# Design of A Streamwise-Lateral Ski-Jump Flow Discharge Spillway

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Physical Structure of the Streamwise-Lateral Discharge Spillway

## 3. Investigation Methods and Definition of the Free Jet Flow Pattern

#### 3.1. Physical Model

^{2}, and the vertical distance from the horizontal floor to the bottom wall of the plunge pool is H

_{0}= 43 m. The length of the horizontal floor L = 55 m, with W = 21 m in width. The straight length of the side wall is L

_{b}= 36.4 m, with the anti-arch radius R = 19 m and the deflection angle θ = 96° in the anti-arch structure. The angle α of the inclined floor ranges from 0° to 60°, as shown in Figure 3. The flow discharge is 4016 m

^{3}/s, with discharge per unit width q

_{w}= 251 m

^{3}/(s·m). The tail water depth is 18 m deep. The scale of a normal model is 1:60 based on the Froude criterion for experimental study. Considering that the roughness coefficient of the concrete in the prototype is 0.0140 and the roughness scale is 60

^{1/6}= 1.979, the theoretical value of the roughness coefficient is 0.0071. The physical model is made of transparent plexiglass with a roughness coefficient of 0.0079. The present physical model is approximately satisfactory for the similarity of boundary roughness between the model and prototype.

^{0.5}is about 3.73 with a Reynolds number Re = Vh/ν of 1.83 × 10

^{5}, where ν is the kinematic water viscosity, g is the gravitational acceleration, and V = 3.2 m/s and h = 0.075 m are the mean velocity and depth of the approach flow on the horizontal floor, respectively. Thus, the present study focuses on the typical pattern of ski-jump flow in a streamwise-lateral discharge spillway, jet trajectory and distribution of mean pressure on the bottom floor of a basin, which satisfies the Froude criterion. The scale effect on air–water properties of supercritical flow cannot be neglected with regards to the surface tension and viscosity effects in high-speed flows [20].

_{m}was determined as p

_{m}= P − P

_{0}.

#### 3.2. Numerical Simulation

- Continuity equation:$$\frac{\partial \rho}{\partial t}+\frac{\partial \rho {u}_{i}}{\partial {x}_{i}}=0$$
- Momentum equation:$$\frac{\partial \rho {u}_{i}}{\partial t}+\frac{\partial \left(\rho {u}_{i}{u}_{j}\right)}{\partial {x}_{j}}=-\frac{\partial p}{\partial {x}_{i}}+\frac{\partial}{\partial {x}_{j}}\left[\right(\mu +{\mu}_{t})\times (\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\left)\right]$$
- k (turbulent kinetic energy) equation:$$\frac{\partial \rho k}{\partial t}+\frac{\partial \left(\rho {u}_{i}k\right)}{\partial {x}_{i}}=\frac{\partial}{\partial {x}_{i}}\left[\right(\mu +\frac{{\mu}_{t}}{{\sigma}_{k}})\times \frac{\partial k}{\partial {x}_{i}}]+{G}_{k}-\rho \epsilon $$
- ε (dissipation rate of turbulent kinetic energy) equation:$$\frac{\partial \rho \epsilon}{\partial t}+\frac{\partial \left(\rho {u}_{i}\epsilon \right)}{\partial {x}_{i}}=\frac{\partial}{\partial {x}_{i}}\left[\right(\mu +\frac{{\mu}_{t}}{{\sigma}_{\epsilon}})\times \frac{\partial \epsilon}{\partial {x}_{i}}]+{C}_{1\epsilon}\rho \frac{\epsilon}{k}{G}_{k}-{C}_{2\epsilon}\rho \frac{{\epsilon}^{2}}{k}$$
_{k}represents the generation of turbulence kinetic energy due to the mean velocity gradients. μ_{t}is the turbulent viscosity which can be deduced for the turbulence intensity k and energy dissipation rate ε:$${\mu}_{t}={C}_{\mu}\frac{{k}^{2}}{\epsilon},\text{}{C}_{1\epsilon}=1.42-\frac{\eta (1-\frac{\eta}{{\eta}_{0}})}{1+\beta {\eta}^{3}},\text{}\eta =\frac{Sk}{\epsilon},\text{}S=\sqrt{2\overline{{S}_{ij}}\text{}\overline{{S}_{ij}}},\overline{{S}_{ij}}=\frac{1}{2}(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}})$$

_{w}is:

_{i}is the velocity components and x

_{i}is the coordinates (i = 1, 2, 3). j is the sum suffix. The air–water interface is tracked by solving the continuity equations. The sum of the water and air volume fraction was one in the controlling body. The ρ

_{w}and ρ

_{a}are the density of water and air, respectively. The μ

_{w}and μ

_{a}are the viscosity of water and air, respectively. When α

_{w}= 0, it means there is full of air; when α

_{w}= 1, it represents the air fraction of fluid is 0. When 0 < α

_{w}< 1, it means the certain controlling volume. The total fraction in a certain controlling body is α

_{w}+ α

_{a}= 1. Note that, although the VOF model can give an aeration result in the free surface area, the air concentration distribution and aeration development along the jet trajectory is different to the real situations due to the complex air–water interaction and bubble diffusion process. In the present study, the simulated results will be used for the analysis on formation of ski-jump flow released from the streamwise-lateral spillway, which is little affected by the aeration. The detailed hydraulic characteristics, including jet trajectory and impact pressure on the pool floor, will be analyzed based on the physical model results. The control volume method is introduced to discretize the partial differential equations, and the SIMPLER method, which has good convergence characteristics, is employed for numerical simulation [22,23].

