Abstract

Discharge coefficients (C0) for ogee weirs are essential factors for predicting the discharge-head relationship. The present study investigates three influences on the C0: effect of approach depth, weir upstream face slope, and the actual head, which may differ from the design head. This study uses experimental data with multiple non-linear regression techniques and Gene Expression Programming (GEP) models that are applied to introduce practical equations that can be used for design. Results show that the GEP method is superior to the regression analysis for predicting the discharge coefficient. Performance criteria for GEP are R2 = 0.995, RMSE = 0.021 and MAE = 0.015. Design examples are presented that show that the proposed GEP equation correlates well with the data and eliminates linear interpolation using existing graphs.

SYMBOLS

The following symbols are used in this paper:

  • C0= discharge coefficient for free (uncontrolled) flow condition with vertical upstream face (m0.5/s);

  • Ci = discharge coefficient for free flow condition, with sloping face in upstream (m0.5/s);

  • g = acceleration due to gravity;

  • H = reprehensive horizontal distance in upstream weir slope;

  • H0 = design head over the crest (m);

  • He = actual head (other than the design head) being considered on the crest (m);

  • h0= upstream water depth above the crest in design discharge (m);

  • I= angle of the upstream face in ogee weir with respect to vertical direction (degrees);

  • L = effective length of the crest (m);

  • Q = design discharge for ogee spillway (m3/s);

  • P = ogee spillway height (m);

  • V= reprehensive vertical distance in upstream weir slope;

  • = approach velocity in m/s

INTRODUCTION

An ogee spillway is one of the weir types that is used in many dam spillway types like diversion, earth, gravity, rock fill, buttress, and arch dams. It is the most common type and is typically constructed from concrete. Spillways can be located over a conventional gravity dam, an arch gravity dam, and a roller compacted concrete (RCC) dam. If the dam is an embankment, then the spillway can be located over the right or left abutments and is typical made from hard material like concrete.

Flow over an ogee spillway is dependent on the discharge coefficient (C0). These spillways are designed based on a specific discharge, referred to as the design discharge (Qdesign).

There are two kinds of ogee spillways: (i) – ogee spillway with gates on the crest that can control the discharge over the spillway and (ii) – ogee spillway without gates. These spillways do not have any control of discharge. The advantages of the un-gated or uncontrolled crest are the elimination of the flow control devices and the lower maintenance and repairs.

The shape of the ogee spillway depends upon the head, the inclination of the upstream face of the overflow section, and the height of the overflow section above the floor of the entrance channel (USBR 1987).

The discharge over an uncontrolled overflow ogee crest is given by Equation (1) (Kim & Park 2005): 
formula
(1)
where Q is the design discharge (m3/s), C0 is the variable discharge coefficient for free (uncontrolled) flow conditions (m0.5/s), L represents the effective length of the crest (m), and H0 symbolizes the design head or actual head being considered (m), including the velocity of approach head, ha (m). In Equation (1), where C0 is the weir coefficient (dimensionless) and g is gravity acceleration (m/s2).

The discharge coefficient, C0, is influenced by a number of factors. These factors are: the effect of the approach depth, the effect of heads different from the design head, the effect of the upstream face slope, the effect of the downstream apron interference, and the effect of the downstream submergence. Influences of the first three factors are the subject of the present study.

USBR (1987) proposed discharge coefficient graphs for ogee crests with different geometries. These graphs (for gated, ungated and various upstream slope conditions) have been used by hydraulic designers for many years. The gap in knowledge is that there is no simple equation to combine these graphs and thus eliminate linear interpolation among several curves. The objective of present study is to propose an equation for estimating discharge coefficients. For this purpose, multiple regression techniques and Gene Expression Programming (GEP) models are applied with dimensionless parameters. The experimental data are from USBR (1987). The information for discharge coefficient includes 202 data points, obtained via experiments on several ogee spillways. The discharge coefficient is calculated using Equation (1), i.e. .

