Discharge coefficients for ogee spillways

ρ

Water density (kg/m³);

μ

Dynamic viscosity of water (N.s/m²);

g

Gravity acceleration (m/s²);

σ

Water surface tension (N/m);

H_e

Water head above the weir crest (m);

V

Water flow velocity (m/s);

h_d

Difference between the downstream head and the ogee spillway crest (m);

h

Water head over weir crest other than the design head (m);

H_des

Difference between spillway upstream invert up to energy line in upstream of spillway (m);

P

Spillway height (m);

W

Spillway width (m);

L_e

Effective length of ogee spillway and normal to flow direction (m);

α

Inclination of ogee upstream face (dimensionless);

d

Difference between the downstream water level and the downstream apron elevation (m);

x, y

Horizontal and vertical co-ordinates (m);

K

Constant coefficient in ogee profile equation (dimensionless);

n

Constant coefficient in ogee profile equation (dimensionless);

R_e

Reynolds number (dimensionless);

W_b

Weber number (dimensionless);

F_r

Froude number (dimensionless).

Generally, storage dams are built for irrigation of agricultural lands and urban and industrial demand purposes. In addition, storage dams have a capability of flood control due to their huge storage potential. In Iran, the total of the water capacity behind storage dams reaches 50.5 × 10⁹ m³ for water supply requirements (Akbari et al. 2019). Overflow spillways are hydraulic structures in dams that are built to pass floods that exceed the dam capacity. Overflow spillways must be resilient, reliable, and effective structures to prevent damage to the dam. Therefore, the design and construction of dam spillways are of great importance.

If the spillway of a dam is properly designed and constructed, river flooding can be prevented with lower risk to the dam and the surrounding region (Boiten 2002). Overflow spillways are made of different types depending on the geological and hydraulic characteristics of the situation. The construction site and most importantly the economic costs affect the choice of spillway. The most common and cheapest spillway that has the ability to pass a large volume of river flow is the ogee spillway. These spillways are often used in diversion dams due to their high hydraulic efficiency (Salmasi & Abraham 2020). In these spillways, an ogee-type profile is such that the weir invert (usually made of concrete) corresponds to the bottom nappe profile of the free water jet from a sharp rectangular-crested weir. The lower end of the spillway can be designed in the form of an inverted curve to more slowly direct the flow to the beginning of the stilling basin invert. This spillway is suitable for wide valleys.

Tullis & Neilson (2008) examined the performance of submerged ogee spillways and their discharge–head relationship. In that study, four relationships for submerged and free-flow conditions were evaluated using laboratory data. Data was used to predict the upstream water level in submergence conditions. Tullis & Neilson (2008) stated that for submergence less than 0.8, the discharge–head relationship can be used from free conditions. The degree of submergence is defined as S = h_d/H_e where h_d is the vertical distance between the downstream water level and the spillway crest and H_e is the water head upstream of the spillway crest. Tullis (2011) evaluated changes in ogee discharge coefficients. The results showed that the weak correlation between the values of the discharge coefficients predicted by the US Bureau of Reclamation (USBR 1987) could be due to the difference in the location of the downstream water-depth measurement. It can be noted that the graphs provided by USBR (1987) are related to the experiments from Bradley (1945).

Shamsi et al. (2019) investigated the discharge coefficients (C_d) of cylindrical weirs and their ability to dissipate energy. The results showed that when H/D ranges between 0.15 and 2, C_d varies between 1 and 1.4. In the same H/D range, other structures dissipate between 150% and 80% of the flow energy. The optimal value of C_d is approximately 1.3, which was achieved in the H/D range between 0.5 and 0.7.

Sheikh Kazemi & Saneie (2014) studied the effect of the approaching channel on the ogee spillway discharge coefficients with axial arches and converging walls. The experiments were carried out on a physical model of a Germy-Chay Dam spillway, with a 1:75 scale (model:prototype). The Germy-Chay Dam is located near Mianeh City, Iran. The results showed that by increasing the width of the approaching channel by 52%, the flow depth on the spillway crest decreases by 11% and the discharge coefficient will increase by 22%. Eshrati et al. (2015) constructed a physical model of the ogee spillway with an arc and an ogee spillway oriented perpendicular to the flow, with similar geometric and hydraulic conditions. The results of the first model showed that by increasing the ratio of the water head to the design head, C_d increased to 1.72. Then, spillway submergence occurred and C_d decreased to 1.23. In addition, for the spillway physical model with normal shape, submergence happens faster and C_d will be less for the same discharge values.

