# Methodological Proposal for the Hydraulic Design of Labyrinth Weirs

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{T}/P ≤ 0.8 and for angles of the cycle sidewall of 6° ≤ α ≤ 20°. The results of the discharge coefficient are presented as a family of curves, which indicates a higher discharge capacity when H

_{T}/P ≤ 0.17. Four aeration conditions are identified with higher discharge capacity when the nappe is adhering to the downstream face of the weir wall and lower discharge capacity when the nappe is drowned. Unstable flow was present when 12° ≤ α ≤ 20°, with a greater presence when the nappe was partially aerated and drowned. The interference of the nappe is characterized and quantified, reaching up to 60% of the length between the apex, and a family of curves is presented as a function of H

_{T}/P in this respect. Finally, a spreadsheet and a flowchart are proposed to support the design of the labyrinth type weir.

## 1. Introduction

_{w}is the width of the weir wall (m), D is the external length of the apex (m), A is the internal length of the apex (m), l

_{C}is the length of the cycle sidewall (m), α is the angle of the cycle sidewall with respect to the flow direction (°), B is the distance between apexes (m), P is the height of the weir (m), h is the piezometric head (m), H

_{T}is the total head (m), and Q is the design flow (m

^{3}s

^{−1}).

_{d}) for the design of labyrinth weirs have been determined from physical models of various prototype structures, examples of which are as follows: Avon and Woronova [2]; Harrezza, Dungo, Keddara, Alijó, Gema, and São Domingo [7]; Hyrum [8]; Ute [9]; Lake Brazos [6]; and Lake Townsend [10]. The evolution of commonly used and documented design methods can be summarized in the following sequence: (a) Hay and Taylor (1970) [11]; (b) Darvas (1971) [2]; (c) Hinchliff and Houston (1984) [12]; (d) Lux (1989) [13]; (e) Magalhães and Lorena (1989) [7]; (f) Tullis, Amanian, and Waldron (1995) [14]; and (g) Crookston and Tullis (2012) [15]. However, the results of Idrees and Al-Ameri (2022) [16] showed that common design equations did not take into account all the parameters that affect the performance of a labyrinth weir such as geometry and flow conditions.

#### 1.1. Discharge Flow Characteristics

_{d}and the efficiency of the weir. However, the main challenge is to know the joint effect of all these factors on the discharge coefficient.

#### 1.1.1. Aeration Conditions

#### 1.1.2. Nappe Instability

#### 1.1.3. Nappe Interference

#### 1.1.4. Drowning

_{T}is the upstream total head when the weir is not drowned (m), and H

_{d}is the downstream total head of the weir in the drowning state (m).

## 2. Materials and Methods

#### 2.1. Description of the Physical Model

#### 2.2. Numerical Solution Method

#### 2.2.1. Computational Fluid Dynamics (CFD)

#### 2.2.2. Grid Convergence Index (GCI)

_{s}is a security factor (taking a value of 3 for comparisons of two grids and 1.25 for comparisons of three or more grids [56]), r is the mesh refinement ratio, p is the order of convergence, and Ꜫ is the error relative to the control variable f

_{i}.

_{C}is the number of cells. Therefore, to perform the calculation, at least three different grid sizes must be selected to determine the value of the control variables ${f}_{i}$(${f}_{1},{f}_{2},{f}_{3})$ considered important for the simulation objective. Then, for ${\lambda}_{1}<{\lambda}_{2}{<\lambda}_{3}$, the mesh refinement factors were determined as ${r}_{21}=\raisebox{1ex}{${\lambda}_{2}$}\!\left/ \!\raisebox{-1ex}{${\lambda}_{1}$}\right.$, ${r}_{32}=\raisebox{1ex}{${\lambda}_{3}$}\!\left/ \!\raisebox{-1ex}{${\lambda}_{2}$}\right.$ and the order of convergence p was calculated with Equation (15) [52].

