## Abstract

Climate change has caused the inefficient operation of a significant number of old weirs to pass large discharges. Therefore, this study aims to increase the discharge capacity of the labyrinth weir. A new approach was proposed by modifying a labyrinth weir structure. The data was obtained from the quarter-round crest and different sidewall angles ranging from 8 to 35°. A conventional labyrinth weir was used for comparison. The results showed that the percentage of the notches area to sidewalls area of the weir (*An*/*Aw*) does not exceed 8%. Also, the percentage of the notches' length to total crest length (*ΔL*/*Lc*) does not exceed 32%. Also, the percentage of the notch depth to the sidewall depth (*ΔP*/*P*) does not exceed 30%. The other parameters are kept constant. These dimensionless terms provided a maximum compound coefficient of discharge of 0.74. Also, the compound discharge coefficient initially increased at low water head ratios and decreased at higher values of water head ratios. The regression empirical equations were generated. The maximum increase in efficiency was 10% for a sidewall angle of 6° when compared to conventional labyrinth weirs. The maximum improvement of the compound coefficient of discharge was 18.8% for a sidewall angle of 8°.

## HIGHLIGHTS

A new approach to the labyrinth weir was proposed and named compound labyrinth weir to improve the discharge capacity of the labyrinth weir.

Derivation of hydraulic equation for compound labyrinth weir has been done.

Establishing the set of curves showing the relationship between the compound coefficient of discharge C

_{dc}against total head over the entire compound labyrinth weir H′_{t}/P′ with a range of sidewall angles.Empirical equations for compound labyrinth weir and compound linear weir were created.

Using the compound labyrinth weir efficiency (έ) as a new tool to provide a guide in choosing a suitable design of the sidewall angle (α).

### Graphical Abstract

## NOTATIONS

The following symbols are utilised in the present study:

*A*Inside apex width

*An*Notch area

*Aw*Sidewall area of the weir

*b*_{1}Bottom width of the notch

*B*Labyrinth weir length in the flow direction

*C*_{dc}Compound coefficient of discharge

*D*Outside apex width

*έ*Cycle efficiency

*g*Acceleration of gravity

*h*Flow depth over the weir crest (high stage)

*h*′Flow depth over the notch (low stage)

*H*_{t}Total head over the weir crest (high stage)

*H*′_{t}Total head over the notch (low stage)

*H*′/_{t}*P*′Headwater ratio

*Lc*Total centreline length of labyrinth weir

*lc*Centreline length of weir side wall

*Lc*-cycleCentreline length for a single labyrinth weir cycle

Length of the labyrinth weir crest after subtracting the length of notches (Δ

*L*), ()*l*′Length of the notch

*N*Number of labyrinth weir cycles

*n*Number of notches

*P*Weir height

*P*′Notch height

*Q*Discharge over the weir

*R*_{crest}Radius of crest shape

*t*_{w}Thickness of weir wall

*V*Average cross-sectional flow velocity upstream of weir

*W*Channel width

*w*A single labyrinth weir cycle width

- Δ
*L* The top width for the notch

- Δ
*P* The notch depth

*α*Sidewall angle

## INTRODUCTION

Climate change and associated extreme weather events are resulting in flood events that occur at higher frequency and magnitude (Fowler & Kilsby 2003; Bruwier *et al.* 2015; Kvočka *et al.* 2016). Climate change caused a change in the hydrological information that was relied upon in the existing weir design. However, the return period of the flood is often updated. This situation could lead to increased water depth upstream of a weir. When this occurs, it leads to an increased risk of failure in the structure that might be linked to the high level of water over the weir crest, which is more than the allowable value of the design water head. Therefore, it has become impossible to discharge flood waves regularly and safely as a result of climate change. Therefore, old weirs need to be upgraded to labyrinth weirs because labyrinth weirs play an important role in ensuring the security of human life as well as safety in developing areas and the natural environment.

However, labyrinth weirs are overflow structures folded in plan-view and consist of a series of linear weirs installed non-perpendicularly to the sidewall of a channel, resulting in a longer crest length compared with the channel width. For sites where the weir width is restricted but a high discharge is demanded, the labyrinth weirs are an effective and economical choice (Khode *et al.* 2011), since they can provide a higher discharge capacity for a given water head. Also, labyrinth weirs have become popular as an alternative rehabilitation measure in the upgrading of existing weirs. For example, in areas where rainfall has resulted in peak flows exceeding the capacity of existing weir systems, the labyrinth weir can provide a practical means of rehabilitation (Vasquez *et al.* 2007). Labyrinth weirs are useful not only for handling larger flow rates but are also used for energy dissipation and self-aeration.

The hydraulic behaviour of labyrinth weirs has received interest from researchers and engineers. Early studies were conducted by Taylor (1968) that provided information on the hydraulic performance and design of the labyrinth weir. Hay & Taylor (1970) developed design criteria based on Taylor's earlier work. Darvas (1971) developed a set of curves to estimate labyrinth spillway performance. Houston (1982) extensively studied a physical model to assess different labyrinth geometries and approach conditions. Lux (1989) mentioned that the aeration condition behind the nappe flow of the labyrinth weir was identified as aerated, transitional, and suppressed. Tullis *et al.* (1995) investigated labyrinth weir performance and obtained design curves relating the coefficient of discharge, upstream head, and the height of the weir for various sidewall angles from 6 to 18°. Savage *et al.* (2004) identified errors of up to ±25% in the proposed method by Tullis *et al.* (1995). Falvey (2003) showed crest flow conditions consist of four conditions of nappe aeration, including pressure, atmospheric, cavity, and sub-atmospheric. Ouamane & Lempérière (2006) studied the rectangular planform of the labyrinth weir and found the rectangular plan shape to be more efficient than the trapezoidal plan shape. Tullis *et al.* (2007) conducted physical model studies and derived relationships of head–discharge for labyrinth weirs. Ghare *et al.* (2008) found that discharge capacity was reduced when the crest length of the labyrinth weir was decreased. A sidewall angle has a significant effect on both the discharge capacity and the labyrinth weir layout. Effective length is increased with a smaller angle, which thereby provides a higher discharge capacity.