Kanyabujinja (2015), studied the hydraulic characteristics of ogee spillways using computational fluid dynamics (CFD). That study used ANSYS FLUENT software and the k-ε realizable model for turbulence modeling. Comparisons between laboratory and numerical models showed that despite significant improvements in numerical models, laboratory models are still superior to numerical models. Salmasi (2018) investigated the effect of invert height and downstream submergence on ogee spillways using laboratory data. In that study, multiple regression and gene expression programming (GEP) were used. The results showed that the GEP method was more accurate than the regression method for predicting the discharge coefficients.

Figure 1 shows a photograph of the ogee spillway of Talegan Dam, near Karaj City, Iran. Talegan Dam is an earth dam with a clay core and its total storage volume is 420 million m³. The useful storage is 329 million m³ and it contains a hydropower generation station with 18 megawatt capacity. The dam height is 109 m from its foundation and the dam crest length is 1,111 m. The water that feeds the lake originates from Shahrud River. This river is as long as 175 km and flows from the heights of the mountains and after joining Alamut River, it eventually flows into the Caspian Sea in the northern region of Iran. The objectives of the Talegan Dam construction are: (i) agricultural water supply of Qazvin plain, (ii) drinking water supply in Tehran and Karaj cities, (iii) groundwater artificial recharge of Qazvin plain and (iv) controlling seasonal floods from the Taleqan River.

A review of the previous studies shows that the effects of ogee spillways with an upstream slope, spillways with an invert elevation downstream, and ogee spillway submergence on the discharge coefficients have not been fully characterized. Therefore, in the present study, by fabricating physical models of ogee spillways at the hydraulic laboratory of the University of Tabriz, and by establishing different discharges over the spillways, the effects of these three factors are investigated. Dimensionless parameters affecting the discharge coefficient are identified. Relations for estimating ogee spillway discharge coefficients and diagrams related to dimensionless parameters in free and submerged conditions are presented and a comparison of the results of experiments with the other studies is provided.

In Equation (3), R_e is the Reynolds number, W_b is the Weber number and F_r is the Froude number. In this study, the depth of water measured on the spillway crest is at least 3 cm, so the effect of surface tension on the spillway is negligible and can be ignored (for example, see Bos 1989). By ignoring the surface tension, the Weber number can also be neglected. Due to the fact that the flow in the channel is generally turbulent and also the thickness of the boundary layer can be neglected, the effect of dynamic viscosity on the hydraulic behavior of the flow can be ignored. Therefore, the Reynolds number can be removed. In addition, the width and the effective length of the spillway are constant and therefore the parameters of W/H_e and L_e/H_e are neglected.

The Froude number (F_r) is defined as and application of it in the general functional relationship leads to an implicit equation in the head–discharge relationship. Consequently, F_r is neglected in dimensional analyses (USBR 1987).

The experiments were carried out in the hydraulic laboratory of the Department of Water Engineering, University of Tabriz, Iran. The water supply system in the laboratory includes an underground tank that is equipped with a submerged vertical 100 hp electric pump. This pump conveys water to an elevated tank with 4.5 m elevation. The water tank provides a constant head of water. This spillway returns the excess water from the elevated tank to the underground tank, thus keeping the height and consequently the discharge constant.

Water passes through a rectangular flume and then the flow is directed to a calibrated triangular spillway for measurement. Finally, the water is redirected to the main underground reservoir.

The laboratory flume is rectangular in cross-section with a length of 10 m, a width of 0.25 m and a height of 0.50 m. It is made of metal with glass walls. This flume is placed on a chassis 1.10 m above the ground. The slope of the flume is constant and equal to 0.0022. The flume is equipped with an upstream lateral spillway with lattice plates at the bottom. In this way, the turbulence of the inlet flow to the flume is reduced and the accuracy of measuring the water height is increased. Excess water overflows when necessary and is returned to the underground tank through pipes. The maximum discharge in the flume is 0.05 m³/s and the discharge can be adjusted by a valve upstream of the flume. Upstream and downstream water depths are measured by a level-gauge mounted on the flume walls. The accuracy of measuring water depth by this gauge is 0.1 mm.