_{1}, h

_{2}, h

_{3}) was considered as a control variable to estimate discretization errors. The flows used were as follows: Q

_{I}= 0.2036 m

^{3}s

^{−1}, Q

_{II}= 0.2003 m

^{3}s

^{−1}, Q

_{III}= 0.2137 m

^{3}s

^{−1}, Q

_{IV}= 0.2170 m

^{3}s

^{−1}, Q

_{V}= 0.2149 m

^{3}s

^{−1}, Q

_{VI}= 0.2036 m

^{3}s

^{−1}, and Q

_{VII}= 0.1915 m

^{3}s

^{−1}for the grids: I, II, III, IV, V, VI, and VII, respectively. The GCI calculations of the numerical solutions are summarized in Table 4; the asymptotic ranges of convergence obtained are approximately equal to 1. Therefore, the numerical solutions are within the asymptotic range. In the present work, the hydraulic head was achieved with a maximum error of up to 2.80%, corresponding to grid II.

#### 2.3. Evaluation of the Computational Model

_{d}) was calculated from the height of the measured hydraulic head on the weir and the general weir equation (Equation (17)).

^{3}s

^{−1}), C

_{d}is the discharge coefficient (dimensionless), g is the acceleration due to gravity (m s

^{2}), L is the characteristic length of the weir (m) (defined as the total length referenced at the center of the weir crest wall thickness), and H

_{T}is the total hydraulic head (m).

^{2}), the relative percentage error (Er), and the mean absolute error (MAE) (Equations (18)–(20)). These criteria assess the agreement between the results of the physical experiment and those from the CFD models.

_{num}are the numerical values, and Y

_{exp}are the experimental values.

_{T}/P are presented.

^{2}= 0.984, which confirms the agreement between the numerically obtained discharge coefficients and the experimental values. In the graph in Figure 6, the numerical results are compared with the experimental results; the dotted diagonal line corresponds to a perfect fit.

#### 2.4. Proposed Sequential Design Method for a Labyrinth Weir

**(i) Stage one:**This stage consists of determining the data necessary for the design and comprises information previously obtained from topographic and hydrological analysis, namely:

- (a)
- The design flow (Q), which represents the design discharge for a given return period;
- (b)
- The upstream head of the weir (H
_{T}), which depends on the channel width (W) and is limited by the freeboard; - (c)
- The downstream head of the weir (H
_{d}) is calculated from the drop height and the flow velocity at the foot of the weir; - (d)
- The weir height (P) corresponds to the height of the storage volume or the Ordinary Maximum Water Level obtained from the topography and the operation of the basin.

**(ii) Stage two:**The topography of the study area allows for selecting the angle α of the weir sidewall (6 ≤ α ≤ 20°). A large angle can be chosen when the number of cycles N and the length between the cycle apexes (B) are limited by topography. The selection of the angle α is also a function of the ratio H

_{T}/P (0.05 ≤ H

_{T}/P ≤ 0.8). For certain values of H

_{T}/P and α, the flow becomes unstable and is a phenomenon to be avoided in the design of the weir, for the safety of the hydraulic structure. Subsequently, the discharge coefficient is calculated. Its calculation is a function of α and the ratio H

_{T}/P, and the value of the discharge coefficient will determine the discharge capacity of the weir.

**(iii) Stage three:**In this stage, the geometric variables of the weir are calculated as follows:

- The length of the weir (L). The selection of the angle α will determine the length of the weir. Its calculation is a function of the discharge coefficient, the hydraulic head, and the design flow.
- The width of the weir wall (t
_{w}) and the internal apex rope (C_{c}) must both be equal to P/8. - The internal and external apex arc (Arc
_{int}, Arc_{ext}) are both functions of t_{w}and α. - The length of the cycle wall (l
_{c}), as a function of L, N, Arc_{int}, and Arc_{ext}. - The length of the platform (B) is a function of L, N, Arc
_{int}, Arc_{ext}, α, and t_{w}.

_{d}(α). The weir efficiency is also a function of the discharge coefficient of a linear weir; its calculation method is described by Crooskton (2010) [5].

_{int}) is calculated from the ratio H

_{T}/P and the angle α. Finally, the type of aeration of the nappe is determined according to the value of H

_{T}/P and the selected angle α.

