A review of hydraulic performance and design methods of labyrinth weirs Open Access

The following symbols are utilised in the present study:

A/w

length of apex geometry to the width of a single cycle/Apex ratio

A

Inside apex width

AI

artificial intelligent

ANFIS

Adaptive neuro-fuzzy inference system

ANN

Artificial neural networks

APE

Average percentage error

Aw

Sidewall area of the weir

B

Length of labyrinth weir

BNN

Bayesian neural network

BPNN

Back propagation neural network

C_d

Coefficient of discharge

CHAID

Chi-squared automatic interaction detector

D

Outside apex width

ELM

Extreme learning machine

Fr

Froude number

g

Gravity acceleration

GA-BPNN

Genetic algorithm and back propagation neural network

GEP

Gene expression programming

GP

Genetic programming

h

Flow depth over the weir crest

H_t /P

Headwater ratio

H_t

Total head over the weir crest

Lc/W

Magnification ratio

Lc

Total centerline length of labyrinth weir

LSSVM

Least square support vector machine

MAE

Mean absolute error

MAPE

Mean absolute percentage error

ME

Mean error

MLPNN

Multilayer perceptron neural network

MLR

Multiple linear regression

N

Number of labyrinth weir cycles

NLR

Nonlinear regression analysis

NN

Neural network

NSC

Nash–Sutcliffe efficiency coefficient

P

Weir height

Q

Discharge over the weir

PSO-BPNN

Particle swarm optimization and back propagation neural network

R²

Determination coefficient

R_crest

Radius of crest shape

RF

Random forests

RMSE

Root mean square error

SAELM

Self-adaptive extreme learning machine

SVM

Support vector machine

T_W

Sidewall thickness

t_w

Thickness of weir wall

w/p

Aspect ratio

w

Single labyrinth weir cycle width

W

Channel width

α/θ

Angle ratio

α

Sidewall angle

θ

Vortex angle

Ghare et al. (2008) showed that as a result of the ingrained benefits of labyrinths regarding magnification of the flow and stability of the structure, it should remain to be utilised to a larger extent for future projects. Therefore, labyrinth weirs have a basic role in guaranteeing the security of human life. In addition, labyrinth weirs save developing regions and natural environment safety. Because of their high efficiency, labyrinth weirs have become increasingly popular in recent years., especially in weirs rehabilitation schemes with limited channel width or for weirs constructed near a populated area. The hydraulics of labyrinth weirs have drawn a great deal of interest. Also, several methods for hydraulic design of the labyrinth weirs have been developed (Paxson et al. 2011).

Labyrinth weirs were utilised as regular structures for flow and also help control aeration, drop structures, and energy dissipation. Emiroglu & Baylar (2005) showed that labyrinth weirs are utilised to improve air entrainment flow proficiency. While Emiroglu et al. (2010) found that the labyrinth weirs are a good choice as side weirs to increase the discharge. There are several types of studies for labyrinth weirs that contributed to providing knowledge regarding hydraulic behaviour and design methods for this type of weir, such as (Lux 1984; Wormleaton & Soufiani 1998; Wormleaton & Tsang 2000; Emiroglu & Baylar 2005; Tullis et al. 2005; Tullis et al. 2007; Crookston & Tullis 2011a, 2011b, 2012a, 2012b, 2012c; Khode et al. 2011; Kumar et al. 2011; Crookston et al. 2012a; Dabling et al. 2013; Gupta & Singh 2013; Akoz et al. 2014; Crookston et al. 2014; Bahrebar et al. 2016; Baeimani & Bajestan 2015; Kanyabujinja 2015; Idrees & Al-Ameri 2022). Table 1 shows a summary of some previous studies related to labyrinth weirs.