The location for measuring the total head of the water upstream of the ogee spillway is a horizontal distance of three to four times the maximum water head on the crest of the spillway. This is to prevent curvature of the water flow near the spillway. The flow, after passing the flume, is directed into the stilling pool and then to the discharge-measuring pool. In the discharge-measuring pool, there is a calibrated triangular weir with a vertex angle of 53 degrees. Flow measurements are obtained by reading the water level inside the well by a level-gauge located next to the flow-measurement pool with an accuracy of 0.1 mm.

In the physical models of the ogee spillways, the ogee profile was designed and fabricated using the USBR (1987) guidelines with a discharge of 0.011 m³/s. The body of the spillway was made from wood and was painted (see Figure 3). The lower invert of the spillway was made of polyethylene and three pieces of medium-density fiberboard (MDF) of thickness 8 mm were used to incline the upstream side of the spillways. The model was placed inside the flume and then sealed with adhesive. In each experiment, after stable conditions were obtained, the upstream water level, downstream water depth and the discharge were determined. Then, using the upstream water head and the discharge, the C_d was determined using Equation (1).

The specifications of the physical models are provided in Table 1 and schematic views of some of the experimental models are given in Figure 4. In Table 1, the notation for the upstream slope refers to the ratio of horizontal distance to vertical distance, for example an upstream slope 2:3 refers to H:V or an angle of 33° with the vertical direction.

To investigate the head–discharge relationship for the ogee spillways under free-flow conditions, Figures 5 and 6 are provided. Figure 5 reflects different upstream slopes and Figure 6 refers to different downstream apron thicknesses. Results show that with increasing discharge, the total upstream water head (H_e) increases nonlinearly. This trend can be justified by the ogee spillway discharge formula (Equation (1)). Based on Figure 5, the effect of upstream slope on the discharge–head relation is low. But for a certain H_e value, the spillway with vertical upstream face has the maximum discharge and the spillway with upstream slope of 45 degrees (slope 3:3) has the minimum discharge. For a given water head over the spillway crest, an upstream slope of 45 degrees results in about 0.003 m³/s less discharge than the spillway with a vertical slope. Probably the convergence of the water nappe passing over the vertical-faced spillway is the cause of this behavior.

Figure 6 shows that for a constant H_e, the discharge (Q) for the ogee spillway with higher downstream apron thickness is lower. In other words, with higher downstream apron thickness, the difference in the hydraulic gradient between the upstream and downstream portions of the spillway decrease. By doubling the height of the apron thickness downstream of the ogee spillway, the discharge decreases by 13.04%.

Figure 7 shows the variation of the discharge coefficient (C_i) against the dimensionless parameter P/H_e for the ogee spillway with a vertical slope, and for 33, 18, and 45 degree upstream faces. The symbol C₀ is used to represent the discharge coefficient for the spillway with a vertical upstream face and the symbol C_i is for spillways with sloping upstream face.

As can be seen in Figure 7, for an ogee spillway with a vertical face, by increasing the ratio of P/H_e the discharge coefficient initially has an upward trend and then reaches a constant value of 2.25. Thereafter, the discharge coefficient becomes independent of P/H_e. This threshold occurs for P/H_e = 1.6. In general, the discharge coefficient for an ogee spillway with a vertical-faced slope (C₀) was higher than the similar condition for spillways with inclined upstream slopes (C_i).

Figure 7 shows that the presence of an upstream slope reduces the discharge coefficient. When the ratio of P/H_e reaches ∼1.2, the discharge coefficient of all three graphs for sloping upstream-faced spillways reaches 1.2–1.9 and becomes independent of P/H_e. Inclination of the upstream face of the ogee spillway reduces the maximum discharge coefficient from 2.25 to about 1.9. In other words, sloping the spillway upstream face reduces its discharge coefficient by an average of 12%. The decrease in discharge coefficient may be due to the increase in the upstream water level which increases the energy loss. In addition, for a vertical-faced spillway, the flow convergence is higher than for the inclined-faced spillways.

Figure 8 provides the variation of the discharge coefficient (C_a) against the dimensionless parameter P/H_e for the ogee spillways with downstream aprons. Figure 8 shows that with increasing P/H_e, the discharge coefficient has an upward trend and then reaches a constant value. In other words, increasing the height of the downstream apron reduces the discharge. Increasing the downstream thickness of the apron results in a decrease in the difference of head between the upstream and downstream regions, and the discharge over the spillway decreases. For practical design, the downstream apron elevation is a function of the tail water depth. Downstream apron elevation is fixed by matching the tail water elevation with water elevation after the hydraulic jump. This provision is needed in order to solve the energy and momentum equations simultaneously. Thus although the downstream apron elevation is calculated by solving the above equations, Figure 8 can be used for a discharge–head relationship (Q–H) for ogee weirs. The Q–H relationship is helpful for operating and maintaining spillways especially in flood conditions and for recording peak discharge.