**(iv) Stage Four:**The last stage of the design method includes the dimensionless head relationships for the drowned weir, which were developed and described by Tullis et al. (2007) [38].

## 3. Results

#### 3.1. Discharge Coefficient, Weir, and Cycle Efficiency

_{T}/P, whose weir cycle sidewall angles vary from 6° to 20° and are compared with the discharge coefficients reported by Crookston and Tullis (2012) [15] for trapezoidal labyrinth weirs. Both weirs have a half-round crest.

_{d}(α) of each weir with a circular apex are presented graphically in Figure 8 for H

_{T}/P ≤ 0.8.

_{T}/P. Weir design methods and curves are mostly generated from empirical equations derived from laboratory experiments [2,5,6,7,8,9]. For example, several researchers reported polynomial equations obtained by non-linear regression to have good fitting [14]. Statistical analysis was used to determine the accuracy of Equation (21) compared to the C

_{d}(α) results obtained from numerical data. The calculated Pearson’s coefficient was very good, varying from 0.999 to 1 for weirs with angles from 6 to 20°. Therefore, Equation (21) provides sufficient accuracy to determine C

_{d}(α). Equations (22)–(27) correspond to the coefficients of Equation (21) as a function of the angle α. The accuracy of predictive Equations (22)–(27) was also evaluated with the numerical results using Pearson’s determination coefficient, obtaining values of 1 for the case of Equations (22)–(26) and 0.996 for Equation (27). Therefore, reliable results can be obtained using coefficients for Equation (21).

_{T}varies between 0.10 and 0.17 times the height of the weir. Table 6 shows the maximum values of the discharge coefficient for each weir as a function of H

_{T}/P.

_{T}/P. Both graphs show that the maximum efficiency occurs for small values of H

_{T}/P and increases with the decreasing sidewall angle.

_{T}/P > 0.1. Figure 10B represents the cycle efficiency of the weir; the dotted line passes through the values of H

_{T}/P where the maximum efficiency of the cycle is present and coincides with the maximum values of C

_{d}(α).

#### 3.2. Nappe Aeration Conditions

#### 3.3. Nappe Instability

_{T}/P and the aeration conditions where instability is generated are presented in Table 8 and are included in the discharge coefficient design curves (Figure 14).

#### 3.4. Nappe Interference

_{int}) were made from the upstream apex to the point (downstream) where the nappe from the sidewall intersects (Figure 16A). The term L

_{int}denotes the projection of B

_{int}on the weir crest that is affected by this phenomenon.

_{int}) in relation to the length of B (perpendicular distance between upstream and downstream apexes) for values of H

_{T}/P ≤ 0.8. The graph allows for the prediction of the length of B

_{int}and indicates that its value can vary from 20% to 60% of the length of B in the drowning condition.

_{T}/P (Equation (30)).

#### 3.5. Application of the Proposed Method

_{N}versus H

_{T}/P, where Q/Q

_{N}is the discharge magnification and Q

_{N}is the discharge over a linear weir. The results reported by Houston (1982) [9] on the weir design are summarized in Table 9 and Figure 18.

## 4. Discussion

#### 4.1. Discussion of Discharge Coefficient, Weir, and Cycle Efficiency

_{d}helps us to understand the hydraulic behavior of a weir and is essential when making decisions during weir design, where its value depends on geometry, aeration conditions, and flow behavior during the discharge. When the ratio H

_{T}/P < 0.2, higher values of C

_{d}(α) up to 0.833 are presented; this is when the nappe is adhered to the weir wall. During the transition of a nappe adhered to the wall becoming partially aerated, there is an accelerated decrease in C

_{d}(α) values at weirs with angles varying from 6° to 10°. The reduction is less abrupt when α ≥ 12° and, as the head H

_{T}on the weir increases, the value of C

_{d}(α) decreases. For values of H

_{T}/P < 0.1, the 8° and 10° weirs exhibit similar behavior in C

_{d}(α), and the C

_{d}(α) of the 12° weir is slightly higher than that of the 10° weir. The higher angle weirs have better discharge capacities. However, lower angle weirs have the advantage of having a longer weir crest length.