Another reason for eliminating linear interpolation among several curves relates to Moody's (1947) diagram. Based on this diagram, friction factor (f) in pipes depend on two dimensionless parameters: Reynolds number (Re) and the relative roughness (e/D). Pipe roughness is denoted with e and pipe diameter is refer with D. However, the mathematical formulation to find f, includes an empirical equation that is well-known as the Colebrook-White equation. This equation is such that the f factor appears on both sides of the equation (Salmasi et al. 2012) and requires iteration for its solution

Effect of approach depth

The approach velocity (Va) for high sharp-crested weirs, is low and the lower nappe from these weirs leads to high contraction. Increasing discharge or reducing weir height causes an increase in Va and this phenomenon reduces lower nappe contraction.

For sharp-crested weirs whose heights are more than one-fifth of the total head (P/H0 > 0.2), the discharge coefficient is constant with a value of about 1.82. For weir heights less than about one-fifth the head, the contraction of the flow becomes increasingly suppressed and the crest coefficient decreases. When the weir height becomes zero, the contraction disappears and the overflow weir becomes a channel or a broad-crested-weir, for which the theoretical discharge coefficient is 1.70. Figure 1 shows variation of discharge coefficient for vertical-faced ogee crests with height-over-head ratio (P/H0).

Figure 1

Vertical-faced discharge coefficients for ogee crests (USBR 1987). Note: values of C0 is based on SI system units (m0.5/s).

Figure 1

Vertical-faced discharge coefficients for ogee crests (USBR 1987). Note: values of C0 is based on SI system units (m0.5/s).

Effect of the heads on discharge coefficient

Higher discharge than the design discharge causes negative pressure over the spillway bottom. This is because the nappe tends to disconnect from the spillway bottom. Lower discharge than the design discharge causes positive pressure over the spillway bottom. This is because the nappe has a tendency to connect to the spillway bottom. Figure 2 provides the changes of the dimensionless C/C0 versus values of He/H0. The symbol He is the actual head.

Figure 2

Discharge coefficients for other than the design head (USBR 1987).

Figure 2

Discharge coefficients for other than the design head (USBR 1987).

Influence of face slope

With small values of P/H0, sloping the upstream face of the ogee spillway increases the discharge coefficient. The effect of upstream slope on discharge coefficient decreases for large ratios of P/H0. Figure 3 compares the ogee spillway discharge coefficient with an inclined face (Ci) to that for a crest with a non-inclined face (Cv) as related to values of P/H0.

Figure 3

Discharge coefficients for ogee spillways including a sloping upstream face (USBR 1987).

Figure 3

Discharge coefficients for ogee spillways including a sloping upstream face (USBR 1987).

Figures 4 and 5 show the application of ogee spillways in storage and diversion dams respectively.

Figure 4

An ogee spillway in Walayar reservoir, India.

Figure 4

An ogee spillway in Walayar reservoir, India.

Savage & Johnson (2001) carried out a study of flow characteristics over a standard ogee-crested spillway by means of physical and numerical models. Observation of pressure and discharge showed proper agreement between the physical and numerical models. Tullis (2011) evaluated the performance of submerged ogee coefficients as related to upstream and downstream weir height and flow discharge. Results showed that the USBR (1987) dimensionless relationships developed by Bradley (1945) underestimated the effect of submergence (S) on Cs for multiple ogee crest weirs. Tullis (2011) reported that the weak relationship between observed and predicted Cs values might be caused by differences between hd measurement positions.

Bradley (1945) measured hd nearer to the weir than in the Tullis (2011) study. Tullis & Neilson (2008) showed that for submergence levels less than 0.70, head-discharge correlations were independent of the tail elevation. On the other hand, for higher submergence, the trends were reversed.

Madadi et al. (2014) investigated flow characteristics of a trapezoidal broad-crested weir. The results showed that a reduction of the upstream slope avoids the progress of a flow separation zone. A decrease in the upstream slope to 21° resulted in an increase in the discharge coefficient. The changes were as large as 10%. Furthermore, the separation relative length and height decreased up to 80% and 95%, respectively.

