# SPHysics Simulation of Experimental Spillway Hydraulics

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}and storage capacity of 426 million m

^{3}. It mainly serves the domestic water supply and agricultural irrigation for the downstream regions and the total project cost nearly 0.2 billion CNY. We plan to use the parallelSPH model to investigate the water surface, and velocity and pressure variations over the stepped spillway aiming to achieve the following purposes: (i) to show how SPH could be used as a practical engineering tool in real industry; (ii) to provide energy dissipation indicators for the different spillway geometries. One distinct feature of the present work is the use of mesh-free numerical model in the spillway hydraulics. This study field has been extensively explored by the pioneering researchers such as Chanson [34,35] due to its practical engineering interest. The paper is structured as follows. Section 2 presents a brief review of the SPH fundamentals and parallelSPHysics [36]. Section 3 presents the engineering background of Dahua Hydraulic Dam and the designs of spillway experiment. Section 4 is dedicated to the validation of the numerical model by using a documented benchmark spillway skimming flow. Section 5 is the SPH modelling of our laboratory experiment with detailed discussions on the simulation results. Finally, Section 6 concludes the whole work.

## 2. paralleSPHysics

#### 2.1. Governing Equations and SPH Formulations

^{3}is the reference density and $\gamma $ = 7 is the polytrophic constant. It should be noted that using the real value of sound speed in water could lead to a very small time step arising from the constraint of Courant-Fredrich-Levy (CFL) condition. In SPH practice, the minimum speed of sound should be taken at least 10 times larger than the maximum bulk flow velocity. This keeps the density variation at less than 1%.

#### 2.2. Free Surface and Boundary Conditions

#### 2.3. Brief Introduction of parallelSPHysics

## 3. Engineering Background and Laboratory Experiment

^{3}with a flood storage of 0.53 million m

^{3}. The reservoir operates at a normal water level of $\nabla $3138.8 m and a designed flood level of $\nabla $3141.0 m, respectively. To discharge the redundant flood water during the rainy seasons, stepped spillways are used on the downstream side of the dam to dissipate flow energy. In order to fully evaluate the complex flow conditions over the step surface as well as to provide useful information for design purposes, a thorough understanding of the spillway hydraulics is required. For such a purpose, a physical laboratory model is thus built. A schematic layout of prototype Dahua Dam and location of the spillway is shown in Figure 1. As shown in the figure, the spillway includes the flow entrance, discharge flume, energy dissipater and outflow channel, with a total length of 216.97 m. The crest elevation of the weir is the same as that of the hydraulic dam at $\nabla $3138.8 m. The spillway takes an ogee shape which has a smoothed streamlined connection with the main dam. The designed flood discharge of the spillway is 48.43 m

^{3}. This is equivalent to a flood recurrence of 50 years. The effective breadth of the spillway is 21 m and the operating water head above the weir crest is 2.2 m. On the safety side, our study considers a 1000-year recurrence flood with a flow discharge of 120.21 m

^{3}/s.

^{3}/s. A kinematic similarity law, i.e., Froude together with Manning’s similarity, has been observed. The water head above the model weir in the experiment is 5.5 cm. The model spillway has a breadth of 0.525 m and the total length is 159.83 cm with an averaged bed slope angle of 33.69°. Different sizes of the spillway steps are used in the experiment, leading to the number of steps at 31, 45 and 62, respectively. The dimension of the steps is provided by their width a multiplied with height h, as 4.5 cm × 3.0 cm, 3.1 cm × 2.07 cm and 2.25 cm × 1.5 cm, for the above different step numbers. All types of the flow are observed in the range of skimming flows.

## 4. Model Validations through Benchmark Spillway Flow

^{−5}s and then dynamically adjusted in the computations to fulfill the CFL condition. The program is run on a 16-core computer cluster (CPU 2 GHz and RAM 32 Gb) (School of Water Resources and Electric Power, Qinghai University, China) and it cost approximately four days of CPU time.

_{c}is the critical flow depth on the critical section over the weir crest where the Froude number equals one. In the current parallelSPHysics simulations, the test condition of h

_{c}= 0.07935 m, 0.0716 m and 0.0635 m as shown in Table 1 in [31] is used. Based on the simulation results, Table 1 presents the experimental [31,41] and parallelSPHysics values of the discharge per unit width (q) for the different critical flow depths (h

_{c}) over the weir crest. It seems a good agreement has been reached.