_{T}/P ≥ 0.1. In addition, the dotted line in Figure 9 indicates the inflection point of each curve, where the discharge coefficients acquire their maximum value (Table 6). In effect, the slopes increase until the nappe is no longer aerated and presents local drowning at the weir apex. When the weirs work in a drowned manner, efficiency decreases, projecting curves with slightly descending slopes at the end.

_{T}/P > 0.8.

#### 4.2. Discussion of Nappe Aeration Conditions

_{T}/P > 0.8. In the latter case, the behavior of the weir is equivalent to that of the linear weir.

_{T}/P ≤ 0.16. On the other hand, it has been observed that, with an increase of the angle α, the presence of this regime increases up to H

_{T}/P ≈ 0.45. However, when α = 20°, the opposite occurs for the case of the aerated regime, i.e., its presence is lower when the angle α increases. The value of the discharge coefficient presents a rapid decrease for angles that vary from 6° to 10°, and this is when the transition from clinging flow to aerated flow occurs. When 15° ≤ α ≤ 20°, the weir has a greater range of flow clinging to the wall, in contrast to the aerated flow condition that is briefly produced by changing to the partially aerated regime. The drowning condition is generated for larger heads, i.e., when H

_{T}/P > 0.49 (α = 6°).

#### 4.3. Discussion of Nappe Instability and Interference

_{T}/P. On the other hand, it should be noted that weirs where α < 10° have a shorter crest length affected by the nappe interference.

#### 4.4. Discussion of Application of the Proposed Method

_{T}/P and the aeration conditions in which the instability originates, which is an important indicator at the time of design.

_{T}/P (from 0.5 to 0.8), sidewall angles from 6° to 20°, and for weirs located in a channel.

## 5. Conclusions

_{T}/P ≤ 0.8 and 6° ≤ α ≤ 20°. The values of the discharge coefficient are presented as a family of curves as function of H

_{T}/P and using a mathematical model of a regressive type (through a fifth-degree polynomial equation found for this purpose). The results indicate a higher discharge capacity of the weir while increasing the angle α. The contrast between the discharge coefficients of circular apex weirs with those of a trapezoidal apex indicate an increase in their value of up to 46% (α = 6°) in relation to the trapezoidal apex weir. The cycle and weir efficiency are presented as a tool in the design procedure. Both parameters indicate that the maximum values occur for H

_{T}/P ≤ 0.17 and the efficiencies are higher with the reduction of the angle α.

_{T}/P for each condition. The relationship between the discharge coefficient and the aeration condition is evident: when the nappe is adhered to the wall, the weir has a higher discharge coefficient value. In addition, its presence is greater when α increases, and the opposite occurs when the nappe is aerated.

_{T}/P were identified when instability occurred. It is necessary not to incur the instability ranges when designing the weir to avoid possible damage to the hydraulic structure.

_{int}/B is presented herein as a function of H

_{T}/P, and a mathematical model was found for its estimation. This model is a second-degree equation. The results show that the length of B

_{int}reaches a maximum of 60% of the length of B.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