Al-Khatib & Gogus (2014) studied discharge in rectangular compound broad-crested weirs using multiple regression equations. In that study, the dependencies of the discharge coefficient and upstream velocity on different operating parameters were investigated based and reported using dimensionless ratios. Results showed when the head in the upstream section is given, the flow discharge can be evaluated with an error of less than ∼ 5%.

Guven et al. (2013) studied flow over broad-crested weirs and through box culverts. The latter's performance was superior for conveying water.

Salmasi (2018) presented equations for estimating discharge coefficients in ogee weirs with consideration of downstream stilling basin elevation and submergence ratio. Salmasi (2018) refers to the ease of these equations compared to traditional charts.

Estimation of discharge coefficient is also important in other hydraulic structures like sluice gates and other weirs. Recent investigations for determining discharge coefficients in radial gates include the study of Salmasi et al. (2019), which refers to the effect of sills on discharge. In another study, Salmasi (2019) found the discharge coefficient for circular labyrinth weirs. Akbari et al. (2019) predicted discharge coefficients for gated piano key weirs using experimental and artificial intelligence (AI) methods.

Some aspects of the hydrodynamics of rectangular broad-crested porous weirs comprise determination of the discharge coefficient. Flow can occur both through and over these weirs and this phenomenon changes the discharge coefficient values (Mohamed 2010; Salmasi & Abraham 2020).

The goal of this study is to create an accurate method for predicting C0 for ogee weirs. To introduce the head-discharge relationship, designers need to use Figures 13 and employ linear interpolation among the slopes in Figure 3. This approach results in estimations of C0 and head-discharge relationship with some error. Here, the effects of three factors: P/H0 ratios, heads that differ from the design head (He/H0), and the effect of the upstream face slope (I) are investigated using regression analysis and GEP modeling. The goal is to provide a methodology that is independent of design charts provided by USBR (1987). To our best knowledge, there is no equation for predicting C0 values in ogee weirs with consideration of the three above factors.

MATERIAL AND METHODS

Geometric and hydraulic variables

Figure 6 denotes hydraulic parameters in an ogee weir with heads greater/less than the design head. In the figure, the symbol El C represents crest elevation from an arbitrary datum, P is ogee weir height, H0 is design head, ha = Va2/2 g is approach velocity head, He1 and He2 are actual heads being considered on the crest, the approach velocity Va is equal to Va = Q/L/(P + H0), L is the weir length and Q is the design discharge. Figure 5 indicates geometric and hydraulic variables in an ogee weir with sloping upstream face. In Figure 7, I is the upstream face of the weir inclination respect to the vertical direction. In the following equations, I is in degrees. The other parameters have been defined previously.

Figure 5

An ogee spillway in a diversion dam in Iran.

Figure 5

An ogee spillway in a diversion dam in Iran.

Figure 6

Geometric and hydraulic variables in an ogee weir with head greater/less than design head (vertical sloping face).

Figure 6

Geometric and hydraulic variables in an ogee weir with head greater/less than design head (vertical sloping face).

Figure 7

Geometric and hydraulic variables in an ogee weir with sloping upstream face in design head.

Figure 7

Geometric and hydraulic variables in an ogee weir with sloping upstream face in design head.

Regression analysis

In this study, the effective parameters are: C0 versus P/H0 (effect of depth of approach), C/C0 versus He/H0 (head differing from the design head) and Ci/Cv versus P/H0 (upstream sloping face effect). A functional relationship among parameters is described in Equation (2) (USBR 1987): 
formula
(2)
where I is the weir upstream face inclination with respect to the vertical direction (in degrees) and other parameters were defined previously.

All test data are from different geometries of ogee spillways. The number of data points used in this study is 202 (Appendix I) and the domains of variation for dimensionless parameters in Equation (2) are: 0 < P/H0 < 3, 0 < I < 45 and <0.05He/H0 < 1.6. Thus, different geometries include different values for P and I parameters and different hydraulic conditions relate to H0 and He.

In the present study, a commercially available computer code (SPSS) software version 22 (SPSS 2013) and CurveExpert Professional (2012) software are used for classical regression analysis.

