Estimating discharge coefficient of triangular free overfall using the GMDH technique

the angle of the brink edge in rad

coefficient of discharge

constants

and

the mean both natural and examined C_d values, respectively

b

canal breadth (L)

C_c

coefficient of contraction

E

specific energy (L)

EDR

end depth ratio

Fr_b

Froude number

g

the acceleration of gravitational forces (L/T²)

GMDH

group method of data handling

H

head over the standard weir (L)

L

crest breadth (L)

MLR

multiple linear regression

MARE

mean absolute relative error

NSE

Nash–Sutcliffe model efficiency coefficient

P

height of the fall (L)

Q_act.

actual discharge (L³/T)

R²

determinant coefficient

RMSE

root mean square error

SI

scatter index

V

speed of flow (L/T)

v_n

the flow normal velocity (L/T)

x_i and y_i

both natural and examined C_d values, respectively

y_b

brink depth (L)

y_bcl

brink depth at the centre line (L)

y_brl

brink depth at the left or right centre line (L)

y_n

the normal depth (L)

upstream face slope

viscosity (M/LT)

the density of the water (M/L³)

Free overfalls or drops are a hydraulic structure widely used in an open channel when there are changes in channel bed levels; they can also use as a measurement device in irrigational channels. Many studies dealt with the hydraulics of free fall with different shapes. Beirami et al. (2006) studied free overfall theoretically for circular, semicircular, triangular, rectangular, and trapezoidal cross-sections to predict end depth ratio (EDR). The results refer to EDR values as 0.7016 in the rectangular shape, 0.8051 in the triangular shape, and 0.7642 in the exponential channel. Dey et al. (2004) investigated free overfall in inverted semicircular canals to predict EDR. The results showed that these values were near 0.695. Rajaratnam & Chamani (1996) and Dey & Kumar (2002) presented energy loss at drop and analysis of EDR in triangular free overfall in two methods. The first depends on the assumption of pseudo-uniform flow, and the second depends on assumed to be similar to the flow over a sharp-crested weir; the results show that the EDR value 0.695. Mohammed (2012) presented the theoretical study and showed equations for EDR and end depth discharge relationship for free overfall with different end shapes. Mohammed (2009) studied the hydraulic characteristics of triangular end lip free overfall with different angles, the EDR and water surface profile (WSP) were studied.

The theoretical data showed a result error of ±5% compared with the experimental. Some of the studies dealt with air entrainment from free overfall, such as Chanson (1993, 2004, 2007). In contrast, other studies refer to the turbulent diffusivity of the air bubble increasing with increasing distance from the point of singularity (Chanson & Toombes 1998; Gualtieri & Chanson 2004; Toombes & Chanson 2005). The studies dealt with erosion and scouring in free overfall and gave a technique for estimating discharge from known end depth equivalent sand roughness (van der Poel & Schwab 1985; Peter 1999). Ghodsian et al. (1999) investigate the effect of Froude number, total head to tailwater depth, and bed material size to tailwater depth parameters on scour backside of a free overfall and predicted equations to scour hole calculation. The effectiveness of free overfall roughness was studied by Dey (2000), Firat (2004), Guo et al. (2006), and Mohammed et al. (2011); these studies show that the EDR was substantially influenced by the spread, intensity, and distribution style of bed materials. The free overfall with various slopes and bed rough spread was investigated in Mohammed (2013), and the EDR and discharge equations with bed rough were predicted. The results showed that EDR values increased when bed rough increased. Other studies dealt with free overfall theoretical analysis (Ferro 1999) and using volume-of-fluid (VOF) (Sousa et al. 2009) as well as using artificial neural network (ANN) (Raikar et al. 2004; Öztürk 2005; Mohammed 2018); all these studies showed a good agreement between theoretical and experimental data. The discharge coefficient of the combined weir and gate structure was studied experimentally and theoretically by Balouchi & Rakhshandehroo (2018). The theoretical model was compared between multi-layer perceptron (MLP) and support vector regression (SVR) methods and given good agreement of C_d values calculated experimentally and theoretically. The ANNs and the M5P model tree application for calculating scour holes at river convergence were studied by Balouchi et al. (2015). The results showed that the MLP model was the most effective in the low scour depth range, while the radial basis function (RBF) model was more effective in the more excellent scour depth range, as measured by various statistical indices (RMSE, MAE, MARE, and R²). Recently, there have been many studies in different fields dealing with group method of data handling (GMDH) to forecasting the behaviours of a complex system, such as Amanifard et al. (2008) and Kaveh et al. (2018) which is studied shear strength predicted for fibre-reinforced plastic (FRP) reinforced concrete, Srinivasan (2008) and Najafzadeh et al. (2013), which they studied the GMDH in energy conservation and engineering geology. Ebtehaj et al. (2015) studied predicting the discharge coefficient in a side weir using GMDH by four input parameters then one output parameter predicted (C_d) experimentally and theoretically, their results lead to an acceptable degree of accuracy (R² 0.779, MAPE 5.263, RMSE 0.038, E 0.757, and SI 0.069). The flowrate coefficient for the piano key lateral weir was investigated using the GMDH and developed group method of data handling (DGMDH) techniques by Mehri et al. (2019), the results of R², MAE, and RMSE values for the GMDH technique are 0.875, 0.0501, and 0.0792, while these values of the DGMDH technique are 0.938, 0.0372, and 0.053, this results given successfully of the DGMDH technique comparison with the GMDH technique. Teng et al. (2017) submitted a study for compared four methods, ANN, GMDH, multiple linear regression (MLR), and support vector machine (SVM), for forecasting China's transport energy demand. The results are compared according to R² and RMSE, and then the GMDH technique gives better performance compared with other techniques. The present study aims to a predicted coefficient of discharge experimentally and compares these results using MLR and GMDH, by using four models with six common algorithms that may be created, then there are 24 outputs. A group of GMDH models are used in this work, with the free overfall modified to triangular end edge shaped, which led to increase the edge length then increased the discharge coefficient and then increased discharge passing over free overfall.