A | internal length apex. |

Arc_{int} | internal apex arc. |

Arc_{ext} | external apex arc. |

A_{x}, A_{y}, A_{z} | fractional area in the x, y, z direction, respectively. |

a | adjustment factor to obtain the discharge coefficient. |

B | length of apron. |

B_{int} | nappe interference length. |

b | adjustment factor to obtain the discharge coefficient. |

c | adjustment factor to obtain the discharge coefficient. |

C_{d} | discharge coefficient. |

C_{d-sum} | submerged weir discharge coefficient. |

C_{d}(α)
| labyrinth weir discharge coefficient |

C_{d}_{(90°)} | linear weir discharge coefficient. |

C_{c} | internal apex rope. |

C_{1}_{Ꜫ}, C_{2}_{Ꜫ}, C_{μ} | constants of the turbulent k-ε model. |

Cov | covariance. |

D | external apex length. |

Dk_{eff} | effective diffusivity for turbulent kinetic energy. |

Dε_{eff} | effective diffusivity for dissipation rate. |

d | adjustment factor to obtain the discharge coefficient. |

e | adjustment factor to obtain the discharge coefficient. |

F_{D} | diffusion term. |

F_{S} | source term. |

F_{s} | security factor. |

f | adjustment factor to obtain the discharge coefficient. |

f_{i} | control variable. |

f_{x}, f_{y}, f_{z} | viscous acceleration in x, y, z direction, respectively. |

G_{k} | turbulent kinetic energy generation due to mean velocity gradients. |

G_{x}, G_{y}, G_{z} | acceleration of the body in the x, y, z direction, respectively. |

g | acceleration gravity. |

H_{d} | downstream total head. |

H_{T} | upstream total head. |

H* | upstream total head of the drowned weir. |

h | piezometric head. |

k | turbulent kinetic energy. |

L | characteristic length of the weir. |

L_{cycle} | cycle length. |

L_{int} | length of the crest affected by the nappe interference. |

l_{C} | centerline length of sidewall. |

M | magnification ratio. |

m | adjustment coefficient to obtain the length B_{int}. |

N | number of cycles. |

N_{C} | number of cells. |

n | adjustment coefficient to obtain the length B_{int}. |

o | adjustment coefficient to obtain the length B_{int}. |

P | weir height. |

p | order of convergence. |

Q | design flow. |

Q_{N} | flow of a linear weir. |

R_{crest} | radius of the weir crest. |

r | mesh refinement ratio. |

S | submergence level. |

S_{ij}^{2} | strain rate tensor. |

t_{w} | weir wall width. |

t_{w}_{−1} | upper crest width. |

t_{w}_{−2} | lower crest width. |

u | velocity component in the x direction. |

V_{F} | fraction volume. |

v | velocity component in y direction. |

v_{t} | turbulent kinematic viscosity. |

W | channel width. |

w | cycle width. |

Y_{exp} | experimental results. |

Y_{num} | numerical results. |

z | velocity component in z direction. |

α | angle of sidewall. |

∆V_{i} | volume of the i-th cell. |

γ | representative cell size. |

ε | dissipation rate. |

ε | relative error. |

ε′ | weir efficiency. |

ε″ | cycle efficiency. |

σ^{2}_{exp} | variance of the experimental results. |

σ^{2}_{num} | variance of the numerical results. |

σ_{Ꜫ} | Prandtl number. |

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**Figure 1.**Geometric parameters of a labyrinth type weir (Mattos-Villarroel et al., 2021 [1]).

**Figure 2.**Submergence limits (adapted from Tullis et al., 2007 [38]).

**Figure 5.**Relative percentage error of the discharge coefficient as a function of H

_{T}/P (Mattos-Villarroel et al., 2021 [1]).

**Figure 6.**Comparison of numerically obtained discharge coefficients with experimental values (Mattos-Villarroel et al., 2021 [1]).

**Figure 8.**Curves of C

_{d}(α) as a function of H

_{T}/P for different values of α for labyrinth weirs.

**Figure 9.**Relationship of the discharge coefficient of the circular apex weir to the trapezoidal apex weir. The dotted line indicates the inflection point on each curve.

**Figure 11.**Nappe aeration conditions: (

**A**) clinging, (

**B**) aerated, (

**C**) partially aerated, and (

**D**) drowned.

**Figure 15.**Effects of the nappe interference. (

**A**) Air contrail and standing waves and (

**B**) local drowning.

**Figure 18.**Shape and dimensions of the 14-cycle labyrinth weir [9].

**Table 1.**Geometric characteristics of the trapezoidal labyrinth weir [15].

α (°) | N | L (m) | A (m) | w (m) | P (m) | W (m) | Crest Profile |
---|---|---|---|---|---|---|---|

15 | 2 | 4 | 0.038 | 0.617 | 0.305 | 1.235 | Quarter-round |

Boundary and Initial Conditions | Solution Method |
---|---|

Domain: inlet | Velocity |

Domain: outlet | Atmospheric pressure |

Domain: weir, sidewalls, and channel platform | Solid, stationary, and non-slip. |