Application of GEP

The software GeneXproTools (2011), version 4.0 (Ferreira 2001a, 2001b) was used in this study for prediction of the discharge coefficient.

The GEP considers crossover and mutation operators to be ‘winners’ (children) and then compete in natural selection. Crossover operations preserve features from one generation to the next. On the other hand, mutations lead to random changes in generations.

Performance criteria

In this study, to evaluate the accuracy of the proposed models (regression and GEP), the three indicators, (1) root mean square error (RMSE), (2) determination coefficient (R2) and (3) mean absolute error (MAE), are used. They are presented in Equations (3)–(5) (Akbari et al. 2019):

  • I.
    Root mean square error (RMSE) 
    formula
    (3)
  • II.
    Determination coefficient (R2) 
    formula
    (4)
  • III.
    Mean absolute error (MAE) 
    formula
    (5)
In these equations, N represents the number of observations; y0 symbolizes observed data; the term yp represents predicted data; is the value of the mean from the observations; and is the mean of the predictions. As previously noted, the total number of data points are 202, of which 141 (70%) are used for training and 61 (30%) for testing. In Appendix I, these data points are presented.

RESULTS AND DISCUSSION

Regression equations

In this study, to derive equations with one dependent variable, CurveExpert Professional (2012) software is used and for equations with two or more dependent variables (multiple regression), SPSS (2013) software and GeneXproTools (2011) software are used. Variation of discharge coefficients (C0) for vertical-faced ogee crest against P/H0 (Figure 1) are presented in Equation (6) (R2 = 0.999): 
formula
(6)
Variation of relative discharge coefficients (C/C0) versus a head other than the design head (Figure 2) is obtained from Equation (7) (R2 = 0.999): 
formula
(7)
Variation in discharge for ogee-shaped crests with an upstream slope (Figure 3) are calculated using Equation (8): 
formula
(8)
Table 1 shows prediction errors for Equations (6)–(8). Equation (8) demonstrates that a non-linear regression equation is not successful in prediction of C0 with R2 = 0.314, RMSE = 1.082 and MAE = 1.075. This implies that more sophisticated methods are needed for prediction of Ci/Cv. Figure 8 presents a scatter plot of training and testing phases based on non-linear regression (Equation (8)).
Table 1

Training and testing phase errors for the relative value of the discharge coefficient (Ci/Cv)

TypeTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Equation (6) 0.998 0.004 0.003 0.996 0.007 0.009 
Equation (7) 0.998 0.007 0.005 0.995 0.056 0.034 
Equation (8) 0.278 0.008 0.005 0.314 1.082 1.075 
TypeTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Equation (6) 0.998 0.004 0.003 0.996 0.007 0.009 
Equation (7) 0.998 0.007 0.005 0.995 0.056 0.034 
Equation (8) 0.278 0.008 0.005 0.314 1.082 1.075 
Figure 8

Scatter plot of training (a) and testing phases (b) based on non-linear regression (Equation (8)).

Figure 8

Scatter plot of training (a) and testing phases (b) based on non-linear regression (Equation (8)).

Performance of GEP

The GEP model (Ci/Cv versus P/H0 and I) was applied using different operators defined in Table 2. GEP performs better than regression type analyses, particularly for option 2 in terms of R2 (0.984), RMSE (0.002) and MAE (0.002) in the testing phase. The derived equation for option 2 using GEP is: 
formula
(9)

The derived GEP model in Equation (9) offers a high-order nonlinear equation that gives good accuracy with relatively low error. Performance of the GEP method is illustrated in Figure 7. According to Figure 9, GEP is well able to forecast the values of the relative discharge coefficient (Ci/Cv).

