# Three-Dimensional Investigation of Hydraulic Properties of Vertical Drop in the Presence of Step and Grid Dissipators

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Turbulence Model

_{i}and ${u}^{\prime}{}_{i}$ are the mean and fluctuating velocities in the direction x

_{i}, where x

_{i}= (x, y, z), U

_{i}= (U, V, W) and ${u}^{\prime}{}_{i}$ = (u’, v’, w’). The symbols ρ, μ, P and g

_{i}are specific gravity, dynamic viscosity, pressure, and gravitational acceleration. The instantaneous velocity is obtained by superposing the time-averaged and instantaneous fluctuating velocity u

_{i}= U

_{i}+ ${u}^{\prime}{}_{i}$ in all three directions. In the software, turbulence can be simulated using the following turbulence models:

- Large Vortex Simulation Model (LES),
- Model of two equations (k-ɛ),
- Model of Re-Normalized Group (RNG),
- Prandtl mixing length.

_{k}is the production of turbulent kinetic energy due to velocity gradient; G

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

_{M}is turbulence dilation oscillation distribution [15,23,24]. In the above equations, α

_{k}= α

_{s}= 1.39, C

_{1s}= 1.42 and C

_{2s}= 1.6 are model constants. The terms S

_{k}and S

_{ε}are source terms for k and ɛ, respectively. The turbulent viscosity is added to the molecular viscosity to obtain μ

_{eff}effective viscosity. The volume of fluid (VOF) method consists of three main components: fluid ratio function, VOF transport equation solution, and boundary conditions at the free surface. The VOF transport equation is expressed by Abbasi, et al. [25]:

#### 2.2. Simulation Specification and Solution Field Network

_{n}and L

_{G}are the grid dissipator length and width, respectively. The amount of L

_{n}is equal to the width of the channel, where the L

_{G}is equal to 40 cm, for which the amount of grid length has been used from the Kabiri-Samani et al. [6] study.

- (i)
- inlet boundary = volume flow rate (VFR),
- (ii)
- outlet boundary = outflow,
- (iii)
- the bottom and side, boundary = Wall, v: the top boundary = symmetry.

#### 2.3. Dimensional Analysis

^{3}T

^{−1}], ρ is density [ML

^{−3}], μ is the kinematic viscosity [ML

^{−1}T

^{−1}], g is the gravitational acceleration [LT

^{−2}], y

_{u}is the upstream depth [L], y

_{d}is the downstream depth [L], y

_{b}represents the edge of the drop depth [L], H is the drop height [L], h is the step height [L], B refers to the channel width [L], a and b are the length and width. The grid dissipators are also annotated [L], E

_{u}is the upstream specific energy [L], E

_{d}is the downstream specific energy [L], y

_{p}refers to the pool depth [L], y

_{c}to the critical depth [L] and ɛ is the grid dissipator porosity [-]. Using the Pi–Buckingham method and considering the parameters ρ, g, and y

_{u}as iterative parameters, dimensional analysis can be performed. Relative energy dissipation and downstream relative depth are extracted as dimensionless parameters according to Equation (7):

_{b}/H were removed because of their lack of influence on the results. The Fr

_{u}parameter, which indicates the upstream Froude number, is subcritical in all models, and it was also eliminated [8] so that:

#### 2.4. Classical Hydraulic Equations and Evaluation Criteria

^{2}and RMSE were used to evaluate the agreement between the calculations and the measurements. RMSE refers to the root mean square errors, R

^{2}is the coefficient of determination, and n is the total number of data points. The metrics above were computed using Equations (12) and (13), respectively. It should be noted that the optimal answer is one in which the RMSE and R

^{2}parameters tend to zero and one, respectively.

#### 2.5. Validation

^{2}are very close to the values of test 5, to reduce the number of meshes, test specification 5 was used to continue the simulation.

## 3. Results and Discussion

#### 3.1. Water Surface Profile

#### 3.2. Relative Downstream Depth

#### 3.3. Relative Energy Dissipation

#### 3.4. Relative Pool Depth

#### 3.5. Equations

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**3D view of a vertical drop equipped with steps and grid structures: (

**a**) h/H = 0.3, (

**b**) h/H = 0.4, and (

**c**) the grid structure.

**Figure 3.**Geometric and hydraulic parameters for a vertical drop equipped with step and grid structures.

**Figure 5.**Comparison of experimental and numerical values of the relative depth downstream of the vertical drop with a step of 7.5 cm.