Viscosity model | k–ε standard |

Multiphasic model | Volume of fluid (VOF) |

Pressure–velocity coupling | SIMPLE |

Spatial discretization scheme | Upwind |

Grid | Scenario | Grid | Scenario |
---|---|---|---|

I | 1–10 | V | 26–30 |

II | 11–15 | VI | 31–35 |

III | 16–20 | VII | 36–40 |

IV | 21–25 |

Grid | r | p | h_{1}(m) | h_{2}(m) | h_{3}(m) | Richardson Extrapolate (m) | Ꜫ_{21} | Ꜫ_{32} | GCI_{21} (%) | GCI_{32} (%) | Asymptotic Range of Convergence |
---|---|---|---|---|---|---|---|---|---|---|---|

I | 1.60 | 1.80 | 0.1622 | 0.1637 | 0.1672 | 0.1611 | 0.0092 | 0.0214 | 0.87 | 2.00 | 0.99 |

II | 1.58 | 1.78 | 0.1182 | 0.1197 | 0.1231 | 0.1170 | 0.0127 | 0.0284 | 1.25 | 2.80 | 0.99 |

III | 1.58 | 1.83 | 0.1302 | 0.1312 | 0.1335 | 0.1294 | 0.0077 | 0.0175 | 0.74 | 1.69 | 0.99 |

IV | 1.59 | 1.76 | 0.1412 | 0.1423 | 0.1448 | 0.1403 | 0.0078 | 0.0176 | 0.77 | 1.73 | 0.99 |

V | 1.65 | 1.79 | 0.1482 | 0.1495 | 0.1527 | 0.1473 | 0.0088 | 0.0214 | 0.75 | 1.83 | 0.99 |

VI | 1.58 | 1.97 | 0.1502 | 0.1519 | 0.1561 | 0.1490 | 0.0113 | 0.0276 | 0.96 | 2.35 | 0.99 |

VII | 1.64 | 1.89 | 0.1622 | 0.1631 | 0.1654 | 0.1616 | 0.0055 | 0.0141 | 0.45 | 1.13 | 0.99 |

Scenario | α (°) | P (m) | L_{cycle}(m) | w/P | N | Q (m ^{3} s^{−1}) | Crest Profile | Apex Shape |
---|---|---|---|---|---|---|---|---|

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | 15 | 0.305 | 2.00 | 2.024 | 1 | 0.0190, 0.0532, 0.0919, 0.1309, 0.1681, 0.2036, 0.2373, 0.2697, 0.3013, 0.3325 | CR | Trapezoidal |

11, 12, 13, 14, 15 | 6 | 0.305 | 3.64 | 2.024 | 1 | 0.0780, 0.1240, 0.2003, 0.2703, 0.3558 | MR | Circular |

16, 17, 18, 19, 20 | 8 | 0.305 | 3.07 | 2.024 | 1 | 0.0429, 0.0750, 0.2137, 0.2873, 0.3744 | MR | Circular |

21, 21, 23, 24, 25 | 10 | 0.305 | 2.65 | 2.024 | 1 | 0.0367, 0.0532, 0.2170, 0.2891, 0.3712 | MR | Circular |

26, 27, 28, 29, 30 | 12 | 0.305 | 2.33 | 2.024 | 1 | 0.0380, 0.0671, 0.1450, 0.2149, 0.3728 | MR | Circular |

31, 32, 33, 34, 35 | 15 | 0.305 | 2.02 | 2.024 | 1 | 0.0532, 0.1308, 0.2036, 0.2697, 0.3325 | MR | Circular |

36, 37, 38, 39, 40 | 20 | 0.305 | 1.59 | 2.024 | 1 | 0.0290, 0.0517, 0.1247, 0.1915, 0.3364 | MR | Circular |

_{cycle}: Cycle length (m).