Table 2

Errors for the training and testing datasets of the relative discharge coefficient with different GEP operators

TypeOperatorTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Option 1 { + , −, *, /} 0.884 0.005 0.004 0.925 0.005 0.004 
Option 2 { + , −, *,/, x2, Exp} 0.984 0.002 0.001 0.984 0.002 0.002 
Option 3 { + , −, *,/, x2, x3, Exp, cube root} 0.903 0.004 0.004 0.927 0.004 0.003 
Option 4 { + , −, *,/, x2, x3, Exp, cube root, 10x, Natural logarithm} 0.857 0.005 0.004 0.935 0.004 0.003 
TypeOperatorTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Option 1 { + , −, *, /} 0.884 0.005 0.004 0.925 0.005 0.004 
Option 2 { + , −, *,/, x2, Exp} 0.984 0.002 0.001 0.984 0.002 0.002 
Option 3 { + , −, *,/, x2, x3, Exp, cube root} 0.903 0.004 0.004 0.927 0.004 0.003 
Option 4 { + , −, *,/, x2, x3, Exp, cube root, 10x, Natural logarithm} 0.857 0.005 0.004 0.935 0.004 0.003 
Figure 9

Scatter plot of training and testing phases based on GEP model (Equation (9)).

Figure 9

Scatter plot of training and testing phases based on GEP model (Equation (9)).

By combining Equations (6) and (8), a new graph (C0 versus P/H0) can be obtained with results that are shown in Figure 10. The figure is comprised of four slopes: 45, 33.69, 18.42 and 0 (vertical face) degrees. From Figure 10, it is seen that an inclination of upstream face of ogee weir increases C0; however, the slope effect is small.

Figure 10

Variation of discharge coefficient (C0) with the ratio of P/H0 for four different upstream ogee weir slopes.

Figure 10

Variation of discharge coefficient (C0) with the ratio of P/H0 for four different upstream ogee weir slopes.

Final equations for predicting C0

Regression equation

By applying Equations (6)–(8), two equations for prediction of C0 were obtained (Equations (12) and (13)). A scatter plot for Equation (11) and errors for both the testing and training data of the discharge coefficient (C0) related with two regression models are presented in Figure 11 and Table 3, respectively. Equation (11) has better performance (R2 = 0.702, RMSE = 0.792 and MAE = 0.803). 
formula
(10)
 
formula
(11)
Table 3

Estimation of the errors for the training and testing datasets of the C0 with different regression models

TypeTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Equation (10) 0.602 0.992 0.959 0.587 0.410 0.705 
Equation (11) 0.724 0.784 0.959 0.702 0.792 0.803 
TypeTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Equation (10) 0.602 0.992 0.959 0.587 0.410 0.705 
Equation (11) 0.724 0.784 0.959 0.702 0.792 0.803 
Figure 11

Comparison between experimental data and regression model (Equation (10)): (a) training phase, (b) testing phase.

Figure 11

Comparison between experimental data and regression model (Equation (10)): (a) training phase, (b) testing phase.

Figure 12

Comparison between experimental data and GEP, the model (option 3): (a) for the training phase, (b) for the testing phase.

Figure 12

Comparison between experimental data and GEP, the model (option 3): (a) for the training phase, (b) for the testing phase.

Figure 13

Contours for estimation of Cd.

Figure 13

Contours for estimation of Cd.

GEP equation

The results from the GEP analysis for C0 are given in Equation (12) and prediction errors are given in Table 4. The performance errors from the preferred GEP were: R2 = 0.995, RMSE = 0.021 and MAE = 0.015 for the testing phase.

Table 4

Estimation of the errors for testing and training of the relative discharge coefficient with different GEP operators

TypeOperatorTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Option 1 { + , −, *, /} 0.994 0.023 0.016 0.987 0.033 0.021 
Option 2 { + , −, *,/, x2, Exp} 0.989 0.032 0.027 0.985 0.035 0.028 
Option 3 { + , −, *,/, x2, x3, Exp, cube root} 0.998 0.016 0.012 0.995 0.021 0.015 
Option 4 { + , −, *,/, x2, x3, Exp, cube root, 10x, Natural logarithm} 0.978 0.047 0.042 0.977 0.047 0.040 
TypeOperatorTraining phase
Testing phase
R2RMSEMAER2RMSEMAE
Option 1 { + , −, *, /} 0.994 0.023 0.016 0.987 0.033 0.021 
Option 2 { + , −, *,/, x2, Exp} 0.989 0.032 0.027 0.985 0.035 0.028 
Option 3 { + , −, *,/, x2, x3, Exp, cube root} 0.998 0.016 0.012 0.995 0.021 0.015 
Option 4 { + , −, *,/, x2, x3, Exp, cube root, 10x, Natural logarithm} 0.978 0.047 0.042 0.977 0.047 0.040 
Figure 12 shows comparison between experimental data and the GEP model (option 3) for training and testing phases. 
formula
(12)