^{2}mesh size was selected to simulate the pattern of ski-jump flow in the air and its diffusion in the still basin. Based on the authors’ previous studies [23,24,25] on the simulation of high-speed flows (flow velocity exceeded over 30 m/s) in hydraulic engineering, these mesh conditions can make sure the maximum uncertainties for the velocity and pressure characteristics in the whole calculated region were approximately smaller 10%, and the uncertainty in most locations was fairly small. The comparison between the experimental and simulated results with regards to the flow characteristics of the streamwise-lateral discharge spillway are shown in Table 2 for α = 45°, where Y = 0 is the lateral cross-section at the straight side wall. The distributions of pressure on the side wall and water depth indicate that the simulation results are well in agreement with those of experimental measurements. The vector profile of the flow jet velocity in Figure 4 shows similar flow patterns for the experiments. According to the typical flow pattern of the streamwise-lateral discharge spillway, it is difficult to acknowledge the interior characteristics through experimental measurement, due to the irregular “curling” nappe shape and the complicated flow jet movement in the diffusion process. Because the computational water profiles and the pressure on the side wall are well in agreement with the test measurements, the flow patterns are similar between the simulation and the experiment and the complex high-speed fluid can be analyzed with the numerically simulated results. The characteristics of the jet profile and the pressure from the physical model were converted to the prototype according to the length scale. The simulated model was built up based on the real dimensions in the prototype, and the velocity distribution inside the ski-jump flow and stilling basin is the same as the prototype’s dimensions.

## 4. Results and Discussion

#### 4.1. Ski-Jump Flow Pattern

#### 4.2. Effect of Inclined Floor on Jet Length

_{1}and the further jet length L

_{2}are introduced, as shown in Figure 7. The further jet length, L

_{2}, is the longest distance in the lateral direction mainly affected by the coupling effect of the inclined floor and the anti-arch side wall of the spillway; and the nearer jet length, L

_{1}, is the lateral jet length distance and is affected mainly by the pressure difference in the horizontal floor. The coefficient β = (L

_{2}− L

_{1})/H

_{0}describes the free jet diffusion in the air, deflecting from the spillway. The increase of β indicates that the diffusion performance improves and the stretch of water jet in the air space becomes fully developed. The angle α of the inclined floor influences L

_{1}and L

_{2}pronouncedly for the otherwise identical conditions. As shown in Figure 8a,b, with the increase of α to 30°, both L

_{1}and L

_{2}remain unchanged, which indicates that the effect of the inclined floor on the jet length is not distinct. As α further increases to 45°, L

_{1}increases while L

_{2}still remains significantly unchanged. The value of β decreases slightly which indicates that the water flow pattern is stable, as shown in Figure 8c. For α > 45°, L

_{1}further increases while L

_{2}decreases pronouncedly. This results in a significant reduction in β, which indicates that the diffusion of the jet in the air is not sufficient in the lateral direction. Besides, the further jet length is around five times greater than the nearer jet length for α < 30°, and when α = 60°, the further jet length is around four times greater than the nearer jet length, indicating that the diffusion of the ski-jump flow nappe remained stable. Based on the effect of the inclined angle on the jet length, it is recommended that α is 30~45° for a good performance of free ski-jump jet diffusion.

#### 4.3. Effect of Inclined Floor on Jet Impact Pressure

_{m}(defined as the difference between the measured and the mean static pressure in the plunge pool), scaled by the tail water depth P

_{0}, is strongly affected by the inclined floor angle. When α = 0°~30°, the increase of p

_{m}/P

_{0}is not obvious. While for α > 30°, the p

_{m}/P

_{0}becomes larger with an increase of α. This is mainly due to the poor effect of the inclined floor constraint on the water flow streamline for a small α condition. Considering that the flow impinging on the anti-arch side wall results in a large amount of energy dissipation, the flow jet’s impact pressure on the bottom floor of the plunge pool is relatively low. For a large α, the water flow is well coupling-constrained by both the inclined floor and the anti-arch side wall, and the streamline varies relatively smoothly when it becomes totally free from the streamwise-lateral discharge spillway. Larger α causes a lower energy dissipation in the spillway part. Consequently, the flow energy is mainly transported into the plunge pool, resulting in a high impact pressure on the bottom floor.