$\left(\mathit{\alpha}\right)$ | H_{T}/P | ${\mathit{C}}_{\mathit{d}}\left(\mathit{\alpha}\right)$ |
---|---|---|

6° | 0.10 | 0.736 |

8° | 0.11 | 0.762 |

10° | 0.13 | 0.771 |

12° | 0.14 | 0.784 |

15° | 0.15 | 0.803 |

20° | 0.17 | 0.833 |

H_{T}/P | ||||
---|---|---|---|---|

$\mathit{\alpha}\left({}^{\xb0}\right)$ | Flow Clinging | Flow Aerated | Flow Partially Aerated | Flow Drowned |

6° | <0.165 | 0.165–0.270 | 0.270–0.487 | >0.487 |

8° | <0.200 | 0.200–0.350 | 0.350–0.500 | >0.500 |

10° | <0.265 | 0.265–0.350 | 0.350–0.540 | >0.540 |

12° | <0.300 | 0.300–0.410 | 0.410–0.550 | >0.550 |

15° | <0.325 | 0.325–0.400 | 0.400–0.600 | >0.600 |

20° | <0.450 | 0.450–0.500 | 0.500–0.600 | >0.600 |

$\mathit{\alpha}\left({}^{\xb0}\right)$ | Instability | Aeration Condition |
---|---|---|

6° | - | - |

8° | - | - |

10° | - | - |

12° | 0.56 ≤ HT/P ≤ 0.8 | Drowned. |

15° | 0.49 ≤ HT/P ≤ 0.8 | Partially aerated, and drowned. |

20° | 0.40 ≤ HT/P ≤ 0.8 | Clinging, aerated, partially aerated, and drowned. |

**Table 9.**Dimensions of the labyrinth weir of the Ute Dam [8].

Concept | Symbol | Value-Unit | Observations |
---|---|---|---|

(i) Initial data | |||

Design flow | Q | 15,574 m^{3}/s | Initially, the design discharge was 16,042 m^{3}/s. |

Weir width | W | 256 m | - |

Weir height | P | 9.14 m | - |

Upstream total head | H_{T} | 5.79 m | - |

(ii) Geometric variables and non-dimensional relationships | |||

Head water ratio | H_{T}/P | 0.63 | 0.05 ≤ H_{T}/P ≤ 1 (upper range is expanded from 0.5 to 1 to use the design curves) |

Flow magnification | Q/Q_{N} | 2.4 | - |

Angle of sidewall | $\alpha $ | 12.1475° | - |

Length magnification | L/W | 4 | 2 ≤ L/W ≤ 8. |

Vertical aspect ratio | w/P | 2 | 2 ≤ w/P ≤ 5 |

Cycle width | w | 18.29 m | - |

Number of cycles | N | 14 | - |

Weir length | L | 1024.24 m | - |

Sidewall length | l_{c} | 34.76 m | - |

Length between apexes | B | 33.99 m | - |

Apex | A | 1.82 m | - |

Crest radius | R_{Crest} | 0.30 | - |

Upper crest width | t_{w}_{−1} | 0.61 m | - |

Lower crest width | t_{w}_{−2} | 1.52 m | - |

Concept | Symbol | Value-Unit | Equations and Limits |
---|---|---|---|

(i) Input data | |||

Design flow | Q | 15,574 m^{3}/s | - |

Weir width | W | 256 m | - |

Weir height | P | 9.14 m | - |

Upstream total head | H_{T} | 5.79 m | - |

(ii) Definition of α and the number of cycles (N) | |||

Head water ratio | H_{T}/P | 0.63 | 0.05 ≤ H_{T}/P ≤ 0.8 |

Angle of sidewall | $\alpha $ | 11.5° | 6°≤ α ≤ 20° |

Nappe stability | - | Stable | Stable/Unstable: Table 8 and Figure 14 |

Labyrinth weir discharge coefficient | ${C}_{d}\left(\alpha \right)$ | 0.483 | ${C}_{d}\left(\alpha \right)=f(HT/P,\alpha ),$ Equations (21)–(27) |

Cycle width | w | 27.42 m | w = 3P |

Number of cycles | N | 9 | N = W/w |

New cycle width | w | 28.44 m | w = W/N |

Vertical aspect ratio | w/P | 3.11 | 2 ≤ w/P ≤ 4 |

(iii) Calculation of geometric variables, weir and cycle efficiencies, nappe interference and aeration condition | |||