Also, Crookston & Tullis (2010) included experimental outcomes for 8° sidewall angles. Khode *et al.* (2011) showed the relationship between coefficients of discharge and *H*/*P* for various sidewall angles. They found that the discharge coefficient decreased as the sidewall angle decreased. Paxson *et al.* (2011) showed that flow conditions downstream of the labyrinth weir could also influence discharge capacity. Khode *et al.* (2012) developed the relationship between the coefficient of discharge and *H*/*P* for a variety of sidewall angles from 8 to 30°. Crookston *et al.* (2012) developed different models with a sidewall angle *α* = 15° and used trapezoidal labyrinth weirs. A significant decrease in cycle efficiency with an increase in the *Ac*/*lc* ratio occurred. The apex had the lowest impact on total discharge efficiency, as it had a larger impact on the total cycle length and caused a decrease in sidewall length, which thereby caused a reduction in total flow efficiency. Carollo *et al.* (2012) studied five groups of physical models and adapted the empirical equation suggested by Ghodsian (2009). Dimensional analysis was used to investigate the discharge over the sharp crest of the labyrinth weir. Kumar *et al.* (2012) investigated discharge capacity using the sharp crest and curving plan shape weir. The coefficients of discharge equations were proposed and the results showed an increase of approximately 40% of the flow capacity over a curved weir compared with a conventional weir. Gupta *et al.* (2013) studied the characteristics of the discharge under free-flow conditions for a sharp-crested, contracted triangular planform weir. The discharge efficiency across the triangular planform weir was greater than across the normal weir. The equation of flow was suggested for the given range of data and was within ±5% of the observed data.

Moreover, Gupta *et al.* (2014) investigated W-planform weirs to determine discharge characteristics. The efficiency of the discharge for W-planform labyrinth weirs was greater than that of the normal weir. The equation of flow was suggested for the given range of data and was within ±5% of the observed data. Gupta *et al.* (2015) used 24 sharp crests to create a labyrinth weir. A rectangular planform weir was utilised to study the impact of the height of the crest on the performance of discharge with different crest lengths. The discharge efficiency of the rectangular labyrinth weir was shown to be higher than that of the conventional weir. Savage *et al.* (2016) showed the design curves that were found by Crookston & Tullis (2013) are suitable for headwater ratios (*H _{t}*/

*P*) > 2.0. Also, Gebhardt

*et al.*(2017) solved navigation requirement problems in the Ilmenau waterway in northern Germany by using a side labyrinth weir. A relationship between the coefficient of discharge and head over various weir heights was obtained. The labyrinth weir is more reliable to operate and safer. Ghaderi

*et al.*(2020a) conducted a numerical and experimental study to investigate the effects of the geometry parameters of trapezoidal–triangular labyrinth weirs (TTLW) on the energy dissipation, discharge coefficient, and downstream flow regime. The results demonstrated that the experimental results agree with the numerical model. Energy dissipation is reduced with an increase in sidewall angle. Daneshfaraz

*et al.*(2020) used the FLOW-3D model to study the effect of channel-bed slope and different channels on the coefficient of discharge of a labyrinth weir. The findings showed that modifying the channel improved the coefficient of discharge. Ghaderi

*et al.*(2020b) used FLOW-3D software to simulate the hydraulic performance of labyrinth weirs. They used the inclining crest edge of the weir and notches in the weir wall. The outcomes showed that modifying the labyrinth weir geometry improved discharge capacity over the labyrinth weir compared with a conventional weir.

Despite the availability of many previous studies about labyrinth weirs’ geometry, evaluation of the performance of the labyrinth weir is still a primary concern and a serious challenge for designers of this type of weir due to major concerns about flood control. This concern has emerged because of climate change and the development of new methods to estimate extreme floods. These methods have demonstrated the insufficiency of a significant number of weirs in handling increasing numbers of extreme flooding events. Consequently, solutions for increasing the capacity of existing weirs and providing more efficient weirs for new projects are being sought. However, due to a lack of knowledge and exemplary ideas, labyrinth weirs need to be investigated further to improve their performance. However, in the present study, a new shape of the labyrinth weir has been proposed. This shape is developed by modifying a labyrinth weir structure (named a compound labyrinth weir). These modifications are made by using trapezoidal notches on the sidewalls of the labyrinth weir. This work provides new knowledge about the new shape of the labyrinth weir. The results of the present study are important to engineers and designers, especially given the current climate change impacts evidenced by possibly higher storm intensities and peak discharges, and where weir rehabilitation is becoming an issue of concern.

## MATERIALS AND METHODS

### Theoretical considerations and hydraulic equation derivation for a compound labyrinth weir

*C*

_{dc}) values are determined using experimental results that are obtained from physical modelling. However, the capacity of discharge of a compound labyrinth weir is a function of the coefficient of discharge, total head over the weir, and crest length. A

*C*

_{dc}depends on the labyrinth weir height, wall thickness, total head, crest shape, apex configuration, and sidewall angle of the labyrinth weir. To simplify the analysis, the impact of surface tension and viscosity could be neglected. These parameters are applied to select a satisfactory magnitude of the model and velocity. According to these assumptions, gravitational acceleration is the only significant parameter. Tullis

*et al.*(1995) recommended a new design method for the labyrinth weirs by depending on a basic design equation for the linear weir. Later, it was developed into a labyrinth weir design. The present study found that the

*C*

_{dc}for compound labyrinths is affected by the same variables that affect conventional labyrinth weirs, such as the sidewall angle and the apex of the labyrinth weir. In addition, some variables are related to notched geometry. To calculate the

*C*

_{dc}for the compound labyrinth weir, the flow pattern consists of two scenarios. First, flow only passes over the notches. Second, the flow passes over the entire labyrinth weir. Figure 1 shows a compound labyrinth weir with common geometry and the following equation determines the theoretical discharge (

*Q*

_{theo.}) over the crest of the labyrinth weir and notches:

*Q*

_{n}_{theo.}is the theoretical discharge only passes over the notches (Henderson 1966) that can be expressed as follows:

*C*

_{dc}is the compound coefficient of discharge. The discharge equation for the compound labyrinth weir is obtained by integrating the flow through notches and the labyrinth weir. The discharge equation for a notch is given in Equation (5). The discharge over the compound labyrinth weir with four notches is calculated by adding the flow through the four notches, as shown in Figure 1(d). The

*C*

_{d}values for the trapezoidal notch are calculated separately from Equation (5). Equation (5) can be used to calculate the flow over the notch when the head over the labyrinth weir (

*h*) is less than the head over the notch (

*h*′) when

*h*= 0 and is determined by as shown in Figure 1(d). In this case, the discharge coefficient

*C*

_{d}corresponding to the notch is used. When the head over the labyrinth weir (

*h*) is greater than the head over the notch (

*h*′), the flow over the compound weir is estimated using Equation (7). Where

*H*is computed by and is the length of the labyrinth weir after subtracting the notches length (Δ

_{t}*L*), (), as shown in Figure 1(d).

*θ*is the side slope angle for the notch (degrees),

*b*

_{1}is the bottom width of the notch, is the crest length of the labyrinth weir,

*g*is the acceleration due to gravity, as shown in Figure 1. Figure 2 shows the hydraulic parameters for flow over the compound labyrinth weir.

### Experimental setup

The experiments were carried out in a rectangular flume of length 7 m, width 0.5 m, and height 0.6 m. The flume walls were made from acrylic panels supported by a steel frame. The flume bed was set horizontally. The water was supplied from a storage tank with a volume of 2.5 m^{3} and recirculated via a 200 mm supply pipe. Two pumps are connected in parallel using a water supply line. The capacity of each pump was 50 L/s giving a total capacity of 100 L/s for both. The flow metre was installed in the main pipeline. The flow metre diameter is 150 mm with an accuracy of 0.5% of the discharge rate. The discharge rates ranged from 10 to 100 L∕s with an accuracy of ± 0.05 to ± 0.5 L/s. A gate valve was utilised in the pipeline to control the flow rate. The flume also contains one regulating gate downstream to control the tailwater elevation. The wave suppressors were provided at upstream of the flume. The wave suppressors were used to control the flow and dissipate the surface disturbances. The water level was measured upstream of the labyrinth weir using a movable pointer gauge with an accuracy of 0.001 m mounted on the flume side rails (allowing longitudinal and transverse movement). As recommended by Dabling (2014), the point gauge was located at a distance of 4 *P* times the weir height. In the present study, approximately 80 cm was measured from the inlet of the flume to the upstream of the physical models to avoid a water drawdown at the weir edge. Water level measurements varied from 5 to 100 mm for the range of flows tested. For each tested weir geometry, 25 readings of head–discharge were carried out. For each run, the water head over the crest of the labyrinth weir was measured about 2–3 times by a point gauge and the average of the readings was recorded to ensure more accuracy.

In the present study, nine models were used to investigate the optimal notch geometry of the compound labyrinth weir. Table 1 shows the physical model test for the optimal geometry of the trapezoidal notch and Figure 1(a) shows the three-dimensional geometry of the compound labyrinth weir. In addition, eight models are used to study the hydraulic characteristics of compound labyrinth weirs, as shown in Table 2. These models consisted of rang sidewall angles (*α*) configurations of 6, 8, 10, 12, 15, 20, 35, and 90° (linear weir for comparison). The quarter-round crest experiments were carried out with the curved edge facing upstream, as shown in Figure 1(b). These models were two cycles (*N* = 2) of compound labyrinth weirs and the total width (*W*) was 0.5 m. All models were fabricated from acrylic sheets with a thickness (*t _{w}*) of 10 mm. The acrylic sheets were cut using a laser machine to obtain precise dimensions and then the parts were assembled using screws. Silicon was used to prevent the leakage of water through all the joints of the models. The walls of the model were fixed on the base of the acrylic with a thickness of 6 mm to assist with installing the weir inside the flume. To avoid the curvature effect, all models were located approximately 1.5 m from the inlet point of the flume.

*An*is the notch area that is defined as

*An*

*=*Δ

*P*(Δ

*L*

*+*

*b*

_{1})/2.

*Aw*is the sidewall area of the weir that is defined as

*Aw*

*=*

*lc P*.

Labyrinth geometry . | Model no. . | Notch geometry . | An/Aw
. | ΔP/P
. | ΔL/Lc
. | ||
---|---|---|---|---|---|---|---|

b_{1} (cm)
. | ΔL (cm)
. | ΔP (cm)
. | |||||

α = 15°P = 20 cmB = 40.05 cmlc-one leg = 41.46 cmLc = 173.87 cmW = 50 cmN = 2D = 3.5 cmA = 2 cmt = 1 cm _{w} | 1 | 15.58 | 17.58 | 2 | 4% | 0.1 | 0.42 |

2 | 6.29 | 10.29 | 4 | 4% | 0.2 | 0.24 | |

3 | 2.52 | 8.529 | 6 | 4% | 0.3 | 0.20 | |

4 | 23.88 | 25.88 | 2 | 6% | 0.1 | 0.62 | |

5 | 10.44 | 14.44 | 4 | 6% | 0.2 | 0.34 | |

6 | 5.29 | 11.29 | 6 | 6% | 0.3 | 0.27 | |

7 | 32.17 | 34.17 | 2 | 8% | 0.1 | 0.82 | |

8 | 14.58 | 18.58 | 4 | 8% | 0.2 | 0.44 | |

9 | 8.05 | 14.05 | 6 | 8% | 0.3 | 0.33 |

Labyrinth geometry . | Model no. . | Notch geometry . | An/Aw
. | ΔP/P
. | ΔL/Lc
. | ||
---|---|---|---|---|---|---|---|

b_{1} (cm)
. | ΔL (cm)
. | ΔP (cm)
. | |||||

α = 15°P = 20 cmB = 40.05 cmlc-one leg = 41.46 cmLc = 173.87 cmW = 50 cmN = 2D = 3.5 cmA = 2 cmt = 1 cm _{w} | 1 | 15.58 | 17.58 | 2 | 4% | 0.1 | 0.42 |

