# Energy Dissipation and Hydraulics of Flow over Trapezoidal–Triangular Labyrinth Weirs

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}/P < 0.4 (h

_{0}the flow head over the weir and P weir height). Also, under submerged conditions, a rectangular labyrinth weir is more sensitive to the linear weir, and its efficiency is reduced by 10% over the linear weir.

## 2. Materials and Methods

#### 2.1. Effective Parameters in the Discharge Coefficient

_{d}is the discharge coefficient, L is the effective length of the weir crest (L = N* × (2D + 2l), H is the hydraulic head relative to the crest level upstream of the weir, and g is the gravity acceleration [6].

_{0}).

_{0}= 0, it follows that these parameters can be eliminated from Equation (3):

#### 2.2. Effective Parameters in the Energy Dissipation

_{max}). According to the special energy curve, as known, the minimum specific energy (E

_{min}) occurs in critical flow conditions (Froude number (Fr) = 1). Thus, the minimum amount of energy remaining downstream of the TTLW will be as follows:

_{c}indicates the critical depth of passing flow the labyrinth weir. The maximum possible energy dissipation is therefore given by:

#### 2.3. Numerical Model

^{®}software package was used to numerically simulate the flow pattern. The software solves the three-dimensional Reynolds averaged Navier–Stokes equations (RANS) with the finite volume method [32,33]:

_{f}is the volume fraction of the fluid in each cell, and (u, v, w) are the Cartesian velocity components. The A symbols are the fractional areas associated with the flow direction. The G terms are local accelerations of the fluid and the f terms are related to frictional forces. The ρ symbol indicates the fluid density, R

_{DIF}is a turbulent diffusion term, and R

_{SOR}is mass source.

_{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 [39,40]. In the above equations, ${\alpha}_{k}$ = ${\alpha}_{s}$ = 1.39, C

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

_{k}and ${S}_{\epsilon}$ are source terms for k and $\epsilon $, respectively. Also $\mu $

_{eff}is the effective viscosity (μ

_{eff}= μ + μ

_{t}), where μ

_{t}is called eddy viscosity or turbulent viscosity. Among the two equation models, the one by Launder and Spalding [41] popularly known as RNG k-ε model, proposes:

_{μ}is an empirical constant equal to 0.09.

^{®}uses an advanced algorithm for tracking free-surface flows, called the volume of fluid (VOF) [42]. The 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 (18):

## 3. Case Study

_{0}= Q/[gW

^{2}(h + P)

^{3}]

^{0.5}, downstream Froude number as Fr

_{1}= V

_{1}/(gy

_{1})

^{0.5}, Reynolds number as R = (gh

^{3})

^{0.5}/υ, critical depth as y

_{c}= (Q

^{2}/Gw

^{2})

^{1/3}, upstream speed as V

_{0}= Q/y

_{0}, and downstream speed as V

_{1}= Q/Wy

_{1}.

#### Computational Mesh and Boundary Conditions

^{+}(Equation (19)) in the range of 5 < y

^{+}< 30 [48]. The ratio of turbulent and laminar influences in a cell, y

^{+}, is defined as:

_{p}is the distance of the first node from the wall, u* is the shear velocity of the wall and υ is the kinematic viscosity.

_{min}) and both of the side boundaries were treated as rigid walls (W). No-slip conditions were applied at the wall boundaries, treated as non-penetrative boundaries. An atmospheric boundary condition was set to the upper boundary of the channel: the flow can enter and leave the domain as null von Neumann conditions are imposed on all variables, except for pressure, set to zero (i.e., atmospheric pressure). Symmetry boundary conditions (S) were used at the inner boundaries as well. Figure 5 shows the computational domain of the present study and the associated boundary conditions.

## 4. Results and Discussions

#### 4.1. The Validity of the FLOW-3D Model

_{d}) is validated with experimental data. To ensure a good agreement between the numerical and experimental data, the error was determined and is presented in Figure 6B.

_{d}increased for a low H/P ratio and decreased for high ratios. A quantitative evaluation of the computed and measured discharge coefficient versus different H/P ratios comparisons were made by using mean absolute relative error (MARE).

_{i}and P

_{i}are measured and computed values, respectively, and N is the total number of data. Table 3 gives the results for MARE at different H/P ratio and weir orientation.