_{c}= 0.0716 m, the computed particle snapshots over the stepped spillway are shown in Figure 4a–d, for the different time instants. It shows that under such a skimming flow condition, the flow passes over the weir crest like a jet and hits the horizontal face of all the downstream steps. One major part of the flow coherently skims over the pseudo bottom formed by the step outer edges. Another minor part is entrapped in the recirculation zone under the main stream and inside the step cavity delimited by the step faces. In this area there exists an extensive exchange with the main flows and the water particles move around a triangular path. These simulation patterns are very similar to those observed in [31]. However, as the computations proceed to a relatively longer time at t = 12 s as shown in Figure 4d, the boundary of two different flows becomes unclear.

_{c}as shown in Table 1. The details of velocity measurement and the instruments used could be found in the original work of [41]. As seen from the figure, in spite of some slight deviations between the numerical and experimental velocities, the general agreement between the two is quite satisfactory. It is shown that the velocity values do not change too much along the flow depth. According to [31,41], this is the region close to the weir crest, where the velocity is relatively low and also nearly constant over the entire flow depth due to the fact that the thickness of boundary layer is quite small. However, as the flow velocity increases towards the downstream of the spillway, the boundary layer will fully develop and then it would be expected to have a more varied velocity profile such as that follows the power law.

## 5. Model Applications through Experimental Spillway Flow

#### 5.1. Computational Settings

^{2}/s is generated, and the water head above the weir crest is 0.22 m. The stepped spillway is 6.4 m long. The water head looks relatively large compared to the length of the spillway, since a more safety design was considered in the study based on a 1000-year recurrence of prototype flood as mentioned in Section 3.

^{−5}s and later automatically changed in the computation following the CFL condition. The simulation time was 20 s and it cost roughly 20–26 days CPU time for the different step numbers on the same computer cluster (but with 48-core) as used in the model validation test in Section 4. To show the stability of inflow generations, Figure 7 shows the time-dependent discharge hydrograph for the case of 62-step spillway. The gauging point is located just upstream of the weir entrance at x = 7.9 m. It shows the increasing flow discharge following the forward motion of the push-paddle. After some very slight oscillations, the flow discharge stabilizes at time t = 8 s.

#### 5.2. Flow Patterns over Stepped Spillways

#### 5.3. Model Validation of Pressure on Spillway Ftep faces

_{s}values, where L and k

_{s}are the distance from the spillway crest to the outer step edge and characteristic roughness height, respectively. There is a good trend of the data consistency between the two results, but some obvious discrepancies do exist in the pressure values due to the difference in the flow/step condition and data sampling point used in the two studies. Nonetheless, the potentials of pressure prediction by the SPH model on the step faces can be well demonstrated.

