Flow Characteristics and Energy Dissipation over Trapezoidal Labyrinth Weirs

Available online 26 November 2022

0955-5986/© 2022 Elsevier Ltd. All rights reserved.

Investigation of flow characteristics and energy dissipation over new shape of the trapezoidal labyrinth weirs

Anees K. Idrees

^a,*

, Riyadh Al-Ameri

^b

aDept. of Environment Engineering, Faculty of Engineering, University of Babylon, Hillah-Najf Road, Babylon, Iraq

bSchool of Engineering, Faculty of Science Engineering & Built Environment, Deakin University, 75 Pigdons Road, Waurn Ponds, VIC, 3220, Australia

A R T I C L E I N F O Keywords:

Compound labyrinth weir Compound coefficient of discharge Energy dissipation

Flat crest Hydraulic jump

A B S T R A C T

Labyrinth weirs are mainly used to increase the discharge capacity. The current study adds a new performance to labyrinth weirs as an energy dissipator. The labyrinth weirs’ zigzag shape and flow behaviour could benefit energy dissipation. Therefore, the present study aims to investigate the hydraulic characteristics and energy dissipation of the compound labyrinth weir. Sixteen models were used for different sidewall angles (α^◦) of 6–35 and 90 (linear weir for comparison). The results demonstrated the highest values of the compound coefficient of discharge, Cdc, for a sidewall angle of 35^◦, and the lowest value of the compound coefficient of discharge for a sidewall angle of 6^◦. The Cdc increased initially at low H՛_t/P՛ values, and the Cdc showed a decreasing trend for higher values of H՛_t/P՛. For sidewall angles (α^◦) ranging from 6 to 35, the compound coefficient of discharge Cdc does not significantly change as it approaches a value of H՛_t/P՛ =1.0. Furthermore, for the range of the relative critical head (yc/P՛) between 0.07 and 0.95, the results showed that the compound labyrinth weirs could dissipate the energy of flow by 93%, 92%, 89%, 85%, 83%, 79%, and 75% for α^◦ =6, 8, 10, 12, 15, 20^◦, and 35, respectively. The amount of improvement in energy dissipation over a compound labyrinth weir was better than a linear weir by 17%, 15%, 14%, 12%, 11%, 10%, and 8% for α^◦=6, 8, 10, 12, 15, 20, and 35, respectively. The residual energy (E1/Emin) at the base of downstream compound labyrinth weirs was closer to the minimum potential amount of residual energy as yc/P՛ increased. For a given value of yc/P՛, the relative residual energy at the base of compound labyrinth weirs increased as the sidewall angle (α) increased. An empirical equation has been provided to predict the compound coefficient of discharge when relative energy dissipation data is available.

1. Introduction

A compound labyrinth weir is a new type of nonlinear weir that is used to increase discharge capacity [1]. The zigzag shape and flow behaviour over labyrinth weirs (e.g., nappe interference and hydraulic jump) could have benefits in energy dissipation. The advantage of knowing the total energy dissipation and flow regime downstream of the compound labyrinth weir is required for designing the downstream channel of the weir.

Several studies were adapted to the assessment of the discharge ca- pacity of labyrinth weirs, such as [2–6]. Kumar et al. [7] used the lab- yrinth weir’s triangle plan to investigate the discharge coefficient. They found that reducing the apex weir angle caused in an increase of the nappe interference zone and a decrease in the discharge coefficient. In addition, they proposed formulas for calculating the coefficient of

discharge at various apex angles. Crookston and Tullis [8] focused on constructing labyrinth weirs into reservoirs and concluded that arched labyrinth weirs performed better due to the orientation of the weir cy- cles. Dimensional analysis was utilised by Carollo et al. [9] to calculate the outflow rate of triangular labyrinth weirs. In an experimental context, Bilhan et al. [10] examined the impact of breakers on the discharge capacity for labyrinth weirs with and without nappe breakers.

They found that the coefficient of discharge for trapezoidal and circular labyrinth weirs with breakers was reduced by up to 4% compared to an unmodified labyrinth weir. Also, Abbaspuor et al. [11] investigated hydraulic passage flow via a triangular labyrinth weir both experi- mentally and numerically. They found that crossflow interference was reduced when increased the angle of the weir apex, which occurred at a low flow. Azimi and Hakim [12] investigated hydraulic flow over a rectangular labyrinth weir in the lab. They demonstrated that a

* Corresponding author.

E-mail address: eng.anees.idrees@uobabylon.edu.iq (A.K. Idrees).

https://doi.org/10.1016/j.flowmeasinst.2022.102276

Received 22 April 2022; Received in revised form 10 November 2022; Accepted 21 November 2022

rectangular labyrinth weir behaved effectively when the ratio ho/P = 0.4, (h_o is the piezometric head over the weir, and P is the weir crest height). Additionally, they demonstrated that, in comparison to a linear weir, a rectangular labyrinth weir had a lower efficiency by 10% under submerged conditions. Furthermore, researchers have made a few at- tempts to study the energy dispersion over the labyrinth weirs. However, Magalh˜aes and Lorena [13] studied trapezoidal labyrinth weirs with the WES-type crest shape. They developed an empirical chart to estimate the relative residual energy. In contrast, Lopes et al. [14] compared the coefficient of discharge and relative residual energy in the base of the labyrinth weir with different magnification ratios (L/W) and crest shapes. They showed that relative residual energy (H1/Ho) at the base of labyrinth weirs is smaller when HT/P values are lower. Also, Lopes et al.

