## Abstract

Discharge over an ogee weir is related to the length of the weir, upstream total head above the weir crest and the discharge coefficient. The discharge coefficient is influenced by several factors; in this study, some parameters that influence discharge coefficients of ogee weirs are investigated. These factors include: the weir upstream slope, apron elevation, and downstream submergence. In this regard, ogee weir physical models were fabricated. These models comprise: an ogee weir with a vertical upstream face, ogee weirs with inclined upstream faces (18, 33, and 45 degrees), ogee weirs with downstream apron thicknesses of 3, 5, 7, and 10 cm in free flow and an ogee weir with a vertical upstream slope under submerged-flow conditions. Results show that for all ogee weirs, the discharge coefficient (*C*) increases with increasing *P*/*H*_{e}, and then remains constant (*P* is the spillway height and *H*_{e} is the head above the weir crest). The value of the discharge coefficient decreases from 2.25 in free-flow conditions to 2.15 with submerged-flow conditions. For a constant head over the ogee weirs (*H*_{e}), the discharge coefficient decreases with increasing downstream apron elevation and submergence. The relative discharge coefficient shows a constant trend initially, with increasing *h*_{d}/*H*_{e}, then later shows a decreasing tendency (*h*_{d} is the difference between the downstream head and the ogee spillway crest). The threshold value for submergence ratio (*h*_{d}/*H*_{e}) is 0.75 in the ogee weir in this study. With increasing submergence ratio from 0.75 to 1, the relative discharge coefficient (*C*_{s}/*C*_{0}) decreases from 0.88 to 0.24.

## HIGHLIGHTS

Some ogee weir physical models were fabricated.

Results show that the discharge coefficient increases with increasing

*P*/*H*_{e}, and then remains constant.The value of the discharge coefficient decreases from 2.25 in free-flow conditions to 2.15 with submerged-flow conditions.

The threshold value for submergence ratio is 0.75 in the ogee weir in this study.

## LIST OF MATHEMATICAL SYMBOLS

*ρ*Water density (kg/m

^{3});*μ*Dynamic viscosity of water (N.s/m

^{2});*g*Gravity acceleration (m/s

^{2});*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).

## INTRODUCTION

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 × 10^{9} m^{3} 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 m^{3}. The useful storage is 329 million m^{3} 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.

## MATERIALS AND METHODS

*Q*(m

^{3}/s) is the discharge through the spillway,

*C*(m

^{0.5}s

^{−1}) is the discharge coefficient,

*L*

_{e}(m) is the effective crest length, and

*H*

_{e}(m) is the total water head over the spillway crest (water depth plus velocity head):

*ρ*is the water density,

*μ*is the dynamic viscosity of water,

*g*is the gravity acceleration,

*σ*is the surface tension,

*V*is the flow velocity,

*h*

_{d}is the difference between the downstream head and the spillway crest,

*P*is the spillway height,

*α*is the upstream slope of the spillway and

*d*is the difference between the downstream water level and the downstream apron elevation (water depth after the hydraulic jump). The other variables were defined previously.

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 m^{3}/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 m^{3}/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.

Physical model . | P (cm)
. | P_{d} (cm)
. | H_{e} (cm)
. | L (cm)
. | α (degrees) . | Length, width and thickness of downstream apron (cm) . | Length and width of the timber used for upstream slope (cm) . |
---|---|---|---|---|---|---|---|

Ogee weir with vertical upstream face | 22 | 0 | 11 | 26.45 | 0 | – | 23.2, 24 |

Ogee weir with 1:3 upstream inclination | 22 | 0 | 11 | 49.65 | 18 | – | 26.4, 24 |

Ogee weir with 2:3 upstream inclination | 22 | 0 | 11 | 52.85 | 33 | – | 29.5, 24 |

Ogee weir with 3:3 upstream inclination | 22 | 0 | 11 | 57.65 | 45 | – | – |

Ogee weir with apron thickness of 3 cm | 22 | 19 | 11 | 86.45 | 0 | 60, 24 and 3 | – |

Ogee weir with apron thickness of 5 cm | 22 | 17 | 11 | 86.45 | 0 | 60, 24 and 5 | – |

Ogee weir with apron thickness of 7 cm | 22 | 15 | 11 | 86.45 | 0 | 60, 24 and 7 | – |

Ogee weir with apron thickness of 10 cm | 22 | 12 | 11 | 86.45 | 0 | 60, 24 and 10 | – |

Physical model . | P (cm)
. | P_{d} (cm)
. | H_{e} (cm)
. | L (cm)
. | α (degrees) . | Length, width and thickness of downstream apron (cm) . | Length and width of the timber used for upstream slope (cm) . |
---|---|---|---|---|---|---|---|

Ogee weir with vertical upstream face | 22 | 0 | 11 | 26.45 | 0 | – | 23.2, 24 |

Ogee weir with 1:3 upstream inclination | 22 | 0 | 11 | 49.65 | 18 | – | 26.4, 24 |

Ogee weir with 2:3 upstream inclination | 22 | 0 | 11 | 52.85 | 33 | – | 29.5, 24 |

Ogee weir with 3:3 upstream inclination | 22 | 0 | 11 | 57.65 | 45 | – | – |

Ogee weir with apron thickness of 3 cm | 22 | 19 | 11 | 86.45 | 0 | 60, 24 and 3 | – |

