# SPHysics Simulation of Experimental Spillway Hydraulics

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^{2}

^{3}

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## Abstract

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## 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

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**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 |

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**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