Furthermore, Mirnaseri & Emadi (2014) tested physical models for rectangular labyrinth weirs and gates. They investigated the effect of the length and height of the labyrinth weir with different bed slopes of the flume on the discharge coefficient. The results demonstrated that the discharge coefficient of the combined model increased with increasing weir height. Data analysis demonstrated that increasing the effective length of the labyrinth weir when H_t/L was less than 0.03 led to an increase in the coefficient of discharge, while an H_t/L ratio of more than 0.03 led to a reduced coefficient of discharge in a combined model. Experimental outcomes for various flume slopes demonstrated that the coefficients of discharge of models increased with increasing flume slopes. The shape of the crest is the basic parameter that influences the hydraulic performance of the labyrinth weirs. Paxson & Savage (2006) showed that a rounded crest weir generally has a coefficient of discharge that is measurably higher than those of sharp and flat crest weirs. Amanian (1987) showed that the quarter-round crest was the more efficient shape. Also, Khode & Tembhurkar (2010) recommended the quarter-round crest. Labyrinth weirs with a half-round crest were more efficient for increasing discharge capacity.

Moreover, the choice of different geometric parameters, orientation, and position of the labyrinth weir affect the capacity of the discharge. Paxson et al. (2011) showed that flow conditions downstream of the labyrinth weir could also influence discharge capacity. The width of the inside apex defines whether the labyrinth weir is trapezoidal or triangular. Kumar et.al. (2012) investigated discharge capacity by using the sharp-crested curve under free-flow conditions. They proposed equations to estimate coefficients of discharge for curve labyrinth weirs. The results showed that discharge capacity was a gain of 40% for a curved weir compared to a normal weir. Kabiri & Javaheri (2012) investigated the flow coefficient for a rectangular labyrinth weir. The analytical method was based on the combination of the equations of a vertical and a lateral weir. This method is based on the determination of the contribution of each cited section (upstream b, downstream a, and lateral L). Gupta et al. (2013) studied the flow characteristics of a sharp-crested triangular labyrinth weir. The discharge efficiency across the triangular labyrinth weir was larger than the normal weir. The equation of flow was suggested for the given range of data and was within ±5% of the observed data. Ouamane & Lempèrière (2013) studied the rectangular labyrinth weir and found the rectangular shape more efficient than the trapezoidal shape. Gupta et al. (2014) studied W-planform weirs to determine discharge characteristics under free-flow conditions. The discharge efficiency of W-planform labyrinth weirs was higher than that of normal weirs. 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 of a labyrinth weir. A rectangular labyrinth weir was used to examine the effect of crest height on the discharge capacity for different crest lengths. The discharge efficiency of the rectangular labyrinth weir was higher than that of the normal weir. Idrees et al. (2016) discussed the effect of a range sidewall angle of 6° to 35° on the discharge coefficient for one cycle of a labyrinth weir. They developed a set of curves for estimating the discharge coefficient. Idrees et al. (2017) examined the hydraulic performance of compound labyrinth weirs. The compound labyrinth weir was located in normal and inverse orientations according to the flow direction. The outcomes demonstrated that the discharge coefficient values do not vary significantly when compared with the labyrinth weir orientations. Idrees et al. (2018) investigated the hydraulic characteristics of compound labyrinth weirs. They found empirical equations with a range of sidewall angles of 6° to 90° for estimating discharge coefficients.

Computational fluid dynamics (CFD) is a program that uses numerical methods for analysing and simulation (Institute for Water & Environmental Engineering 2012). Currently, CFD modelling has been widely utilised in water resources engineering applications by combining it with physical modelling. Versteeg & Malalasekera (1995) explained that CFD modelling is essential to the simulation of the problem rather than providing a final result. The CFD model can analyse hydraulic structures more cheaply than a physical model. Studies have investigated using CFD for verifying and refining the design of the labyrinth weir. Savage et al. (2004) compared physical models carried out by the USBR. They showed that current design approaches were deficient for different values of weir height (P). This resulted from using the design methods of Tullis et al. (1995) that depend on the H_T/P ratio for finding the rating curve, which is influenced by the various P values. Savage et al. (2004) also observed that physical models were useful for verifying discharge and suggested using modelling of the CFD as an alternative method for reducing cost and time. They found good agreement with the physical model. Bhuiyan & Hey (2007) simulated experimental results for U-, V- and W-weir shapes. A positive correlation resulted from the calculated patterns of flow and the characteristics of the flow. Paxson et al. (2008) used CFD modelling to simulate the spillway for the Lake Townsend Dam. The results showed that the labyrinth weir was a suitable choice as a replacement gate for the spillway. A modified Tullis et al. (1995) method was applied to develop a discharge relationship for this proposed weir geometry, and the discharge coefficient estimated by this method was shown to be underestimated for low H/P ratios (less than 0.3).

