# On the Effect of Block Roughness in Ogee Spillways with Flip Buckets

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## Abstract

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## 1. Introduction

^{®}software to simulate the flow with regular mesh. Steiner et al. [7] investigated the ski jump hydraulics in a laboratory model study related to circular bucket geometry. Kermannejad et al. [8] worked on dynamic pressure due to the impact of a ski jump out of a flip bucket downstream of a chute spillway model. Their results showed that the pressure coefficient is highly sensitive to horizontal and vertical distances from the impact location and impact angle. Zhenvwei and Zhiyan [9], and Parsaie et al. [10] investigated flow characteristics over chutes and the cavitation phenomenon using computational-fluid-dynamics (CFD) approaches on a spillway’s flip bucket. Daneshfaraz and Ghaderi [11] studied the effect of a reverse arch of an ogee spillway on pressure on the spillway. Yamini et al. [12] experimentally investigated the effect of entrance flow conditions on pressure fluctuation on the bed of compound flip buckets of the Gotvand dam in Iran.

## 2. Methods

#### 2.1. Energy-Dissipation Analysis

_{0}is the relative energy dissipation, H

_{dam}is the dam height, ρ is the water density, μ is the dynamic viscosity of water, g is the gravitational acceleration, V the approaching flow velocity, y the flow depth, R the bucket’s radius, θ the take-off angle, σ the surface tension, and a the dimension of the roughness block. According to the Buckingham π theorem, the general relationship below can be obtained:

_{dam}is the relative critical flow depth, Fr is the approach flow Froude number to the bucket, We is the Weber number, and Re is the Reynolds number of the flow approaching the bucket, assuming the same value in all experiments (Re > 1 × 10

^{5}), thus not influencing Equation (2) [16]. Moreover, the effect of surface tension can be neglected: when flow depth on the spillway was more than 0.05 m (in the experiments, the least flow depth on the spillway was 0.053 m), inertia force dominated, and the effects of the Weber number (We = ρV

^{2}L/σ) could be neglected. In this research, longitudinal dimension and roughness height a were fixed. Thus, the final equation can be summarized as:

#### 2.2. Multiphase (Water/Air) Hydraulic Model

^{®}computational package, able to solve complex fluid dynamic problems, was used. This software shows high performance in modeling unsteady free-surface flows: it utilizes the finite-volume method for structured meshes to solve the three-dimensional Reynolds-averaged Navier–Stokes equations of fluid motion. These equations can be written in a Cartesian coordinate system (x, y, z) as follows [17,18]:

_{F}the fractional volume open to flow in the fractional area/volume obstacle representation (FAVOR) method, p the pressure, and R the source term.

_{k}is the production of turbulent kinetic energy that arises due to mean velocity gradient, G

_{b}is turbulent kinetic energy production from buoyancy, Y

_{M}is the fluctuating dilation in compressible turbulence, and α

_{k}and α

_{ε}are inverse effective Prandtl numbers for turbulent kinetic energy and its dissipation, respectively [28]. In the above equations, ${\alpha}_{k}$ = ${\alpha}_{s}$ = 1.39, C

_{1s}= 1.42, and ${C}_{2\epsilon}$= 1.6 are model constants. All constants were explicitly derived in the RNG procedure. Terms S

_{k}and ${S}_{\epsilon}$ are sources terms for $k$ and $\epsilon $, respectively. In addition, R

_{ε}is a proper term of the RNG model with respect to the k–ε one. The following equations provide details on how effective viscosity is determined, considering molecular and turbulent effects:

_{t}is turbulence viscosity and $\mu $

_{eff}is effective viscosity.

## 3. Experimental and Numerical Test Cases

#### 3.1. Experiment Facilities

^{3}/s, connected to two rotameters with ±2% accuracy [34,35]. To eliminate turbulence in the entrance region, a planar mesh was added. At the inlet of the flume, there was a screen that eliminated the flow turbulence, and flow slowly entered the laboratory flume; to ensure steady flow, spillway models were installed 1.5 m downstream of the inlet tank. The physical model of the ogee spillway, based on the procedure reported in USBR (1987) [36], was fabricated from dense polyethylene and located 0.7 m far from the inlet of the flume (Figure 1), with a scale of 1:33 (height of 0.27 m, length of 0.4 m, and width of 0.3 m) on the basis of the Froude similarity criterion between model and prototype. It includes the whole ogee spillway and chute, which is 0.4 m long and connected with the flip bucket at the exit. After some pre-experiments, two types of flip buckets with take-off angles θ = 35° and 52 °C, and radii R = 19 and 12 cm, respectively, were considered. Block roughness with length and width of 1.5 cm, and height of 0.5 cm was considered in the bed of the ogee spillway.