2 | 6.29 | 10.29 | 4 | 4% | 0.2 | 0.24 | |

3 | 2.52 | 8.529 | 6 | 4% | 0.3 | 0.20 | |

4 | 23.88 | 25.88 | 2 | 6% | 0.1 | 0.62 | |

5 | 10.44 | 14.44 | 4 | 6% | 0.2 | 0.34 | |

6 | 5.29 | 11.29 | 6 | 6% | 0.3 | 0.27 | |

7 | 32.17 | 34.17 | 2 | 8% | 0.1 | 0.82 | |

8 | 14.58 | 18.58 | 4 | 8% | 0.2 | 0.44 | |

9 | 8.05 | 14.05 | 6 | 8% | 0.3 | 0.33 |

α (°)
. | P (cm)
. | B (cm)
. | Lc (cm)
. | A (cm)
. | D (cm)
. | Notch geometry . | ΔP/P
. | ΔL/lc
. | Shape of crest . | ||
---|---|---|---|---|---|---|---|---|---|---|---|

b_{1} (cm)
. | ΔL (cm)
. | ΔP (cm)
. | |||||||||

6 | 20 | 100.8 | 413.6 | 2 | 3.8 | 52.8 | 60.8 | 4 | 0.2 | 0.6 | QR |

8 | 20 | 75.6 | 313.5 | 2 | 3.7 | 37.8 | 45.8 | 4 | 0.2 | 0.6 | QR |

10 | 20 | 60.4 | 253.5 | 2 | 3.6 | 28.8 | 36.8 | 4 | 0.2 | 0.6 | QR |

12 | 20 | 50.2 | 213.6 | 2 | 3.6 | 22.8 | 30.8 | 4 | 0.2 | 0.6 | QR |

15 | 20 | 40.0 | 173.8 | 2 | 3.5 | 16.8 | 24.8 | 4 | 0.2 | 0.6 | QR |

20 | 20 | 29.6 | 134.3 | 2 | 3.4 | 10.9 | 18.9 | 4 | 0.2 | 0.6 | QR |

35 | 20 | 15.6 | 84.5 | 2 | 3.0 | 3.4 | 11.4 | 4 | 0.2 | 0.6 | QR |

90 | 20 | – | 50 | – | – | 22 | 30 | 4 | 0.2 | 0.6 | QR |

α (°)
. | P (cm)
. | B (cm)
. | Lc (cm)
. | A (cm)
. | D (cm)
. | Notch geometry . | ΔP/P
. | ΔL/lc
. | Shape of crest . | ||
---|---|---|---|---|---|---|---|---|---|---|---|

b_{1} (cm)
. | ΔL (cm)
. | ΔP (cm)
. | |||||||||

6 | 20 | 100.8 | 413.6 | 2 | 3.8 | 52.8 | 60.8 | 4 | 0.2 | 0.6 | QR |

8 | 20 | 75.6 | 313.5 | 2 | 3.7 | 37.8 | 45.8 | 4 | 0.2 | 0.6 | QR |

10 | 20 | 60.4 | 253.5 | 2 | 3.6 | 28.8 | 36.8 | 4 | 0.2 | 0.6 | QR |

12 | 20 | 50.2 | 213.6 | 2 | 3.6 | 22.8 | 30.8 | 4 | 0.2 | 0.6 | QR |

15 | 20 | 40.0 | 173.8 | 2 | 3.5 | 16.8 | 24.8 | 4 | 0.2 | 0.6 | QR |

20 | 20 | 29.6 | 134.3 | 2 | 3.4 | 10.9 | 18.9 | 4 | 0.2 | 0.6 | QR |

35 | 20 | 15.6 | 84.5 | 2 | 3.0 | 3.4 | 11.4 | 4 | 0.2 | 0.6 | QR |

90 | 20 | – | 50 | – | – | 22 | 30 | 4 | 0.2 | 0.6 | QR |

The current research is based mainly on laboratory experiments to study the effect of the sidewall angle, which contains different notch geometries for each angle. The trapezoidal shape is considered a suitable choice for the notch shape because the trapezoidal shape is best for hydraulic sections. In the present study, to maintain the notches’ shape as a trapezoid, the geometry of the notches was adopted as 0.1 ≤ Δ*P*/*P* ≤ 0.3 and 4% ≤ *An*/*Aw* ≤ 8%. In other words, if the notches scale is outside this range, it will change to a triangular shape and thus it will be outside the limitations of the present study. Also, the height of the notch from the bottom of the channel to the notch crest should be at least two times the maximum estimated head of water level above the notch crest. This is essential to reducing the velocity of approach (Walkowiak 2006).

Water level data was collected with respect to the weir crest elevation with 0.05 < *H _{t}*′/

*P*′ < 0.80 (for the range of flow rate (

*Q*) tested) by measuring the water level over the low stage (

*h*′) and high stage (

*h*) weirs. The measurements were taken only after the flow conditions had reached approximately steady conditions (Crookston 2010), where

*Q*,

*h*, and

*h*′ were measured and recorded. The total heads,

*H*and

_{t}*H*′ were computed by adding the velocity head (

_{t}*V*

^{2}/2

*g*) to

*h*and

*h*′

*.*The velocity of flow was calculated by dividing the actual discharge value, which was measured by a flow metre, by the cross-sectional area of flow based on the width of the weir and flow height. All the measurements were taken at the centre of the width of the flume.

### Scale effect

*et al.*(1999) indicated that prototype behaviour can be represented by a scale ratio of 1:20 or more and showed that the scale ratio of 1:15 was optimal. Furthermore, Boes & Hager (2003) stated that the scale ratio should be from 1:10 to 1:20 if smaller scale models can provide safe design information. In the present study, the scale was set to 1:20 to limit possible scale effects. The purpose of utilising a scaled model is to find values for use at a prototype scale. Chanson

*et al.*(2002) recommended adopting a Reynolds number greater than (10

^{5}) to avoid scale effects. As shown in the Moody's diagram, the Reynolds number is independent of the friction factor in a wholly rough turbulent flow. In other words, energy dissipation is independent of the Reynolds number in a roughly turbulent flow. Reynolds number was defined by Equation (8). For the present study, the flow adopted a minimum Reynolds number of 102,850 for a sidewall angle of 6° and reached a maximum of 277,778 for a sidewall angle of 15°. Afterwards, the Reynolds number is large enough to avoid scale effects. Because the Reynolds number is greater than 10

^{5}, the flow conditions are referred to as turbulent flow:where

*V*is the velocity of the flow,

*L*is the length of the labyrinth weir in the flow direction, and

*Ѵ*is the kinematic viscosity.