#### 4.2. Hydraulics of Free Flow

_{d}versus H/P for different weir geometries. It can be noted that C

_{d}decreases by decreasing the sidewall angle due to the collision of the falling jets, for high value of H/P. It can be noted that for the low value of H/P, the C

_{d}value is high, due to the fact that the collision of jets is not so severe. As a result, by increasing the sidewall angle, the decrease rate of C

_{d}with H/P increases.

_{d}decreases by increasing the weir heights due to nappe interaction. In fact, the nappe interaction increases with discharge, leading to a decrease in discharge coefficient C

_{d}of the TTLW. For TTLW, nappe interference originates at the upstream apex and can produce wakes downstream of the apex (Figure 8). It can be stated that for triangular shape, downstream of the apex, nappe interactions do not happen.

_{d}with H/P for the weirs of different orientations, obtained from experimental and numerical results, is shown in Figure 9. It can be noted that the trend of discharge coefficient for normal and inverse orientation is the same and C

_{d}decreases by decreasing the sidewall angle.

#### 4.3. Energy Charachterization and Flow Regime

_{1}. Variation of Fr

_{1}with the relative critical depth y

_{c}/E

_{0}for experimental results is shown in Figure 11 for different sidewall angles and height of the weirs. As highlighted in this figure, in most of the cases, the downstream flow regime of TTLW was supercritical: Froude number was larger than one with an increase of y

_{c}/E

_{0}at different weir heights.

_{c}/E

_{0}), Fr

_{1}tends to be 1 but in some models with a high sidewall angle Fr

_{1}< 1 (subcritical flow); this is the result of weak hydraulic jump occurrence, due to circulating flows behind the nappes and collision of supercritical flows at the base of nappes (see Figure 12). Also, in the same discharge values, the submerged flow was earlier with a low sidewall angle.

_{0}versus the relative critical depth y

_{c}/E

_{0}is presented. Energy dissipation results of vertical drops are compared with the results of Rajaratnam and Chamani [50] and Chamani et al. [51], indicating that TTLW approximately dissipates the maximum amount of energy. Increasing the sidewall angle and weir height (from 8 to 12) and subsequently increasing the dimensionless ratio of y

_{c}/E

_{0}, TTLW reduces more energy with respect to the vertical case. In fact, collision of nappes happens in the upstream apexes of TTLW, generating a localized hydraulic jump condition near it that leads therefore to energy dissipation. Moreover, with the creation of a pool behind the nappes in the upstream apexes of TTLW, circulating flow (in the pool) enhances the turbulent mixing and dissipates a large portion of energy (see Figure 12). It should also be noted that a weak hydraulic jump downstream of TTLW contributes to dissipating a higher and significant portion of energy.

_{1}downstream of weir models is normalized with the minimum theoretical minimum residual energy E

_{min}, and results are then presented with the normalized critical depth y

_{c}/E

_{0}(Figure 14). This result indicates that TTLW has energetic dissipation performance that leads to value of residual energy very close to the minimum possible value of residual energy. The energy dissipation of TTLW is largest compared to vertical drop and has the least possible value of residual energy as flow increases.

## 5. Conclusions

- It can be concluded that the FLOW-3D software can accurately predict characteristics of flow over TTLW. In comparisons between the calculated and measured free surface profiles, appropriate mesh with 2,118,270 elements by relative error and RMSE of 3.05%, 0.43 cm and maximum aspect ratio 1.07, was selected for calculations. The maximum and minimum value MARE errors are 3% and 0.77%.
- In the high value of H/P due to the collision of the falling jets, the discharge coefficient decreases by decreasing the sidewall. While for the low of H/P, the collision of jets is not so severe, hence in high values for C
_{d}. Also, by increasing the sidewall angle, the decrease of C_{d}with H/P increases. Increasing the weir heights decreases the discharge coefficient due to nappe interactions occurring when two or more nappes collide, and the nappe interaction increases as discharge increases. - In low discharge, hydraulics of flow over the TTLW has free flow conditions and for high discharge it has submerged conditions. In all models, the downstream Froude number is larger than one with an increase of relative critical depth y
_{c}/E_{0}at different weir heights. - TTLW approximately dissipates the maximum amount of energy and increases the sidewall angle, weir height, and relative critical depth y
_{c}/E_{0}, it reduces more energy dissipation. Circulating flows behind the nappes enhances the turbulent mixing and dissipates a large portion of energy in the upstream apexes of TTLW. It must be noted that a weak hydraulic jump downstream of TTLW contributes to the more energy dissipation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**(

