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Hydraulic Characteristics and Data-Driven Prediction of Discharge

Coefficient in Rectangular Labyrinth Side Weirs

Mohd Amir, Mujib Ahmad Ansari, Ajmal Hussain*

Department of Civil Engineering, Zakir Hussain College of Engineering & Technology, A.M.U.,

Aligarh 202002, India

DOI:

https://doi.org/10.51583/IJLTEMAS.2026.1501000106

Received: 30 January 2025; Accepted: 07 February 2026; Published: 19 February 2026

ABSTRACT

Rectangular labyrinth side weirs are widely used as flow-diversion structures in irrigation and drainage systems

due to their ability to enhance discharge capacity by increasing the effective crest length within limited channel

widths. In the present study, laboratory experiments were conducted under subcritical flow conditions to

investigate the discharge characteristics of a rectangular labyrinth side weir installed in a straight channel. The

influence of key hydraulic and geometric parameters on the discharge coefficient (C

) was systematically

examined. A statistical F-test was applied to identify the relative significance of the governing dimensionless

parameters affecting C

. Based on the experimental dataset, a nonlinear regression model was developed to

establish a predictive relationship between C

and the dominant flow variables. The developed regression

model showed good agreement with experimental data, with an overall correlation coefficient (R = 0.8572)

and a low RMSE value of 0.1282 respectively, along with reasonable prediction accuracy reflected by low

RMSE values. The outcomes of this study improve the understanding of discharge behaviour of rectangular

labyrinth side weirs and provide a practical regression-based formulation that can assist in hydraulic design

and performance evaluation of flow-diversion structures.

Keywords: Rectangular labyrinth side weir; Discharge coefficient; Three-cycle configuration; Nonlinear

regression; Flow diversion

Symbols

Main channel section width (m)

Rectangular labyrinth side weir discharge coefficient

Elevation of the side weir crest from the channel bed (m)

Total effective length of the side weir opening (m)

h₁

Water depth along the centreline upstream of the side weir in the main channel (m)

Upstream flow velocity near the side weir (m/s)

Constant gravity acceleration

Q₁

Upstream discharge in the main channel (m³/s)

Discharge diverted through the side weir (m

/s)

Fr Upstream Froude number

, c

Coefficients

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, n

Exponents

R Correlation coefficients

INTRODUCTION

Side weirs are widely used as hydraulic control and flow-diversion structures in open-channel systems such as

irrigation canals, urban drainage networks, flood-control works, and water and wastewater treatment facilities.

Flow over a side weir is classified as spatially varied flow with decreasing discharge, in which the main-channel

flow continuously reduces along the weir length. Due to this complex hydraulic behaviour, accurate estimation

of diverted discharge is essential for reliable hydraulic design.

The classical formulation of side-weir flow was proposed by De Marchi (1934) under the assumption of

constant specific energy along the crest. Subsequent experimental studies demonstrated that the discharge

coefficient is not constant but depends strongly on flow conditions and geometric characteristics of the structure

(Subramanya and Awasthi, 1972; El-Khashab and Smith, 1976; Ramamurthy et al., 1986; Swamee et al., 1994).

These investigations also highlighted the limited discharge capacity of conventional straight-crested side weirs

due to restricted crest length.

To enhance diversion capacity, labyrinth side weirs were developed by folding the crest into multiple cycles,

thereby increasing the effective overflow length without increasing channel width. Experimental investigations

have shown that labyrinth configurations significantly improve discharge performance under identical

upstream head conditions (Khode et al., 2012; Dey et al., 2013; Thompson et al., 2016; Sangsefdi et al., 2017).

Among different configurations, rectangular labyrinth side weirs are particularly attractive because of their

structural simplicity and suitability for irrigation and drainage channels.

Recent studies based on experimental observations and regression analysis have attempted to describe the

discharge behaviour of labyrinth side weirs. Shariq et al. (2019) reported strong dependence of the discharge

coefficient on geometric parameters, while Norouzi et al. (2019) demonstrated the effectiveness of regression-

based relationships for predicting discharge characteristics of trapezoidal labyrinth weirs. More recently, Wan

et al. (2024) experimentally investigated stepped labyrinth side weirs and emphasized the influence of crest

geometry on diversion efficiency. However, experimental investigations focusing on multi-cycle rectangular

labyrinth side weirs remain limited, and the influence of governing dimensionless parameters has not yet been

fully clarified.