_{r}is the relative energy dissipation, E

_{0}is the total energy of upstream flow, and E

_{1}is the current energy in the base of the spillway. Figure 2 and Table 1 illustrate the dimensions of the ogee spillway model with the roughness block, and list the geometric and hydraulic conditions, respectively (parameter a in Figure 2 is the dimension of the roughness block). Considering the 1:33 scale of physical model, approach flow Froude number Fr = V/(gh)

^{0.5}was about 1.29–3.62, with Reynolds number Re = Vh/ν being about 2.07 × 10

^{4}–3.895 × 10

^{4}, where ν is the kinematic water viscosity, g is the gravitational acceleration, and V and h are the mean velocity and depth of approach flow, respectively.

#### 3.2. Numerical Domain

^{®}software was used to build up the geometry of the models with a stereolithography (STL) file. According to the experiment conditions, the following boundary conditions were employed:

- inlet boundary condition was set as discharge flow rate (Q);
- outflow (O) boundary condition was used downstream (at a sufficiently far location to prevent boundary effects on the results);
- the bottom and side boundaries were treated as a rigid wall (W), and no-slip conditions were applied at the wall boundaries;
- an atmospheric boundary condition was set to the upper boundary of the channel, which allowed for the flow to enter and leave the domain; and
- symmetry boundary condition (S) was also used at the inner boundaries.

_{min}boundary was evaluated to be 4H

_{dam}= 1.1 m. In fact, monitoring the water-surface variations at different stations upstream of the ogee spillway revealed that 1 m was sufficient for an undisturbed approach flow to be established. Figure 3A shows the computational domain of the present study and associated boundary conditions.

## 4. Results

#### 4.1. Verification of Numerical Model and Laboratory Results

^{®}and experimental data were calculated according to Equation (14):

_{SE(Exp)}is the experimental free-surface elevation, and F

_{SE(Num)}is the numerical counterpart. Figure 5 and Table 3 show free-surface profiles and errors. Good agreement was found between numerical and experimental data, and both of them represent a similar trend. The maximal difference between the data was related to X = 0.25 m and is equal to ~7%, and the overall mean values of the relative errors was ~2%, which confirmed the ability of the numerical model to predict flow specifications over the ogee spillway. A summary of the overall mean values of the relative errors generated for other discharges is shown in Table 4.

#### 4.2. Flow Pattern Downstream of Ogee Spillway with Block Bed and Different Buckets

#### 4.3. Energy Dissipation in Ogee Spillway with Block Bed and Different Buckets

#### 4.4. Jet Length in Ogee Spillway with Block Bed and Different Buckets

^{2}= 0.95 via ± 7% relative error and R

^{2}= 0.96 via ± 7.4% relative error, respectively. A scatter of points relative to the linear regression indicated that the experimental and computational values were in good agreement.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Comparison of free-surface profiles from FLOW-3D software with laboratory data; Q = 0.0085 m

^{3}/s.

**Figure 6.**Streamlines on x–y plane. (

**A**) Without buckets; (

**B**) buckets with θ = 32°; (

**C**) buckets with θ = 52°.

**Figure 7.**Streamlines on x–y plane. (

**A**) Without buckets and smooth bed; (

**B**) without buckets and block bed.

**Figure 8.**Difference in relative energy dissipation versus Froude number in block bed with different take-off angles.

**Figure 9.**Difference in relative energy dissipation in smooth and block beds. (

**A**) Without bucket; (

**B**) bucket with θ = 32° (

**C**) bucket with θ = 52°.

**Figure 11.**Difference in jet length versus Froude number in flip bucket with different take-off angles.

**Figure 12.**Difference in jet length in smooth and block beds. Bucket with (

**A**) θ = 32° and (

**B**) θ = 52°.

**Figure 13.**Comparison between observed and computational relative energy dissipation and jet length.

Range | Q (m^{3}/s) | y (cm) | R (cm) | $\mathsf{\Theta}\text{}\left(\mathbf{Deg}\right)$ | Fr | Re |
---|---|---|---|---|---|---|

Min | 0.005 | 5.3 | 12 | 32 | 1.29 | 20,700 |

Max | 0.014 | 11.7 | 19 | 52 | 3.62 | 38,950 |

Test No. | Coarser Cell Size (cm) | Finer Cell Size (cm) | Total Mesh Number | MAPE (%) * $100\times \frac{\mathbf{1}}{\mathbf{n}}{\displaystyle {\displaystyle \sum}_{\mathbf{1}}^{\mathbf{n}}}\left|\frac{{\mathbf{X}}_{\mathbf{exp}}-{\mathbf{X}}_{\mathbf{num}}}{{\mathbf{X}}_{\mathbf{exp}}}\right|$ | ** RMSE (cm) $\sqrt{\frac{\mathbf{1}}{\mathit{n}}{\displaystyle \sum _{\mathbf{1}}^{\mathit{n}}({\mathbf{X}}_{\mathbf{exp}}}-{\mathbf{X}}_{\mathbf{num}}{)}^{\mathbf{2}}}$ |
---|---|---|---|---|---|

T1 | 1.2 | 0.55 | 944,125 | 19.24 | 4.6 |

T2 | 1.1 | 0.50 | 1,335,000 | 8.89 | 1.94 |

T3 | 0.95 | 0.45 | 1,769,834 | 8.45 | 1.75 |

T4 | 0.80 | 0.40 | 2,541,311 | 2.42 | 0.39 |

T5 | 0.70 | 0.35 | 3,906,163 | 1.90 | 0.33 |

_{exp}: experimental value of X; X

_{num}: numerical value of X; n: data count. ** Root mean square error; X

_{exp}: experimental value of X; X

_{num}: numerical value of X; n: data count.