*σ*is the surface tension and

*ρ*is the fluid density. The water temperature was 22 °C during the physical tests.

## RESULTS AND DISCUSSION

### Determination of the optimal geometry for the notch

In the present study, a new approach has been proposed for the geometry of the labyrinth weir. The new proposal focused on making notches on the sidewalls of the labyrinth weir. The benefit of using notches was to decrease the water level upstream of the weir during peak flood events. In addition, this type of labyrinth weir can provide sufficient flow capacity to deliver greater storm events. The low stage weir is designed to pass moderate-flow events, while the higher stage weir is designed to add sufficient capacity to deliver the more extreme weir design flood. Therefore, it was important to determine the optimal geometry of the notches and the geometry effect on the *C*_{dc} under free-flow conditions.

*C*

_{dc}and

*H*′/

_{t}*P*′ for various Δ

*P*/

*P*, various

*An*/

*Aw*and the sidewall angle

*α*= 15° of a compound labyrinth weir. The other dimensions of the notch were constant. Δ

*P*/

*P*represents the percentage notch depth to weir height, while

*An*/

*Aw*represents the percentage notch area to the sidewall area of the weir. All the parameters have been presented in Figure 1.

According to Figure 3, the impact of notch depth on the *C*_{dc} has been observed when the ratio of dropped height increases (Δ*P*/*P* from 0.1 to 0.3). *C*_{dc} values differed significantly at low *H _{t}*′/

*P*′, but

*C*

_{dc}values converged as

*H*′/

_{t}*P*′ values increased. This difference affects the notch depth in low discharge, but this effect reduces at high discharge. When

*H*′/

_{t}*P*′ > 0.5, the flow over the compound labyrinth weir becomes submerged; therefore, the effect of notch depth is neglected. When

*An*/

*Aw*= 4% (Figure 3(a)), the

*C*

_{dc}for Δ

*P*/

*P*= 0.3 was about 3 and 7% greater than the

*C*

_{dc}values for Δ

*P*/

*P*= 0.2 and 0.1, respectively. While

*An*/

*Aw*= 6% (Figure 3(b)), the

*C*

_{dc}value for Δ

*P*/

*P*= 0.3 was 4.5 and 8% greater than the

*C*

_{dc}values for Δ

*P*/

*P*= 0.2 and 0.1, respectively. Furthermore, when

*An*/

*Aw*= 8% (Figure 3(c)), the

*C*

_{dc}value for Δ

*P*/

*P*= 0.3 was 5 and 10% higher than the

*C*

_{dc}values for Δ

*P*/

*P*= 0.2 and 0.1, respectively.

*C*

_{dc}and

*H*′/

_{t}*P*′ for various Δ

*P*/

*P*values and various Δ

*L*/

*Lc*values for the sidewall angle

*α*= 15° of a compound labyrinth weir. In Figure 4, the effect of notch length on the

*C*

_{dc}has been investigated. The models for each dropped height (Δ

*P*/

*P*) with different lengths of the notch crest (Δ

*L*/

*Lc*) were tested. These tests investigated the impact of the notch length on the weir's efficiency. The behaviour of the compound labyrinth weir in various Δ

*L*/

*Lc*cases was similar to the behaviour of the compound labyrinth weir for various Δ

*P*/

*P*values.

Moreover, Figure 4 shows that the difference in *C*_{dc} values was significant at low *H _{t}*′/

*P*′ and that the

*C*

_{dc}values converged when

*H*′/

_{t}*P*′ values increased. This difference is attributed to the effect of notch length in low discharge, but this effect decreases with high discharge. When

*H*′/

_{t}*P*′ > 0.5, the flow over the labyrinth weir may become submerged and, therefore, the effect of the notch length is neglected. When Δ

*P*/

*P*= 0.1 (Figure 4(a)), the average

*C*

_{dc}for (Δ

*L*/

*Lc*= 0.82,

*An*/

*Aw*= 8%) was 4.5 and 8% higher than the

*C*

_{dc}values for (Δ

*L*/

*Lc*= 0.62,

*An*/

*Aw*= 6%) and (Δ

*L*/

*Lc*= 0.42,

*An*/

*Aw*

*=*4%), respectively. While Δ

*P*/

*P*= 0.2 (Figure 4(b)), the average

*C*

_{dc}for (Δ

*L*/

*Lc*= 0.45,

*An*/

*Aw*= 8%) was about 5 and 9% greater than the

*C*

_{dc}values for (Δ

*L*/

*Lc*= 0.35,

*An*/

*Aw*= 6%) and (Δ

*L*/

*Lc*= 0.25,

*An*/

*Aw*= 4%), respectively. Also, in case Δ

*P*/

*P*= 0.3 (Figure 4(c)), the average

*C*

_{dc}value for (Δ

*L*/

*Lc*= 0.34,

*An*/

*Aw*= 8%) was about 4 and 11% greater than the

*C*

_{dc}values for (Δ

*L*/

*Lc*= 0.27,

*An*/

*Aw*= 6%) and (Δ

*L*/

*Lc*= 0.21,

*An*/

*Aw*= 4%), respectively.

Using a trapezoidal notch at the sidewall of a labyrinth weir has a good effect on the *C*_{dc} when using various *An*/*Aw* percentages (4, 6, and 8%) and different values of Δ*P*/*P* (0.1, 0.2, and 0.3). From the results, the optimum geometry of the notch to yield the optimum *C*_{dc} was noted. For the present study, the *An*/*Aw* does not exceed 8%, Δ*L*/*Lc* does not exceed 0.32, and Δ*P*/*P* does not exceed 0.3 when the other parameters are constant. These parameters have also been given a maximum *C*_{dc} of 0.74.