[15] studied a physical model to investigate the relative residual energy at the downstream area of the trapezoidal labyrinth weir with a quarter-round crest shape (α^◦=30, w/P =2). Relative residual energy for H/P ≥0.4 was within 5%, while for H/P <0.4, the energy was less than 10%. Bieri et al. [16] studied France’s Gloriettes Concrete Arch Dam to rehabilitate the dam’s spillway. A Piano Key Weir (PKW) was proposed and designed as an additional spillway. Steps downstream of the PKW and a stilling basin were recommended for energy dissipation downstream of the PKW [17]. Lopes and Melo [18] evaluated the flow characteristics and the energy dissipation downstream of the labyrinth weir. The results showed that the chute flow is essentially three-dimensional in the locality of the labyrinth weir. For the tested range of H/p and L/W, the current study found that the relative residual energy is within 10% of ref [13]. Silvestri et al. [19] compared the re- sidual energy at the toe of a stepped spillway of different lengths. They considered four lengths of a spillway for a classical ogee-crested weir, and two different PKWs were investigated at the top of the structure for a wide range of discharge. Maatooq [20] studied a physical model of a stepped spillway labyrinth weir. They suggested changing the path width for each step to labyrinth weir length (L) instead of the width of the weir (W) and found that the labyrinth flow path induces interlocking between the mainstream channel and spreads laterally.

Mohammadzadeh-Habili et al. [21] studied a physical model to inves- tigate the energy dissipation and downstream flow regime for trape- zoidal and triangular labyrinth weirs and linear weirs. Merkel et al. [22]

investigated the characteristics of flow and energy dissipation down- stream of the labyrinth weir. They developed a relation between discharge, drop height, and energy dissipation. Experimental and nu- merical investigations were conducted by Ghaderi et al. [23] to inves- tigate the effects of the geometry parameters of trapezoidal-triangular labyrinth weirs (TTLW) on the downstream flow regime, energy dissi- pation, and discharge coefficient. Their outcomes demonstrated that the experimental results agree with the numerical model. Energy dissipation is reduced with an increase in sidewall angle. Ghaderi et al. [24] utilised FLOW-3D software and modified labyrinth weir geometry to simulate the free flow surface. They found that the flow rate and the flow magnification ratio decreased when the HT/P ratio increased. Modifying the labyrinth weir improved hydraulic performance at low HT/P ratios (HT/P <0.2). Haghiabi et al. [25] investigated the energy dissipation of the trapezoidal and triangular labyrinth weirs. They used one and two key cycles of labyrinth weir models. They found that energy dissipation by labyrinth weir was between 85% and 70% when the ho/P was be- tween 0.1 and 1.0.

In contrast, some studies examined energy dissipation using different weir shapes, such as stepped weirs, drop weirs, and cascading weirs [26–30]. Shamsi et al. [31] investigated a cylindrical weir’s energy dissipation and hydraulic performance. The results demonstrated that the coefficient of discharge ranged from 1.0 to 1.4 for the relative head (H/D) range between 0.15 and 2.0. Also, they showed that cylindrical weirs could dissipate energy between 80% and 15%. Parsaie et al. [32]

investigated the finite crested stepped spillway’s coefficient of discharge (Cd) and energy dissipation. The outcomes showed that the coefficient of discharge varies between 0.9 and 1.2. They found that the main

parameter of the ratio of the upstream head to the length of the crest hup/Lc affects the coefficient of discharge. The energy dissipation of the finite crested stepped spillway changes between 95% and 40%.

Moreover, Parsaie et al. [33] investigated the energy dissipation and discharge coefficient of the circular crested stepped spillway (CCSS).

They found that the coefficient of discharge of the circular crested stepped spillway ranges between 0.9 and 1.4. Also, the energy dissipa- tion by CCSS between 90% and 40%. Parsaie et al. [34] utilised an artificial neural network to predict the energy dissipation over stepped spillways. They demonstrated that the most influential parameters on energy dissipation over stepped spillways were the Froude number, ratio of critical depth to the height of steps, and drop number.

However, available information in the literature is still lacking in the downstream flow regime region and compound labyrinth weirs’ energy dissipation. Therefore, the energy dissipation over compound labyrinth weirs has not yet been investigated. The air entrainment and super- critical waves downstream of the labyrinth weir produce a complex flow pattern. The significance of the present study is to determine the type of flow (e.g., critical flow, subcritical, and supercritical) downstream of the compound labyrinth weir. Therefore, in the present study, the Froude number (Fr) is used as an index to determine the type of flow. This index provides a clearer understanding of hydraulic behaviour downstream of the weir. In the present study, a compound labyrinth weir has been used.

A compound labyrinth weir is a classical labyrinth weir with notches. It is important to highlight the process of conveying the flow from up- stream to downstream of the labyrinth weir. This process should be achieved without causing dangerous harm to the river, environment, dam, or weir structure. Therefore, determining the amount of energy dissipation and residual energy will help engineers to determine the type and size of an energy dissipator, which often uses a stilling basin, and then reduce the cost of construction. The energy dissipator must be designed to safely satisfy requirements for a broad range of flow rates. In addition, the current study provides an empirical equation for predicting the compound coefficient of discharge when relative energy dissipation data is available. However, the aim of the current study is to investigate the ability of the compound labyrinth weir to dissipate energy and hy- draulic performance.

