# Numerical Study of Fluctuating Pressure on Stilling Basin Slab with Sudden Lateral Enlargement and Bottom Drop

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Physical Model and Problem Description

#### 2.1. Experimental Setup

#### 2.2. Large Eddy Simulation (LES) Model

^{−8}s. According to the sampling theorem, the sampling interval time was 0.02 s. Consequently, the total calculation completion time was set to 52.5 s. After the calculation was stable, the simulation time was 30–50.48 s for analysis. In order to analyze the fluctuating pressure in the time domain, the total sampling time was 20.48 s, and 1024 sets of discrete fluctuating pressure data points were accumulated.

#### 2.3. Computational Domain and Boundary Conditions

## 3. Numerical Methodology and Model Validation

#### 3.1. Grid Sensitivity Analysis

#### 3.2. Comparison of Flow Pattern, Water Surface Profile, and Velocity

**I**), impinging zone (

**II**), and wall-attached jet zone (

**III**). Combined with Figure 4, it can be concluded that because the sluice and the bottom plate are connected by a falling drop, there is a certain thickness of water cushion in the downstream of the stilling basin.

_{C}represents calculation velocity and V

_{T}represents test velocity.

**II**), there is a velocity stagnation point (Velocity = 0) at x = 60 cm. While the water flows into the stilling basin, the downstream flow gradually descends along the way, forming a horizontal vortex at the bottom of the submerged jet zone. At the same time, due to the influence of the horizontal vortex, the velocity gradient is large, causing an underflow swirling zone in the front of the stilling basin. The flow velocity in zone (

**I**) is negative because of the horizontal clockwise vortex with an elliptical shape underneath the main jets, and the flow velocity of the wall-attached jet zone in the middle and rear part changes smoothly.

#### 3.3. Verification of Pressure

**II**), the main flow of the submerged jet impacted the bottom of the stilling basin, so the time-average hydrodynamic pressure acting on the bottom of the stilling basin slab (

**II**) increased sharply. In the underflow swirling zone, the RMS of fluctuating pressure increased along the way. The pressure at the 4th measuring point (x = 68.3 cm) reached the maximum value and then gradually decreased and remained stable. Both fluctuating and maximum time-average pressure occurred at the 4th measuring point. The comparison of RMS of maximum and minimum pressure is shown in Figure 8b. The calculated maximum pressure is slightly smaller than the experimental maximum pressure. However, the minimum pressure result is the opposite. The reason for this phenomenon is that actual water current fluctuates more sharply.

## 4. Results and Discussion

#### 4.1. Qualitative Analysis of RMS of Fluctuating Pressure

^{3}/s in physical model, Q = 16,652 m

^{3}/s in prototype) is the largest of three working conditions.

#### 4.1.1. Qualitative Analysis of Longitudinal Distribution

**I**) and impinging zone (

**II**), the RMS value gradually increases and reaches the peak value. In the stilling basin, the bottom velocity decreases and its distribution is more uniform. Therefore, the RMS value of fluctuating pressure gradually decreases and tends to be low. Results indicated that fluctuating pressure increase with x/Ls (Ls = stilling basin length), when it reaches the highest point, comes down. Near the sill, the water depth increases and the time-average pressure increases gradually along the way, according to the distribution of hydrostatic pressure. Because the residual flow of water has a certain impact on the tail sill of the stilling basin, the RMS value at the end of the stilling basin increases slightly, and the value of RMS value increases, with decreasing inflow Froude number, which is clearly similar to the experimental results of Yan et al. [25].

#### 4.1.2. Qualitative Analysis of Horizontal Distribution

#### 4.1.3. RMS of Fluctuating Pressure Distribution at Different Flow Rates

_{max}) and the location of occurrence under three operating conditions are given in Table 3.

#### 4.2. Quantitative Analysis of Fluctuating Pressure

#### 4.2.1. Mathematical Model

#### 4.2.2. Quantitative Analysis of Longitudinal Distribution

#### 4.2.3. Cross-Sectional Distribution Quantitative Analysis

_{max}point of the stilling basin is y = 60 cm, which is located in the core area of the vertical vortex. The distribution of fluctuating pressure is mainly affected by the vorticity. Therefore, quantitative analysis of the RMS distribution law of fluctuating pressure on the stilling basin slab with sudden lateral enlargement and bottom drop needs to be comprehensively considered with multiple influencing factors. The difference of flow patterns in different regions will inevitably affect the distribution of fluctuating pressure.

#### 4.3. Discussion

_{1}(y

_{1}is the inflow water depth entering the jump, θ is the angle of entering flow) along with horizontal slab of stilling basin for different inlet Froude number of 5.0 [33] and 5.3 (numerical model in this research). The pressure fluctuation beneath free jump in the spillway with the entering flow at an angle of 10° has been subject to several experimental researches by Gunal. In Figure 13, η = 0 represents the center line of the stilling basin slab, and η = 0.5 represents the half of slab width direction (y = 40 cm).