**A**) Comparison of discharge coefficient obtained from numerical solution with experimental data; (

**B**) Determination of error percentage.

**Figure 7.**Experimental results obtained for the effects of weir geometry on variations of discharge coefficient C

_{d}versus H/P, (

**a**) P = 8 cm, (

**b**) P = 10 cm, (

**c**) P = 12 cm.

**Figure 8.**The collision of nappes from adjacent sidewalls of TTLW in (

**A**) normal orientation and (

**B**) inverse orientation.

**Figure 11.**Downstream Froude number (Fr

_{1}) versus the relative critical depth y

_{c}/E

_{0}, (

**a**) P = 8 cm, (

**b**) P = 10 cm, (

**c**) P = 12 cm.

**Figure 12.**The collision of nappes and circulating flows in the pool at upstream apexes of the TTLW.

**Figure 13.**Energy dissipation on TTLW with vertical drop and maximum energy dissipation [31] (∆E/E

_{0}versus y

_{c}/E

_{0}), (

**a**) P = 8 cm, (

**b**) P = 10 cm, (

**c**) P = 12 cm.

**Figure 14.**Comparison of the residual energy downstream of TTLW with a vertical drop, and minimum residual energy (E

_{1}/E

_{min}versus y

_{c}/E

_{0}), (

**a**) P = 8 cm, (

**b**) P = 10 cm, (

**c**) P = 12 cm.

Models | Q (l/s) | α (°) | l (cm) | w (cm) | P (cm) | l/w | w/p | Fr_{0} |
---|---|---|---|---|---|---|---|---|

Range | 3–15 | 10–30 | 27–72 | 15 | 8–12 | 1.8–4.8 | 1.87–1.25 | 0.08–0.16 |

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

T1 | k-ε (RNG) | 1.10 | 0.65 | 711,758 | 50 | 19.39 | 3.44 |

T2 | k-ε (RNG) | 1.00 | 0.55 | 1,302,587 | 85 | 11.12 | 1.96 |

T3 | k-ε (RNG) | 0.90 | 0.45 | 1,997,425 | 120 | 6.07 | 0.85 |

T4 | k-ε (RNG) | 0.80 | 0.35 | 2,703,458 | 175 | 3.75 | 0.56 |

T5 | k-ε (RNG) | 0.70 | 0.30 | 3,587,624 | 220 | 3.32 | 0.44 |

**Table 3.**MARE Values for the discharge coefficient versus different H/P ratios on NO and IO orientation.

Model | H/P | C_{d}-NO-Measured Values | C_{d}-NO-Computed Values | C_{d}-IO-Measured Values | C_{d}-IO-Computed Values | MARE (%)-NO | MARE (%)-IO |
---|---|---|---|---|---|---|---|

Trapezoidal–Triangular Labyrinth Weir (TTLW) | 0.176 | 0.799 | 0.789 | 0.804 | 0.790 | 1.29 | 1.70 |

0.256 | 0.779 | 0.757 | 0.764 | 0.772 | 2.87 | 1.10 | |

0.338 | 0.716 | 0.722 | 0.706 | 0.719 | 0.77 | 1.90 | |

0.417 | 0.670 | 0.684 | 0.662 | 0.642 | 2.02 | 3.00 | |

0.509 | 0.606 | 0.613 | 0.600 | 0.585 | 1.14 | 2.50 | |

0.589 | 0.575 | 0.560 | 0.570 | 0.559 | 2.55 | 2.00 | |

0.688 | 0.525 | 0.510 | 0.521 | 0.529 | 2.72 | 1.60 | |

Mean | 1.91 | 1.97 |

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**MDPI and ACS Style**

Ghaderi, A.; Daneshfaraz, R.; Dasineh, M.; Di Francesco, S.
Energy Dissipation and Hydraulics of Flow over Trapezoidal–Triangular Labyrinth Weirs. *Water* **2020**, *12*, 1992.
https://doi.org/10.3390/w12071992

**AMA Style**

Ghaderi A, Daneshfaraz R, Dasineh M, Di Francesco S.
Energy Dissipation and Hydraulics of Flow over Trapezoidal–Triangular Labyrinth Weirs. *Water*. 2020; 12(7):1992.
https://doi.org/10.3390/w12071992

**Chicago/Turabian Style**

Ghaderi, Amir, Rasoul Daneshfaraz, Mehdi Dasineh, and Silvia Di Francesco.
2020. "Energy Dissipation and Hydraulics of Flow over Trapezoidal–Triangular Labyrinth Weirs" *Water* 12, no. 7: 1992.
https://doi.org/10.3390/w12071992