# Investigation of a Negative Step Effect on Stilling Basin by Using CFD

^{*}

## Abstract

**:**

## 1. Introduction

_{2}/h

_{1}, in terms of the approaching Froude number, F

_{1}, and the relative pressure head on the step, h

_{d}/h

_{1}. Hager and Bretz [6] showed that the shape of the negative step did not have a significant effect on the transition of the flow regime, but the relative step height had a significant effect on the sequent depth ratios and the relative energy dissipation. The relative length of the roller can be given as a function of F

_{1}. Eroglu and Tastan [11] investigated the local energy losses at positive and negative steps in subcritical open channel flows. They found empirical equations for local energy losses depending on the Froude number and the relative step height. However, in the above studies, only low-head flows were considered.

_{1}= 6. The computed velocity, pressure obtained by this method were in good agreement with the laboratory measurements. Valero et al. [14] applied the same method to simulate the flow field of a USBR Type III hydraulic jump stilling basin under the design and adverse conditions for stepped and smooth spillways. The fine grid convergence index (GCI) as suggested by Celik et al. [16] was used in their study to perform a mesh sensitivity analysis. Zhou and Wang [17] simulated the 3D flow field among a compound stilling basin using the commercial CFD package FLOW-3D. The simulation results of four turbulence models (standard k-ε, RNG k-ε, realizable k-ε, and large eddy simulation turbulence models) were compared, and the RNG k-ε turbulence model showed the most successful agreement with the experimental results. Macian-Perez et al. [18] investigated the performance of a USBR Type II stilling basin using a CFD numerical model. The sequent depth ratio, the roller and hydraulic jump lengths, and the energy dissipation efficiency were analyzed in their study. The previous research showed that CFD complements experimental studies and the inherent measurement limitations, providing additional insight into complex flows [14]. In particular, when properly applied, RANS-based one- and two-equation turbulence models have been shown to provide accurate averaged force, distribution, and velocity fields [14]. Therefore, CFD has become an effective tool to study the hydraulic performance of stilling basins.

## 2. Experiment Setup

## 3. Numerical Simulations

#### 3.1. Flow Equations

#### 3.2. Numerical Simulation Runs and Solution Domain

_{1}, u

_{1}, and F

_{1}are the mean flow depth, mean velocity, and Froude number at the toe section, respectively; and h

_{t}is the tailwater depth. Stilling basins with six different negative step heights (d = 0 cm, 2.5 cm, 5 cm, 7.5 cm, 10 cm, 15 cm) and five different incident angles (θ = 0°, 5°, 10°, 15°, 20°) were investigated. It is important to note that the stilling basin matches the physical model dimensions. All CFD tests were performed under the design discharge Q = 0.09 m

^{3}/s (discharge per unit width is q = 0.18 m

^{2}/s), and the tailwater depth was 0.388 m.

#### 3.3. Numerical Settings

#### 3.4. Meshing and Boundary Conditions

#### 3.5. Discretization Error Analysis

_{1}= 95,900, N

_{2}= 54,985, and N

_{3}= 34,015, (fine mesh, medium mesh, and coarse mesh), were tested. The fine-grid convergence index can be computed as

## 4. Results

#### 4.1. Validation of the Numerical Model

#### 4.2. Flow Pattern

_{t}is the tailwater depth. The simulation results of the submergence parameter E in this study are given in Table 4. It can be seen that the value of E was in the range of 0.29 to 0.51 and E did not show a regular variation with the increase in the step height and incident angle. This is mainly due to the pulsating position of the jump toe, which has a great influence on the measurement accuracy.

#### 4.3. Velocity Profile

#### 4.4. Roller Length and Reattachment Length

#### 4.5. Energy Dissipation Efficiency

## 5. Discussion

## 6. Conclusions

- (1)
- The influence of step height on the flow regime is significant and the influence of incident angle on the flow regime is insignificant. Increasing the step height will lead to an increase in the flow depth in the basin, so the negative-step stilling basin needs a higher sidewall than the traditional stilling basin. However, the step height should be less than the critical height to avoid the surface jump in the basin.
- (2)
- Increasing the step height will significantly reduce the bottom flow velocity, and increasing the incident angle will significantly increase the critical bottom flow velocity. Therefore, the design of the negative-step stilling basin needs to consider the influence of the step height and the incident angle to achieve the most ideal design geometry.
- (3)
- Using a higher step height can reduce the roller length of the jump in the negative-step stilling basin, thus shortening the length of the dissipation basin. The reattachment length maintains a good linear relationship with both the step height and incident angle.
- (4)
- Within the scope of this study, changing the step height and incident angle did not have a strong effect on the total energy dissipation efficiency of the negative-step stilling basin.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

