OPEN ACCESS
River and open channel flow measurement is regarded as an important element in improving water system management. An experimental analysis in a laboratory flume has been executed to obtain formulas which can be used to predict the discharge quantity which crosses over the sharp crested weir and under the gate in the composite structure which is a tool of flow measurement. These formulas are applied for the free flow condition. Different configurations of composite structure have been used. These configurations have regular and irregular shapes. The derivation of the discharge quantity equation is based on the dimensional analysis which is employed to analyze the factors that control the operation of the composite structure. The family of formulas consists of eight formulas that are divided into two groups. The first group consists of four formulas which are derived based on the variable that dominates the hydraulic behavior of the composite structure, while the second group consists of four formulas based on the same variables of the first group but taking into consideration the effect of weir and gate shape by adopting the hydraulic radius of weir and gate. Based on the evaluation process, all formulas can be used perfectly well to predict the discharge quantity, regardless of the shape of weir and gate. On the other hand, based on the validation process, the formula that considered the effects of shape geometry, flow and fluid properties has given the best prediction of the discharge quantity.
composite structure, discharge quantity, free flow, gate, weir
Weir gate hydraulic structure may be simply defined as the irrigation structure which is spread wide and used in hydraulic system water work. This system accomplishes several tasks such as distributing, diverting, monitoring and measuring water for free flow and submerging flow conditions, respectively, under a reasonable operation [1]. Several researchers dealt with this hydraulic structure owing to its important role in water work. The discharge coefficient of combined hydraulic structure is examined. The stated that the hydraulic structure, consisting of a rectangular notch, has various widths and a gate with a semicircular section that has a constant diameter. In this study, the vertical distance between weir and gate is varied. In this respect, a multiregression model is employed to calculate the combined discharge coefficient of the hydraulic structure [2]. The combined discharge coefficient for free flow of the combined hydraulic structure is examined. With this regard, the weir of combined structure comprises three portions of the rectangular notch, located over the triangular notch and the trapezoidal notch existed between them while the gate has a semicircular shape. Also, they obtained a formula to predict the combined discharge coefficient of the hydraulic structure. They found that the discharge coefficient of the combined weirgate structure has range from 0.358 to 0.426 with an average value equal to 0.392 [3]. An experimental study is carried out to appreciate the hydraulic characteristics of notches and combined the hydraulic structure by adopting various shapes of the notch and the combined weirgate structure. This study illustrated that the combined semicircular notch over the semicircular gate shows a preferable discharge coefficient [4]. The combined discharge coefficient of the hydraulic structure which comprised the combined weir and the semicircular gate is examined. They deduced that the increase in discharge coefficient is associated with the hydraulic variables increasing (H/D, H_{1}/D, H_{2}/D, H_{3}/D). At the constant value of (X), any increase in (W) will lead to the increase in the combined discharge coefficient and when the value of (Y,Z) increases, it will be considered as a factor showing the decrease of the combined discharge coefficient. Based on this study, they also found a semi empirical formula to predicate the discharge coefficient. Where H: total head, H_{1}, H_{2}, H_{3}: water depth over the first, the second and third notch of the combine weir, D: gate diameter, W: width of the second and third notch of the combine weir. B: flume width. Z: height of the second notch of combine weir. Y: vertical distance between weir and gate [5]. The flow of the hydraulic characteristics over the notch with trapezoidal section and under the gate with rectangular section is examined by performing experimental work to evaluate the impact of water depth at upstream, depth of water flow above the sharp crested notch, and the notch angle of trapezoidal section on the combined discharge coefficient of hydraulic structure, respectively. Also, by utilizing the dimensional analysis with the STATISTICA software, they predicted an empirical equation to calculate the combined discharge coefficient. The dimensional analysis was dealt with the following: upstream water depth, water depth above weir, water depth at the gate and trapezoidal weir angle [6]. The fluent software is utilized to appreciate the combined weirgate hydraulic characteristics. From the