_{1}= 0) due to the high turbulence intensity formed by hydraulic jump. In Figure 13, it is observed that fluctuating pressure coefficient first tends to decrease in the range 0 < x/y

_{1}< 2.5, then increases gradually. For the stilling basin with bottom drop, ${{\mathrm{C}}_{\mathrm{p}}}^{\prime}$ can reach the maximum value at the position of about 5.0 < x/y

_{1}< 6.5 within the impinging zone (

**II**). The results are in accordance with by Yan et al. [25]. and Naseri et al. [21].

## 5. Conclusions

- The flow pattern, velocity distribution, time-average pressure, root mean square (RMS) of fluctuating pressure, maximum and minimum pressure of a stilling basin slab of the water flow obtained by numerical simulation are in good agreement with the experimental results, indicating that it is advisable to use large eddy simulation to study the fluctuating pressure of stilling basin slab.
- Due to the superposition of the horizontal and vertical vortex, the turbulence and mixing of water in the front of the stilling basin and the extension of the side wall of the vent are severe, resulting in a large RMS of fluctuating pressure in this area, which requires attention. With the increase of per-unit width discharge, the peak point of σ deviates from the center line of the stilling basin and approaches the side wall line. Both sudden lateral enlargement and bottom drop will result in the difference distributions of spatial hydraulic jumps compared with those of equivalent classical hydraulic jumps.
- The RMS of fluctuating pressure longitudinally changes along the center line of the stilling basin, first increasing, then decreasing, and finally increasing slightly. The submerged jet zone is mainly affected by the vortex body, and the impinging zone is affected by the fluctuating velocity and the vortex body. The wall-attached jet zone is mainly caused by the fluctuating velocity. The horizontal direction from the front of the stilling basin along the center to the side wall shows a trend of first decreasing, increasing, and then decreasing, which is highly correlated with the vorticity distribution, but has little correlation with the fluctuating velocity distribution.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Flow pattern of operating condition 1: (

**a**) Physical model flow diagram; (

**b**) Simulation model flow.

**Figure 8.**Comparison of pressure at 10 pressure measurement points: (

**a**) Comparison of root mean square (RMS) of fluctuating and time-average pressure; (

**b**) Comparison of maximum and minimum pressure.

**Figure 9.**The distribution diagram of time-average pressure and RMS of fluctuating pressure of bottom of stilling basin under three cases.

**Figure 10.**Speed vector in stilling basin at z = 40 cm horizontal plane and bottom plate plane in three cases.

**Figure 11.**Distribution of fluctuating velocity and vorticity in the horizontal section of the stilling basin slab: (

**a**) Distribution of fluctuating velocity; (

**b**) Distribution of fluctuating vorticity.

**Figure 12.**Distribution of root time-average square fluctuating pressure of x = 50 cm cross-section of bottom plate: (

**a**) Distributions of fluctuating velocity and fluctuating pressure; (

**b**) Distributions of fluctuating vorticity and fluctuating pressure.

Condition | Flow Discharge (m ^{3}/s) | Inflow Froude Number | Inflow Velocity (m/s) | Inflow Water Depth (m) |
---|---|---|---|---|

1 | 0.942 | 5.295 | 5.611 | 0.114 |

2 | 0.643 | 4.545 | 4.489 | 0.097 |

3 | 0.232 | 4.227 | 3.018 | 0.052 |

Grid | Containing Block Cell Size (m) | Nested Block Cell Size (m) | Discharge (m ^{3}/s) | Relative Error (%) |
---|---|---|---|---|

1 | 0.050 | 0.025 | 0.990 | 5.10 |

2 | 0.040 | 0.020 | 0.969 | 2.70 |

3 | 0.030 | 0.015 | 0.956 | 1.49 |

4 | 0.020 | 0.010 | 0.952 | 1.06 |

Condition | σ_{max} (Pa) | σ_{max} Point Coordinates (cm) |
---|---|---|

1 | 2139 | (50,60) |

2 | 1253 | (35,55) |

3 | 932 | (30,35) |

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**MDPI and ACS Style**

Lu, Y.; Yin, J.; Yang, Z.; Wei, K.; Liu, Z.
Numerical Study of Fluctuating Pressure on Stilling Basin Slab with Sudden Lateral Enlargement and Bottom Drop. *Water* **2021**, *13*, 238.
https://doi.org/10.3390/w13020238

**AMA Style**

Lu Y, Yin J, Yang Z, Wei K, Liu Z.
Numerical Study of Fluctuating Pressure on Stilling Basin Slab with Sudden Lateral Enlargement and Bottom Drop. *Water*. 2021; 13(2):238.
https://doi.org/10.3390/w13020238

**Chicago/Turabian Style**

Lu, Yangliang, Jinbu Yin, Zhou Yang, Kebang Wei, and Zhiming Liu.
2021. "Numerical Study of Fluctuating Pressure on Stilling Basin Slab with Sudden Lateral Enlargement and Bottom Drop" *Water* 13, no. 2: 238.
https://doi.org/10.3390/w13020238