2. Experimental setup

The present work used a rectangular flume of 0.5 m wide, 7 m long, and 0.6 m deep. Both sides of the flume were acrylic sheets supported by a steel framework to observe the flow pattern along the flume section visually. The flume has a screw mechanical jack on both sides to provide a slope to the flume. The flume’s bed slope in the current study was horizontal. The physical models were located on a horizontal platform base. To avoid water leakage, screws and silicon were utilised. The platform base dimensions have a width of 50 cm and a variable length depending on the length of the labyrinth weir. A platform base was also made of an acrylic sheet with a thickness of 6 mm. Water supply was provided from a storage tank of 2.5 m ×1 m ×1 m. Two pumps con- nected in parallel were used to recirculate the water through the flume.

A regular gate was used to control the tailwater elevation downstream of the flume. The flume was supplied with wave suppressors upstream of the flume. Wave suppressors consisted of a set of pipes (each pipe 20 mm diameter) that were used to dissipate the surface disturbances. Also, to maintain the uniform flow, metal mesh with 15 mm square cells is used at the entrance of the stilling tank to suppress large-scale flow distur- bances at the flume entrance, as shown in Fig. 1. A gate valve regulated the flow rates for each test. The total capacity of both pumps was 80 L/s (40 L/s for each pump). The flow rates were measured using a flow metre. The flow rates had an accuracy of ±1 L/s and varied from 2 L∕s to 80 L∕s. The diameter of the flow meter is 150 mm with a maximum discharge of 138.9 L/s and a minimum discharge of 1.667 L/s. A dia- gram of the test facility is presented in Fig. 1.

A two-pointer gauge mounted on the flume side rails (allowing

longitudinal and transverse movement) has been utilised to determine water depth upstream and downstream of the labyrinth weir. The ac- curacy of both pointer gauges was ±0.2 mm. All the measurements were taken at the centre of the width of the flume. The present study used 16 models, as shown in Fig. 2 and Table 1. The physical models were placed at a distance of 1.5 m from the wave suppressors, as shown in Fig. 1, to investigate the compound labyrinth weirs’ energy dissipation and discharge coefficient. The SPSS software was used for filtering the experimental data.

These models consisted of rang sidewall angle (α^◦) configurations of 6, 8, 10, 12, 15, 20, 35, and 90 (linear weir for comparison). All the models were tested for flat crest shape. These models were two cycles (N

=2) of a trapezoidal compound labyrinth weir, and the total width (W) was 0.5 m. Physical models were manufactured using acrylic sheets with a 10 mm thickness. A laser machine has been used to cut the acrylic sheets to obtain precise dimensions. The physical models were assem- bled by screws. Silicon was used to avoid water leaking through the models’ joints.

3. Theoretical considerations

Energy dissipation was calculated by subtracting the initial flow energy (Eo) upstream of the compound labyrinth weir from the residual energy (E1) downstream that remained at the toe of the compound Fig. 1.Schematic of the rectangular flume test facility, all dimension in centimetre.

Fig. 2. Compound labyrinth weir: (a) plan view of common geometry. (b) The model during operation: α^◦=8, Q =0.043 m³/s. (c) Longitudinal section a-a of the weir. (d) Linear weir, α^◦=90.

labyrinth weir (i.e., before the hydraulic jump) (see Fig. 2(b)). The initial energy of flow (Eo) was calculated using a specific energy Eq. (1) after neglecting potential energy due to taking the bed of the flume as a reference for measurements.

Eo=ho+α^{՛ q}

2

2gh²_o (1)

where q is the unit width of flow measured using a flowmeter, ho is the flow depth measured at a distance of 1.5 m from the flume outlet toward upstream of the weir to avoid the curvature effect [38], and α^′is the kinetic correction coefficient with a value of 1.0. Given that the distri- bution of kinetic energy is non-uniform at the base of the compound labyrinth weir, α^′was 1.1, as recommended by Refs. [35,36]. Likewise, E1 can be determined, as shown in Eq. (2).

E1=h1+α՛ q²

2gh²₁ (2)

where h1 is the water depth at the toe of the compound labyrinth weir before the hydraulic jump (initial depth of jump). Directly measuring h₁ is impossible due to the highly turbulent flow, whereas measuring the downstream sequent depth, h2, and for a horizontal channel, h1 can be undertaken assuming momentum conservation Eq. (3) [37].

h1=h2

2

⎧

⎨

⎩

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

1+8 (q²

8h³₂ )2

√ ⎫

⎬

⎭− 1 (3)

To get more stable flow and high accuracy during testing, a second point gauge was located downstream of the physical model at a distance of 5P (approximately 1 m). This position is used to avoid the effect of turbulent flow and the visual position [38]. After calculating the initial and residual energy, the difference in energy between the upstream and downstream of the weir was expressed in Eq. (4).

ΔE=Eo− E1 (4)

The energy dissipation efficiency (%) ^△E_Eo was calculated through Eq.

(5).

%△E Eo

=

(Eo− E1

Eo

)

x100 (5)

Estimating energy dissipation of a compound labyrinth weir was ach- ieved by comparing the total energy dissipation ΔE with the maximum energy dissipation ΔEmax. Mohammadzadeh et al. [39] noted that depending on the specific energy curve, Emin is the minimum value of particular energy that can take place in the case of critical flow.

Therefore, the minimum potential amount of residual energy E1min in the downstream region of a compound labyrinth weir can be determined through Eq. (6).

E1min=Emin=1.5yc (6)

yc=

̅̅̅̅̅

q² g

3

√

(7) where yc is the critical depth of total flow over the weir, q is flow per unit meter, and g is the acceleration due to gravity (m/s²).

Eqs. (8) and (9) represent the Froude number (F_r1) and the minimum Froude number (Fr1(min)) downstream of the weir, respectively.

Where v1 is the flow velocity downstream, h1 is the headwater downstream, and g is the acceleration due to gravity (m/s²).