Furthermore, to maintain safety at the Lake Holiday Dam in the State of Virginia, Ackers et al. (2011) utilised the hydraulic design equation that was recommended by Tullis et al. (1995) and examined the flow behaviour by utilising the CFD program. Holmquist-Johnson (2011) simulated the effect of alternative solutions for U-weir. The results explained that using U-weir geometry and flow patterns including stress, velocity distributions, bed shear and upstream flow depth, significant variations could be made. In addition, the author developed stage–discharge relationships by using results from the numerical model simulations. To compare with physical models, Torres et al. (2018) used two numerical models, including the ANSYS Fluent program and OpenFOAM. The results showed that the prediction data generated by the fluent model were more closely related to the physical model than those obtained by OpenFOAM. Carrillo et al. (2019) compared three different turbulence models through experimental work and CFD modelling. There was a positive correlation between the capacity of discharge in the cases of free discharge and submerged conditions. Idrees et al. (2022) simulated the flow behaviour of the compound labyrinth weir within three regions of flow depth. They used the CFD program for this purpose. The results showed that the streamlines were divided into two parts. Also, negative pressure behind the nappe flow was produced by air bubbles.

Recently, machine learning methods have been used to predict the discharge coefficient of weirs. These methods have also been used in modelling complex problems of hydraulic structures (Emiroglu & Kisi 2013). Machine learning techniques were investigated by several researchers, e.g. Roushangar et al. (2017) used gene expression programming (GEP) and adaptive neuro-fuzzy inference system (ANFIS) methods to predict the coefficient of discharge for labyrinth weirs. The results demonstrated that the ANFIS method was the best model for predicting the discharge coefficient. Roushangar et al. (2018) built a model to predict the coefficients of discharge for inverted and normal direction labyrinth weirs. They utilise support vector machine (SVM)-based models for this purpose. The results demonstrated that discharge coefficient prediction by the SVM-based model of arced labyrinth weir (ALW) was better solution when dimensionless parameters, e.g., angle ratio (α/θ), headwater ratio (H_T/P), and magnification ratio (L_c/W) were used as input parameters. Furthermore, sensitivity analysis demonstrated that both parameters of the Froude number and headwater ratio were the most effective on the coefficient of discharge for normal labyrinth weirs, while for arced labyrinth weirs, both parameters of the angle ratio (α/θ) and headwater ratio were the most effective on coefficients of discharge. Karami et al. (2018) applied three models of intelligence techniques, including genetic programming (GP), extreme learning machine (ELM), and artificial neural networks (ANN) to predict the coefficient of discharge. The ELM model showed the best results for predicting discharge coefficients in comparison with the GP and ANN models. Salazar & Crookston (2019) applied the machine learning method to the analysis of arced labyrinth weirs geometrically. They used random forests (RF) and neural networks (NN) to estimate the coefficient of discharge. They demonstrated that RF and NN can be utilised as a unique representation for curve fitting. Norouzi et al. (2019) applied three models of the radial basis function networks, multilayer perceptron (MLP) networks, and support vector machines with various kernel functions to predict the coefficient of discharge (C_d) of labyrinth weirs. Moreover, the MLP model was more appropriate to predict the discharge coefficient. Dutta et al. (2020) built predictive models to compute the capacity of discharge for circular arc weirs and a multi-cycle labyrinth weir. They used an artificial neural network (ANN), multiple linear regression (MLR), and SVM. The results confirmed that the prediction made by the SVM regression model was more accurate than the ANN and MLR approaches. Norouzi et al. (2020) used the self-adaptive extreme learning machine (SAELM) artificial intelligence model to simulate the coefficient of discharge for a labyrinth weir. The results identified that dimensionless parameters such as (A/w) and (H_T/P) were the most effective as input parameters. Wang et al. (2022) applied four methods of nonlinear regression analysis (NLR), the back propagation neural network (BPNN), the genetic algorithm (GA) BPNN (GA-BPNN), and the particle swarm optimization (PSO) BPNN (PSO-BPNN) to estimate the discharge coefficient of labyrinth weirs. The results showed that the predictions by the GA-BPNN and PSO-BPNN were better than the BPNN. The maximum error values with the discharge coefficient prediction by PSO-BPNN and GA-BPNN were 2.6% and 4.4%, respectively. They compared the results of discharge prediction to previous studies and showed that the mean absolute percentage errors (MAPE) for predicting the discharge coefficients by PSO-BPNN and GA-BPNN were 1.225% and 1.504%, respectively. Table 2 shows a summary of some previous studies related to machine learning techniques for predicting discharge coefficients (C_d).