### Discharge rating curves

*C*

_{dc}and

*H*′/

_{t}*P*′ for a quarter-round crest. Different sidewall angles (

*α*) of compound labyrinth weir (6° ≤

*α*≤ 35°) were used. For comparison, the linear weir (

*α*= 90°) has been included. In general, the flow over the compound labyrinth weir has two different scenarios: first, the flow passes through notches, as shown in Figure 6(a). Second, flow passes over the entire labyrinth weir, as shown in Figure 6(b).

Figure 5 shows that *C*_{dc} initially behaves similarly for all models when *H _{t}*′/

*P*′ are low and the flow only passes through the notch. When

*H*′/

_{t}*P*′ values are low, the

*C*

_{dc}begins to rise because the nappe flow is still non-aerated with little to no interference nappe, as shown in Figure 6(a). At a low value of H′

_{t}/P′, the flow touches the sidewalls of the compound labyrinth wear. This case is analogous to the linear weir because losses at the weir crest still dominate. Also, obviously, nappe interference is absent. When the flow passes over the entire labyrinth weir,

*C*reaches a maximum value because the flow is compound. The discharge over the notches is aerated and the nappe flow becomes more effective. At the same time, the discharge over the entire weir is non-aerated, and nappe flow does not interfere. Figure 5 also shows that

_{dc}*C*

_{dc}reaches a maximum value before decreasing and tending towards an asymptotic value when the flow is large values of

*H*′/

_{t}*P*′

*.*This is different from linear weirs, where

*C*

_{dc}reaches a maximum and approaches

*C*

_{dc}asymptotically after a slight decrease. When

*H*′/

_{t}*P*′ is large, increasing

*H*′/

_{t}*P*′ has a marginal effect on

*C*

_{dc}. When the weir is fully submerged, interference effects between the discharges from the individual weir elements become insignificant. The clinging condition finished at H′

_{t}/P′∼0.07 for α = 6, 8, 12, and 20°, whereas it ended at H′

_{t}/P′ ∼0.06 for α = 10 and 15°. In comparison to linear weirs, the results showed that the

*C*

_{dc}decreases significantly after its peak. A significant decrease is because labyrinth weirs are subject to the nappe interference of the falling jets (Crookston & Tullis 2011), which is absent in linear weirs. In addition to the sudden removal of the air cavities behind the nappe, nappe interference becomes less significant as

*α*increases; therefore,

*C*

_{dc}increases as

*α*increases for a given upstream head.

*et al.*2014). Equations (10) and (11) are valid for 0.05 ≤

*H*′/

_{t}*P*′ < ∼0.75–0.8, and the coefficients of A, B, C, and D are tabulated in Table 3. After applying Equation (10) for compound labyrinth weir and Equation (11) for compound linear weir, the data display a well-behaved nature when the

*C*

_{dc}(

*α°*) curves have been generalised to

*H*′/

_{t}*P*′ = 1.0, as shown in Figure 5 with dashed lines. Equations (10) and (11) were selected instead of polynomial formulas because the data have a good correlation representation (

*R*

^{2}≥ 0.98). In addition, extrapolation performance remains well-behaved up to

*H*′/

_{t}*P*′

*≤*2.0. Equations (10) and (11) reflect a good choice for the designer in the case of using compound labyrinth weirs with different sidewall angles:

Crest shape . | Fitting coefficients . | Sidewall angle (α) . | |||||||
---|---|---|---|---|---|---|---|---|---|

6° . | 8° . | 10° . | 12° . | 15° . | 20° . | 35° . | 90° . | ||

Quarter-round crest | A | 0.601 | 0.653 | 0.697 | 0.732 | 0.749 | 0.765 | 0.543 | 2.716 |

B | 0.254 | 0.268 | 0.28 | 0.346 | 0.376 | 0.466 | 0.804 | 0.061 | |

C | 0.353 | 0.384 | 0.446 | 0.435 | 0.475 | 0.544 | 0.642 | 1.127 | |

D | 3.262 | 2.923 | 2.886 | 3.313 | 3.255 | 4.916 | −4.25 | 0.632 | |

R^{2} | 0.985 | 0.986 | 0.991 | 0.992 | 0.982 | 0.979 | 0.978 | 0.987 |

Crest shape . | Fitting coefficients . | Sidewall angle (α) . | |||||||
---|---|---|---|---|---|---|---|---|---|

6° . | 8° . | 10° . | 12° . | 15° . | 20° . | 35° . | 90° . | ||

Quarter-round crest | A | 0.601 | 0.653 | 0.697 | 0.732 | 0.749 | 0.765 | 0.543 | 2.716 |

B | 0.254 | 0.268 | 0.28 | 0.346 | 0.376 | 0.466 | 0.804 | 0.061 | |

C | 0.353 | 0.384 | 0.446 | 0.435 | 0.475 | 0.544 | 0.642 | 1.127 | |

D | 3.262 | 2.923 | 2.886 | 3.313 | 3.255 | 4.916 | −4.25 | 0.632 | |

R^{2} | 0.985 | 0.986 | 0.991 | 0.992 | 0.982 | 0.979 | 0.978 | 0.987 |

### Labyrinth weir efficiency (*ε*)

*H*′ value). A compound labyrinth weir is a complex structure; consequently, optimising the design could be difficult. Therefore, the efficiency of the compound labyrinth weir provides guidance for choosing a suitable design. As shown in Figure 5, although the

_{t}*C*

_{dc}for labyrinth weirs is less than the

*C*

_{dc}for linear weirs, the increase in weir length for labyrinth weirs results in a higher overall discharge capacity.

*C*

_{dc}varies for a given water depth in labyrinth weirs. As a result, the alternative analysis for the

*C*

_{dc}of labyrinth weirs takes into account the total width of the labyrinth weir by analysing the efficiency, as follows:Here,

*Q*is the flow over the compound labyrinth weir and

*W*is the width of the compound labyrinth weir.