Fr1 = V1

̅̅̅̅̅̅̅̅

√gh1 (8)

Fr1(min)=0.43+0.36yc

Eo (9)

The derivation of the hydraulic discharge equation for the trape- zoidal compound labyrinth weir is carried out by first considering the discharge Q through the trapezoidal notches [40]:

Q=cdn

(8 15x ̅̅̅̅̅

√2g

x tanθ

2x H՛^1.5_t +2 3x ̅̅̅̅̅

√2g

x b1x H՛^1.5_t )

(10) where Cdn =coefficient of the discharge over notches,b1 =bottom width of the notch, H՛_t =total head over the notch (lower stage). The basic equation for discharge (Q) over a labyrinth weir under free-flow con- ditions is given by Ref. [1]:

Q=Cd ∗ 2 3

̅̅̅̅̅

2g

√

x L՛cx H^1.5_t (11)

where Cd =coefficient of the discharge, Ht =total head over the whole weir (high stage), L՛_c = length of the labyrinth weir crest after sub- tracting the length of notches (ΔL), (L՛_c =Lc- n* ΔL), g =acceleration due to gravity and n =number of notches (in present study n =4). The discharge equation for the trapezoidal compound labyrinth weir is ob- tained through a combination of Eqs. (10) and (11) [1].

Q=Cdc

[ n

(8 15x ̅̅̅̅̅

√2g x tanθ

2x H՛^1.5_t +2 3x ̅̅̅̅̅

√2g

x b1xH՛^1.5_t )

+2 3x ̅̅̅̅̅

√2g

x L՛cx H^1.5_t ]

(12) where Cdc is the compound discharge coefficient. As shown in Fig. 3, H՛_t is the total head over the notch (lower stage) and was determined by H՛_t

=h՛ +(V²/2g) while Ht is the total head over the whole labyrinth weir crest (high stage) and was calculated by Ht =ho +(V²/2g).

Energy dissipation data was collected when adjusting the discharge (Q). A point gauge measured the piezometric head, (h_o) upstream and (h2) downstream, of the weir. For calculating the compound coefficient of discharge, the data were collected by modifying the flow rate (Q) and measuring the piezometric head (h՛) over the notch (lower stage) up- stream of the weir and measuring (ho) over the whole labyrinth weir crest (high stage). Measurements were taken after at least 4 min to ensure steady flow conditions, as recommended by Ref. [41]. Q, ho, h՛, and h2 were measured and recorded, and these values were recorded Table 1

Details of the test program.

α ₍^o_{) P (cm) B (cm) Lc (cm) A (cm) D (cm)} Notch Geometry ΔP/P ΔL/lc

b1 (cm) ΔL (cm) ΔP (cm)

6 20 100 413.6 2 3.8 52.8 60.8 4 0.2 0.6

8 20 75.6 313.5 2 3.7 37.8 45.8 4 0.2 0.6

10 20 60.4 253.5 2 3.6 28.8 36.8 4 0.2 0.6

12 20 50.2 213.6 2 3.6 22.8 30.8 4 0.2 0.6

15 20 40.0 173.8 2 3.5 16.8 24.8 4 0.2 0.6

20 20 29.6 134.3 2 3.4 10.9 18.9 4 0.2 0.6

35 20 15.6 84.5 2 3.0 3.4 11.4 4 0.2 0.6

90 20 – 50 – – 22 30 4 0.2 0.6

directly into an Excel spreadsheet and logged in a notebook.

4. Scale effects

When a scaled model is applied to a prototype, the accuracy of the findings it generates may be impacted. To avoid scale effects, Chanson et al. [42] suggested using a Reynolds number more than 10⁵. In the present study, the minimum value of the Reynolds number was calcu- lated for α^◦=12, with a range between 105,828 and 2777778. As a result of the Reynolds number being above 10⁵. Therefore, Reynolds number is large enough to avoid scaling effects.

The Weber number is a dimensionless parameter that represents the relationship between the forces of inertia and surface tension, according to Falvey [43]. Since surface tension forces can have a considerable impact in models with low depths, the Weber number is utilised to avoid their impact. A minimal Weber number of 11 was suggested by Novak and Cabelka [44]. In the current study, the Weber number’s minimum value was estimated for α^◦=10 with a range of 13–2610. As a result, surface tension’s impact enables it to be disregarded.

Surface tension force and water viscosity both affect scaling effects.

Scale-up to prototype’s influence is therefore not particularly signifi- cant. Since the prototype and model both utilised the same fluid, the viscosity was the same. Surface tension force of a prototype can be ignored because it is too low [45].

Additionally, Heller [46] demonstrated that the Reynolds and Weber numbers were not congruent as a result of Froude’s similitude. Typi- cally, these numbers continue to be high enough to prevent severe scaling errors. Froude similarity or gravitational force predominates in open channel flow. Researchers have generally supported a variety of scale ratios. Pegram et al. [47] demonstrated that a scale ratio of 1:15 was ideal and that behaviour from a prototype could be replicated at a scale ratio of 1:20 or more. According to Boes and Hager [48], if smaller scale models can deliver reliable design information, the scale ratio should be between 1:10 and 1:20. In the current investigation, to avoid any scale effects, the scale was fixed at 1:20.

