# Numerical Investigation on the Hydraulic Properties of the Skimming Flow over Pooled Stepped Spillway

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation

^{®}, Canonsburg, PA, USA) was utilized to investigate the flow characteristics over stepped spillway with different pool weir configurations. The numerical model consists of 4 parts: the upstream water tank, the broad-crested weir with an upstream rounded corner (radius = 0.08 m), the stepped spillway and the downstream channel. The length, L

_{crest}, and width, W, of the broad-crested weir are 1.01 m and 0.52 m, respectively. The spillway has 10 steps with vertical step height, h, and horizontal step length, l, of 0.1 m and 0.2 m, respectively. The channel slope is 26.6° and constant in each configuration. Moreover, the height, d, and the thickness, l

_{w}, of pool weir are, respectively, 0.09 m and 0.015 m. The schematics of the spillways are shown in Figure 1.

#### 2.1. Volume of Fluid Method

#### 2.2. Turbulence Model

#### 2.3. Boundary Conditions

- (a)
- Inflow boundary: Velocity of the flow calculated from the discharge, which is 0.113 m
^{3}/s in all cases, was given as the inlet boundary; - (b)
- Outflow boundary: Pressure outlet boundary was selected at the outlet and there was no normal gradient for all variables;
- (c)
- Free surface: Pressure inlet was employed and its value was the standard atmospheric pressure; and
- (d)
- Wall boundary: No-slip velocity boundary and wall function were chosen for the wall surfaces and near-wall regions, respectively.