d | Step height |

θ | Incident angle |

Q | Flow discharge |

q | Unit discharge |

h_{1} | Mean depth at the jump toe |

u_{1} | Mean velocity at the jump toe |

F_{1} | Froude number at the jump toe |

h_{t} | Tailwater depth |

GCI | Grid convergence index |

x_{1} | Jump toe position |

L_{1} | Roller length |

h_{1} | Flow depth at jump toe |

u_{b} | Bottom velocity |

E | Submergence parameter |

u_{max} | Maximum velocity |

L_{2} | Reattachment length |

η | Energy dissipation efficiency |

d_{c} | Critical step height |

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**Figure 1.**Schematic diagram of hydraulic jump stilling basins: (

**a**) traditional stilling basin; (

**b**) negative-step stilling basin.

**Figure 4.**Vertical velocity profiles at x = 0.5 m (Run M3): (

**a**) results of different meshes; and (

**b**) fine-mesh solution, with discretization error bars.

**Figure 5.**Validation of the numerical model: (

**a**) mean free-surface longitudinal profile; and (

**b**) bottom velocity profile (y = 2.0 cm).

**Table 1.**Representative high-head projects in China using the negative-step stilling basin [4].

No. | Name | Dam Height (m) | Design Discharge (m ^{3}/s) | Step Height (m) |
---|---|---|---|---|

1 | Xiangjiaba | 162.0 | 41,200 | 9.0 |

2 | Huangjinping | 95.5 | 5650 | - |

3 | Jin’anqiao | 160.0 | 11,668 | - |

4 | Guanyinyuan | 159.0 | 16,900 | 7.5 |

5 | Liyuan | 155.0 | 11,361 | 15.8 |

6 | Guandi | 168.0 | 14,000 | 6.5 |

7 | Tingzikou | 110.0 | 34,500 | 8.0 |

Run | d (cm) | θ (°) | Q (m^{3}/s) | q (m^{2}/s) | h_{1} (m) | u_{1} (m/s) | F_{1} | h_{t} (m) |
---|---|---|---|---|---|---|---|---|

M1 | 0 | 0 | 0.09 | 0.18 | 0.033 | 5.386 | 9.4 | 0.388 |

M2 | 5 | 0 | 0.09 | 0.18 | 0.034 | 5.262 | 9.1 | 0.388 |

M3 | 10 | 0 | 0.09 | 0.18 | 0.037 | 5.091 | 8.6 | 0.388 |

M4 | 12.5 | 0 | 0.09 | 0.18 | 0.035 | 5.087 | 8.6 | 0.388 |

M5 (S1) | 15 | 0 | 0.09 | 0.18 | 0.036 | 4.945 | 8.3 | 0.388 |

S2 | 15 | 5 | 0.09 | 0.18 | 0.038 | 4.782 | 7.9 | 0.388 |

S3 | 15 | 10 | 0.09 | 0.18 | 0.034 | 5.367 | 9.4 | 0.388 |

S4 | 15 | 15 | 0.09 | 0.18 | 0.034 | 5.301 | 9.2 | 0.388 |

S5 | 15 | 20 | 0.09 | 0.18 | 0.035 | 5.126 | 8.7 | 0.388 |

y (m) | ${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | ${\mathit{\varphi}}_{3}$ | ${\mathit{E}}_{21}$ | ${\mathit{E}}_{32}$ | ${\mathit{r}}_{21}$ | ${\mathit{r}}_{32}$ | P | ${\mathrm{GCI}}_{21}^{\mathrm{fine}}$ |
---|---|---|---|---|---|---|---|---|---|