5. Results and discussions 5.1. Flow regimes

Determining the flow type (e.g., critical flow, subcritical, and su- percritical) downstream of the compound labyrinth weir is crucial. The Froude number (Fr) has been used to determine the flow type and pro- vide a more precise understanding of hydraulic behaviour downstream of the labyrinth weir. The flow conditions of the compound labyrinth weir are developed from clinging, aerated, partially aerated, and sub- merged conditions. This development occurs with increasing discharge over the compound labyrinth weir. The hydraulic jump appearance downstream of the compound labyrinth weir can be determined by computing the Froude number (Fr1) downstream of the weir. Energy dissipation can be analysed by sequent depth analysis of the hydraulic jump downstream of the weir. For each test, the Froude number

downstream of the hydraulic jump Fr1 can be calculated from Eq. (8).

Fig. 4 presents the results of Fr1 against the relative critical depth of yc/P՛ for the flat crest. For comparison, Fig. 4 included a minimum limit of Froude numbers downstream of the compound labyrinth weir models (Fr1(min)) and which is calculated from Eq. (9). Also, the Froude number downstream of the linear weir is included.

Fig. 4 shows that the flow regime downstream of the labyrinth weir can vary significantly from subcritical to supercritical. Subcritical flow occurs downstream of a compound labyrinth weir when the depth of water downstream is greater than the critical depth. Fig. 4 showed that the compound labyrinth weir had some hydraulic conditions that were subcritical, as shown in Table 2. However, the collision of oblique su- percritical flows at the base of nappes results in supercritical flow con- ditions downstream of the compound labyrinth weir, creating a hydraulic jump. For each sidewall angle (α), Fr1 (Eq. (8)) increases when yc/P՛ increases. Also, Fr1 increases when (α) increases. The results of the Froude number downstream of the weir (Fr1) were more significant than the minimum Froude number downstream (Fr1(min)). Thus, it caused the generation of the hydraulic jump.

Fig. 5(a) depicted a longitudinal view of a robust hydraulic jump downstream of the compound labyrinth weir. There is highly turbulence in the flow between the hydraulic jump position and the downstream face of the compound labyrinth weir. Therefore, the initial depth of the hydraulic jump cannot be measured. However, the findings revealed that a significant proportion of upstream energy was lost. Additionally, because supercritical flows collide in the base of the nappes and circu- lating flows occur behind the nappes, the flow depth in the base of the nappes increases.

Consequently, the inflow Froude number of hydraulic jumps de- creases. A weak hydraulic jump occurs when the inflow Froude number is small. The difference between the initial depth before the jump and Fig. 3. Sketch of hydraulic parameters for flow on: (a) lower stage; (b) upper stage.

Fig. 4. Froude number (Fr1) downstream of the examined weir models versus the relative critical depth to height of the low crest (yc/P՛).

the sequent depth of the hydraulic jump is not very significant, as shown in Fig. 5(b)).

5.2. Discharge rating curves

Labyrinth weirs are efficient in improving weir discharge because the crest length is increased beyond the width of the channel. Fig. 6 shows the relationship between the compound coefficient of discharge (Cdc) against H՛_t/P՛ for various values of α (6 =α^◦≤35). Data for a linear weir (α^◦=90) are included for comparison.

A linear weir was used for comparison because labyrinth weirs consist of a series of linear weirs installed non perpendicularly to the direction of flow in a channel. The flow over the compound labyrinth weir generally has two distinct scenarios, as observed in the current study. First, discharges can be segregated to the lower stage segment, as shown in Fig. 7(a). Second, discharges over the entire weir length were engaged, as shown in Fig. 7(b). Fig. 6 shows that Cdc initially behaves similarly when H՛_t/P՛ is small when the flow passes only through the notch. When H՛_t/P՛ values are low, the C_dc begins to rise because the nappe flow is still nonaerated with little to no interference nappe, as shown in Fig. 7(a).

The flow situation in the compound labyrinth weir when H՛_t/P՛ _is small is analogous to the linear weir because losses at the crest dominate for small values of H՛_t/P՛, and obviously, nappe interference is absent.

When the flow passes over the entire labyrinth weir, C_dc reaches a maximum value because the flow is compound. The discharge over notches is aerated, and the nappe flow becomes more effective. Simul- taneously, the discharge across the entire weir is nonaerated, and the

nappe flow does not interfere, as illustrated in Fig. 7(b).

Fig. 6 also showed that C_dc reaches a maximum value before decreasing and tending towards an asymptotic value for large values of H՛_t/P՛. It is different from linear weirs, where Cdc reaches a maximum and approaches C_dc asymptotically after a slight decrease. When H՛_t/P՛ _is significant, increasing H՛_t/P՛ has a marginal effect on Cdc. When the weir is fully submerged, interference effects between the discharge from the individual weir elements become insignificant. This situation when H՛_t/ P՛ is large is analogous to the linear weir case because Cdc approaches an asymptotic value. After all, the effects of the weir crest no longer dominate.

Compared to linear weirs, the results showed that the Cdc decreases significantly after reaching its peak. This significant decrease is because, in addition to the abrupt removal of the air cavities behind the nappe, labyrinth weirs are subject to nappe interference from the falling jets, which is absent in linear weirs [8]. Nappe interference becomes less significant when α is larger; therefore, Cdc increases as α increases for a given upstream head.

5.3. Energy dissipation

The relative energy dissipation ΔE/E_o is plotted in Fig. 8 as a function of yc/P՛ for the flat crest shape and the range of sidewall angles. Eq (4) was used to compute the energy difference (ΔE) between the upstream and downstream sides of the weir, and the initial energy of flow (Eo) was calculated by Eq. (1). By comparing the results, the theoretical maximum energy dissipation rate ΔEmax on compound labyrinth weirs is found by utilising the specific energy curve. The energy dissipation for a vertical drop weir was collected from the literature [49] and Eq. (13).