results obtained, they deduced that any rise in the combined discharge coefficient will lead to a rise in the discharge quantity. In this work, the combined hydraulic structure comprises the rectangular gate and rectangular weir [7]. An experimental study is performed to examine the combined discharge coefficient of the weirgate structure. The weir consists of two portions of rectangular notches over the trapezoidal notch with various gate shapes (rectangular gate, semicircular gate and triangular gate). The study concentrated on the influence of geometrical dimension and hydraulic characteristics on the combined discharge coefficient of the hydraulic structure. Out of this study, they obtained three nonlinear formulas to estimate the discharge coefficient, depending on the geometrical and hydraulic variables [8]. The effect of the dimensional parameters and nondimensional parameters on the hydraulic behavior of the combined hydraulic structure is investigated by employing three various shapes of combine hydraulic structures (triangular gate with triangular notch, rectangular gate with rectangular notch and trapezoidal gate with trapezoidal notch) [9]. The interference between the under flow and the over flow for the combined hydraulic structure comprising of parabolic gate with the parabolic weir is investigated. This study deals with the interaction between the geometrical dimensions and the hydraulic characteristics [10]. Further, the effect of utilized combined hydraulic structure consisting of a gate with nonregular shape (half ellipse) and with various weir shapes (parabolic weir, triangular weir, and rectangular weir), regarding the hydraulic variables and the geometrical variables is investigated. From the results they found that the used of gate with nonregular shape with any shape of weir in the composite hydraulic structure is better. Also, they found that the use of nonregular shape for both weir and gate in the composite structure give a noticeable influence on the hydraulic variables. They found good results in using a nonregular gate with a nonregular weir [11]. The vital role of the major and minor factors which were prevalent on the weirgate hydraulic structure behavior is explained, like the discharge coefficient [12]. A comparative study depending on the impact of bed flume with or without the contraction on the obtained values of composite hydraulic structure discharge coefficient is made. This study showed the influence of the hydraulic variables on the discharge coefficient, like discharge quantity, flow velocity, Froude number, Reynolds number, and downstream water depth average [13]. The main objective of the current work is based on finding out general discharge formula of the combined weirgate hydraulic structure which is used to calculate the total discharge regardless of the shape of weir and gate, whether the shape is regular or irregular.
A set of 86 experimental works were carried out in a flume at the hydraulic laboratory of Basra Engineering Technical College. The flume is 200cm long, has 15cm depth and is 7.5cm wide with a horizontal bed. Eight groups of composite hydraulic structure are used to drive the discharge quantity. The depth of the water is measured by using the point gauge, while the discharge quantity is measured by using the method of volume. Table 1 illustrates the measurements of the model which is manufactured from wood material. Models are made of 5 mm thick wood sheet beveled along all edges at 45 with sharp edges of 1 mm thick. Models are fixed to flume using supports made of plexiglass. The choice of model material and model flume was based on the laboratory facilities available. Here, the experimental works are classified into two parts. The first part includes a selection of 66 runs, randomly chosen for evaluation and the second part includes the other 20 runs for validation.
Figure 1. The definition of combined flow over weir and under gate used in the present study
Table 1. The Dimensions and details for different shapes of weir and gate
Shape 
h_{u }(cm) 
y (cm) 
d (cm) 
b (cm) 
b1 (cm) 
H (cm) 

Weir 
Gate 

Parabolic 
Parabolic 
1 to 3 
2 to 6 
2 to 4 
 
 
7 to 9 
Vnotch 
Triangle 
1 to 3 
3 to 5 
2 to 3 
2 to 2.8 
1.07 to 2.8 
8 to9 
Rectangle 
Rectangle 
1 to 3 
2 to 4 
2 to 4 
2 
2 
7 to 9 
Trapezoidal 
Trapezoidal 
1 to 3 
2 to 5 
2 to 4 
2 
2 
8 to 9 
Triangle 
Ellipse 
1 to 3 
3 to 4.5 
2.5 to 3 
2.59 
1.5 
7 to 9 
Rectangle 
Ellipse 
1 to 3 
3 to 4.5 
2.5 to 3 
2.31 
1.5 
7 to 9 
Parabolic 
Ellipse 
1 to 3 
3 to 4.5 
2.5 to 3 
3.46 
1.25 
7 to 9 
Rectangle 
Triangle 
1 to 3 
2.34 to 3.34 
3.34 
2.5 to 5 
5.00 
7 to 9 
The variables that used in Table 1 can defined as follow:
H: water depth at upstream (H= d + y + h_{u})
h_{u}: weir water head
y: vertical distance between weir and gate
d: gate opening depth
b: gate width
b1: weir width
The derivation of general equation is based on the assumption that both weir and gate have the same discharge coefficient under the free flow condition. Figure 1 shows all the details of the adopted weirgate structure in the present study.