Ogee weir with apron thickness of 5 cm | 22 | 17 | 11 | 86.45 | 0 | 60, 24 and 5 | – |

Ogee weir with apron thickness of 7 cm | 22 | 15 | 11 | 86.45 | 0 | 60, 24 and 7 | – |

Ogee weir with apron thickness of 10 cm | 22 | 12 | 11 | 86.45 | 0 | 60, 24 and 10 | – |

### Evaluation of accuracy

*R*

^{2}) and the relative error (RE), which can be calculated by the following (Di Nunno

*et al.*2021):where

*O*are the discharge coefficients that result from regression equations,

_{i}*P*are the experimental results, terms with an over-bar are mean values and

_{i}*n*is number of data.

## RESULTS AND DISCUSSION

### Free-flow conditions

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 m^{3}/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 *C*_{0} 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 (*C*_{0}) 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}/*C*_{0} 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}/*C*_{0} increases and reaches a constant value of unity. In this case, the relative value of *C*_{a}/*C*_{0} 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.

*R*

^{2}) for Equation (7) was calculated to be 0.91. It should be noted that Equation (7) is valid for (

*h*

_{d}+

*d*)/

*H*

_{e}in the range of 2.25–4.65. In addition, for the dimensionless parameter (

*h*

_{d}+

*d*)/

*H*

_{e}, the numerator represents the downstream invert level (Figure 1).

In Equation (7), *C*_{a}/*C*_{0} 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}/*C*_{0} 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}/*C*_{0} along with the determination coefficient (*R*^{2}) and relative error (RE). Based on Table 2, the RE values for individual regression equations are less than 1.4%.

Relative error, RE (%) . | Determination coefficient, R^{2}
. | Fitted equation . | Ogee spillway models . |
---|---|---|---|

0.442 | 0.526 | Spillway with upstream inclined angle α = 18° | |

1.399 | 0.718 | Spillway with upstream inclined angle α = 33° | |

0.763 | 0.725 | Spillway with upstream inclined angle α = 45° | |

3.4 | 0.625 | Spillways with upstream inclined angle α = 18°, 33°, and 45° |

Relative error, RE (%) . | Determination coefficient, R^{2}
. | Fitted equation . | Ogee spillway models . |
---|---|---|---|

0.442 | 0.526 | Spillway with upstream inclined angle α = 18° | |

1.399 | 0.718 | Spillway with upstream inclined angle α = 33° | |

0.763 | 0.725 | Spillway with upstream inclined angle α = 45° | |

3.4 | 0.625 | Spillways with upstream inclined angle α = 18°, 33°, and 45° |

### Submerged-flow conditions

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 *C*_{0}) 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 *C*_{0} 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}).

*C*

_{s}/

*C*

_{0}) against the dimensionless parameter (

*h*

_{d}/

*H*

_{e}) for the ogee spillway with a vertical upstream face. It is observed that by increasing

*h*

_{d}/

*H*

_{e}, the

*C*

_{s}/

*C*

_{0}at first has a constant value and then has a downward trend. For values of

*h*

_{d}/

*H*

_{e}in the range 0.3–0.75,

*C*

_{s}/

*C*

_{0}is ∼0.88. For

*h*

_{d}/

*H*

_{e}> 0.75,

*C*

_{s}/

*C*

_{0}decreases and the threshold value for

*h*

_{d}/

*H*

_{e}is about 0.75, which is indicative of the submergence ratio. This threshold refers to a submergence ratio value such that if the ratio is increased, a significant change in the decreasing of the discharge coefficient happens. The fitted curve for the test points is given in Equation (8). The coefficient of determination (

*R*

^{2}) of Equation (8) was calculated to be 0.805.

In Equation (8), *C*_{s}/*C*_{0} 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.

### Comparison with previous studies

Figure 15 shows a comparison between *C*_{s}/*C*_{0} 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}/*C*_{0} has a constant value. For values of *h*_{d}/*H*_{e} in the range 0.75–1, the ratio of *C*_{s}/*C*_{0} shows a downward trend. A good correlation between the present study and the Tullis (2011) study is observed.

## DISCUSSION

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}/*C*_{0}) 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 (*C*_{0}) 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.

*x*and

*y*are the horizontal and vertical co-ordinates with the dam crest as the origin and the symbols

*K*and

*n*are coefficients. In addition to the USBR (1987) equation, there are other formulae for the design of ogee profiles. For example, Hager (1991) presented the following equation for design of the ogee profile in dam spillways. Equation (10) is valid for:

*H*

_{des}is the difference between the spillway upstream invert up to the energy line in the upstream of the spillway (Figure 16).

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).

## CONCLUSIONS

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}/*C*_{0}) 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 (*R*^{2}) higher than 0.7, which indicates the high accuracy of the relationships and data obtained.

## FUNDING

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

## CONFLICT OF INTEREST

The authors declare that they have no conflict of interest.

## 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.

## DATA AVAILABILITY STATEMENT

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

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*MSc thesis*

*CFD Modelling of Ogee Spillway Hydraulics and Comparison with Physical Model Tests*

*Master of Engineering thesis*

*(The form of flow over weirs)*. L'Energia Elettrica