Application example

To demonstrate a practical application of the discussed method, it is used to calculate the discharge coefficient (C0) of an ogee weir due to the existence of an upstream sloping face and a discharge over it that differs from the design discharge with the following characteristics:

The design discharge for the weir (with a vertical face) is 63.917 m3/s. The overflow dam spillway height is 5 m. The length of the overflow spillway is 10 m and the design head is 2 m. The upstream face of the weir is inclined at 0, 25, 45 and 60 degrees. The values of C0 are computed using Equation (12) and Q is computed with Equation (1). The results are presented in Table 5.

Table 5

Prediction of the discharge coefficient for an ogee weir with inclined in upstream face (I) and with head (He) that differs from the design head (H0)

L (m)P (m)H0 (m)P/H0I (Degrees)He (m)He/H0C0Q (m3/s)Q inclined/Q vertical
10 2.5 0.2 0.10 1.785 1.597 
10 2.5 0.4 0.20 1.862 4.710 
10 2.5 0.6 0.30 1.932 8.980 
10 2.5 0.8 0.40 1.997 14.289 
10 2.5 0.50 2.056 20.556 
10 2.5 1.2 0.60 2.108 27.715 
10 2.5 1.4 0.70 2.155 35.700 
10 2.5 1.6 0.80 2.196 44.444 
10 2.5 1.8 0.90 2.231 53.874 
10 2.5 1.00 2.260 63.917 
10 2.5 2.2 1.10 2.283 74.490 
10 2.5 2.4 1.20 2.300 85.509 
10 2.5 2.6 1.30 2.311 96.882 
10 2.5 2.8 1.40 2.316 108.513 
10 2.5 1.50 2.315 120.301 
10 2.5 3.2 1.60 2.308 132.142 
10 2.5 25 0.2 0.10 1.790 1.601 1.003 
10 2.5 25 0.4 0.20 1.872 4.735 1.005 
10 2.5 25 0.6 0.30 1.948 9.051 1.008 
10 2.5 25 0.8 0.40 2.017 14.435 1.010 
10 2.5 25 0.50 2.081 20.811 1.012 
10 2.5 25 1.2 0.60 2.139 28.117 1.015 
10 2.5 25 1.4 0.70 2.191 36.291 1.017 
10 2.5 25 1.6 0.80 2.237 45.269 1.019 
10 2.5 25 1.8 0.90 2.277 54.983 1.021 
10 2.5 25 1.00 2.311 65.359 1.023 
10 2.5 25 2.2 1.10 2.339 76.320 1.025 
10 2.5 25 2.4 1.20 2.361 87.784 1.027 
10 2.5 25 2.6 1.30 2.377 99.660 1.029 
10 2.5 25 2.8 1.40 2.387 111.857 1.031 
10 2.5 25 1.50 2.392 124.275 1.033 
10 2.5 25 3.2 1.60 2.390 136.811 1.035 
10 2.5 45 0.2 0.10 1.794 1.605 1.005 
10 2.5 45 0.4 0.20 1.880 4.757 1.010 
10 2.5 45 0.6 0.30 1.960 9.111 1.015 
10 2.5 45 0.8 0.40 2.034 14.557 1.019 
10 2.5 45 0.50 2.102 21.024 1.023 
10 2.5 45 1.2 0.60 2.165 28.454 1.027 
10 2.5 45 1.4 0.70 2.221 36.786 1.030 
10 2.5 45 1.6 0.80 2.271 45.961 1.034 
10 2.5 45 1.8 0.90 2.315 55.911 1.038 
10 2.5 45 1.00 2.354 66.567 1.041 
10 2.5 45 2.2 1.10 2.386 77.854 1.045 
10 2.5 45 2.4 1.20 2.412 89.689 1.049 
10 2.5 45 2.6 1.30 2.433 101.988 1.053 
10 2.5 45 2.8 1.40 2.447 114.659 1.057 
10 2.5 45 1.50 2.456 127.605 1.061 
10 2.5 45 3.2 1.60 2.458 140.724 1.065 
10 2.5 60 0.2 0.10 1.798 1.608 1.007 
10 2.5 60 0.4 0.20 1.887 4.774 1.014 
10 2.5 60 0.6 0.30 1.970 9.157 1.020 
10 2.5 60 0.8 0.40 2.048 14.652 1.025 
10 2.5 60 0.50 2.119 21.190 1.031 