The weirgate structure operation is affected by several variables. These variables are adopted in derivative of discharge quantity equation when the hydraulic structure is operated under the free flow condition. The discharge quantity (Q) is written as a function of the geometry variables, flow variables, and fluid variables. The geometry variables are represented by water flow that crosses the gate (A_{g}), water flow that crosses the weir (A_{w}), channel width or flume width (B), and the vertical distance between weir and gate (y), while the flow variables are represented by upstream water flow velocity (V_{u}), water depth at upstream (H) and acceleration due to gravity (g). Also, the fluid variables are represented by water density (ρ) and absolute viscosity (μ). Then, from the dimensional analysis the following function is obtained:
$Q=f({{A}_{g}},y,\,{{A}_{w}},\,H,B,\,{{V}_{u}},\,g,\,\rho ,\,\mu )$ (1)
where, f represents the unknown function and the functional relationship which describes the discharge quantity that is normalized with the distance between weir and gate (y). The flow velocity is written as:
$\frac{Q}{{{V}_{u}}\,{{y}^{2}}}=f(\frac{{{A}_{g}}}{{{y}^{2}}},\,\frac{{{A}_{w}}}{{{y}^{2}}},\,\frac{H}{y},\,\frac{B}{y},\frac{y\,g}{{{V}_{u}}^{2}},\,\frac{\mu }{\rho \,{{V}_{u}}y})$ (2)
It is observed that Eqns. (1) and (2) do not include the influence of weir and gate shape. The shape effect is added by the adoption of the hydraulic radius for weir and gate, therefore, the water flow that crosses the gate (Ag) and the water flow that crosses the weir (Aw) are replaced by Hydraulic radius of gate (R_{g}) and Hydraulic radius of weir (R_{w}), respectively. So, from the dimensional analysis the following function is obtained:
$Q=f({{R}_{g}},y,\,{{R}_{w}},\,H,B,\,{{V}_{u}},\,g,\,\rho ,\,\mu )$ (3)
The functional relationship which describes the discharge quantity is normalized with the distance between weir and gate (y). The flow velocity is written as:
$\frac{Q}{{{V}_{u}}\,{{y}^{2}}}=f(\frac{{{R}_{g}}}{y},\,\frac{{{R}_{w}}}{y},\,\frac{H}{y},\,\frac{B}{y},\frac{y\,g}{{{V}_{u}}^{2}},\,\frac{\mu }{\rho \,{{V}_{u}}y})$ (4)
The relationship between the discharge quantity that crosses the composite structure and the various variables that govern the composite structure's hydraulic behavior is not an easy task. To obtain an effective formula between the discharge quantity and the different variables that are mentioned in dimensional analysis and the curve fitting between the discharge quantity and those variables should be achieved. Linear, nonlinear, exponential and power functions can be described by referring to the relationship between these variables and the discharge quantity. Depending on the mathematical behavior that emerges from the curve that matches between the discharge quantity and any variable, it is seems possible to derive the equation family that describes the rational relationship between the discharge quantities and the dominates variables. Depending on the nature of the resulting relationships between the discharge quantity and the influencing variables individually and entering the considering data (66 sets of data) into the Curve Expert Professional program software, a group of formulas can be obtained to predict the discharge quantity, passing through the composite hydraulic structure. For the purpose of checking the efficiency of the formulas, an evaluation was made for each extracted formula, using the correlation coefficient (r), NashSutcliffe (E), mean square error (MSE), and standardization root means squire error (RSR) see [1417]. The r value is an indicator of capability of correlation between the measured and the modeled values. When the value of r gets closer to 1 or 1, the regression is now perfect.