*ε*for the sidewall angles range from 6 to 35° and the quarter-round crest are shown in Figure 7. In general,

*ε*is larger when

*α*is small. According to total capacity, labyrinth weirs with larger sidewall angles are less efficient. For each

*α*, the labyrinth weir efficiency (

*ε*) increases with increasing

*H*′/

_{t}*P*′. This is because nappe interference and aeration behind the nappe flow decrease when

*H*′/

_{t}*P′*increase. For comparison between the efficiency of the compound and conventional labyrinth weirs, the percentage of the improvement in efficiency was 10, 8.9, 7.8, 7.2, 6.6, 5, and 4% for

*α*= 6, 8, 10, 12, 15, 20, and 35°, respectively. The best curve-fit coefficients of Equation (13) and the coefficient of determination (

*R*

^{2}) are presented in Table 4. These equations are valid for 0.07 ≤

*H*′/

_{t}*P′*< ∼ 0.85. Equation (13) assists in determining the efficiency of the compound and conventional labyrinth weirs:

. | Compound labyrinth weir . | Conventional labyrinth weir . | ||||
---|---|---|---|---|---|---|

α(°)
. | a
. | b
. | R^{2}
. | a
. | b
. | R^{2}
. |

6 | 0.37 | 1.41 | 0.99 | 0.36 | 1.49 | 0.99 |

8 | 0.33 | 1.51 | 0.99 | 0.32 | 1.65 | 0.99 |

10 | 0.30 | 1.60 | 0.99 | 0.30 | 1.80 | 0.99 |

12 | 0.27 | 1.81 | 0.99 | 0.26 | 2.03 | 0.99 |

15 | 0.25 | 1.99 | 0.99 | 0.24 | 2.22 | 0.99 |

20 | 0.24 | 2.15 | 0.99 | 0.24 | 2.36 | 0.99 |

35 | 0.23 | 2.16 | 0.99 | 0.233 | 2.91 | 0.99 |

. | Compound labyrinth weir . | Conventional labyrinth weir . | ||||
---|---|---|---|---|---|---|

α(°)
. | a
. | b
. | R^{2}
. | a
. | b
. | R^{2}
. |

6 | 0.37 | 1.41 | 0.99 | 0.36 | 1.49 | 0.99 |

8 | 0.33 | 1.51 | 0.99 | 0.32 | 1.65 | 0.99 |

10 | 0.30 | 1.60 | 0.99 | 0.30 | 1.80 | 0.99 |

12 | 0.27 | 1.81 | 0.99 | 0.26 | 2.03 | 0.99 |

15 | 0.25 | 1.99 | 0.99 | 0.24 | 2.22 | 0.99 |

20 | 0.24 | 2.15 | 0.99 | 0.24 | 2.36 | 0.99 |

35 | 0.23 | 2.16 | 0.99 | 0.233 | 2.91 | 0.99 |

### Nappe aeration conditions

The results showed that when increasing *H _{t}*′ over a compound labyrinth weir, the nappe flow condition changes from clinging aerated to partially aerated to finally drowned. However, all aeration conditions do not certainly take place for all geometries of the compound labyrinth weir. The nappe aeration condition is one of the factors that influence the efficiency of flow over a compound labyrinth weir. Aeration conditions also characterise nappe behaviour. The nappe flow may initially be relatively tranquil and then develop into an unstable condition because of the fluctuating pressure produced on the weir wall. This pressure causes nappe vibrations and noise. In general, the compound labyrinth weir is more efficient when the nappe is clinging (Figure 8(a)) than when it is aerated (Figure 8(b)). This difference is because sub-atmospheric pressures develop on the downstream face of the weir. A partially aerated nappe (Figure 8(c)) occurs at higher

*H*′/

_{t}*P′*values. In this case, the air cavities behind the nappe are unstable and vary both temporally and spatially. The air cavities fluctuate between labyrinth weir apexes. Also, the air cavities may be completely removed when the nappe flow condition is submerged. This process reappears as the unsteady flow and turbulent levels behind the nappe fluctuate. The air cavities are highly dynamic and cause oscillating pressures on the downstream face of the weir.

The results showed that nappe trajectories are unstable for the partially aerated nappe condition based on flow conditions and weir geometry. For a stable nappe, the partially aerated condition had less effect on the nappe trajectory. In addition, increasing *H _{t}*′/

*P′*causes the nappe to transition from partially aerated to drowned (Figure 8(d)). The drowned nappe appears as a thick nappe without an air cavity.

The clinging condition finished at *H _{t}*′/

*P′*∼ 0.07 for

*α*= 6, 8, 12, and 20° and at

*H*′/

_{t}*P′*∼ 0.06 for

*α*= 10°, 15°. The compound labyrinth weir with

*α*= 35° was observed to change directly to an aerated or partially aerated nappe condition. The nappe aeration condition was observed when 0.12 <

*H*′/

_{t}*P′*< 0.22 for

*α*= 35°. Depending on

*α*, the largest range of aeration nappe was observed for

*α*= 6–20°. The nappe condition changes from aerated to partially aerated at 0.23 ≤

*H*′/

_{t}*P′*≤ 0.43 for

*α*= 6°. Depending on

*α*, the largest range of partially aerated nappe was observed for

*α*= 8–35°. The nappe drowned condition was observed when

*H*′/

_{t}*P′*> 0.43 for

*α*= 6°. Also, when

*α*increased, the nappe drowned condition increased. The range of

*H*′/

_{t}*P′*observed for each nappe aeration condition of compound labyrinth weirs is shown in Table 5. This table will assist the designer to avoid using undesirable ranges of

*H*′/

_{t}*P′*in the design process.