The data for the linear weir was collected from the present study using Eqs. (1), (2) and (4).

△E Eo

=1− h+_2h^yc32

P+1.5yc (13)

Fig. 8 shows that ΔE/E0 decreases as yc/P՛ increases, where Eo is the total upstream energy, as nappe interference is weak under fully sub- merged conditions, resulting in less energy dissipation. Also, for a given yc/P՛, relative energy dissipation (ΔE/Eo) increases with decreased sidewall angle (α). Smaller values of α allow a more significant potential for nappe collision from adjacent sidewalls closer together as the side- wall angle decreases. Hence, there is greater energy dissipation at the base of the compound labyrinth weir, especially under low-discharge conditions.

For the range of the relative critical head (yc/P՛) between 0.07 and 0.95, the results showed that the compound labyrinth weirs could dissipate the energy of flow by 93%, 92%, 89%, 85%, 83%, 79%, and 75% for α^◦=6, 8, 10, 12, 15, 20, and 35, respectively. Furthermore, the energy loss in a compound labyrinth weir is significantly more than that of a vertical drop weir or a compound linear weir. The energy dissipation over a compound labyrinth weir was more significant than a linear weir by 17%, 15%, 14%, 12%, 11%, 10%, and 8% for α^◦=6, 8, 10, 12, 15, 20, and 35, respectively. Therefore, it has been shown that trapezoidal compound labyrinth weirs are excellent energy dissipating structures.

The effective energy dissipation rates in trapezoidal labyrinth weirs are due to three main reasons. First, a nappe collision occurs near the upstream apexes. Our experimental observations and those of Crookston and Tullis [50] showed that nappe collision could produce a local quasi-oblique hydraulic jump condition close to the apexes. This hy- draulic jump causes energy dissipation, as shown in Fig. 9.

Second, a standing pool develops behind the nappe because to tur- bulent mixing. The overflow from the weir crest resembles an oblique jet that strikes the channel bed, part of which is deflected against the downstream face of the weir, creating a turbulent recirculating pool behind the nappe (see Fig. 9), giving rise to energy dissipation because of turbulent mixing. In comparison to vertical drop weirs, linear weirs Table 2

Type of flow downstream of the compound labyrinth weir.

α^◦ Subcritical Supercritical

yc/P՛ yc/P՛

6 ≤0.30 >0.30

8 ≤0.28 >0.28

10 ≤0.22 >0.22

12 ≤0.18 >0.18

15 ≤0.15 >0.15

20 ≤0.15 >0.15

35 – >0.10

Fig. 5.A longitudinal view shows the existence of a hydraulic jump down- stream of the compound labyrinth weir for α^◦=20 and flat crest: (a) strong hydraulic jump and (b) weak hydraulic jump.

dissipate more energy. The pool’s circulating water is the primary cause of the energy loss on the vertical drop weir and linear weir [49,51].

The linear weir has a larger recirculating flow in the pool because the nappe flow is deflected more, giving rise to higher energy dissipation by linear weirs. Third, Hydraulic jumps happen downstream of the com- pound labyrinth weir when the flow is supercritical because of the

collision of oblique flow at the foot of the nappe, which causes a considerable amount of energy to be lost. According to the set of curves presented in Fig. 8, the dashed line represents the best curve-fit equation given by Eq. (14) that is obtained from the experimental data. Co- efficients of Eq. (14) and coefficient of determination (R²) are given in Table 3. These equations are valid for 0.07 ≤yc/P՛ <~0.95. Eq. (14) Fig. 6. Compound Cdc versus H՛_t/P՛ for flat crest of trapezoidal compound labyrinth weirs. The Symbols represent experimental works, dashed lines represent curves fitting.

Fig. 7. Discharge over compound labyrinth weir for α^◦=12. (a) Discharge pass over notches only, (b) discharge pass over entire labyrinth weir.

assists in estimating the amount of energy dissipation downstream of the compound labyrinth weir.

%△E Eo

=a e^b (

yc P՛

)

(14) The air entrainment and supercritical waves downstream of the compound labyrinth weir produce complex flow patterns. Also, the flow depth position measured downstream of the weir may not match the exact position of the water jet impact. Instead, the sidewall angle (α) and

the flow rate determine where the water jet impacts. The results showed that a compound labyrinth weir has a more remarkable energy dissi- pation ability than a liner weir. Therefore, a compound labyrinth weir is considered economical because this type of weir does not need an energy dissipator downstream of the weir.

5.4. Residual energy

Design engineers need to accurately predict the kinetic energy of the Fig. 8. Comparison of the energy dissipation on compound labyrinth weir for range sidewall angle (α^◦) (6–35) with linear weir, vertical drop weir.

Fig. 9. Nappe collisions produce turbulent mixing that forms a pool behind the nappes and localised quasi-oblique hydraulic jump close to the apexes for α^◦=20 and the flat crested of a compound labyrinth weir.

flow downstream of the weir to design the size of the energy dissipator.

The remaining energy estimates the amount of energy loss in a down- stream energy dissipator. Energy dissipation downstream of weirs is usually conducted by using a stilling basin. The flow type changes from supercritical to subcritical conditions as a result of a hydraulic jump that is produced, dissipating the flow energy. The high-velocity water jet begins from the weir crest and collects downstream into a submerged pool. Because the water jet continuously hits a submerged pool pro- duced behind nappe flow, the kinetic energy is dissipated through tur- bulent recirculation. A wide range of flow rates must be supported in the energy dissipator’s design for safe operation. Therefore, the compound labyrinth weir can be selected as the energy dissipator.