The E value indicates how well the plot of measured versus predicted value fits the 1:1 line. As the value of E gets closer to 1 the performance will be perfect. The MSE is the average square deviation of the predicted value from the measured value. The range of MSE lies between zero (perfect fit) and infinity. (RSR) were used to measure the quality of fit employed to assess model estimation. Whenever the value of RSR getting close to zero, this indicates the efficiency of the model is perfect because it is a measure of the extent of errors in the regression results. The value of RSR is calculated from dividing the root mean square error (RMSE) by using standard deviation of the measured data.
Also, the validity of the results was checked by using (20 sets of data) and by finding the (r^{2}) and (RSR). The value of r^{2} lies in between 0 and 1. This value details shows how the value of the measured dispersion correlates to the previous prediction. If the value is zero, then that means it was not at all related, and if the value is 1, then that means it was 100% equal in dispersion. However, r^{2} is regarded an imperfect system if it is the only value taken into account, regardless of the value of the prediction, we can get r^{2} values that are close to 1 even if the predictions were far off from the measured value. Thus, predictions for the measured dispersion must be taken into account, in addition to r^{2}, gradient (a1) and intercept (a) that are used. In this system, the intercept should be close to zero and the gradient should be close to one.
For results gained from the experimental study, the 66 runs are employed to predict the family formulas which are used to estimate the discharge quantity of composite structure and 20 runs are employed to verify from the suitability of the derived formulas.
Figure 2. The relationship between the dimensionless value of discharge quantity Q/V_{u}y^{2} and the dimensionless values of state variables
5.1 Behavior of the variables with quantity of discharge
Figures 2a to 2f illustrate the relationship between the dimensionless value of quantity of discharge, represented by the ratio Q/V_{u}y^{2} and the dimensionless values of all variables, represented by the area of gate ratio A_{g}/y^{2}, area of weir ratio A_{w}/y^{2}, upstream water head ratio H/y, width of water surface at upstream region ratio B/y, distance between weir and gate ratio $y g / V_{u}^{2}$, and properties of fluid ratio μ/ρV_{u}y. It is obvious from these figures that the ratio Q/V_{u}y^{2} has positive proportional relationship with all variables except with the ratio $y g / V_{u}^{2}$. Here, y is considered constant and the upstream water flow velocity (V) has minor effect on the relationship. So, any increase in cross sectional area of flow that crosses the gate and/or weir leads to an increase in the discharge of the gate and/or weir, and this will be reflected on the discharge quantity of the composite structure. Also, any increase in upstream water head ratio H/y or width of water surface at upstream region ratio B/y leads to an increase in the quantity of discharge of composite structure. This is due to the fact that any increase in upstream water head or width of water surface at upstream region will be reflected on the cross sectional areas of flow which crosses the gate and weir. So, this will lead to an increase in the quantity of discharge of composite structure. Furthermore, it is observed that when the ratio μ/ρV_{u}y increased the ratio, Q/V_{u}y^{2 }will also be increased but with small increment, therefore, the influence of the increase can be considered less significant as compared with other ratios. On the other hand, it is obvious that when the ratio $y g / V_{u}^{2}$ increased, the ratio Q/V_{u}y^{2} will decreased. Here, the upstream water flow velocity (V_{u}) has a minor effect on this relationship and when the vertical distance between the weir and the gate (y) increases it leads to the decrease in quantity of discharge of composite structure owing to the large quantity of water that will be confined behind the distance (y). Figures 2g and 2h illustrate the relationship between the quantity of discharge ratio Q/V_{u}y^{2} and the shape of gate ratio R_{g}/y and shape of weir ratio R_{w}/y. It is obvious from figures that when the ratio R_{g}/y and ratio R_{w}/y increased the ratio Q/V_{u}y^{2} will be increased. As it has been mentioned above that y is considered constant and the upstream water flow velocity (V_{u}) has minor effect on the relationship, so, any increase in cross sectional area of flow that crosses the gate and/or weir as well as the wetted perimeter of gate and/or weir, will lead to increase in hydraulic radius of the gate and weir and this will be reflected on quantity of discharge of composite structure. The type of relationship between dimensionless values of quantity of discharge represented by the ratio Q/V_{u}y^{2} and the dimensionless values of all state variables are determined. It is observed that the variables A_{g}/y^{2}, μ/ρV_{u}y, and R_{g}/y display nonlinear function relationship as is shown in Figures 2a, 2f and 2g, while the variables A_{w}/y^{2}, H/y, B/y, $y g / V_{u}^{2}$, and R_{w}/y display power function relationship as is shown in Figures 2b, 2c, 2d, 2e and 2h. Also, the variables A_{g}/y^{2}, H/y, and B/y give very good and perfect correlation with Q/V_{u}y^{2 }respectively. The variables A_{w}/y^{2}, $y g / V_{u}^{2}$, R_{g}/y, and R_{w}/y give fair relationship with Q/V_{u}y^{2}, while the variable μ/ρV_{u}y gives very poor relationship.
