined with a bucket type stilling basin at Yantan dam, where the bucket region velocity and downstream scouring could be significantly reduced [18]. Simultaneously, the FGP concept was again used with stepped spillways (hereafter termed FGP-SS) and a bucket type stilling basin to save construction time and project investment at Shuidong dam, where slight damages were observed along the first steps behind the FGP after a flood in May 1994, with an unit discharge of around 90 m

^{2}·s

^{−1}[19,20]. Several model tests in Chinese laboratories showed that FGP can horizontally squeeze and vertically stretch the nappe, inducing larger air contact areas, intensive impact between the jet and downstream water, and significant three-dimensional momentum exchange in the stilling basin, thereby dissipating large amounts of kinetic energy [18,21,22].

^{2}·s

^{−1}unit design discharge. They revealed that the air concentration on the stepped surface for Y-shaped FGP was 10% to 15%, and 3% to 5% for X-shaped FGP. The reference case without FGP featured an air concentration of 1%, and prototype air concentrations at Dachaoshan dam under identical hydraulic conditions were approximately two times the scale model values. Wang et al. [24] provided a relationship considering the first step height and the stepped spillway slope to estimate the cavity size at the steps below X-shaped flaring gate piers. Zhang et al. [25] simulated the two-phase flows along the stepped spillway below X-shaped FGP under a unit discharge of 195 m

^{2}·s

^{−1}, without any particular air entrainment model and bubble transport model, and presented general distributions of the velocity and pressure on the steps below FGP. Li et al. [26] simulated skimming flow over a pooled stepped spillway with four types of pool weirs, but no air entrainment was considered. Koh et al. [27] adopted the two-phase consistent particle method (CPM) for the simulation of a large dam break, where a thermodynamically consistent compressible solver was used by employing the ideal gas law for the compressible air phase. They successfully reproduced the air pocket trapped by overturning water and its cushion effect, obtaining better agreement with the experimental data than the single-phase simulations (both CPM and smoothed particle hydrodynamics-SPH). Wan et al. [28] numerically studied the hydrodynamics and aeration on a stepped spillway using the SPH method and obtained consistent velocity and dissolved oxygen distributions, in agreement with experimental data. Zhang et al. [29] simulated high turbulent air–water two-phase flows in a drop shaft using an air entrainment model, resulting in wall air concentrations that slightly deviated from the experimental data. Chen [30] successfully simulated the flow over typical stepped spillways using a k-ε turbulence model and the volume of fluid (VOF) method. However, investigations on the two-phase flows for FGP-SS remain insufficient for practical design and operation, especially in view of large-scale effects between the laboratory model and prototype, particularly for air concentration [31,32,33]. Therefore, numerical simulation could be an alternative to tackle this problem.

^{®}, which incorporates the 3-D Reynolds-averaged Navier–Stokes (RANS) equations, including the RNG k-ε turbulence model, VOF method, and sub-grid models for air entrainment, density evaluation, and drift-flux. The results were compared with the experimental data from scale model tests obtained in the laboratory. Furthermore, the step failure at Ludila dam was analyzed in prototype scale based on the numerical results and an optimization of the corresponding bottom aeration was further studied.

## 2. Prototype Site and Investigated Sub-Model

_{d}= 18.2 m) according to the US Army Corps of Engineers’ Hydraulic Design Criteria, followed by a power curve (y = 0.0425x

^{1.85}) and a linear slope of 1:0.854, ending at 1179.00 m a.s.l. The stepped spillway consists of 50 steps that are 0.9 m wide and 1.2 m high, except for the first step, which is 2.22 m high. The end of the spillway is connected to a bucket (1114 m a.s.l.) followed by the stilling basin (1115 m a.s.l.). A continuous tail sill is located 145.24 m downstream of the dam axis. More details can be found in the longitudinal profile as shown in Figure 2.

^{3}/s).