Therefore, the present study experimentally investigates the discharge characteristics of a three-cycle

rectangular labyrinth side weir operating under subcritical flow conditions. Based on the experimental dataset,

a nonlinear regression equation is developed to relate the discharge coefficient to the dominant dimensionless

parameters, and a statistical F-test is applied to identify their relative significance. The outcomes of this study

aim to contribute to improved understanding of rectangular labyrinth side-weir hydraulics and to support

reliable design and performance assessment.

Dimensional Analysis

Discharge coefficient (C

) of labyrinth side weirs can be expressed as a function of average velocity of flow

over the cross section of the channel (V), depth over the crest of rectangular labyrinth side weir (h1) upstream

depth of flow in channel (y1), acceleration due to gravity (g), width of side weir (L), width of main channel

(B), crest height of rectangular labyrinth side weir (p), number of cycles (N), dynamic viscosity of water (μ)

and density of water (ρ).

Applying the Buckingham-π theorem, non-dimensional equations in functional forms can be written as below:

= f (Fr, l/L, B/y

, h

/p, N (1)

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To see the effect of various parameters on coefficient of discharge, C

and to establish generalized relationship

among the dependent and independent parameters of Eq. (1), experimental programmes are carried out in

present study.

Experimental set up and Program

The experimental investigation was carried out in the Advanced Hydraulics Laboratory, Department of Civil

Engineering, Aligarh Muslim University, Aligarh. The experiments were performed in a horizontal rectangular

flume of 9.74 m length, 0.50 m width, and 0.38 m depth, operating under a recirculating flow system. Water

was supplied to the flume through a 0.10 m diameter inlet p ipe, and the flow depth in the main channel was

regulated using a sluice gate installed at the downstream end. A rectangular labyrinth side weir was installed

along the right wall of the main channel at a distance of 7.30 m from the inlet. The diverted flow passed into a

secondary collection channel of 4.29 m length, 0.22 m width, and 0.38 m depth. A sharp-crested rectangular

measuring weir (Weir-A) was provided at the downstream end of the diversion channel to measure the side-

weir discharge. The remaining flow in the main channel, together with the diverted discharge, was conveyed

to a return channel where another sharp-crested weir (Weir-B) was installed to measure the total discharge.

To ensure stable and uniform flow conditions, splitter plates and wave suppressors were installed upstream of

the test section to reduce large-scale turbulence and surface disturbances. All experiments were conducted

under free-flow conditions with fully ventilated nappes. The flow in the vicinity of the labyrinth side weir

remained subcritical throughout the experimental program. Experiments were performed on a three-cycle

rectangular labyrinth side weir with a chord length of 0.50 m. Three different crest heights, namely 11 cm, 14

cm, and 18 cm, were investigated. For each crest height, approximately 50–60 experimental runs were

conducted by varying the discharge and controlling the upstream flow depth through the downstream sluice

gate. Water surface elevations in the main channel and heads over the measuring weirs were recorded using a

point gauge with a least count of 0.1 mm. The experiments covered a discharge range of approximately 0.01–

0.60 m³/s. A photographs of the experimental setup and three cycle labyrinths are presented in Fig. 1 and Fig

2 shows the schematic diagram of three cycle labyrinth side weir.

Fig 1. Photographic view of experimental set up and three cycle labyrinth side weir

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Fig 2. Schematic view of three cycle labyrinth side weir

Relative Importance of various parameters

An F-test was conducted to evaluate the relative importance of the independent variables in predicting the

discharge coefficient (C

). This test helps identify the contribution of each parameter to the dependent variable.

Parameters with lower F-values were considered to have negligible influence.

The effect of the dimensionless parameters Fr, l/L, B/y

, h

/p, and N on C

was examined, as shown in Fig. 3.

The results indicate that B/y1 has the highest influence on the discharge coefficient, whereas the number of

cycles (N) exhibits the lowest effect and may therefore be neglected compared to the other parameters.

Fig. 3. F-test results for input parameter importance on discharge coefficient (C

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Effect of dimensionless parameter on C

The influence of different dimensionless parameters on the discharge coefficient (C

) was examined in the

present study, and the corresponding variations are illustrated in Fig. 4(a), (b) and (c).

The results show that C

increases with an increase in the upstream Froude number (Fr) when other parameters

remain constant, which can be attributed to enhanced lateral momentum transfer and stronger secondary flow

toward the labyrinth side weir at higher flow intensities.