5.2 Formulas based on first option
Nonlinear regression analysis of 66 experimental runs is performed by the adoption of the professional expert software to derive family of formulas. This family consists of four equations which are derived from the first option represented by Eq. (2) of dimensional analysis. The first formula represents the general equation which includes all proposed dimensionless variables of the first option.
$\begin{align} & \frac{Q}{{{V}_{u}}\,{{y}^{2}}}=5.0+0.0073{{(\frac{{{A}_{g}}}{{{y}^{2}}})}^{2}}+{{(\,\frac{{{A}_{w}}}{{{y}^{2}}})}^{0.064}}+{{(\,\frac{H}{y})}^{1.59}} \\ & +{{(\,\frac{B}{y})}^{1.62}}+{{(\frac{y\,g}{{{V}_{u}}^{2}})}^{0.18}}0.035{{(\,\frac{\mu }{\rho \,{{V}_{u}}y})}^{2}} \\ \end{align}$ (5a)
Based on the behavior of the variables with the quantity of discharge that has been discussed above, formula (5a) includes some limits which have significant and strong effect on the quantity of discharge. Also this formula includes some limits which have minor effect. So to refine this formula, the limit μ/ρV_{u}y is eliminated to obtain the following formula:
$\begin{align} & \frac{Q}{{{V}_{u}}\,{{y}^{2}}}=4.63+0.035{{(\frac{{{A}_{g}}}{{{y}^{2}}})}^{2}}+{{(\,\frac{{{A}_{w}}}{{{y}^{2}}})}^{0.08}}+{{(\,\frac{H}{y})}^{1.6}} \\ & +{{(\,\frac{B}{y})}^{1.54}}+{{(\frac{y\,g}{{{V}_{u}}^{2}})}^{0.1}} \\ \end{align}$ (5b)
Another refinement could be made by eliminating the limit A_{w}/y^{2} in addition to the elimination of the limit μ/ρV_{u}y from the formula (5a) to obtain the following formula:
$\begin{align} & ~\frac{Q}{{{V}_{u}}\,{{y}^{2}}}=3.63+0.035{{(\frac{{{A}_{g}}}{{{y}^{2}}})}^{2}}+{{(\,\frac{H}{y})}^{1.58}} \\ & +{{(\,\frac{B}{y})}^{1.56}}+{{(\frac{y\,g}{{{V}_{u}}^{2}})}^{0.12}} \\ \end{align}$ (5c)
A final modification can be obtained by eliminating the limits μ/ρV_{u}y, A_{w}/y^{2 }and yg/V^{2} from formula (5a) to obtain the formula below. The elimination of the limits depends on the effectiveness of these limits on the quantity of discharge of composite structure.
$\frac{Q}{{{V}_{u}}\,{{y}^{2}}}=1.92+0.026{{(\frac{{{A}_{g}}}{{{y}^{2}}})}^{2}}+{{(\,\frac{H}{y})}^{1.57}}+{{(\,\frac{B}{y})}^{1.54}}$ (5d)
5.3 Formulas based on second option
Nonlinear regression analysis of 66 experimental runs is performed by using professional expert software to derive family formulas. This family consists of four formulas which are derived based on second option (represented by Eq. (4)) of dimensional analysis. The first formula represents the general equation of second option which includes all proposed dimensionless variables of second option.
