Discharge coefficients for ogee weirs including the effects of a sloping upstream face

The following symbols are used in this paper:

• C₀= discharge coefficient for free (uncontrolled) flow condition with vertical upstream face (m^0.5/s);

• C_i = discharge coefficient for free flow condition, with sloping face in upstream (m^0.5/s);

• g = acceleration due to gravity;

• H = reprehensive horizontal distance in upstream weir slope;

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

• H_e = 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

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 (Q_design).

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 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 (R_e) 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

The approach velocity (V_a) 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 V_a 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).

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 H_e/H₀. The symbol H_e is the actual head.

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 (C_i) to that for a crest with a non-inclined face (C_v) as related to values of P/H₀.

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

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 C_s for multiple ogee crest weirs. Tullis (2011) reported that the weak relationship between observed and predicted C_s values might be caused by differences between h_d measurement positions.

Bradley (1945) measured h_d 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 1–3 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 (H_e/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.

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, h_a = V_a²/2 g is approach velocity head, H_e1 and H_e2 are actual heads being considered on the crest, the approach velocity V_a is equal to V_a = 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.

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 <0.05H_e/H₀ < 1.6. Thus, different geometries include different values for P and I parameters and different hydraulic conditions relate to H₀ and H_e.

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.

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.

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)

  

  (3)
• II.
  Determination coefficient (R²)

  

  (4)
• III.
  Mean absolute error (MAE)

  

  (5)

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 (C_i/C_v).

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.

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.

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 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/H_e and H_e/H₀) against dependent variable C_d. These contours represent values for C_d estimation based on the 202 data points.

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

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

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 H_e/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 (H_e/H₀).

There is no funding for this article.

The authors declare that they have no conflict of interest.

None.

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

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

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