# Analysis of the Flow in a Typified USBR II Stilling Basin through a Numerical and Physical Modeling Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}was used to model the spillway and stilling basin case study. This commercial software, widely used for hydraulic engineering, has proved to successfully reproduce hydraulic jumps as well as the flow taking place in spillways and energy dissipation structures [3,21,22,23]. On the other hand, a reduced-scale physical model with Froude similarity of the case study was built in the Hydraulics Laboratory of the Institute of Hydraulic Engineering and Water Resources Management (Technische Universität Wien, Wien, Austria), following the limiting criteria to avoid significant scale effects proposed by Heller [5].

_{1}= 1.33 m) whereas, for the end sill, the width and spacing of the blocks is 1.67 m.

## 2. Materials and Methods

#### 2.1. Numerical Model

^{®}[31]. In particular, the prototype scale dimensions were considered for the numerical model, whose characteristics are presented in the forthcoming subsections.

#### 2.1.1. Flow Equations and General Settings

#### 2.1.2. Free Surface Modeling

#### 2.1.3. Turbulence Modeling

#### 2.1.4. Air Entrainment

#### 2.1.5. Meshing and Boundary Conditions

#### 2.2. Physical Model

#### 2.2.1. Digital Image Processing

#### 2.2.2. Turbine Velocity Meter

^{®}). The working principle of this device is based on a turbine whose rotation frequency depends on the water velocity. Then, this rotation frequency is converted into an analog output that gives a measure of the flow velocity. The velocity profiles obtained in the physical model covered the following positions: the flow in the spillway, right before the entrance to the stilling basin, the flow in the stilling basin, close to the end sill, and the flow downstream of the end sill. Measures with the turbine velocity meter were taken during 180 s for each of the points forming the profiles, with an acquisition frequency of 2 Hz. The relatively low acquisition frequency of the device prevented taking velocity measures in the hydraulic jump roller, where velocity fluctuations are more intense.

#### 2.2.3. Pressure Transmitters

#### 2.2.4. Optical Fiber Probe

## 3. Results and Discussion

_{1}= 4.95 is considered for the numerical and physical models.

#### 3.1. Free Surface Profile

_{2}) and the upstream (y

_{1}) flow depth to the hydraulic jump. According to the values shown in Table 3, the sequent depth ratios were 6.25 and 6.07 for the numerical and the physical model, respectively. The sequent depth ratio was also calculated with the expression proposed by Hager and Bremen [52], based on Bélanger’s equation [53], which was developed for the particular case of a classical hydraulic jump (CHJ). The result obtained is 6.30. Therefore, both models provide slightly lower ratios. With reference to the bibliographical value, the numerical model has an accuracy of 99.2% and the physical model, 96.2%. It should be remarked, though, that these lower values of sequent depth ratios are in good agreement with the research presented by Hager and Li [54] and Padulano et al. [30] regarding USBR II stilling basins.

_{1}values used in [30] with respect to the ones above mentioned.

^{2}[58] as a measure of the accuracy of the modeled profiles, the FLOW-3D model achieves R

^{2}values of 0.979 and 0.977 when compared with Bakhmeteff and Matzke [56] and Wang and Chanson [8], respectively. For the physical model R

^{2}is 0.937 when compared to Bakhmeteff and Matzke [56] and 0.944 in relation to the profile by Wang and Chanson [8].

#### 3.2. Velocity Profiles

#### 3.3. Pressure Analysis

#### 3.4. Void Fraction Distribution

#### 3.4.1. Theoretical Development

#### 3.4.2. Void Fraction Analysis—Case study

_{1}3.7 and 6.3, respectively (Figure 7 and Figure 8).

^{2}values are 0.966 and 0.928 for the physical and numerical model, respectively. For the physical model, the rate of decrease for ${C}_{max}$ is very close to those reported by Murzyn et al. [18] and Chanson and Brattberg [17]. For the numerical model, though, such a rate is slightly slower, as shown in Figure 7a.