Fig. 10 shows the residual energy E₁ (Eq. (2)) downstream of examined models based on the minimum potential amount of residual energy Emin (Eq. (6)). The results are plotted versus the critical depth of y_c/P՛ for flat crest shape and the different sidewall angles (α^◦) of 6–35 and 90 (linear weir) for comparison. For comparing the results, the

of the weir when increasing the sidewall angle (α) for a given yc/P՛_. Furthermore, this tendency may be described by the fact that greater energy dissipation is required for a given relative critical head and height of the weir (yc/P՛) with a smaller sidewall angle (α). Hence, compound labyrinth weirs are highly effective in energy dissipation.

For a given value of yc/P՛, residual energy from smallest to largest is noticed as follows: compound labyrinth weir, linear weir, and vertical drop weir, respectively. The reason could be attributed to the hydraulic behaviour of a compound labyrinth weir with three dimensions. In contrast, the compound linear weir is free-falling without nappe inter- ference and vertical drops in the weir condition [43]. Furthermore, as α increases, the differences in residual energy between the compound labyrinth weir and the compound linear weir or the vertical drop weir are reduced. It may be because larger discharges are required for a given yc/P՛ and bigger α. According to the set of curves presented in Fig. 10, the solid line represents the best curve-fit equation given by Eq. (15) that is obtained from the experimental data. Coefficients of Eq. (15) and co- efficient of determination (R²) are presented in Table 4. These equations are valid for 0.07 ≤yc/P՛ <~0.95. Eq. (15) assists in determining the minimum residual energy downstream of the compound labyrinth weir.

35 83.76 − 1.768 0.99

90 78.759 − 1.921 0.98

Fig. 10.Comparison of the residual energy downstream of compound labyrinth weir by vertical drop weir, linear weir, and minimum limit of residual energy.

E1

Emin

=a (yc

p^′ )_b

(15)

5.5. Effect of relative energy dissipation on compound coefficient of discharge

Determining the amount of energy dissipation over the labyrinth weir assists the designer in selecting a suitable Cdc. The complex shape of the compound labyrinth weir makes it behave hydraulically differently from other structures such as vertical drops, overflow weirs, and stepped weirs. Fig. 11 shows the variation of the compound discharge coefficient (C_dc as a dimensionless term) against relative energy dissipation (ΔE/E_o

%). As mentioned above, Cdc decreased when increasing H՛_t/P՛_{. While} relative energy dissipation (ΔE/Eo%) decreased when increasing H՛_t/P՛_. However, Fig. 11 has presented families of curves based upon sidewall angle. The data for a set of curves showed a similar trend. However, Fig. 11 showed that the compound coefficient of discharge (Cdc) increased as the relative energy dissipation (ΔE/Eo%) increased. It may

be due to variations in geometric parameters such as crest length (Lc), weir height (P), crest shape (overhanging profile), and cycle geometry. A Cdc is greatest at the lowest water levels, and a compound labyrinth weir is less Cdc at higher water levels. On the other hand, ΔE/Eo% is most significant at the lowest water levels, while it is less effective at higher water levels. However, the labyrinth weir is lower Cdc when the laby- rinth weir is lower ΔE/Eo%. It is attributed to the gradual change in the nappe condition and local submergence [50]. Also, nappe condition transitioned from aerated to partially aerated and then drowned [8].

Furthermore, nappe interference is weak under fully submerged condi- tions, resulting in lesser energy dissipation and a lower compound co- efficient of discharge.

For comparison between sidewall angles (α), Fig. 11 illustrated that a labyrinth weir with a smaller sidewall angle had a lower coefficient of discharge and higher relative energy dissipation. Smaller values of α allow a more significant potential for nappes collision from adjacent sidewalls. Therefore, nappe interference in the upstream apexes in- creases as α decreases. Generation of a local hydraulic jump near apexes caused high energy dissipation and a reduction in Cdc. The narrower outlet cycles dissipate incredible energy and obtain lesser Cdc than broader labyrinth cycles. The cause could be attributed to the creation of a pool behind the nappes in the upstream apexes of the compound labyrinth weir. The pool’s circulating flow causes significant energy loss while enhancing turbulent mixing. A weak hydraulic jump located downstream of the compound labyrinth weir contributed to the dissi- pation of a higher amount of energy [52].

A logarithmic function with the relative energy dissipation can be utilised to fit the compound coefficient of discharge. The compound coefficient of discharge (Cdc) can be predicted using Fig. 11 as a function of relative energy dissipation (ΔE/Eo%) for a range of sidewall angles of 35 =α^◦≥6 with a flat crest. Eq. (16) is presented to predict C_dc in terms of ΔE/Eo% for a compound labyrinth weir.

Table 4

Coefficients of Eq. (15) and coefficient of determination (R²) for residual energy downstream of compound labyrinth weir validated for 0.07 ≤yc/P՛ <~0.95.

α^◦ _{a b R}²

6 1.0281 − 0.026 0.82

8 1.0955 − 0.025 0.87

10 1.1137 − 0.066 0.85

12 1.148 − 0.089 0.83

15 1.1658 − 0.122 0.89

20 1.2278 − 0.163 0.84

35 1.2996 − 0.208 0.95

90 1.3914 − 0.219 0.95

Fig. 11.Compound coefficient of discharge (Cdc) against relative energy dissipation (ΔE/Eo%) for α^◦=6, 8, 10, 12, 15, 20, 35.