The other parameters measured in this study are water depth over the brink of the free overfall (y_b) at channel centre (y_bcl), water depth at the mid-distance from the centre of the model referring to the angle of the template lip (y_brl), and normal water level over the free overfall (y_n).

First the free overfall was fixed with 60° end lip edge with flow direction (Figure 2(a)) and five flow discharges then measured typical depth, brink depth at centre, and brink depth left and right of the centre line, then another model in the opposite flow direction was fixed (Figure 2(b)) and measured the previous parameters, then repeat procedures with 75° and 90°.

According to prior research by Ferro (1999), flow through an overfall in a rectangular channel is similar to flow over a thin weir for free rectangular overfall.

The conception of the runoff over a thin weir assumes a zero-pressure variation and parallel streamlines at the edge, ignoring the nappe contraction.

The factors that influence the flow of water at a free overfall are: y_b is the water depth over the edge (L); Q is flowrate (L³/T); y_n is the consistent height of flow (L); P is the height of the fall (L); v_n is the normal velocity of flow (L/T); g is the acceleration due to gravity (L/T²); is the viscosity in a dynamic state (M/LT); is the water's density (M/L³); and angle of the brink edge in rad.

Regression analysis is engaged in many applications. These applications' linear and non-linear analysis depends on the variables involved for instances of these equations obtained when more than one variable is used in the MLR model. Several experiments with several equation models were studied using the SPSS user guide to get a general equation for the coefficient of discharge in free overfall with a triangular edge.

The percentage error for C_d values is calculated using Equation (11), and experimental values do not exceed 10%.

As case studies of the generalized architecture, six standard algorithms may be created:

• (1)

  Only the middle components of the multilayered iterative algorithm (MIA) contain pairs.

• (2)

  The middle and beginning factors occur in pairs in the relaxation iterative algorithm (RIA).

• (3)

  CIA stands for combined iterative algorithm, which allows for pairing middle and starting factors.

• (4)

  Iterative-combinational multilayered (MICA).

• (5)

  Iterative-combinational relaxation (RICA).

• (6)

  Combined iterative-combinational (CICA) (practically equivalent to GIA).

These four selected models can be expressed as

mod11 < - gmdh.combi(X = x, y = y, G = 2, criteria = "ICOMP")

mod21 < - gmdh.combi.twice(X = x, y = y, G = 2, criteria = "ICOMP")

mod36 < - gmdh.mia(X = x, y = y, prune = 150, criteria = "PRESS")

mod46 < - gmdh.mia(X = x, y = y, prune = 200, criteria = "PRESS")

where,

G = 2: polynomial degree.