(H′/_{t}P′). | ||||
---|---|---|---|---|

α (°)
. | Clinging . | Aerated . | Partially aerated . | Drowned . |

6 | <0.073 | 0.073–0.23 | 0.23–0.43 | >0.43 |

8 | <0.067 | 0.067–0.31 | 0.31–0.48 | >0.48 |

10 | <0.064 | 0.064–0.32 | 0.32–0.5 | >0.5 |

12 | <0.074 | 0.074–0.29 | 0.29–0.54 | >0.54 |

15 | <0.06 | 0.06–0.28 | 0.28–0.58 | >0.58 |

20 | <0.07 | 0.07–0.26 | 0.26–0.60 | >0.60 |

35 | – | 0.12–0.22 | 0.22–0.62 | >0.62 |

(H′/_{t}P′). | ||||
---|---|---|---|---|

α (°)
. | Clinging . | Aerated . | Partially aerated . | Drowned . |

6 | <0.073 | 0.073–0.23 | 0.23–0.43 | >0.43 |

8 | <0.067 | 0.067–0.31 | 0.31–0.48 | >0.48 |

10 | <0.064 | 0.064–0.32 | 0.32–0.5 | >0.5 |

12 | <0.074 | 0.074–0.29 | 0.29–0.54 | >0.54 |

15 | <0.06 | 0.06–0.28 | 0.28–0.58 | >0.58 |

20 | <0.07 | 0.07–0.26 | 0.26–0.60 | >0.60 |

35 | – | 0.12–0.22 | 0.22–0.62 | >0.62 |

### Comparison of the compound labyrinth weir and conventional labyrinth weir

*C*

_{dc}of the proposed compound labyrinth weirs is compared with research in the literature on conventional labyrinth weirs. The available data for traditional labyrinth weir were collected by Willmore (2004) and Crookston & Tullis (2013). The comparisons demonstrated a considerable increase in

*C*

_{dc}for the compound labyrinth weir over the data of Willmore (2004) and Crookston & Tullis (2013) for

*α*ranging from 6 to 35°. The results showed that the existing notches on the sidewall of the labyrinth weir have significantly improved the discharge capacity of the labyrinth weir. In Figure 9, comparing the present study with Willmore (2004), the maximum value of the improved percentage was for

*α*= 6°, with an improvement of 20.5% and the minimum value of the improved percentage was for

*α*= 8°, with an improvement of 17.3%. While comparing the present study with Crookston & Tullis (2013) data, the maximum value of the improved percentage was for

*α*= 8°, with an improvement of 18.8%, and the minimum value of the improved percentage was for

*α*= 35°, with an improvement of 15.3%.

## CONCLUSIONS

The purpose of the present study is to improve the discharge capacity of the labyrinth weir. The labyrinth weir reflects an effective method for increasing storage capacity. The results have shown that dimensionless terms of notches area ratio (*An*/*Aw*), notches length ratio (Δ*L*/*Lc),* and notches depth ratio (Δ*P*/*P*) do not exceed 8, 32, and 3%, respectively, when setting the other parameters as constants. These dimensionless terms have been provided with a maximum *C*_{dc} of 0.74. For small values of *H _{t}*′/

*P*′

*, C*

_{dc}increases with increased

*H*′/

_{t}*P*′, while

*C*

_{dc}reduces with increased

*H*′/

_{t}*P*′ and decreases towards a constant value as a function of

*α*when

*H*′/

_{t}*P*′ is large. This reduction in

*C*

_{dc}is observed in the case of a compound labyrinth weir with a range of sidewall angles (

*α*) (from 6 to 35°) but not in the linear weir. The results also showed that the highest values of the

*C*

_{dc}were with a sidewall angle of 35° and the lowest value of the

*C*

_{dc}was with a sidewall angle of 6° for all examined models. The outcomes demonstrated relationships between

*C*

_{dc}values as a function of

*H*′/

_{t}*P*′

*.*Empirical equations relating to

*C*

_{dc}and

*H*′/

_{t}*P*′ were created, and the data have a good correlation representation (

*R*

^{2}≥ 0.98). Empirical equations have been used to predict

*C*

_{dc}values. The set of curves was well-behaved up to

*H*′/

_{t}*P*′ ≤ 2.0.

Furthermore, the efficiency (*ε*) is larger when the sidewall angle (*α*) is small. For each *α* angle, the efficiency (*ε*) decreases with increasing *H _{t}*′/

*P*′

*.*The compound labyrinth weir has higher efficiency when compared with conventional labyrinth weirs, the percentage of the improvement in efficiency was 10, 8.9, 7.8, 7.2, 6.6, 5, and 4% for

*α*= 6, 8, 10, 12, 15, 20, and 35°, respectively. Moreover, the results demonstrated that nappe flow conditions pass through four phases for ranges of

*H*′/

_{t}*P*′

*.*These phases were clinging, aerated, partially aerated, and drowned. In addition, the discharge capacity of the compound labyrinth weir was more efficient with a clinging nappe condition than an aerated, partially aerated, or drowned nappe for all tested models. The maximum value of the improved percentage in the

*C*

_{dc}value, when comparing the present results with a traditional labyrinth weir that was done by Willmore (2004), was as large as 20.5% for

*α*= 6°, while the minimum value of the improved percentage was as large as 17.3% for

*α*= 8°. While comparing the present study with Crookston & Tullis (2013), the maximum value of the improved percentage was as large as 18.8% for

*α*= 8°, and the minimum value of the improved percentage was as large as 15.3% for

*α*= 35°.

Although the methods and information for compound labyrinth weirs have been obtained in the present study, we recommend that a compound labyrinth weir design be verified with numerical and/or physical model studies. This information should be taken into account for site-specific conditions that may be outside the limitations of the present study. Also, the results of the present study will give valuable insights into the operation and performance of the compound labyrinth weir.

## ACKNOWLEDGEMENTS

The authors would like to express their sincere thanks and gratitude to the government and the Ministry of Higher Education and Scientific Research in Iraq for providing financial support for this study. They also express their sincere thanks to the school of engineering at Deakin University for the use of the new test facility. They appreciate the technical support that is provided by laboratory staff at the School of Engineering (Deakin University).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Labyrinth Weirs*

*Ph.D. Dissertation*

*Nonlinear Weir Hydraulics*

*M.Sc. Thesis*

**IOCRSEM – 14**

*Report No. GR-82-7*. U.S. Bureau of Reclamations, Denver, CO, USA

*The Performance of Labyrinth Weirs*

*Hydraulic Characteristics of Labyrinth Weirs*