## 3. Mathematical Models and Simulation Setup

^{®}[34] was used to numerically solve the 3D Reynolds-averaged Navier–Stokes (RANS) equations for one fluid, including the RNG k-ε turbulence model [35] and the TurVOF method [36] for interface tracking. Sediment transport and cavitation processes were not considered in the current study.

#### 3.1. Mass Continuity Equation

#### 3.2. Momentum Equation

**g**is the gravitational acceleration. $\mathit{\tau}$ can be calculated by the effective kinematic viscosity, ${\vartheta}_{eff}$, the turbulent kinetic energy, k, and the identity matrix,

**I**, using the Boussinesq hypothesis:

#### 3.3. RNG k-ε Turbulence Model

#### 3.4. VOF Model

#### 3.5. Air Entrainment Model

^{®}is based on the assumption that air entrainment at the free surface will occur when instabilities due to turbulence (expressed by force P

_{t}in Equation (8)) overcome the stabilizing forces, P

_{d}, originating from gravity and surface tension. Consequently, air with volume, δV, may be entrained into the fluid, which can be described by the governing equations as follows:

^{®}, the air entrainment is combined with a mixture model for the single-phase fluid, where air is added to the fluid as passive tracer, i.e., without directly affecting the fluid flow (e.g., no voids due to bubbles, no momentum transfer) but changes the density of the fluid depending on the air concentration. This approach is reliable only if the entrained air concentration in the computational cells is less than 10% [37]. To consider additional physical processes of air transport in the water, bulking and buoyancy effects were taken into account. These can be implemented by using the density evaluation and drift-flux models introduced hereafter.

#### 3.6. Density Evaluation Model

#### 3.7. Drift-Flux Model

^{−5}kg m

^{−1}s

^{−1}since the operating temperature was considered to be 15 °C. Finally, the critical air volume fraction that controls the air turning from dispersed to continuous was defined as 1, which means water will always be in the continuous phase as suggested by FLOW-3D. More details about the drift-flux and dynamic bubble size sub-model can be found in [39,40].

#### 3.8. Simulation Setup

^{6}to 3 × 10

^{6}(calculated by the velocity varying between 10 and 15m/s and mesh scale at 0.17 to 0.2 m) and therefore the air phase effect only occurs within a very small region near the wall and has an insignificant effect on the main flow. Three mesh blocks were used, namely a containing mesh block and two nested blocks, as shown in Figure 5. Two nested mesh blocks were arranged for the steps and the radial gate to resolve the geometry in the model. The size of the containing block (mesh block 1) was 1.2 × 1.2 × 1.2 m, and the size of the nested mesh blocks 2 and 3 were 0.3 × 0.67 × 0.4 m and 0.1 × 0.1 × 0.1 m, respectively, with respect to the x-y-z coordinate system, as shown in Figure 5.

## 4. Calculation Results and Discussion

#### 4.1. Mesh Sensitivity Analysis and Model Validation

#### 4.2. Pressure Distribution

#### 4.3. Velocity Magnitude

#### 4.4. Step Surface Cavitation Index

#### 4.5. Air Concentration

#### 4.6. Aerator Optimization

## 5. Conclusions

^{®}. In doing so, the 3-D Reynolds-averaged Navier–Stokes equations were solved, including the RNG k-ε turbulence model and a VOF method, to capture the free surface. A sub-grid air entrainment model was used in combination with a density evaluation model and drift-flux model to reproduce air entrainment and transport in water flow. The resulting pressure, velocity, and especially air concentration data above the steps were used for further data analysis and validation with data from laboratory experiments. As a case study, the stepped spillway at Ludila dam in southwest China was considered, which underwent damage to the steps in 2015. Finally, the effect of the height of the first step of the spillway on the air concentration along the stepped surface was addressed. The findings of the present research can be concluded as follows:

- The reliability of the involved models is considered satisfying, as the results are in good agreement with laboratory data.
- The velocity close to the step tips was about 15 m/s, indicating potential cavitation occurrence for all investigated discharges. The negative pressure close to the step tips is a key factor that may cause cavitation erosion, since both analytical evidence of a low cavitation index and practical real-world step failure were discovered.
- The calculated air concentration data is rather reliable, considering the air concentration on the stepped surface behind the X-shaped FGP of roughly 5% to 6% and the reported scaled physical model value of 3% to 5% in [14]. In addition, the simulated air concentration and derived cavitation potential are plausible with regard to the prototype step failure. It was found that for a high unit discharge of 166 m
^{2}/s and flow depths up to 30 m (25 times the step height), free surface self-entrained air fails to reach the stepped surface before step 45. A bottom-aeration thus becomes very important but insufficient if a threshold of 7% for the air concentration is considered necessary for cavitation erosion avoidance. - The height of the first step of the spillway may affect the air concentration on the stepped surface to some degree. An increased air concentration along the steps for a larger first step height was found in the case study of Ludia dam spillway, especially upstream of step 35. Therefore, for aerator optimization of built projects, heightening the first step could be an economic and efficient measure to prevent cavitation erosion.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

${A}_{p}$ | cross sectional area of air bubble (m^{2}) |

A_{s} | area of surface disturbance (m^{2}) |

${C}_{air}$ | air concentration (–) |

${C}_{d}$ | user-defined drag coefficient (–) |

CNU | coefficient equal to 0.09 (–) |

${d}_{p}$ | bubble diameter (m) |

${D}_{k}$ | effective diffusivity of k (kg m^{−1} s^{−1}) |

${D}_{\epsilon}$ | effective diffusivity of $\epsilon $ (kg m^{−1} s^{−1}) |

e | opening of radial gate (m) |

f | water volume fraction (–) |

g_{n} | component of gravity normal to the free surface (m s^{−2}) |

H_{d} | design head of weir (m) |

I | identity matrix |

$K$ | drag coefficient (kg m^{−3} s^{−1}) |

${K}_{p}$ | drag coefficient for single particle (kg s^{−1}) |

${k}_{air}$ | coefficient of proportionality (–) |

k | turbulent kinetic energy (m^{2} s^{−2}) |

${k}_{RZ}$ | Richardson–Zaki coefficient multiplier (-) |

L_{T} | turbulent length (m) |

P | gauge pressure in flow field (Pa) |

${P}_{amb}$ | ambient pressure (Pa) |

${P}_{k}$ | generation of k due to mean velocity gradient (kg m^{−1} s^{−3}) |

${P}_{vp}$ | vapor pressure (Pa) |

P_{d} | disturbance energy per unit volume (N m^{−2}) |

P_{t} | destabilization force per unit volume (N m^{−2}) |

q | unit width discharge (m^{2} s^{−1}) |

${R}_{p}$ | bubble radius (m) |

${\mathit{u}}_{\mathit{m}}$ | mixture velocity (m s^{−1}) |

${\mathit{u}}_{\mathit{r}}$ | relative/slip velocity (m s^{−1}) |

${\mathit{u}}_{\mathit{r}}^{\mathit{e}\mathit{f}\mathit{f}}$ | effective relative velocity (m s^{−1}) |

${\mathit{U}}_{\mathit{r}}$ | magnitude of ${u}_{r}$ (m s^{−1}) |

$\vartheta $ | kinematic viscosity related to the turbulence Schmidt number (m^{2} s^{−1}) |