Values:

Criteria = "PRESS"; Predicted Residual Error Sum of Squares. It calculates without processing a system per reference node and considering all data into a standard sample.

ICOMP: Index of Informational Complexity. Like PRESS, it calculates without the need to recalculate the system.

Prune:

The number of prior predictors is the suggested minimum value for this integer.

The amount of RAM available will determine the highest values. It advises working at the highest value, although it can be time-consuming and computationally.

From layer i, the number of neurons pruned to layer i + 1 is the number chosen. The resultant layer i + 1 contains prune (prune − 1)/2 neurons; for example, if prune = 190 use, the causing neurons are 11.175. Table 1 shows the resulting layers used in all GMDH models. Given many fitted models to select the best algorithm and model with the highest fitting accuracy, Table 1 is the most concise description of the inner layers and internal functions used in the fits.

Table 2 describes the parameters for all models using the GMDH technique.

The formulas for computation free overfall before the coefficient of runoff studies are presented in Table 3.

Table 3

Formulas for computation of free overfall discharge coefficient in the prior lecturer

Authors .	Equations .	.
Bos (1985)		13
Fritz & Hager (1998)		14
Goodarzi et al. (2012)		15

Authors .	Equations .	.
Bos (1985)		13
Fritz & Hager (1998)		14
Goodarzi et al. (2012)		15

is the upstream face slope; P is the weir height; l is the crest length, h is a depth of flow over the crest.

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According to Figure 6, the residuals of both models are minimal, and the choice between these two models depends on other conditions such as difficulty.

Below is a calculation and definition of the subjective output of the accessible equations in terms of (R²) determination coefficient, root mean square error (RMSE), scatter index (SI), and mean absolute relative error (MARE).

where and are the corresponding typical and investigated C_d values; and are the average of the measured amounts of C_d and the typical values, respectively.

E = 1 corresponds to an ideal combination of the expected coefficient of discharge with the data gathered; E = 0 denotes that the model is just as potent as the current data's average; and −∞ < E < 0 indicates that the identified median value is preferable than the model, meaning that the findings are undesirable.

Table 4 compares performance parameters and percentage error of GMDH, MLR model, and available equation of free overfall coefficient of discharge in literature.

GMDH and MLR models predicted outcomes that are adequately related to the accessible equations of C_d for free overfall. The qualitative performance of the present MLR has low RMSE (0.06332), MARE (0.05970), SI (0.06624), NSE (−3.0419), and R² (15.431%), respectively. The GMDH model has lower RMSE (6.08 × 10⁻¹⁷), MARE (2.65 × 10⁻¹⁷), S.I. (6.00 × 10⁻¹⁷), higher NSE and R² (1.000) and (100.00%), respectively, this demonstrates that it performs better than other available predictors.

This study conducted the experimental analysis of C_d for triangular end-shape free overfall using 180 collected data based on data collected. Numerical models (MLR and GMDH) apply to predicting outcomes. Data were collected for two models (with and opposite) flow directions with three different angles (60°, 75°, and 90°). Results show that RMSE, MARE, SI, R², and NSE values for testing data in GMDH technique mod36 are 1.93E-14, 1.48E-14, 1.90E-14, 100.00%, and 1.00, respectively, while these values for MLR are 0.06332, 0.05970, 0.06624, 15.431%, and −3.0419, respectively. From these results, the performance of the GMDH technique gives excellent indications for using this technique in the future because of its accuracy in calculating C_d. Even though the MLR model has an incredible ability to fit and predict, in contrast to the GMDH method, MLR has a much lower power in explaining the response variable. The results show that all four selected GMDH models have much higher levels of prediction. Model 46 took 28 h to run and gave the best results, while model 36 took 18 h, so model 36 was recommended because it took less time and gave better values than the other 24 scenarios. Also, the results show that the worst GMDH estimate has far better results than the best MLR.

A.Y.M. would like to thank the staff of the hydraulic laboratory in the dams and water resources department for their support in laboratory work.

All authors have read and agreed to the published version of the manuscript.

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

The authors declare there is no conflict.