0.017 | 2.742 | 2.799 | 2.804 | 0.021 | 0.002 | 1.3 | 1.3 | 9.45 | 0.7 |

0.035 | 2.667 | 2.744 | 2.890 | 0.029 | 0.053 | 1.3 | 1.3 | 2.42 | 10.8 |

0.052 | 2.292 | 2.427 | 2.764 | 0.059 | 0.139 | 1.3 | 1.3 | 3.48 | 11.3 |

0.069 | 1.824 | 1.947 | 2.367 | 0.067 | 0.215 | 1.3 | 1.3 | 4.69 | 6.3 |

0.087 | 1.415 | 1.476 | 1.919 | 0.043 | 0.300 | 1.3 | 1.3 | 7.59 | 1.2 |

0.104 | 1.054 | 1.108 | 1.416 | 0.051 | 0.277 | 1.3 | 1.3 | 6.62 | 1.5 |

0.122 | 0.767 | 0.788 | 0.835 | 0.027 | 0.059 | 1.3 | 1.3 | 3.14 | 2.0 |

0.139 | 0.477 | 0.513 | 0.266 | 0.075 | 0.481 | 1.3 | 1.3 | 7.36 | 0.8 |

0.156 | 0.233 | 0.291 | 0.170 | 0.248 | 0.416 | 1.3 | 1.3 | 2.82 | 6.6 |

0.174 | 0.067 | 0.156 | 0.190 | 1.337 | 0.219 | 1.3 | 1.3 | 3.66 | 6.9 |

0.191 | 0.196 | 0.193 | 0.294 | 0.015 | 0.525 | 1.3 | 1.3 | 13.51 | 0.0 |

0.208 | 0.341 | 0.301 | 0.411 | 0.115 | 0.363 | 1.3 | 1.3 | 3.91 | 2.7 |

0.226 | 0.472 | 0.459 | 0.521 | 0.027 | 0.136 | 1.3 | 1.3 | 6.05 | 0.4 |

0.243 | 0.590 | 0.561 | 0.613 | 0.048 | 0.092 | 1.3 | 1.3 | 2.30 | 4.3 |

0.261 | 0.681 | 0.674 | 0.679 | 0.011 | 0.008 | 1.3 | 1.3 | 1.34 | 2.2 |

0.278 | 0.739 | 0.739 | 0.723 | 0.000 | 0.022 | 1.3 | 1.3 | 24.00 | 0.0 |

0.295 | 0.785 | 0.796 | 0.752 | 0.014 | 0.055 | 1.3 | 1.3 | 5.35 | 0.4 |

0.313 | 0.807 | 0.824 | 0.769 | 0.022 | 0.067 | 1.3 | 1.3 | 4.43 | 1.0 |

0.330 | 0.804 | 0.834 | 0.779 | 0.038 | 0.067 | 1.3 | 1.3 | 2.33 | 4.5 |

Run | d (cm) | θ (°) | ${\mathit{x}}_{1}\left(\mathbf{m}\right)$ | ${\mathit{y}}_{1}\left(\mathbf{m}\right)$ | E |
---|---|---|---|---|---|

M1 | 0 | 0 | −0.380 | 0.196 | 0.41 |

M2 | 5 | 0 | −0.383 | 0.241 | 0.29 |

M3 | 10 | 0 | −0.377 | 0.235 | 0.30 |

M4 | 12.5 | 0 | −0.365 | 0.229 | 0.32 |

M5 (S1) | 15 | 0 | −0.219 | 0.164 | 0.51 |

S2 | 15 | 5 | −0.248 | 0.182 | 0.46 |

S3 | 15 | 10 | −0.227 | 0.191 | 0.43 |

S4 | 15 | 15 | −0.149 | 0.191 | 0.43 |

S5 | 15 | 20 | −0.153 | 0.206 | 0.39 |

Run | ${\mathit{x}}_{\mathrm{bmin}}\left(\mathbf{m}\right)$ | ${\mathit{u}}_{\mathrm{bmin}}(\mathbf{m}/\mathbf{s})$ | ${\mathit{x}}_{\mathrm{bmax}}\left(\mathbf{m}\right)$ | ${\mathit{u}}_{\mathrm{bmax}}(\mathbf{m}/\mathbf{s})$ |
---|---|---|---|---|

M1 | - | - | 0.041 | 5.293 |

M2 | 0.041 | −0.239 | 0.245 | 3.473 |

M3 | 0.163 | −0.892 | 0.612 | 2.954 |

M4 | 0.204 | −1.110 | 0.694 | 2.784 |

M5 | 0.204 | −1.356 | 0.531 | 2.365 |

S2 | 0.204 | −1.148 | 0.531 | 2.448 |

S3 | 0.163 | −0.932 | 0.490 | 2.588 |

S4 | 0.163 | −0.781 | 0.449 | 2.785 |

S5 | 0.122 | −0.169 | 0.408 | 2.944 |

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**MDPI and ACS Style**

Jiang, L.; Diao, M.; Wang, C.
Investigation of a Negative Step Effect on Stilling Basin by Using CFD. *Entropy* **2022**, *24*, 1523.
https://doi.org/10.3390/e24111523

**AMA Style**

Jiang L, Diao M, Wang C.
Investigation of a Negative Step Effect on Stilling Basin by Using CFD. *Entropy*. 2022; 24(11):1523.
https://doi.org/10.3390/e24111523

**Chicago/Turabian Style**

Jiang, Lei, Minjun Diao, and Chuan’ai Wang.
2022. "Investigation of a Negative Step Effect on Stilling Basin by Using CFD" *Entropy* 24, no. 11: 1523.
https://doi.org/10.3390/e24111523