The labyrinth weir is a complex structure and the flows over labyrinth weirs are three-dimensional. The labyrinth weir has many parameters affecting discharge capacity. Paxson et al. (2011) showed that the success of a project will depend on factors that influence its design. Ghare et al. (2008) explained that a design should be sought that has a sound hydraulic operation and is economically viable. Although the labyrinth weirs are a favoured design option, they still face difficulties in reaching the optimal design for given sites and location conditions.

The set of curves developed by Tullis et al. (1995) for a trapezoidal labyrinth weir. The angle of the crest was an effective parameter on the curves. Falvey (2003) showed that a set of curves obtained by Tullis et al. (1995) are beneficial for calculating discharge coefficients for both trapezoidal and triangular labyrinth weirs. Empirical equations for different sidewall angles, α, were developed by Tullis et al. (1995) to assist engineers in labyrinth weir design.

There are numerous types of studies on labyrinth weirs that have been accomplished through numerous experimental works and the construction of labyrinth weirs all over the world. However, a more in-depth understanding of some of the subjects is still required to adequately address the problems associated with them. Experiments with modifying the geometry and planform of the labyrinth weirs are being conducted in order to increase the discharge capacity and are being investigated. Although few authors have studied the floating debris risk, it needs further investigation. The construction methods require further investigation, and the produced reinforcement concrete can be an interesting solution. The main issues that are influencing a project, such as costs and delays, Therefore, we can seek new methods for the rapid construction of labyrinth weirs. These methods depend on hydrologic data and a minimum number of parameters that could be further taken up for research. The structural properties, as well as the materials used in the construction of the labyrinth weirs as composite structures, can be an area of further research. Also, relating labyrinth weirs to other kinds of spillways with the lowest cost is an area of further investigation. The use of numerical modelling to conduct research rather than physical models, which are frequently more time-consuming and expensive, can be an inspection. The energy dissipation downstream of the labyrinth weirs is one of the interesting issues in labyrinth weirs that needs further study. Different energy dissipators can be further investigated for effective energy dissipation with minimum cost. Machine learning methods need more study to provide significant and reliable equations to assist the designers in predicting discharge over labyrinth weirs.

The labyrinth weirs play an essential role in guaranteeing the security of human life, in addition to protecting developing areas. In recent years, labyrinth weirs have been widely used because of their high efficiency. Although labyrinth weirs have complex hydraulic behaviour, there have been studied by researchers for a long time. Many studies demonstrated that labyrinth weirs are not applicable, particularly under high relative overflow depth, high submergence, and poor plan geometry. The labyrinth weir is a complex structure. Therefore, it needs a large number of experimental tests to find empirically the impact of the different parameters on the capacity of discharge. Therefore, some parameters still need to be further investigated. Ideally, the hydraulically most efficient design needs to be considered during the design and construction of a labyrinth weir, but this is not always possible because many parameters affect the weir design. In addition, many factors may affect discharge capacities, such as increasing construction costs and the nature of the site regarding geological, topographic, and facility limitations. In addition, common design equations do not take into account all parameters affecting labyrinth weir performance, such as labyrinth weir geometries, flow conditions, site conditions, and scale effect. The design of labyrinth weir methods depends mainly on empirical equations derived from experimental work and generalised performance curves. Physical models still need more investigation. A labyrinth weir has a complex geometry. Therefore, it is a fascinating challenge for future numerical and experimental research. Also, labyrinth weir development may be enhanced with this historical outlook for further improvement both in economic and hydraulic features.

The authors would like to express their special thanks to the technician staff in the laboratory of the school of engineering at Deakin University, who supported them to complete this work.

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

The authors declare there is no conflict.