_{m}, in each area of the plunge pool describes the pressure and water discharge rate distributions through the stretched flow jet. For different inclined floor angles, the pressure distribution is obviously different, owing to the non-uniform flow jet stretching, as shown in Figure 10. For α < 30°, the pressure profiles distribute smoothly in the whole plunge pool without obvious impact pressure on the bottom floor. The flow jets mainly drop into Area 1, Area 2 and Area 3, which are located in the center of the plunge pool. When α increases to 45°, the flow dropping areas move to Area 1, Area 2 and Area 6, with an obvious increase of impact pressure. This indicates that more water discharge moves in a lateral direction, which is affected by the inclined floor. For α > 45°, the pressure is extremely non-evenly distributed, with greater impact pressure in Area 6. When α = 50°~60°, the local impact pressure is near 1.5~2.0 times greater than the mean static pressure in the plunge pool, and this will cause the bottom floor structure to be affected by negative unbalanced force conditions and may cause structural damage with flood discharge.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Typical pattern of the streamwise-lateral discharge spillway: (

**a**) physical model from the downstream view; (

**b**) physical model from the lateral view; (

**c**) ski-jump flow.

**Figure 4.**Comparison of numerical simulation with experimental flow: (

**a**) a sample of computational mesh in the spillway area; (

**b**) velocity vector profile; (

**c**) experimental image.

**Figure 5.**Jet diffusion of streamwise-lateral ski-jump flow (simulated results for α = 45°): (

**a**) water profile at different elevations; (

**b**) velocity vector profile at different cross-sections.

**Figure 6.**Velocity vector distributions of free jet flow: (

**a**) Z = −17 m; (

**b**) Z = −22 m; (

**c**) Z = −27 m; (

**d**) Z = −32 m.

**Figure 7.**Sketch of free jet flow deflecting from the streamwise-lateral discharge spillway: (

**a**) plane view; (

**b**) lateral view (A–A).

**Figure 9.**Flow jet impact in the plunge pool: (

**a**) sketch plane view; (

**b**) experimental image (α = 60°).

**Figure 11.**Velocity vector distributions in the plunge pool: (

**a**) Z = −38 m; (

**b**) Z = −44 m; (

**c**) Z = −50 m; (

**d**) Z = −58 m.

Parameters | η_{0} | β | C_{μ} | C_{1ε} | C_{2ε} | σ_{k} | σ_{ε} |
---|---|---|---|---|---|---|---|

Value | 4.38 | 0.012 | 0.0845 | 1.42 | 1.68 | 0.7179 | 0.7179 |

**Table 2.**Comparison of tested and calculated data on the spillway flood discharge (α = 45°, q

_{w}= 251 m

^{2}/s).

Pressure on the Sidewall of Spillway (m) | Water Depth along the Side Wall of Spillway (m) | |||||||
---|---|---|---|---|---|---|---|---|

Y = 15 m | Y = 18 m | |||||||

H | Simulation | Test | H | Simulation | Test | X | Simulation | Test |

8.9 | 23.7 | 23.4 | 6.7 | 25.4 | 24.3 | 4.1 | 11.1 | 10.3 |

11.3 | 17.0 | 17.6 | 9.2 | 19.6 | 19.1 | 14.1 | 10.5 | 10.3 |

12.4 | 13.6 | 12.6 | 11.7 | 13.4 | 13.6 | 23.8 | 11.3 | 11.2 |

14.8 | 8.5 | 9.7 | 15.3 | 6.3 | 5.6 | 33.1 | 13.0 | 13.0 |

17.1 | 4.8 | 4.5 | 17.8 | 2.4 | 2.7 | 41.5 | 15.4 | 14.6 |

19.5 | 1.9 | 1.6 | 21.5 | 1.4 | 0.9 | 49.5 | 19.7 | 19.3 |

23.0 | 0.5 | 0.6 | 22.8 | 1.0 | 0.9 | 55.1 | 25.5 | 25.4 |

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## Share and Cite

**MDPI and ACS Style**

Deng, J.; Wei, W.; Tian, Z.; Zhang, F.
Design of A Streamwise-Lateral Ski-Jump Flow Discharge Spillway. *Water* **2018**, *10*, 1585.
https://doi.org/10.3390/w10111585

**AMA Style**

Deng J, Wei W, Tian Z, Zhang F.
Design of A Streamwise-Lateral Ski-Jump Flow Discharge Spillway. *Water*. 2018; 10(11):1585.
https://doi.org/10.3390/w10111585

**Chicago/Turabian Style**

Deng, Jun, Wangru Wei, Zhong Tian, and Faxing Zhang.
2018. "Design of A Streamwise-Lateral Ski-Jump Flow Discharge Spillway" *Water* 10, no. 11: 1585.
https://doi.org/10.3390/w10111585