10 2.5 60 1.2 0.60 2.184 28.716 1.036 
10 2.5 60 1.4 0.70 2.244 37.171 1.041 
10 2.5 60 1.6 0.80 2.298 46.498 1.046 
10 2.5 60 1.8 0.90 2.345 56.633 1.051 
10 2.5 60 1.00 2.387 67.507 1.056 
10 2.5 60 2.2 1.10 2.422 79.046 1.061 
10 2.5 60 2.4 1.20 2.452 91.171 1.066 
10 2.5 60 2.6 1.30 2.476 103.798 1.071 
10 2.5 60 2.8 1.40 2.494 116.837 1.077 
10 2.5 60 1.50 2.506 130.193 1.082 
10 2.5 60 3.2 1.60 2.511 143.765 1.088 
L (m)P (m)H0 (m)P/H0I (Degrees)He (m)He/H0C0Q (m3/s)Q inclined/Q vertical
10 2.5 0.2 0.10 1.785 1.597 
10 2.5 0.4 0.20 1.862 4.710 
10 2.5 0.6 0.30 1.932 8.980 
10 2.5 0.8 0.40 1.997 14.289 
10 2.5 0.50 2.056 20.556 
10 2.5 1.2 0.60 2.108 27.715 
10 2.5 1.4 0.70 2.155 35.700 
10 2.5 1.6 0.80 2.196 44.444 
10 2.5 1.8 0.90 2.231 53.874 
10 2.5 1.00 2.260 63.917 
10 2.5 2.2 1.10 2.283 74.490 
10 2.5 2.4 1.20 2.300 85.509 
10 2.5 2.6 1.30 2.311 96.882 
10 2.5 2.8 1.40 2.316 108.513 
10 2.5 1.50 2.315 120.301 
10 2.5 3.2 1.60 2.308 132.142 
10 2.5 25 0.2 0.10 1.790 1.601 1.003 
10 2.5 25 0.4 0.20 1.872 4.735 1.005 
10 2.5 25 0.6 0.30 1.948 9.051 1.008 
10 2.5 25 0.8 0.40 2.017 14.435 1.010 
10 2.5 25 0.50 2.081 20.811 1.012 
10 2.5 25 1.2 0.60 2.139 28.117 1.015 
10 2.5 25 1.4 0.70 2.191 36.291 1.017 
10 2.5 25 1.6 0.80 2.237 45.269 1.019 
10 2.5 25 1.8 0.90 2.277 54.983 1.021 
10 2.5 25 1.00 2.311 65.359 1.023 
10 2.5 25 2.2 1.10 2.339 76.320 1.025 
10 2.5 25 2.4 1.20 2.361 87.784 1.027 
10 2.5 25 2.6 1.30 2.377 99.660 1.029 
10 2.5 25 2.8 1.40 2.387 111.857 1.031 
10 2.5 25 1.50 2.392 124.275 1.033 
10 2.5 25 3.2 1.60 2.390 136.811 1.035 
10 2.5 45 0.2 0.10 1.794 1.605 1.005 
10 2.5 45 0.4 0.20 1.880 4.757 1.010 
10 2.5 45 0.6 0.30 1.960 9.111 1.015 
10 2.5 45 0.8 0.40 2.034 14.557 1.019 
10 2.5 45 0.50 2.102 21.024 1.023 
10 2.5 45 1.2 0.60 2.165 28.454 1.027 
10 2.5 45 1.4 0.70 2.221 36.786 1.030 
10 2.5 45 1.6 0.80 2.271 45.961 1.034 
10 2.5 45 1.8 0.90 2.315 55.911 1.038 
10 2.5 45 1.00 2.354 66.567 1.041 
10 2.5 45 2.2 1.10 2.386 77.854 1.045 
10 2.5 45 2.4 1.20 2.412 89.689 1.049 
10 2.5 45 2.6 1.30 2.433 101.988 1.053 
10 2.5 45 2.8 1.40 2.447 114.659 1.057 
10 2.5 45 1.50 2.456 127.605 1.061 
10 2.5 45 3.2 1.60 2.458 140.724 1.065 
10 2.5 60 0.2 0.10 1.798 1.608 1.007 
10 2.5 60 0.4 0.20 1.887 4.774 1.014 
10 2.5 60 0.6 0.30 1.970 9.157 1.020 
10 2.5 60 0.8 0.40 2.048 14.652 1.025 
10 2.5 60 0.50 2.119 21.190 1.031 
10 2.5 60 1.2 0.60 2.184 28.716 1.036 
10 2.5 60 1.4 0.70 2.244 37.171 1.041 
10 2.5 60 1.6 0.80 2.298 46.498 1.046 
10 2.5 60 1.8 0.90 2.345 56.633 1.051 
10 2.5 60 1.00 2.387 67.507 1.056 
10 2.5 60 2.2 1.10 2.422 79.046 1.061 
10 2.5 60 2.4 1.20 2.452 91.171 1.066 
10 2.5 60 2.6 1.30 2.476 103.798 1.071 
10 2.5 60 2.8 1.40 2.494 116.837 1.077 
10 2.5 60 1.50 2.506 130.193 1.082 
10 2.5 60 3.2 1.60 2.511 143.765 1.088 