#### 5.4. Analysis of Energy Dissipation Efficiency of Spillway with Different Geometries

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Khatsuria, R.M. Hydraulics of Spillways and Energy Dissipators; Marcel Dekker: New York, NY, USA, 2005. [Google Scholar]
- Boes, R.M.; Hager, W.H. Hydraulic design of stepped spillways. J. Hydrol. Eng.
**2003**, 129, 671–679. [Google Scholar] [CrossRef] - Chanson, H.; Toombes, L. Air-water flows down stepped chutes: Turbulence and flow structure observations. Int. J. Multiph. Flow
**2002**, 28, 1737–1761. [Google Scholar] [CrossRef] - Bung, D.B. Non-intrusive detection of air-water surface roughness in self-aerated chute flows. J. Hydraul. Res.
**2013**, 51, 322–329. [Google Scholar] [CrossRef] - Monaghan, J.J. Simulating free surface flows with SPH. J. Comput. Phys.
**1994**, 110, 399–406. [Google Scholar] [CrossRef] - Gómez-Gesteira, M.; Dalrymple, R.A. Using a 3D SPH method for wave impact on a tall structure. J. Waterw. Port Coast. Ocean Eng.
**2004**, 130, 63–69. [Google Scholar] [CrossRef] - Khayyer, A.; Gotoh, H.; Shao, S.D. Corrected incompressible SPH method for accurate water-surface tracking in breaking waves. Coast. Eng.
**2008**, 55, 236–250. [Google Scholar] [CrossRef] [Green Version] - Khayyer, A.; Gotoh, H. Wave impact pressure calculations by improved SPH methods. Int. J. Offshore Polar Eng.
**2009**, 19, 300–307. [Google Scholar] - Rudman, M.; Cleary, P.W. The influence of mooring system in rogue wave impact on an offshore platform. Ocean Eng.
**2016**, 115, 168–181. [Google Scholar] [CrossRef] - Amicarelli, A.; Albano, R.; Mirauda, D.; Agate, G.; Sole, A.; Guandalini, R. A Smoothed Particle Hydrodynamics model for 3D solid body transport in free surface flows. Comp. Fluids
**2015**, 116, 205–228. [Google Scholar] [CrossRef] - Colagrossi, A.; Landrini, M. Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J. Comput. Phys.
**2003**, 191, 448–475. [Google Scholar] [CrossRef] - Hu, X.Y.; Adams, N.A. An incompressible multi-phase SPH method. J. Comput. Phys.
**2007**, 227, 264–278. [Google Scholar] [CrossRef] - Albano, R.; Sole, A.; Mirauda, D.; Adamowski, J. Modelling large floating bodies in urban area flash-floods via a Smoothed Particle Hydrodynamics model. J. Hydrol.
**2016**, 541, 344–358. [Google Scholar] [CrossRef] - Canelas, R.B.; Crespo, A.J.C.; Dominguez, J.M.; Ferreira, R.M.L.; Gómez-Gesteira, M. SPH-DCDEM model for arbitrary geometries in free surface solid-fluid flows. Comp. Phys. Commun.
**2016**, 202, 131–140. [Google Scholar] [CrossRef] - Lind, S.J.; Stansby, P.K.; Rogers, B.D. Incompressible-compressible flows with a transient discontinuous interface using smoothed particle hydrodynamics (SPH). J. Comput. Phys.
**2016**, 309, 129–147. [Google Scholar] [CrossRef] - Akbari, H.; Namin, M. Moving particle method for modeling wave interaction with porous structures. Coast. Eng.
**2013**, 74, 59–73. [Google Scholar] [CrossRef] - Ren, B.; Wen, H.J.; Dong, P.; Wang, Y.X. Numerical simulation of wave interaction with porous structures using an improved smoothed particle hydrodynamic method. Coast. Eng.
**2014**, 88, 88–100. [Google Scholar] [CrossRef] - Losada, I.J.; Lara, J.L.; del Jesus, M. Modeling the interaction of water waves with porous coastal structures. J. Watery. Port Coast. Ocean Eng.
**2016**, 142. [Google Scholar] [CrossRef] - Shen, H.T.; Su, J.S.; Liu, L.W. SPH simulation of river ice dynamics. J. Comput. Phys.
**2000**, 165, 752–770. [Google Scholar] [CrossRef] - Shen, H.T.; Gao, L.; Kolerski, T.; Liu, L.W. Dynamics of ice jam formation and release. J. Coast. Res.
**2008**, SI(52), 25–31. [Google Scholar] [CrossRef] - Kolerski, T.; Shen, H.T.; Kioka, S. A numerical model study on ice boom in a coastal lake. J. Coast. Res.
**2013**, 29, 177–186. [Google Scholar] [CrossRef] - Gotoh, H.; Shibahara, T.; Sakai, T. Sub-particle-scale turbulence model for the MPS method–Lagrangian flow model for hydraulic engineering. Comput. Fluid Dyn. J.
**2001**, 9, 339–347. [Google Scholar] - Violeau, D.; Issa, R. Numerical modelling of complex turbulent free-surface flows with the SPH method: An overview. Int. J. Numer. Methods Fluids
**2007**, 53, 277–304. [Google Scholar] [CrossRef] - Liu, X.; Lin, P.Z.; Shao, S.D. An ISPH simulation of coupled structure interaction with free surface flows. J. Fluids Struct.
**2014**, 48, 46–61. [Google Scholar] [CrossRef] - Hosseini, S.M.; Feng, J.J. Pressure boundary conditions for computing incompressible flows with SPH. J. Comput. Phys.
**2011**, 230, 7473–7487. [Google Scholar] [CrossRef] - Federico, I.; Marrone, S.; Colagrossi, A.; Aristodemo, F.; Antuono, M. Simulating 2D open-channel flows through an SPH model. Eur. J. Mech. B/Fluids
**2012**, 34, 35–46. [Google Scholar] [CrossRef] - Vacondio, R.; Rogers, B.D.; Stansby, P.K. Accurate particle splitting for smoothed particle hydrodynamics in shallow water with shock capturing. Int. J. Numer. Methods Fluids
**2012**, 69, 1377–1410. [Google Scholar] [CrossRef] - Chang, T.J.; Chang, K.H. SPH modeling of one-dimensional nonrectangular and nonprismatic channel flows with open boundaries. J. Hydraul. Eng.
