# Flow Turbulence and Pressure Fluctuations in a Hydraulic Jump

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## Abstract

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## 1. Introduction

- (1)
- To quantify the spatial variations in the flow characteristics and pressure fluctuation distribution under hydraulic jump flow conditions;
- (2)
- To determine the relationship between flow property variables (horizontal and vertical velocities, horizontal and vertical turbulence intensities, water level, and water level fluctuation) and pressure fluctuation;
- (3)
- To propose an empirical formula for estimating the pressure fluctuation coefficient at the bottom of the structure using dimensionless numbers.

## 2. Materials and Methods

#### 2.1. Experimental Setup and Conditions

#### 2.2. Measurement Method

## 3. Results

#### 3.1. Validation of Velocity, Water Level, and Pressure Distribution

#### 3.2. Analysis of Correlation between Flow Properties and Pressure Fluctuations

## 4. Discussions

## 5. Conclusions

- The water level fluctuation had a minimal relationship with the pressure fluctuation at the bottom of the structure and the location of the maximum pressure fluctuation was identified at points where $(x-{x}_{1})/({H}_{d}-{y}_{0})\approx $ 2.3.
- A comprehensive correlation analysis between the flow properties and pressure fluctuations was performed by dividing the flow region into upstream and downstream areas from the maximum pressure fluctuation point. The analysis results indicated that the water level and turbulence intensity were the main factors influencing the pressure fluctuations. A linear relationship between the turbulence intensity and pressure fluctuation was demonstrated, and the horizontal turbulence intensity was consistently larger than the vertical one.
- An empirical formula was proposed for estimating the pressure fluctuation at the bottom of the structure using the novel dimensionless number. It was suggested that the pressure fluctuation under the influence of weight force and turbulence may undergo a critical regime shift and the trends of the pressure fluctuation may be significantly changed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagram of the experimental channel and the instrumental setup for the measurements: ${y}_{0}$ = the incoming supercritical flow before the jump; ${x}_{0}$ = the jump toe position; ${u}_{0}$ = the velocity of the incoming supercritical flow; ${H}_{u}$ = the water level upstream of the weir; and ${H}_{d}$ = the water level downstream of the hydraulic jump.

**Figure 4.**Comparison of the overlapping data between the PIV and BIV measurements in case 2: (

**a**) the locations of the measurements; (

**b**) the vertical profiles of the time-averaged velocity in the horizontal direction; (

**c**) the vertical profiles of the time-averaged velocity in the vertical direction [20].

**Figure 5.**Flow characteristics of the hydraulic jump with Fr = 7.3: (

**a**) the horizontal turbulence intensity; (

**b**) the vertical turbulence intensity; (

**c**) the horizontal velocity; (

**d**) the vertical velocity.

**Figure 7.**Longitudinal profiles of the depth-averaged flow properties in case 1: (

**a**) the water level, (

**b**) the pressure on the bottom.

**Figure 8.**Relationships between the dimensionless variables of the pressure fluctuations and the flow properties: (

**a**) the water level; (

**b**) the water level fluctuation; (

**c**) the horizontal velocity; (

**d**) the horizontal turbulence intensity; (

**e**) the vertical velocity; (

**f**) the vertical turbulence intensity.

**Figure 10.**Dimensionless pressure fluctuations with the variations in the dimensionless water levels (h*) and the vertical turbulence intensities (v*).

Case | Discharge (m^{3}/s) | H_{u} (m) | H_{d} (m) | y_{0} (m) | u_{0} (m/s) | x_{0} (m) | $\mathit{F}\mathbf{r}$ ($=\frac{{\mathit{u}}_{0}}{\sqrt{\mathit{g}{\mathit{y}}_{0}}}$) | $\mathit{R}\mathbf{e}$ ($=\frac{{\mathit{u}}_{0}{\mathit{y}}_{0}}{\mathit{\nu}}$) |
---|---|---|---|---|---|---|---|---|

1 | 0.0063 | 0.355 | 0.0900 | 0.0085 | 2.47 | 0.125 | 8.6 | $2.1\times {10}^{4}$ |

2 | 0.0893 | 0.216 | ||||||

3 | 0.0047 | 0.347 | 0.0766 | 0.0078 | 2.01 | 0.125 | 7.3 | $1.6\times {10}^{4}$ |

4 | 0.0754 | 0.214 | ||||||

5 | 0.0035 | 0.338 | 0.0604 | 0.0068 | 1.71 | 0.280 | 6.6 | $1.2\times {10}^{4}$ |

6 | 0.0019 | 0.327 | 0.0355 | 0.0053 | 1.19 | 0.365 | 5.2 | $6.3\times {10}^{3}$ |

**Table 2.**Correlation coefficients of the dimensionless variables of the pressure fluctuations and the flow properties upstream and downstream of the locations of the pressure fluctuation peaks.

Flow Properties | $\mathit{h}/({\mathit{u}}_{0}^{2}/2\mathit{g})$ | ${\mathit{h}}_{\mathit{r}\mathit{m}\mathit{s}}/({\mathit{u}}_{0}^{2}/2\mathit{g})$ | $\mathit{u}/{\mathit{u}}_{0}$ | ${\mathit{u}}_{\mathit{r}\mathit{m}\mathit{s}}/{\mathit{u}}_{0}$ | $\mathit{v}/{\mathit{u}}_{0}$ | ${\mathit{v}}_{\mathit{r}\mathit{m}\mathit{s}}/{\mathit{u}}_{0}$ |
---|---|---|---|---|---|---|

Entire section | −0.2980 | 0.3931 | 0.7904 | 0.6419 | 0.4063 | 0.7783 |

Developing air–water flow region | 0.8745 | 0.7936 | 0.1595 | −0.7126 | 0.3050 | −0.4968 |

Developed air–water flow region | −0.8421 | 0.3897 | 0.9534 | 0.9125 | 0.6955 | 0.9760 |

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**MDPI and ACS Style**

Kim, H.S.; Choi, S.; Park, M.; Ryu, Y.
Flow Turbulence and Pressure Fluctuations in a Hydraulic Jump. *Sustainability* **2023**, *15*, 14246.
https://doi.org/10.3390/su151914246

**AMA Style**

Kim HS, Choi S, Park M, Ryu Y.
Flow Turbulence and Pressure Fluctuations in a Hydraulic Jump. *Sustainability*. 2023; 15(19):14246.
https://doi.org/10.3390/su151914246

**Chicago/Turabian Style**

Kim, Hyung Suk, Seohye Choi, Moonhyeong Park, and Yonguk Ryu.
2023. "Flow Turbulence and Pressure Fluctuations in a Hydraulic Jump" *Sustainability* 15, no. 19: 14246.
https://doi.org/10.3390/su151914246