#### 2.4. Grid Testing and Model Verification

## 3. Results and Analysis

#### 3.1. Flow Pattern

#### 3.2. Velocity Distribution

#### 3.3. Pressure Distribution

#### 3.4. Residual Head and Energy Dissipation

## 4. Conclusions

- The flow pattern in FP-FP presented no significant change in the transverse direction and was similar on different steps. But changing half of the fully pooled steps into partially pooled steps (two-sided pooled or central pooled) can result in three dimensional flow motion. In FP-TP, the maximum vortex intensity on the fully pooled steps and partially pooled steps occurred at the axial plane and sidewalls, respectively, while the opposite phenomenon can be seen in FP-CP. When replacing all the fully pooled steps by staggered configuration of two-sided pooled and central pooled steps (TP-CP), the flow was characterized by highly three dimensional motion, and the vortex development in the transverse direction showed a unique pattern with the maximum intensity of vortex occurring at Z/W = 0.25. Besides, TP-CP created more flow instabilities and turbulent structures.
- The velocity distributions over the steps in FP-FP highlighted the similar pattern with the minimum and maximum values occurring at y/y
_{max}= 0.3 and 0.9, respectively. In FP-TP, FP-CP and TP-CP, the velocity distributions on the odd-number steps was different from that of the even-number steps, but the maximum velocity in all configurations indicated no difference. In the transverse direction, the velocity distribution in FP-FP showed the smallest variation, while TP-CP presented the greatest change. - The pressure on the horizontal step surfaces of FP-FP, FP-TP and FP-CP showed U-shaped variations with the maximum pressures occurring at the downstream end of the step surface, and TP-CP highlighted the greatest pressure fluctuation in both streamwise and transverse direction. The pressure distribution patterns on the fully pooled steps in FP-TP and on the central pooled steps in FP-CP were similar; the pressure distributions on the two-sided pooled steps in FP-TP and on the fully pooled steps in FP-CP demonstrated the similar patterns. For the vertical steps, the maximum pressures on odd-number step surfaces were in the following order: FP-CP > TP-CP > FP-TP > FP-FP, while for the even-number steps, it was TP-CP > FP-FP > FP-TP > FP-CP. There was no negative pressure on the step surfaces.
- From the largest to the smallest, the energy dissipation rates were in the following order: TP-CP > FP-CP > FP-TP > FP-FP. Specifically, the energy loss for FP-TP and FP-CP was quite close, but slightly higher than FP-FP, and the energy dissipation rate for TP-CP was 1.5 times larger than FP-FP. Thus, changing half of the fully pooled steps into central pooled or two-sided pooled steps presented no obvious effect on the energy dissipation ratio, while shifting all the fully pooled steps into the combination of central pooled and two-sided pooled steps can significantly improve the energy dissipation performance.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Boes, R.M.; Hager, W.H. Hydraulic design of stepped spillways. J. Hydraul. Eng.
**2002**, 129, 671–679. [Google Scholar] [CrossRef] - Chanson, H. Hydraulic Design of stepped spillways and downstream energy dissipators. Dam Eng.
**2001**, 11, 205–242. [Google Scholar] - Chanson, H.; Luke, T. Hydraulics of stepped chutes: The transition flow. J. Hydraul. Res.
**2004**, 42, 43–54. [Google Scholar] [CrossRef] [Green Version] - Chanson, H.; Toombes, L. Experimental investigations of air entrainment in transition and skimming flows down a stepped chute. Can. J. Civ. Eng.
**2002**, 29, 145–156. [Google Scholar] [CrossRef] [Green Version] - Felder, S.; Chanson, H. Turbulence, dynamic similarity and scale effects in high-velocity free-surface flows above a stepped chute. Exp. Fluids
**2009**, 47, 1–18. [Google Scholar] [CrossRef] [Green Version] - Wang, J.; Fu, L.; Xu, H.; Jin, Y. Numerical study on flow over stepped spillway using Lagrangian method. In Proceedings of the International Conference on Energy Engineering and Environmental Protection, Sanya, China, 20–22 November 2017. [Google Scholar]
- Felder, S. Air-Water Flow Properties on Stepped Spillways for Embankment Dams: Aeration, Energy Dissipation and Turbulence on Uniform, Non-Uniform and Pooled Stepped Chutes. Ph.D. Thesis, University of Queensland, Brisbane, Australia, 2013. [Google Scholar]
- Felder, S.; Chanson, H. Energy dissipation down a stepped spillway with non-uniform step heights. J. Hydraul. Eng.