Geometric variables | |||

Total centerline length of weir | L | 783.20 m | $L=1.5Q/\left[{C}_{d}\right(\alpha \left){HT}^{1.5}{\left(2g\right)}^{0.5}\right]$ |

Wall width | t_{w} | 1.14 m | _{tw} ≈ P/8 |

Internal apex rope | C_{c} | 1.14 m | C_{c} = t_{w} |

Internal apex arc | Arc_{int} | 1.60 m | ${Arc}_{int}={t}_{w}\pi (90-\alpha )/\left(180cos\alpha \right)$ |

External apex arc | Arc_{ext} | 1.16 m | ${Arc}_{ext}={t}_{w}\pi (2cos\alpha +1)(90$ − α)/(180 cos α) |

Centerline length of sidewall | l_{c} | 42.14 m | $lc=L/\left(2N\right)-({Arc}_{int}+{Arc}_{ext})/2$ |

Length of apron | B | 44.28 m | $B=[L/(2N)-({Arc}_{int}+{Arc}_{ext})/2]cos\alpha +2{t}_{w}+{t}_{w}[1-sen\alpha (1+cos\alpha \left)\right]/cos\alpha $ (or input data) |

Weir and cycle efficiency | |||

Magnification ratio | M | 3.17 | $M=L/\left(wN\right)$ |

Linear weir coefficient discharge | C_{d}_{(90°)} | 0.754 | ${C}_{d\left(90{}^{\xb0}\right)}=1/[-8.609+22.65HT/P+1.812/HT/P]+0.6375$ [5] |

Cycle efficiency | ${\ua72a}^{\u2033}$ | 0.74 | ${\ua72a}^{\u2033}={C}_{d}\left(\alpha \right)M$ |

Weir efficiency | ${\ua72a}^{\prime}$ | 1.96 | ${\ua72a}^{\prime}={C}_{d}\left(\alpha \right)M/{C}_{d\left(90{}^{\xb0}\right)}$ |

Nappe interference length and aeration condition | |||

Nappe interference length | B_{int} | 10.89 m | Equations (30)–(33) |

Aeration condition | - | Drowned | Table 7 and Figure 12 |

(iv) Submergence (Tullis et al., 2007 [38]) | |||

Downstream total head | H_{d} | 1.22 m | - |

Head ratio | H_{d}/H_{T} | 0.21 | - |

Submergence upstream total head | H* | 5.84 m | Equations (1)–(3) and Figure 2 |

Submergence level | S | 0.20 | S = H_{d}/H*; 0 ≤ S ≤ 1 |

Submerged weir discharge coefficient | ${C}_{d-sum}$ | 0.476 | ${C}_{d-sum}={C}_{d}(\alpha ){({H}^{*}/HT)}^{1.5}$ |

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## Share and Cite

**MDPI and ACS Style**

Mattos-Villarroel, E.D.; Ojeda-Bustamante, W.; Díaz-Delgado, C.; Salinas-Tapia, H.; Flores-Velázquez, J.; Bautista Capetillo, C.
Methodological Proposal for the Hydraulic Design of Labyrinth Weirs. *Water* **2023**, *15*, 722.
https://doi.org/10.3390/w15040722

**AMA Style**

Mattos-Villarroel ED, Ojeda-Bustamante W, Díaz-Delgado C, Salinas-Tapia H, Flores-Velázquez J, Bautista Capetillo C.
Methodological Proposal for the Hydraulic Design of Labyrinth Weirs. *Water*. 2023; 15(4):722.
https://doi.org/10.3390/w15040722

**Chicago/Turabian Style**

Mattos-Villarroel, Erick Dante, Waldo Ojeda-Bustamante, Carlos Díaz-Delgado, Humberto Salinas-Tapia, Jorge Flores-Velázquez, and Carlos Bautista Capetillo.
2023. "Methodological Proposal for the Hydraulic Design of Labyrinth Weirs" *Water* 15, no. 4: 722.
https://doi.org/10.3390/w15040722