Cdc=A ln (ΔE

E0

)

+B (16)

where: Cdc is compound coefficient of discharge, (^ΔE_Eo) is relative energy dissipation, A and B are curving fit coefficients. Coefficients for Eq. (16) are given in Table 5, which also contains four statistical measures of relative error (RE), determination coefficient (R²), root mean square error (RMSE), and mean absolute error (MAE). These statistical mea- sures were used to evaluate the results’ accuracy. The RE, RMSE, and MAE are defined as:

RE=100 N

∑^N

i=1

⃒⃒

⃒⃒

⃒

(Cdc)_exp.− (Cdc)_pre.

(Cdc)_exp

⃒⃒

⃒⃒

⃒ (17)

RMSE=

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

1 N

∑^N

i=1

(

(Cdc)_exp. − (Cdc)_pre.

)₂

√√

√√ (18)

MAE=1 N

∑^N

i=1

⃒⃒

⃒(Cdc)_exp.− (Cdc)_pre.

⃒⃒

⃒ (19)

where; (Cdc)exp is the experimental compound discharge coefficient, (Cdc)pre is the compound discharge coefficient that is computed by Eq.

(16), N is the total number of measurements, and i is the number of points. In Table 5, the difference between the experimental and pre- dicted data of the compound discharge coefficient has been presented.

The maximum relative error was 5.86% for α^◦ = 6. Therefore, all models’ RMSE and MAE indices were closer to zero, which signifies better agreement.

6. Conclusions

One of the primary essential concepts in hydraulic structure engi- neering is the energy dissipation efficiency of these structures. Based on the results and observations of the tests, the following conclusions are drawn:

The outcomes demonstrated that the flow regime downstream of the labyrinth weir could vary significantly from subcritical to supercritical for all sidewall angles. For each sidewall angle (α), Fr1 increased when y_c/P՛ increased. Furthermore, for a given y_c/P՛_{, Fr1} increased as y_c/P՛ increased. The Cdc increased initially at low H՛_t/P՛ values, and the Cdc

showed a decreasing trend for higher values of H՛_t/P՛. The highest values of the Cdc had a sidewall angle of 35^◦, and the lowest value of the Cdc had a sidewall angle of 6^◦. For sidewall angles (α^◦) ranging from 6 to 35, the

respectively. Moreover, the average energy dissipation over a compound labyrinth weir was more significant than a linear weir by 17%, 15%, 14%, 12%, 11%, 10%, and 8% for α^◦=6, 8, 10, 12, 15, 20, and 35, respectively.

The residual energy (E₁/E_min) at the base of downstream compound labyrinth weirs was closer to the minimum potential amount of residual energy as yc/P՛ increased. For a given value of yc/P՛, the relative residual energy at the base of compound labyrinth weirs (E₁/E_min) increased as the sidewall angle (α) increased. For a given value of yc/P՛, the residual energy downstream of the compound labyrinth weir was less than the residual energy of the linear weir and vertical drop weir. The compound labyrinth weir provided lower Cdc when the labyrinth weir was lower ΔE/Eo%. A compound labyrinth weir with a smaller sidewall angle achieved a lower coefficient of discharge and higher relative energy dissipation. The narrower outlet cycles have dissipated incredible en- ergy and obtained lesser Cdc than broader labyrinth cycles. An empirical equation proposed to predict the compound discharge coefficient when energy dissipation data are given. The maximum relative error was 5.86% for α^◦=6. Therefore, all models’ RMSE and MAE indices were nearer to zero, which signifies better agreement.

Although the methods and data were obtained in the present study, future work is recommended to investigate the effect of different crest shapes on energy dissipation, such as half-round and quarter-round crests.

CrediT authors statement

Anees K Idrees: Writing - original draft, Conceptualisation, Investi- gation, Methodology. Riyadh Al-Ameri: Supervision, Methodology, Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Acknowledgements

The authors are grateful to the laboratory staff at the School of En- gineering (Deakin University) for provision of the test facilities.

Notations

The following symbols have been used in present paper A Inside apex width (L)

B Labyrinth weir length in the flow direction (L) b1 Bottom notch width (L)

CCSS Circular crested stepped spillway D Outside apex width (L)

Eo Total upstream energy (L) E₁ Residual energy (L)

Emin Minimum specific energy value (L)

E1min Minimum potential value of residual energy (L) F_r1 Froude number at downstream of the weir g Acceleration due to gravity (L T⁻²) ho Upstream flow depth (L)

h1 downstream flow depth (L)

Lc Total centerline length of labyrinth weir (L) lc Centerline length of the weir side wall (L)

´l Length of the notch (L)

L՛cLength of the labyrinth weir crest after subtracting the length of notches (ΔL), (L՛c =Lc– n x ΔL) (L)MAE Length of the labyrinth weir crest after subtracting the length of notches (ΔL), (L՛c =Lc– n x ΔL) (L)MAEMean absolute error

N Number of labyrinth weir cycles P Weir height (L)

Pˊ Notch height (L) PKW Piano key weir

TTLW Trapezoidal–triangular labyrinth weirs Q Flow rate (L3T-1)

Rcrest Crest shape radius (L) R2 Coefficient of determination RE Relative error

RMSE Root mean square error tw Thickness of weir wall (L) W Labyrinth weir width (L)

w Width of a single labyrinth weir cycle (L) ΔL: Top notch width (L)

ΔP The notch depth (L)

Vo Upstream mean velocity (LT-1) V1 Downstream mean velocity (LT-1) yc Critical depth (L)

α^′ Energy correction factor (degree) α Sidewall angle (degree)

ΔE Total dissipated energy

ΔEmax Maximum possible amount of energy dissipation ΔP/P = Ratio of notch height to labyrinth weir height

ΔL/lc-one leg = Notch length to the length of one leg of the weir ratio

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