The discharge coefficient also increases with increasing relative channel width (B/y

) indicating improved

diversion efficiency under lower upstream flow depths. In contrast, an increase in the relative head over the

crest (h

/p) results in a reduction of C

mainly due to intensified flow interference, local energy losses, and

stronger flow separation near the side-weir boundary.

Overall, the findings confirm that the discharge behaviour of a rectangular labyrinth side weir is significantly

governed by upstream hydraulic conditions and relative geometric parameters.

Fig.4(a) Variation of coefficient of discharge with upstream Froude number for 3 cycle rectangular labyrinth

side weir.

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Fig.4(b) Variation of coefficient of discharge with B/y

for 3 cycle rectangular labyrinth side weir.

Fig.4(c) Variation of coefficient of discharge with h

/p for 3 cycle rectangular labyrinth side weir.

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Model Development

Based on the dimensional analysis and established formulations, the discharge coefficient (C

) was expressed

in a generalized power-law form as

 

1 2 3 4

C c c c c Fr



   





(2)

where c

–c

are regression constants and n

-n

are the associated exponents. The coefficients were determined

using the experimental dataset after removing outliers exceeding three standard deviations through MATLAB

routines. Approximately 80% of the data was used for model calibration and the remaining 20% for validation,

following commonly adopted practices in hydraulic modelling studies. The resulting empirical relationship is

expressed as

 

0.25

0.53

1.64

1.5 0.60 1.56 0.49

C Fr

   







(3)

The predictive performance of Eq. (3) was evaluated using the validation dataset, as shown in Fig. 4 (a) and

(b). The regression model provided satisfactory agreement with the experimental data; however, its predictive

capability is constrained by the complex and highly nonlinear hydraulic behaviour associated with flow over

labyrinth side weirs.

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Fig. 5 Comparison between observed and calculated Cd values obtained from the regression model for (a)

training and (b) validation datasets.

Based on the experimental dataset, a nonlinear regression equation was developed to estimate the discharge

coefficient of the rectangular labyrinth side weir. The regression model showed satisfactory agreement between

observed and predicted C

values, yielding an overall correlation coefficient of (R = 0.8572) and a low root

mean square error (RMSE = 0.12). These results indicate that the proposed regression equation can reasonably

represent the discharge behaviour of the rectangular labyrinth side weir within the investigated range of

hydraulic and geometric conditions. However, its applicability is limited to the experimental domain

considered in the present study

RESULTS AND DISCUSSION

The experimental results indicate that the discharge coefficient (C

) of three cycle rectangular labyrinth side

weirs is significantly influenced by hydraulic and geometric parameters. An increase in the upstream Froude

number (Fr) leads to higher C

values due to enhanced secondary flow and increased lateral momentum

transfer toward the side weir, which improves flow diversion efficiency.

A positive relationship is also observed between C

and the dimensionless parameter B/y₁ under constant

geometric conditions (B/L = 1.0, l/L = 2.8, and p = 14 cm). Since the weir width remains unchanged, a

reduction in upstream flow depth increases approach velocity, thereby strengthening lateral discharge and

resulting in higher discharge coefficients.

In contrast, C

decreases with increasing relative upstream head (h₁/p). Higher h₁/p values intensify turbulence,

flow separation, and local energy losses near the crest region, reducing the effectiveness of lateral overflow.

The combined influence of Fr, B/y₁, and h₁/p on C

is illustrated in Fig 3(a) (b) and (c) respectively.

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CONCLUSIONS

The present study experimentally examined the discharge characteristics of a rectangular labyrinth side weir

installed in a straight rectangular channel. The results indicate that the discharge coefficient (C

) decreases

with increasing relative upstream head (h₁/p), whereas higher values of B/y₁ and Froude number (Fr) lead to

an increase in C

. These trends demonstrate the strong influence of hydraulic conditions and geometric

parameters on the lateral diversion behaviour of labyrinth side weirs. Based on the experimental data, a

nonlinear regression equation was developed to estimate the discharge coefficient. The regression model

showed satisfactory agreement with the measured values within the investigated range of flow and geometric

conditions, indicating its applicability for practical estimation of C

. Overall, the findings of this study provide

useful insight into the hydraulic performance of rectangular labyrinth side weirs and offer a reliable regression-

based approach that can support preliminary design and performance evaluation in irrigation and drainage

systems.

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