$\begin{align} & \frac{Q}{{{V}_{u}}\,{{y}^{2}}}=4.0+0.071{{(\frac{{{R}_{g}}}{y})}^{2}}0.25{{(\,\frac{{{R}_{w}}}{y})}^{2}}+{{(\,\frac{H}{y})}^{1.58}} \\ & +{{(\,\frac{B}{y})}^{1.65}}+{{(\frac{y\,g}{{{V}_{u}}^{2}})}^{0.2}}0.039{{(\,\frac{\mu }{\rho \,{{V}_{u}}y})}^{2}} \\ \end{align}$ (5e)
Based on the behavior of variables with quantity of discharge that was discussed above, formula (5e) includes some limits which have significant and strong effect on the quantity of discharge. Also, this formula includes some limits which have minor effect. So, to refine this formula the limit μ/ρV_{u}y is eliminated to obtain the following formula:
$\begin{align} & \frac{Q}{{{V}_{u}}\,{{y}^{2}}}=3.72+0.065{{(\frac{{{R}_{g}}}{y})}^{2}}0.2{{(\,\frac{{{R}_{w}}}{y})}^{2}}+{{(\,\frac{H}{y})}^{1.58}} \\ & +{{(\,\frac{B}{y})}^{1.59}}+{{(\frac{y\,g}{{{V}_{u}}^{2}})}^{0.14}} \\ \end{align}$ (5f)
Another refinement could be made by eliminating the limit R_{w}/y from formula (5f) to obtain the following formula:
$\begin{align} & \frac{Q}{{{V}_{u}}\,{{y}^{2}}}=3.72+0.58{{(\frac{{{R}_{g}}}{y})}^{2}}+{{(\,\frac{H}{y})}^{1.58}} \\ & +{{(\,\frac{B}{y})}^{1.59}}+{{(\frac{y\,g}{{{V}_{u}}^{2}})}^{0.14}} \\ \end{align}$ (5g)
A final modification can be obtained by eliminating the limit yg/V^{2} from formula (5g) to obtain the following formula:
$\frac{Q}{{{V}_{u}}\,{{y}^{2}}}=1.95+0.116{{(\frac{{{R}_{g}}}{y})}^{2}}+{{(\,\frac{H}{y})}^{1.57}}\,+{{(\,\frac{B}{y})}^{1.57}}$ (5h)
The elimination of limits depends on the effectiveness of these limits on the quantity of discharge of composite structure.
The output details of statistical analysis for the suggested equations are illustrated in Table 2. These details were included in the following; correlation coefficient, NashSutcliffe, mean square error, root mean square error and standardization root means squire error. It is clear from Table 2 that the values of the correlation coefficient and NashSutcliffe are nearest to the unity. Also, there is no limited value for MSE or RMSE. The lower the value the better and 0 means the model is perfect. Since there is no limited value, the MSE and RMSE basic value are in selecting one prediction model over another. It is observed from Table 2 that the value of MSE, RMSE as well as RSR are closer to zero. This implies a reasonable result for all the derived equations.
Table 2. Statistical characteristics and comparison between the predicted equations
Equation No. 