**Figure 6.**Flow over a vertical drop with h/H = 0.3: (

**a**) drop equipped with a step, (

**b**) drop equipped with both a step and grid structures.

**Figure 11.**Comparison of predictions and measured results: (

**a**) relative downstream depth, (

**b**) relative energy dissipation, (

**c**) relative pool depth.

Dissipator | N4 × 4 | N6 × 6 |
---|---|---|

a (cm) | 4 | 6 |

b (cm) | 4 | 6 |

L_{n} (cm) | 60 | 60 |

L_{G} (cm) | 40 | 40 |

a/b | 1.0 | 1.0 |

ɛ (%) | 44 | 56 |

h/H | Dissipator Size (cm) | Variables | ||||
---|---|---|---|---|---|---|

y_{d} (cm) | y_{c} (cm) | Q (Lit/s) | Fr_{d} | Re_{u} (×10^{4}) | ||

Without step | - | 2.00–6.00 | 6.00–12.50 | 27.60–64.50 | 3.83–5.20 | 16–39.9 |

0.3 | - | 2.60–6.00 | 2.90–3.85 | |||

0.4 | 2.45–6.00 | 2.90–3.83 | ||||

0.3 | 6 × 6 | 4.65–7.00 | 1.46–2.00 | |||

0.4 | 4.73–8.60 | 1.43–2.00 | ||||

0.3 | 4 × 4 | 3.45–7.30 | 2.20–2.90 | |||

0.4 | 3.95–7.80 | 1.87–2.36 |

Test No. | Turbulence Model | Cell Size (cm) | Total Number of Cells | RMSE (%) | R^{2} |
---|---|---|---|---|---|

Test 1 | RNG (k-ɛ) | 2.5 | 46,080 | 6.504 | 0.74 |

Test 2 | RNG (k-ɛ) | 1.0 | 720,000 | 3.251 | 0.776 |

Test 3 | RNG (k-ɛ) | 0.95 | 836,136 | 2.186 | 0.802 |

Test 4 | RNG (k-ɛ) | 0.83 | 1,856,200 | 1.136 | 0.89 |

Test 5 | RNG (k-ɛ) | 0.60 | 3,350,000 | 0.718 | 0.993 |

Test 6 | RNG (k-ɛ) | 0.50 | 5,253,000 | 0.708 | 0.995 |

h/H | Grid Cell Size (mm) | Froude Number (Downstream) |
---|---|---|

Without step | Without dissipator | 3.83–2.50 |

0.3 | Without dissipator | 2.90–3.85 |

0.4 | Without dissipator | 2.90–3.83 |

0.3 | 60 × 60 | 1.46–2.00 |

0.4 | 1.43–2.00 | |

0.3 | 40 × 40 | 2.20–2.90 |

0.4 | 1.87–2.36 |

Dependent Parameters | Constant Parameters | Criteria Parameters | |||||
---|---|---|---|---|---|---|---|

a | b | c | d | e | RMSE | R^{2} | |

$\frac{{y}_{d}}{H}$ | 0.4565 | −0.2808 | 0.3487 | 1.2620 | 1.4209 | 0.0223 | 0.980 |

$\frac{\Delta E}{{E}_{u}}$ | 0.3226 | 0.8063 | 0.6159 | 0.3669 | 1.3921 | 0.0228 | 0.973 |

$\frac{{y}_{p}}{H}$ | 5.325 | 1.370 | 1.796 | - | 0.0124 | 0.993 |

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**MDPI and ACS Style**

Daneshfaraz, R.; Aminvash, E.; Ghaderi, A.; Kuriqi, A.; Abraham, J.
Three-Dimensional Investigation of Hydraulic Properties of Vertical Drop in the Presence of Step and Grid Dissipators. *Symmetry* **2021**, *13*, 895.
https://doi.org/10.3390/sym13050895

**AMA Style**

Daneshfaraz R, Aminvash E, Ghaderi A, Kuriqi A, Abraham J.
Three-Dimensional Investigation of Hydraulic Properties of Vertical Drop in the Presence of Step and Grid Dissipators. *Symmetry*. 2021; 13(5):895.
https://doi.org/10.3390/sym13050895

**Chicago/Turabian Style**

Daneshfaraz, Rasoul, Ehsan Aminvash, Amir Ghaderi, Alban Kuriqi, and John Abraham.
2021. "Three-Dimensional Investigation of Hydraulic Properties of Vertical Drop in the Presence of Step and Grid Dissipators" *Symmetry* 13, no. 5: 895.
https://doi.org/10.3390/sym13050895