${\vartheta}_{eff}$ | effective kinematic viscosity (m^{2} s^{−1}) |

We | Weber number (–) |

Z_{up} | upstream water elevation (m) |

Z_{down} | downstream water elevation (m) |

$\delta V$ | volume of air entrained to the flow (m^{3}) |

$\Delta h$ | Heightened height when extend curve end 0.9 m downstream (m) |

$\epsilon $ | dissipation rate of k (m^{2} s^{−3}) |

${\mu}_{a}$ | dynamic viscosity of air (Pa s^{−1}) |

${\xi}_{0}$ | Richardson–Zaki coefficient (–) |

${\rho}_{a}$ | density of air, 1.225 kg m^{−3} |

${\rho}_{m}$ | macroscopic mixture density (kg m^{−3}) |

${\rho}_{w}$ | density of water, 1000 kg m^{−3} |

${\sigma}_{sur}$ | coefficient of surface tension (N m^{−1}) |

$\mathit{\tau}$ | Reynolds stress tensor (kg m^{−1} s^{−2}) |

## Appendix A

**Figure A1.**Assessment of the convergence to a steady state for case 3: the red line indicates the variation of the total volume of fluid in the domain (V

_{f}), the blue solid line indicates the average of V

_{f}from 90 to 130 s of simulation time; the shaded region (last 10 s of simulation) represents when the numerical data was acquired.

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**Figure 1.**Schematic of the general flow pattern and aeration mechanism of the stepped spillway combined with X-shaped FGP: (

**a**) 3D air-side view and (

**b**) side view.

**Figure 3.**(

**a**) Photo of damaged steps and (

**b**) measured destruction width after the 2015 flood at the Ludila dam spillway.

**Figure 5.**Geometry and mesh blocks considered for the numerical simulations, including details of the grid resolution at the radial gate lip and spillway steps.

**Figure 6.**Velocity magnitude (U) and turbulent kinetic energy (k) profiles perpendicular to the pseudo-bottom at step 30 in the axis of the longitudinal section.

**Figure 7.**Comparison of experimental and numerical step surface pressure at the middle of the corresponding steps’ vertical faces. Experimental 1 data was measured by rubber piezometric pipe while Experimental 2 data was obtained from a high-speed pressure transmitter.

**Figure 9.**Velocity distribution above steps at the axial longitudinal section for load case 3 with details of steps 35 and 36.

**Figure 10.**Air concentration at step surface and corresponding cavitation index along steps 21 to 50 for three load cases: the main damage region between steps 30 and 45 is colored grey and the slash line region stands for cavitation index and air concentration below threshold values of 0.6 and 7%, respectively.

**Figure 11.**Air concentration distribution (

**a**) at the axial longitudinal section and (

**b**) at the stepped surface for load case 3.

**Figure 13.**Air concentration distribution at the axial longitudinal section for (

**a**) the original design and optimized designs (

**b**) M1 and (

**c**) M2 with an increased first step.

**Figure 14.**Air concentration on the steps’ surface for the original design and optimized designs M1 and M2 with an increased first step.

Load Case No. | e (m) | q (m^{2} s^{−1}) |
---|---|---|

1 | 4.75 | 80 |

2 | 9.5 | 120 |

3 | 19 | 166 |

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**MDPI and ACS Style**

Dong, Z.; Wang, J.; Vetsch, D.F.; Boes, R.M.; Tan, G. Numerical Simulation of Air–Water Two-Phase Flow on Stepped Spillways behind X-Shaped Flaring Gate Piers under Very High Unit Discharge. *Water* **2019**, *11*, 1956.
https://doi.org/10.3390/w11101956

**AMA Style**

Dong Z, Wang J, Vetsch DF, Boes RM, Tan G. Numerical Simulation of Air–Water Two-Phase Flow on Stepped Spillways behind X-Shaped Flaring Gate Piers under Very High Unit Discharge. *Water*. 2019; 11(10):1956.
https://doi.org/10.3390/w11101956

**Chicago/Turabian Style**

Dong, Zongshi, Junxing Wang, David Florian Vetsch, Robert Michael Boes, and Guangming Tan. 2019. "Numerical Simulation of Air–Water Two-Phase Flow on Stepped Spillways behind X-Shaped Flaring Gate Piers under Very High Unit Discharge" *Water* 11, no. 10: 1956.
https://doi.org/10.3390/w11101956