r 
E 
MSE 
RMSE 
RSR 
5a 
0.9993 
0.9936 
0.0326 
0.1805 
0.0424 
5b 
0.9988 
0.9968 
0.0561 
0.2369 
0.0557 
5c 
0.9989 
0.9969 
0.0548 
0.2340 
0.0550 
5d 
0.9984 
0.9969 
0.0547 
0.2339 
0.0549 
5e 
0.9983 
0.9906 
0.1660 
0.4075 
0.0958 
5f 
0.9988 
0.9975 
0.0435 
0.2087 
0.0490 
5g 
0.9988 
0.9978 
0.0377 
0.01942 
0.0456 
5h 
0.9988 
0.9966 
0.0593 
0.2435 
0.0572 
Figure 3. The coefficient of determination and the values of (a) and (a1) coefficients for the a: derived formula (a) and e: derived formula (e)
Table 3. Data of validation (g=981cm/sec^{2}, B=7.5 cm, μ/ ρ =10^{2} cm^{2}/sec)
Q_{act }cm^{3}/sec 
V_{u }cm/sec 
y cm 
A_{g }cm^{2} 
A_{w }cm^{2} 
H cm 
R_{w }cm 
R_{g }cm 
728.16 
10.79 
4.0 
6.93 
0.54 
9.00 
0.183 
0.655 
750.00 
11.11 
4.0 
4.20 
2.00 
9.00 
0.354 
0.512 
779.22 
11.54 
3.0 
4.20 
4.20 
9.00 
0.512 
0.512 
410.02 
7.81 
4.0 
4.00 
2.00 
7.00 
0.500 
0.667 
801.42 
13.36 
4.0 
6.00 
6.00 
8.00 
0.857 
0.857 
738.92 
12.32 
3.0 
6.00 
4.00 
8.00 
0.667 
0.750 
774.53 
12.91 
3.0 
9.00 
6.00 
8.00 
0.857 
1.000 
952.38 
18.14 
2.0 
12.0 
3.00 
7.00 
0.600 
1.091 
762.71 
12.71 
5.0 
6.07 
2.59 
8.00 
0.599 
0.933 
768.57 
11.39 
4.0 
6.07 
10.47 
9.00 
1.203 
0.933 
802.14 
15.28 
3.0 
10.47 
2.59 
7.00 
0.599 
1.203 
1045.30 
15.49 
2.0 
15.60 
10.47 
9.00 
1.203 
1.437 
786.71 
13.11 
2.0 
9.76 
3.63 
8.00 
0.050 
0.208 
591.72 
8.77 
5.0 
6.44 
1.15 
9.00 
0.114 
0.714 
620.26 
10.34 
3.0 
6.44 
3.63 
8.00 
0.050 
0.429 
728.74 
11.19 
3.3 
8.33 
5.00 
8.68 
0.769 
0.998 
534.12 
7.913 
3.5 
4.909 
3.897 
9.00 
0.596 
0.791 
801.42 
11.873 
4.0 
7.069 
4.620 
9.00 
0.732 
0.973 
790.17 
10.536 
4.0 
7.069 
6.928 
10.0 
0.989 
0.973 
799.29 
10.657 
3.0 
7.069 
6.928 
10.0 
0.794 
0.973 
Table 4. Validation of formula (5e)
R_{g}/y^{2 }(1) 
R_{w}/y^{2 }(2) 
H/y (3) 
B/y (4) 
y.g/V_{u}^{2 }(5) 
$\frac{\mu }{\rho \,Vy}$(6) 
Q/V_{u}. Y^{2 }(7) 
Q/V_{u}. Y^{2 }(8) 
0.163 
0.045 
2.2 
1.87 
33.68 
2.31 
4.218 
4.235 
0.127 
0.088 
2.2 
1.87 
31.75 
2.25 
4.21 
4.22 
0.170 
0.170 
3 
2.5 
22.06 
2.88 
7.50 
7.73 
0.166 
0.12 
1.75 
1.87 
64.26 
3.20 
3.28 
3.14 
0.214 
0.214 
2 
1.87 
21.97 
1.71 
3.75 
3.52 
0.250 
0.222 
2.67 
2.50 
19.38 
2.70 
6.66 
6.76 
0.333 
0.285 
2.67 
2.50 
17.64 
2.58 
6.66 
6.74 
0.545 
0.300 
3.5 
3.75 
5.95 
2.75 
13.12 
13.22 
0.186 
0.119 
1.6 
1.50 
30.33 
1.57 
2.40 
1.93 
0.233 
0.300 
2.25 
1.87 
30.23 
2.19 
4.21 
4.19 
0.401 
0.199 
2.34 
2.50 
12.59 
2.18 
5.83 
5.82 
0.718 
0.617 
4.50 
3.75 
8.17 
3.22 
16.87 
16.68 
0.104 
0.025 
4.00 
3.75 
11.40 
3.81 
15.00 
14.85 
0.142 
0.022 
1.80 
1.50 
63.76 
2.28 
2.70 
2.57 
0.142 
0.016 
2.67 
2.50 
27.51 
3.22 
6.66 
6.78 
0.298 
0.230 
2.59 
2.24 
26.12 
2.67 
5.83 
5.95 
0.225 
0.170 
2.57 
2.14 
54.77 
3.61 
5.51 
5.67 
0.243 
0.183 
2.25 
1.87 
27.80 
2.10 
4.21 
4.19 
0.243 
0.247 
2.50 
1.87 
35.31 
2.37 
4.68 
4.88 
0.324 
0.264 
3.34 
2.50 
25.88 
3.12 
8.33 
8.76 
Table 5. Validation results of all formulas
Equation No. 