**2013**, 139, 1142–1149. [Google Scholar] [CrossRef] - Chern, M.; Syamsuri, S. Effect of corrugated bed on hydraulic jump characteristic using SPH method. J. Hydraul. Eng.
**2013**, 139, 221–232. [Google Scholar] [CrossRef] - Jonsson, P.; Andreasson, P.; Hellström, J.; Jonsén, P.; Lundström, T.S. Smoothed Particle Hydrodynamic simulation of hydraulic jump using periodic open boundaries. Appl. Math. Model.
**2016**, 40, 8391–8405. [Google Scholar] [CrossRef] - Husain, S.M.; Muhammed, J.R.; Karunarathna, H.U.; Reeve, D.E. Investigation of pressure variations over stepped spillways using smooth particle hydrodynamics. Adv. Water Resour.
**2014**, 66, 52–69. [Google Scholar] [CrossRef] - Lee, E.S.; Violeau, D.; Issa, R.; Ploix, S. Application of weakly compressible and truly incompressible SPH to 3-D water collapse in waterworks. J. Hydraul. Res.
**2010**, 48, 50–60. [Google Scholar] [CrossRef] - Saunders, K.; Prakash, M.; Cleary, P.W.; Cordell, M. Application of Smoothed Particle Hydrodynamics for modelling gated spillway flows. Appl. Math. Model.
**2014**, 38, 4308–4322. [Google Scholar] [CrossRef] - Chanson, H. Hydraulics of skimming flows over stepped channels and spillways. J. Hydraul. Res.
**1994**, 32, 445–460. [Google Scholar] [CrossRef] - Chanson, H. Turbulent air-water flows in hydraulic structures: Dynamic similarity and scale effects. Environ. Fluid Mech.
**2009**, 9, 125–142. [Google Scholar] [CrossRef] [Green Version] - Rogers, B.D.; Dalrymple, R.A.; Gómez-Gesteira, M.; Crespo, A.J.C. User Guide for the parallelSPHysics Code Using MPI (Version 2.0); SPHYSICS: Manchester, UK, 2011; Available online: http://www.sphysics.org (accessed on 6 June 2016).
- Bonet, J.; Lok, T.S.L. Variational and momentum preservation aspects of Smoothed Particle Hydrodynamic formulations. Comput. Methods Appl. Mech. Eng.
**1999**, 180, 97–115. [Google Scholar] [CrossRef] - Dalrymple, R.A.; Knio, O. SPH modelling of water waves. In Proceedings of the Fourth Conference on Coastal Dynamics, Lund, Sweden, 11–15 June 2001; ASCE: Reston, VA, USA, 2001; pp. 779–787. [Google Scholar]
- Gómez-Gesteira, M.; Rogers, B.D.; Crespo, A.J.C.; Dalrymple, R.A.; Narayanaswamy, M.; Dominguez, J.M. SPHysics—Development of a free-surface fluid solver—Part 1: Theory and Formulations. Comput. Geosci.
**2012**, 48, 289–299. [Google Scholar] [CrossRef] - Ren, L.; Gu, S. Numerical simulation on practical weir flow based on Smoothed Particle Hydrodynamics (SPH). Hydraul. Hydroelectr. Tech.
**2017**. in press (In Chinese) [Google Scholar] - Meireles, I.; Matos, J. Skimming flow in the nonaerated region of stepped spillways over embankment dams. J. Hydraul. Eng.
**2009**, 135, 685–689. [Google Scholar] [CrossRef] - Amador, A.; Sánchez-Juny, M.; Dolz, J. Developing flow region and pressure fluctuations on steeply sloping stepped spillways. J. Hydraul. Eng.
**2009**, 135, 1092–1100. [Google Scholar] [CrossRef] - Dodaro, G.; Tafarojnoruz, A.; Stefanucci, F.; Adduce, C.; Calomino, F.; Gaudio, R.; Sciortino, G. An experimental and numerical study on the spatial and temporal evolution of a scour hole downstream of a rigid bed. In Proceedings of the River Flow 2014—The 7th International Conference on Fluvial Hydraulics, Lausanne, Switzerland, 3–5 September 2014. [Google Scholar]
- Dodaro, G.; Tafarojnoruz, A.; Sciortino, G.; Adduce, C.; Calomino, F.; Gaudio, R. Modified Einstein sediment transport method to simulate the local scour evolution downstream of a rigid bed. J. Hydraul. Eng.
**2016**, 142. [Google Scholar] [CrossRef] - Felder, S.; Chanson, H. Simple design criterion for residual energy on embankment dam stepped spillways. J. Hydraul. Eng.
**2016**, 142. [Google Scholar] [CrossRef] - Chanson, H.; Bung, D.B.; Matos, J. Energy Dissipation in Hydraulic Structures; CRC Press: Leiden, The Netherlands, 2015; pp. 45–64. [Google Scholar]
- Bung, D.B. Developing flow in skimming flow regime on embankment stepped spillways. J. Hydraul. Res.
**2011**, 49, 639–648. [Google Scholar] [CrossRef] - Calomino, F.; Tafarojnoruz, A.; De Marchis, M.; Gaudio, R.; Napoli, E. Experimental and numerical study on the flow field and friction factor in a pressurized corrugated pipe. J. Hydraul. Eng.
**2015**, 141. [Google Scholar] [CrossRef] - De Padova, D.; Mossa, M.; Sibilla, S. SPH numerical investigation of the velocity field and vorticity generation within a hydrofoil-induced spilling breaker. Environ. Fluid Mech.
**2016**, 16, 267–287. [Google Scholar] [CrossRef] - De Padova, D.; Mossa, M.; Sibilla, S. SPH modelling of hydraulic jump oscillations at an abrupt drop. Water
**2017**, 9, 790. [Google Scholar] [CrossRef]