X(m) | Fs_{E} (m)_{Exp} | Fs_{E} (m)_{Num} | Relative Errors (%) | RMSE (cm) |
---|---|---|---|---|

−0.28 | 0.326 | 0.330 | 1.22 | 0.04 |

−0.20 | 0.327 | 0.331 | 1.22 | 0.04 |

−0.15 | 0.325 | 0.330 | 1.53 | 0.05 |

−0.10 | 0.324 | 0.328 | 1.22 | 0.04 |

−0.05 | 0.324 | 0.325 | 0.38 | 0.01 |

0.00 | 0.322 | 0.323 | 0.38 | 0.01 |

0.05 | 0.309 | 0.313 | 1.22 | 0.04 |

0.15 | 0.217 | 0.223 | 2.76 | 0.06 |

0.25 | 0.116 | 0.124 | 6.89 | 0.08 |

0.35 | 0.074 | 0.076 | 2.07 | 0.02 |

0.45 | 0.083 | 0.088 | 1.53 | 0.05 |

0.55 | 0.133 | 0.139 | 2.76 | 0.06 |

0.60 | 0.144 | 0.148 | 1.22 | 0.04 |

0.65 | 0.146 | 0.149 | 2.03 | 0.03 |

Mean | 1.88 | 0.04 |

Q(m^{3}/s) | Mean Relative Errors (%) | Mean RMSE(cm) |
---|---|---|

0.005 | 1.42 | 0.02 |

0.007 | 1.21 | 0.04 |

0.0085 | 1.88 | 0.04 |

0.010 | 1.56 | 0.03 |

0.011 | 1.42 | 0.04 |

0.013 | 1.26 | 0.03 |

0.014 | 1.47 | 0.04 |

Take-Off Angles (θ) | Bed | Q (m^{3}/s) | |||||||
---|---|---|---|---|---|---|---|---|---|

0.005 | 0.007 | 0.0085 | 0.010 | 0.011 | 0.013 | 0.014 | |||

∆E_{r} | Without flip bucket | Smooth | 0.66 | 0.61 | 0.59 | 0.59 | 0.58 | 0.57 | 0.54 |

Blocked | 0.78 | 0.72 | 0.67 | 0.65 | 0.64 | 0.61 | 0.58 | ||

32° | Smooth | 0.75 | 0.70 | 0.67 | 0.66 | 0.66 | 0.61 | 0.58 | |

Blocked | 0.81 | 0.78 | 0.75 | 0.71 | 0.69 | 0.65 | 0.64 | ||

52° | Smooth | 0.83 | 0.79 | 0.74 | 0.73 | 0.71 | 0.68 | 0.65 | |

Blocked | 0.83 | 0.81 | 0.78 | 0.76 | 0.75 | 0.71 | 0.69 |

Take-Off Angles (θ) | Bed | Q (m^{3}/s) | |||||||
---|---|---|---|---|---|---|---|---|---|

0.005 | 0.0070 | 0.0085 | 0.010 | 0.011 | 0.013 | 0.014 | |||

L | 32° | Smooth | 20.5 | 24.5 | 29.5 | 32.7 | 37.7 | 40.3 | 42.8 |

Blocked | 8.6 | 14.0 | 20.5 | 32.0 | 32.0 | 35.9 | 39.5 | ||

52° | Smooth | 12.4 | 15.5 | 19.2 | 31.0 | 31.0 | 38.5 | 43.0 | |

Blocked | 7.4 | 12.5 | 15.5 | 23.5 | 23.5 | 30.7 | 36.0 |

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**MDPI and ACS Style**

Daneshfaraz, R.; Ghaderi, A.; Akhtari, A.; Di Francesco, S.
On the Effect of Block Roughness in Ogee Spillways with Flip Buckets. *Fluids* **2020**, *5*, 182.
https://doi.org/10.3390/fluids5040182

**AMA Style**

Daneshfaraz R, Ghaderi A, Akhtari A, Di Francesco S.
On the Effect of Block Roughness in Ogee Spillways with Flip Buckets. *Fluids*. 2020; 5(4):182.
https://doi.org/10.3390/fluids5040182

**Chicago/Turabian Style**

Daneshfaraz, Rasoul, Amir Ghaderi, Aliakbar Akhtari, and Silvia Di Francesco.
2020. "On the Effect of Block Roughness in Ogee Spillways with Flip Buckets" *Fluids* 5, no. 4: 182.
https://doi.org/10.3390/fluids5040182