r^{2} 
a 
a1 
RMSE 
RSR 
9a 
0.997 
0.0388 
0.998 
0.2009 
0.0503 
9b 
0.997 
0.0006 
0.998 
0.1955 
0.0489 
9c 
0.997 
0.0208 
1.004 
0.2050 
0.0513 
9d 
0.997 
0.08 
0.996 
0.2184 
0.0546 
9e 
0.998 
0.020 
1.004 
0.1907 
0.0476 
9f 
0.996 
0.066 
0.985 
0.2448 
0.0612 
9g 
0.996 
0.035 
0.999 
0.2302 
0.0576 
9h 
0.996 
0.125 
0.993 
0.2475 
0.0619 
5.4 Validation
Table 3 contains the data of 20 experimental works which is used to feed the formula 5a to 5h in order to estimate the quantity of discharge of composite structure. While Table 4 contain the information of formula 5e in addition to the quantity of discharge which is calculated from formula 5e, quantity of discharge which is calculated from experimental work. For Table 4, column (7) represents the result of experimental work while column (8) represents the result which is obtained from formula 5e. So the formula can be used without any confusion or conflict. Table 5 show the values of the coefficient of determination and the values of (a), (a1), RMSE, RSR parameters for the derived equations. Also, Figures 3a and 3e show the values of the coefficient of determination, value of (a), and (a1) for arbitrary case. It is clear from Table 5 that the values of (a) are nearest to zero and the values of (a1) are nearest to the unity. The values of (a) and (a1) imply that the derived equation can give a suitable prediction for estimating the quantity of discharge. It is observed that the value of coefficient (a) of formula (9e) is closer to zero and the value of coefficient (a1) of the same formula is almost equal to one. On the other hand, it is observed that the minimum values of RMSE and RSR are achieved in formula (9e), therefore, it is expected that the formula (9e) will give the best prediction to the quantity of discharge.
The flow below gate and over weir in composite structure was investigated experimentally in flume or open channel as a measurement device. So, the main points to conclude from this work are:
Reasonable formulas are derived and based on the mathematical trends among the quantity of discharge and the variables which control the hydraulic behavior of composite structure.
The hydraulic radius can be used to express the influence of weir and gate shape.
The water’s physical properties have minor effect on the quantity of discharge of composite structure.
The statistical analysis can be considered as an important subject in the description of the suitability of family formulas.
The upstream velocity has minor effects on the quantity of discharge of composite structure.
The vertical distance between the weir and the gate has a significant role in estimating the quantity of discharge.
The fluid variables have less effect on estimating the discharge quantity and the best formula is the one that considered the shape geometry, flow and fluid variables.
B 
Flum Width, L 
H 
Water depth at upstream, L 
g 
Gravitational acceleration, L.T^{2} 
y 
Vertical distance between weir and gate, L 
d 
Gate opening depth, L 
b 
Gate width, L 
b1 
Weir width, L 
hu 
Weir water head, L 
A 
Area, L^{2} 
V 
Velocity, L/T 
R 
Hydraulic radius, L 
Q 
Discharge, L^{3}/T 
r 
Correlation coefficient 
r^{2} 
Coefficient of determination 
MSE 
Mean square error 
RMSE 
Root mean square error 
RSR 
Standardization root means squire error 
E 
Nash Sutcliffe coefficient 
Greek symbols 

$\rho$ 
Fluid density, M/L^{3} 
$\mu$ 
dynamic viscosity, M. L^{1}.T^{1} 
Subscripts 

g 
gate 
w 
weir 
u 
upstream 
d 
downstream 
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