Figure 9 provides the effect of downstream apron on discharge coefficients. The variation of C_a/C₀ versus (h_d + d)/H_e is presented. As shown in the figure, for an ogee spillway with a smaller apron thickness, by increasing the ratio of (h_d + d)/H_e, the value of C_a/C₀ increases and reaches a constant value of unity. In this case, the relative value of C_a/C₀ becomes independent of (h_d + d)/H_e. Physically, with increasing height of the downstream invert, the difference of water head between the upstream and downstream decreases and the flow capacity or the discharge coefficient is reduced.

In Equation (7), C_a/C₀ is the relative discharge coefficient (dimensionless), H_e (m) is the water head above the weir crest, h_d (m) is the difference between the downstream head and the spillway crest and d (m) is the difference between the downstream water level and the downstream apron elevation (water depth after the hydraulic jump).

Figure 10 shows a scatter plot of the observed data compared with the calculations of Equation (7). All data points are near the bisection line y = x and thus demonstrate good agreement between the observed data and the fitted regression equation.

Figure 11 shows the effect of a spillway upstream face inclination on discharge coefficients. For simplicity, the variation in dimensionless variable C_i/C₀ with the dimensionless variable P/H_e as the x axis is provided for three inclination angles of 18, 33 and 45 degrees. An ogee spillway with upstream inclination of 45° results in the lowest discharge coefficients. Meanwhile for α = 18°, high discharge coefficients are observed.

In Figure 11, error bars (standard deviation error) are provided for distinguishing uncertainty. The standard deviation ranges reflect the degree of uncertainty in measurements of the independent variables.

In Table 2, regression equations show the correlation between P/H_e and C_i/C₀ along with the determination coefficient (R²) and relative error (RE). Based on Table 2, the RE values for individual regression equations are less than 1.4%.

Figure 12 shows the head–discharge relation for both submerged and free-flow conditions. The rate of decrease in discharge for a constant water head is evident for submerged-flow conditions. Due to the reduction of discharge through the spillway for submerged-flow conditions, this type of flow is not suitable for spillway performance. In addition, submerged-flow conditions exert hydrodynamic forces on the spillway downstream face and this not conducive for spillway stability. In Figure 12, the spread of the data in the submerged state is due to the high sensitivity of the submerged flow to the downstream water depth. With a slight change in the downstream depth, a large change in the upstream depth of water occurs. Therefore, errors measuring the downstream depth will have a large impact on the results. At the same time, the downstream depth experiences oscillating conditions.

Figure 13 shows the changes in discharge coefficients (C_s and C₀) versus P/H_e for ogee spillways with upstream vertical faces for both submerged and free-flow conditions. As the vertical axis, C_s refers to discharge coefficients in submerged-flow conditions and C₀ refers to discharge coefficients in free-flow conditions. It can be seen that the discharge coefficients decrease in submerged conditions in comparison with free-flow conditions. This decrease is large for low ratios of P/H_e but gradually decreases with increasing P/H_e so that the maximum discharge coefficient of 2.25 in free-flow conditions declines to 2.15 in submerged-flow conditions. In other words, with submerged-flow conditions, higher upstream head (H_e) leads to lower discharge coefficients (C_s).

In Equation (8), C_s/C₀ is the relative discharge coefficient (dimensionless), H_e (m) is the water head above the weir crest and h_d (m) is the difference between the downstream head and the spillway crest. As discussed earlier, uncertainty ranges (standard deviation) are provided for measurements of the independent variables. The uncertainty is indicated by the uncertainty bars in the image.

Figure 15 shows a comparison between C_s/C₀ in the present study with that in the Tullis (2011) study. For spillways with submerged conditions with/without downstream aprons, in the range of h_d/H_e = 0.3–0.75, C_s/C₀ has a constant value. For values of h_d/H_e in the range 0.75–1, the ratio of C_s/C₀ shows a downward trend. A good correlation between the present study and the Tullis (2011) study is observed.