Table 5 shows that the inclination of the upstream facecauses an increase in the discharge coefficient (C0). This can be seen from comparison of I = 0 degrees with I = 25, 45 and 60 degrees. The last column in Table 5 demonstrates that the ratio of Q inclined/Q vertical exceeds 1 for I values greater than 0.

Figure 13 provides additional information about the relation among independent variables (P/He and He/H0) against dependent variable Cd. These contours represent values for Cd estimation based on the 202 data points.

The relation between He/H0 and I shows that for a constant value of He/H0, increases in I causes decreases in Cd. In addition, for a constant value for I, increasing He/H0 results in increases in Cd.

The relationship between P/H0 and He/H0 shows that for a constant value for P/H0, increasing He/H0 causes an increase in Cd. For a constant value of He/H0, increasing P/H0 has no effect on Cd except for P/H0 < 0.25.

CONCLUSIONS

This study creates a predictive model that accurately quantifies the discharge coefficient (C0) from ogee weirs with upstream sloping faces and with discharge other than the design discharge. Regression analysis and GEP were carried out. Parameters such as weir upstream face inclination (I), ratio of P/H0 that reflects weir height over the design head, and the ratio of He/H0 are the input variables, while discharge coefficient (C0) was an output. The GEP technique was more capable than regression analysis in predicting C0. Performance errors from the preferred GEP analysis were R2 = 0.995, RMSE = 0.021 and MAE = 0.015, while for a non-linear regression equation they were: R2 = 0.702, RMSE = 0.792 and MAE = 0.803. Finally, examples were presented to show the application of the suggested equations. These examples accounted for 0, 25, 45 and 60 degree upstream weir slopes and included different ratios of actual heads (He/H0).

FUNDING

There is no funding for this article.

CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

DECLARATIONS OF INTEREST

None.

ACKNOWLEDGEMENTS

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

ETHICAL APPROVAL

This article does not contain any studies with human participants or animals performed by any of the authors.

SUPPLEMENTARY MATERIAL

The Supplementary Material for this paper is available online at https://dx.doi.org/10.2166/ws.2020.064.

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Supplementary data