**2011**, 137, 1543–1548. [Google Scholar] [CrossRef] - Felder, S.; Chanson, H. Aeration, flow instabilities, and residual energy on pooled stepped spillways of embankment dams. J. Irrig. Drain. Eng.
**2013**, 139, 880–887. [Google Scholar] [CrossRef] - Zhang, J.M.; Chen, J.G.; Wang, Y.R. Experimental study on time-averaged pressures in stepped spillway. J. Hydraul. Res.
**2012**, 50, 236–240. [Google Scholar] [CrossRef] - Zhang, G.; Chanson, H. Effects of step and cavity shapes on aeration and energy dissipation performances of stepped chutes. J. Hydraul. Eng.
**2018**, 144, 04018060. [Google Scholar] [CrossRef] - Peng, Y.; Zhou, J.G.; Burrows, R. Modelling the free surface flow in rectangular shallow basins by lattice Boltzmann method. J. Hydraul. Eng.
**2011**, 137, 1680–1685. [Google Scholar] [CrossRef] - Peng, Y.; Zhou, J.G.; Burrows, R. Modelling solute transport in shallow water with the lattice Boltzmann method. Comput. Fluids
**2011**, 50, 181–188. [Google Scholar] [CrossRef] - Peng, Y.; Mao, Y.F.; Wang, B.; Xie, B. Study on C-S and P-R EOS in pseudo-potential lattice Boltzmann model for two-phase flows. Int. J. Mod. Phys. C
**2017**, 28, 1750120. [Google Scholar] [CrossRef] - Peng, Y.; Wang, B.; Mao, Y.F. Study on force schemes in pseudopotential lattice Boltzmann model for two-phase flows. Math. Probl. Eng.
**2018**, 6496379. [Google Scholar] [CrossRef] - Baylar, A.; Unsal, M.; Ozkan, F. The effect of flow patterns and energy dissipation over stepped chutes on aeration efficiency. KSCE J. Civ. Eng.
**2011**, 15, 1329–1334. [Google Scholar] [CrossRef] - Dong; Zhi-yong; Joseph; Hun-wei. Numerical simulation of skimming flow over mild stepped channel. J. Hydrodyn.
**2006**, 18, 367–371. [Google Scholar] [CrossRef] - Zhenwei, M.U.; Zhang, Z.; Zhao, T. Numerical simulation of 3-D flow field of spillway based on VOF method. Procedia Eng.
**2012**, 28, 808–812. [Google Scholar] [CrossRef] - Tabbara, M.; Chatila, J.; Awwad, R. Computational simulation of flow over stepped spillways. Comput. Struct.
**2005**, 83, 2215–2224. [Google Scholar] [CrossRef] - André, S. High Velocity Aerated Flows on Stepped Chutes with Macro-Roughness Elements. Ph.D. Thesis, EPFL, Lausanne, Switzerland, 2004. [Google Scholar]
- Kökpinar, M.A. Flow over a stepped chute with and without macro-roughness elements. Can. J. Civ. Eng.
**2004**, 31, 880–891. [Google Scholar] [CrossRef] - Thorwarth, J. Hydraulics of Pooled Stepped Spillways—Self-Induced Unsteady Flow and Energy Dissipation. Ph.D. Thesis, University of Aachen, Aachen, Germany, 2008. [Google Scholar]
- Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Li, X.; Wang, Y.R.; Zhang, J.M. Numerical simulation of an offset jet in bounded pool with deflection wall. Math. Probl. Eng.
**2017**, 2017, 1–11. [Google Scholar] [CrossRef] - Zhang, J.M.; Chen, J.G.; Xu, W.L.; Wang, Y.R. Three-dimensional numerical simulation of aerated flows downstream sudden fall aerator expansion-in a tunnel. J. Hydrodyn.
**2011**, 23, 71–80. [Google Scholar] [CrossRef] - Chen, Q.; Dai, G.; Liu, H. Volume of Fluid Model for Turbulence Numerical Simulation of Stepped Spillway Overflow. J. Hydraul. Eng.
**2002**, 128, 68–688. [Google Scholar] [CrossRef] - Yakhot, V.; Orszag, S.A. Renormalization group analysis of turbulence. I. Basic theory. J. Sci. Comput.
**1986**, 1, 3–51. [Google Scholar] [CrossRef] - Bai, Z.L.; Zhang, J.M. Comparison of different turbulence models for numerical simulation of pressure distribution in V-shaped stepped spillway. Math. Probl. Eng.
**2017**, 2017, 1–9. [Google Scholar] [CrossRef] - Morovati, K.; Eghbalzadeh, A.; Javan, M. Numerical investigation of the configuration of the pools on the flow pattern passing over pooled stepped spillway in skimming flow regime. Acta Mech.
**2015**, 227, 1–14. [Google Scholar] [CrossRef] - Carvalho, R.F.; Rui, M. Stepped spillway with hydraulic jumps: Application of a numerical model to a scale model of a conceptual prototype. J. Hydraul. Eng.
**2014**, 135, 615–619. [Google Scholar] [CrossRef] - Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J. Procedure of estimation and reporting of uncertainty due to discretization in CFD applications. J. Fluids Eng.
**2008**, 130, 078001–078004. [Google Scholar] [CrossRef] - Felder, S.; Guenther, P.; Hubert, C. Air-Water Flow Properties and Energy Dissipation on Stepped Spillways: A Physical Study of Several Pooled Stepped Configurations; The University of Queensland: Brisbane, Australia, 2012; ISBN 9781742720555. [Google Scholar]
- Hunt, J.C.R.; Wray, A.A.; Moin, P. Eddies, streams and convergence zones in turbulent flows. In Proceedings of the Summer Program 1988; Stanford University: Stanford, CA, USA, 1988. [Google Scholar]