**Figure 2.**Site photo of laboratory spillway experiment: (

**a**) Global view of facility; (

**b**) Enlarged view of spillway with pressure measurement.

**Figure 4.**SPH particle snapshots of flow over the stepped spillway at different time t: (

**a**) 1.2 s; (

**b**) 2 s; (

**c**) 8 s; (

**d**) 12 s.

**Figure 5.**Comparisons between experimental and parallelSPHysics velocity profiles at x = 8.1 m for the critical flow depth h

_{c}: (

**a**) 0.0635 m; (

**b**) 0.0716 m; (

**c**) 0.07935 m.

**Figure 8.**Computed particle snapshots of 62-step spillway flow at different time instants at: (

**a**) t = 2 s; (

**b**) t = 8 s; (

**c**) t = 12 s; (

**d**) t = 15 s.

**Figure 9.**Experimental data gauging points along the spillway for the 31-step case: (

**a**) Velocity point; (

**b**) Pressure point.

**Figure 10.**Pressure distributions on: (

**a**) Horizontal; (

**b**) Vertical step faces, for the 31-step spillway case.

**Figure 11.**Pressure distributions on: (

**a**) Horizontal; (

**b**) Vertical step faces, for the 45-step spillway case.

**Figure 12.**Energy dissipation curves along the downstream direction of spillway for different step numbers: (

**a**) 62-step; (

**b**) 45-step; (

**c**) 31-step.

h_{c} (m) | q_{EXP} (m^{2}/s) | q_{SPH} (m^{2}/s) | Relative Error (%) |
---|---|---|---|

0.07935 | 0.07 | 0.0683 | −2.43 |

0.0716 | 0.06 | 0.0578 | −3.67 |

0.0635 | 0.05 | 0.0523 | +4.60 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gu, S.; Ren, L.; Wang, X.; Xie, H.; Huang, Y.; Wei, J.; Shao, S.
SPHysics Simulation of Experimental Spillway Hydraulics. *Water* **2017**, *9*, 973.
https://doi.org/10.3390/w9120973

**AMA Style**

Gu S, Ren L, Wang X, Xie H, Huang Y, Wei J, Shao S.
SPHysics Simulation of Experimental Spillway Hydraulics. *Water*. 2017; 9(12):973.
https://doi.org/10.3390/w9120973

**Chicago/Turabian Style**

Gu, Shenglong, Liqun Ren, Xing Wang, Hongwei Xie, Yuefei Huang, Jiahua Wei, and Songdong Shao.
2017. "SPHysics Simulation of Experimental Spillway Hydraulics" *Water* 9, no. 12: 973.
https://doi.org/10.3390/w9120973