## 4. Conclusions

_{1}= 4.95. This case study could obviously be redesigned, changing the flow conditions and consequently adapting the structure to cover a wider range of Froude numbers. This will certainly provide a wider perspective of the possible hydraulic jump types taking place and their properties.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Case study: Creager profile spillway and United States Bureau of Reclamation Type II (USBR) type II stilling basin with the basic information regarding flow conditions and dimensions.

**Figure 3.**Physical model of the case study at the TUWien Hydraulics Laboratory with the profiles used to measure the void fraction in the experimental campaign (VF1–VF6).

**Figure 5.**Vertical profiles of streamwise normalized velocity measured upstream and downstream the end sill of the USBR II stilling basin: (

**a**) Physical model, (

**b**) numerical model.

**Figure 6.**Measured void fraction in a vertical profile with the adjusted expressions for the upper and the lower regions: (

**a**) Physical model (x/y

_{1}= 20.57), (

**b**) numerical model (x/y

_{1}= 20.64).

**Figure 7.**Parameters of the void fraction distribution in the hydraulic jump (lower region): (

**a**) Maximum void fraction (${C}_{max}$), (

**b**) normalized height for the maximum void fraction (${\xi}_{Cmax}$), (

**c**) diffusion coefficient ($D$).

**Figure 8.**Parameters of the void fraction distribution in the hydraulic jump (upper region): (

**a**) Normalized height at which the void fraction is 0.95 (${\xi}_{\mathrm{C}95}$), (

**b**) normalized height at which the void fraction is 0.5 (${\xi}_{C0.5}$), (

**c**) normalized height of the boundary between regions (${\xi}_{*}$), (

**d**) diffusion coefficient ($D$).

Mesh | Nested Block Cell Size | Containing Block Cell Size |
---|---|---|

1 | 0.400 m | 0.800 m |

2 | 0.250 m | 0.500 m |

3 | 0.180 m | 0.360 m |

4 | 0.135 m | 0.270 m |

Mesh Combination | Model Apparent Order (p) | Grid Convergence Index (GCI) |
---|---|---|

1-2-3 | 2.78 | 6.0% |

2-3-4 | 5.13 | 6.4% |

1-3-4 | 133.23 | 3.6% |

1-2-4 | 2.51 | 13.6% |

Model | Supercritical Flow Depth (y_{1}) | Subcritical Flow Depth (y_{2}) | Unit Discharge (q) | Inflow Froude Number (Fr_{1}) |
---|---|---|---|---|

Numerical model | 1.520 m | 9.500 m | 29.143 m^{2}/s | 4.97 |

Physical model | 0.061 (1.525) ^{1} m | 0.370 (9.250) ^{1} m | 0.233 (29.143) ^{1} m^{2}/s | 4.93 |

^{1}In parenthesis: values at prototype scale applying the scale factor.

Model | x/y_{1} | |||||||
---|---|---|---|---|---|---|---|---|

Numerical model | 1.32 | 5.84 | 11.18 | 20.64 | 26.56 | 33.96 | ||

Physical model | 1.31 | 5.82 | 11.14 | 20.57 | 26.47 | 33.85 |

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**MDPI and ACS Style**

Macián-Pérez, J.F.; García-Bartual, R.; Huber, B.; Bayon, A.; Vallés-Morán, F.J.
Analysis of the Flow in a Typified USBR II Stilling Basin through a Numerical and Physical Modeling Approach. *Water* **2020**, *12*, 227.
https://doi.org/10.3390/w12010227

**AMA Style**

Macián-Pérez JF, García-Bartual R, Huber B, Bayon A, Vallés-Morán FJ.
Analysis of the Flow in a Typified USBR II Stilling Basin through a Numerical and Physical Modeling Approach. *Water*. 2020; 12(1):227.
https://doi.org/10.3390/w12010227

**Chicago/Turabian Style**

Macián-Pérez, Juan Francisco, Rafael García-Bartual, Boris Huber, Arnau Bayon, and Francisco José Vallés-Morán.
2020. "Analysis of the Flow in a Typified USBR II Stilling Basin through a Numerical and Physical Modeling Approach" *Water* 12, no. 1: 227.
https://doi.org/10.3390/w12010227