**Figure 1.**Sketch of pooled stepped spillway configurations. (

**a**) Fully pooled steps (FP-FP); (

**b**) fully pooled and two-sided pooled steps (FP-TP); (

**c**) fully pooled and central pooled steps (FP-CP); and (

**d**) two-sided pooled and central pooled steps (TP-CP).

**Figure 4.**Comparison of numerical and experimental data. Notes: V

_{c}: critical flow velocity; y represents the distance normal to the pseudo-bottom formed by the pool edges with y = 0 at the pool edges.

**Figure 5.**Flow pattern at the axial plane in different configurations. (

**a**) FP-FP; (

**b**) FP-TP; (

**c**) FP-CP; and (

**d**) TP-CP.

**Figure 6.**Instantaneous streamlines in the transeverse direction. (

**a**) FP-FP; (

**b**) FP-TP; (

**c**) FP-CP; and (

**d**) TP-CP.

**Figure 7.**Distributions of lumps of the iso-surface of Q = 100. (

**a**) FP-FP; (

**b**) FP-TP; (

**c**) FP-CP; and (

**d**) TP-CP.

**Figure 9.**Velocity distributions above the step surfaces in each configuration. (

**a**) Z/W = 0.5; and (

**b**) Z/W = 0.25.

$${\eta}_{0}$$
| 438 |

$$\beta $$
| 0.012 |

$${C}_{\mu}$$
| 0.085 |

$${C}_{2}{}_{\mu}$$
| 1.68 |

$${\sigma}_{k}$$
| 0.7179 |

$${\sigma}_{\epsilon}$$
| 0.7179 |

Reference | Step Geometry |
---|---|

Morovati et al. [29] | Fully pooled steps: h = 0.1 m, l = 0.2 m, d = 0.09 m |

In-line configuration (pooled and flat steps in-line): h = 0.1 m, l = 0.2 m, d = 0.09 m | |

Staggered configuration (pooled and flat steps staggered): h = 0.1 m, l = 0.2 m, d = 0.09 m | |

Two-sided pooled steps: h = 0.1 m, l = 0.2 m, d = 0.09 m | |

Central-pooled steps: h = 0.1 m, l = 0.2 m, d = 0.09 m | |

Felder et al. [32] | Fully pooled steps: h = 0.1 m, l = 0.2 m, d = 0.031 m |

In-line configuration (pooled and flat steps in-line): h = 0.1 m, l = 0.2 m, d = 0.031 m | |

Staggered configuration (pooled and flat steps staggered): h = 0.1 m, l = 0.2 m, d = 0.031 m | |

Present study | FP-FP: h = 0.1 m, l = 0.2 m, d = 0.09 m |

FP-TP: h = 0.1 m, l = 0.2 m, d = 0.09 m | |

FP-CP: h = 0.1 m, l = 0.2 m, d = 0.09 m | |

TP-CP: h = 0.1 m, l = 0.2 m, d = 0.09 m |

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**MDPI and ACS Style**

Li, S.; Zhang, J.
Numerical Investigation on the Hydraulic Properties of the Skimming Flow over Pooled Stepped Spillway. *Water* **2018**, *10*, 1478.
https://doi.org/10.3390/w10101478

**AMA Style**

Li S, Zhang J.
Numerical Investigation on the Hydraulic Properties of the Skimming Flow over Pooled Stepped Spillway. *Water*. 2018; 10(10):1478.
https://doi.org/10.3390/w10101478

**Chicago/Turabian Style**

Li, Shicheng, and Jianmin Zhang.
2018. "Numerical Investigation on the Hydraulic Properties of the Skimming Flow over Pooled Stepped Spillway" *Water* 10, no. 10: 1478.
https://doi.org/10.3390/w10101478