Due to the reduction of discharge through the spillway for submerged-flow conditions, this type of flow is not suitable. In addition, submerged-flow conditions exert hydrodynamic forces on the spillway downstream face and this is not conducive for spillway stability. The submergence ratio is highly sensitive to the downstream water depth. At the same time, the downstream depth experiences oscillating conditions. With increasing submergence ratio (h_d/H_e), from 0.75 to 1, the relative discharge coefficient (C_s/C₀) decreases from 0.88 to 0.24.

Thus, it is suggested that more experiments for determining the threshold of submergence conditions be performed. This threshold refers to a submergence ratio value such that if the ratio is increased, significant changes in the decreasing of the discharge coefficient happen.

The presence of a downstream apron reduces the discharge coefficient (C_d). For a constant P/H_e, C_d decreases by about 18% when the apron thickness increases from 3 to 10 cm. For practical design, the downstream apron elevation is a function of the tail water depth. Downstream apron elevation is fixed by matching tail water elevation with water elevation after the hydraulic jump. This facility is needed in order to solve the energy and momentum equations simultaneously.

In general, the discharge coefficient for an ogee spillway with a vertical-faced slope (C₀) was higher than the similar condition for spillways with inclined upstream slopes (C_i). In other words, sloping the spillway upstream face reduces its discharge coefficient by an average of 12%. The decrease in discharge coefficient may be due to the increase in the upstream water level, which increases the energy loss. In addition, for a vertical-faced spillway. the flow convergence is higher than for the inclined-face spillways.

Today, application of computational fluid dynamics (CFD) in flow around hydraulic structures has increased in research studies (Abraham et al. 2021; Gorman et al. 2021). Implementation of CFD simulation provides more spatial details of the flow than is available from experiments. The CFD studies of ogee weirs upstream of inclinations on C_d is highly recommended. This helps for understanding of flow convergence upstream of ogee weirs. The fabricated physical models have had smaller dimensions than the physical models of Tullis (2011). Although Figure 15 shows a good correlation between the present study and the Tullis (2011) study, an expanded investigation using CFD is suggested for future efforts.

In future studies it is recommended to test ogee curves other than that of the USBR (1987) equation.

When the ogee spillway profile is constructed according to the design discharge or design head (H_e), the amount of water pressure on the surface of the ogee profile will be zero. If the head increases to exceed the design head, negative pressures are created on the ogee profile. This is because at high water head over the spillway crest, the water jet tends to separate from the surface of the ogee curve. Figure 17 shows the average negative pressure on the ogee weir based on the USBR (1987) report for H_e/h = 0.75. Here H_e denotes the design head (m) and h is the water head over the weir crest other than the design head (m). This negative pressure diagram has been obtained by ignoring the approach velocity in the upstream of the spillway. Investigation of negative pressures on ogee spillway profile would be another aspect for future studies. Negative pressure increases the discharge coefficient (positive effect), but may result in cavitation damage at the ogee weir surface (negative effect).

In the present study, the effect of the three factors of spillway upstream face slope, downstream thickness of the apron, and downstream submergence on discharge coefficients was investigated. Physical models of ogee spillways were fabricated. Some important results are summarized here.

The discharge coefficient changes with P/H_e in all spillways; first as an increasing trend and then reaching an almost constant value. However, this increase was less for spillways with a downstream apron or spillways with an upstream slope face. Results indicate that in the presence of a downstream apron, upstream slope and downstream submergence, the discharge coefficients (C_d) are reduced. Spillways including an apron with a height of 3 cm have the smallest decrease in C_d whereas spillways with an apron height of 10 cm have the highest decrease of C_d. In other words, for a constant P/H_e, C_d decreases by about 18% when the apron thickness increases from 3 to 10 cm. The discharge coefficients decrease as the ogee spillway reaches submergence conditions downstream. This decrease is large for low ratios of P/H_e, but decreases gradually with increasing P/H_e. The maximum value of C_d decreases from 2.25 in free-flow conditions to 2.10 in submerged-flow conditions. The variation of relative discharge coefficient (C_s/C₀) with h_d/H_e in an ogee spillway with a vertical upstream face and with submerged conditions shows a constant value at first and then a downward trend. The submergence threshold for these ogee spillways is 0.75. The presence of a downstream apron reduces the discharge coefficient (C_d). The reason for their use in ogee spillways is due to the stabilization of the hydraulic jump in the stilling basins. The relationships obtained from the experiment observations had a coefficient of determination (R²) higher than 0.7, which indicates the high accuracy of the relationships and data obtained.

This paper is the outcome of a research project supported by the University of Tabriz research affairs office.

The authors declare that they have no conflict of interest.

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.

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