# Flow at an Ogee Crest Axis for a Wide Range of Head Ratios: Theoretical Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{d}, is a measure of the spillway’s hydraulic efficiency and is defined here as:

^{2}/s), H is the upstream head, and g is the gravitational acceleration [1]. On the other hand, the concern for crest pressures is a matter of safety, since a pressure below the atmospheric pressure can favor flow detachment or can cause cavitation damage if it decreases close to the vapor pressure of water [2]. Therefore, the design of ogee crests aims for a high discharge coefficient while avoiding too small pressures. This is achieved by designing the crest shape according to the trajectory of the lower nappe of a fully ventilated flow over a sharp-crested weir—the head, H

_{d}, of this design flow being the design head of the spillway [1,3].

_{d}[4,5,6]. As a result, a better understanding of the connections between the crest shape, the operating head, the discharge coefficient, and the crest pressures is a relevant task [7].

_{d}, both flows are almost the same, despite the difference in the lower boundary condition (atmospheric pressure versus crest invert); in particular, the discharge coefficient is the same, and the pressures on the ogee crest equal atmospheric pressure. For H ≠ H

_{d}, the Froude similarity states that the thickness of the jet flow is identical to that of the design flow scaled by H/H

_{d}and that the discharge coefficient remains the same. In comparison to this jet flow, the flow over the ogee crest is required to have a lower boundary with a different curvature, i.e., the curvature of the fixed spillway crest (nappe separation is not considered here). This difference suggests that the discharge coefficient and crest pressures must be different from those of the design flow.

_{sub}(ξ) and the supercritical h

_{sup}(ξ) branches of the curve defined by Equation (2) and shown in Figure 2, which are the only mathematical solutions that are physically acceptable, depending on the upstream and downstream boundary conditions of the flow.

_{c}, h

_{c}) if

_{c}, h

_{c}) is a critical point of function f [31]. Such a singular point can be a point where several branches of a curve intersect [30]. This is the case for curves defined by Equation (2): (ξ

_{c}, h

_{c}) is the intersection point of the subcritical and supercritical branches. This implies that any flow ranging from the large upstream to the small downstream water depths h needs to intersect this point (Figure 2), which, in the hydraulic sense, is the critical section of the flow. Its position, ξ

_{c}, and value, h

_{c}, are given by Equation (3) and are dependent on parameter H. Inserting ξ

_{c}and h

_{c}in Equation (2) gives the function q

_{c}(H), which leads to the presence of a singular point on the curve, or, in other words, the head–discharge function of an overflow structure such as a spillway crest. Note that, according to Equation (3) and Figure 2, q

_{c}is the largest discharge that can flow through the critical section, ξ

_{c}.

^{n}h/dξ

^{n}(n = 1, 2, …) [7]. These spatial derivatives are included in order to increase the quality of the model by taking into account the flow velocities perpendicular to the main flow direction. However, at the same time, these derivatives lead to the loss of the property that the critical section is a singular point whose position and value are known apart from the rest of the curve. Only an iterative approach or the use of lower order approximations make it possible to continue to use critical flow theory with such models [7].

_{d}= 3.7 [7], which confirms the relevance of the building blocks he used. Castro-Orgaz [34] followed a similar approach but considered the constant parameter of Jaeger’s model to be dependent on the water depth. This model gives good results for head ratios up to 2 [7,34]. Note, however, that ogee crests do not match Jaeger’s hypothesis of a quasi-circular weir in a strict sense, which challenges his definition of the critical section.

## 2. Model Development

#### 2.1. Assumptions

#### 2.1.1. Curvilinear Coordinate System

_{b}(ξ) and z

_{b}(ξ) be two-times differentiable.

_{b}be its radius of curvature. These quantities are given by:

_{b}is positive when the reference curve is locally convex and negative when it is locally concave.

_{b}is negative, Equation (10) is always true within the flow. Note that r

_{b}can be measured along the η-axis (see Figure 4, where r

_{b}is negative).

#### 2.1.2. Conservation Principles

#### 2.1.3. Velocity and Pressure Distributions

_{b}is u at the lower flow boundary, r

_{b}the radius of curvature of the lower flow boundary, and

#### 2.1.4. Upper and Lower Boundary Conditions

- A lower boundary whose pressure is atmospheric, that is:$${p}_{\mathrm{b}}=p\left(0\right)=0$$
- A lower boundary whose shape is imposed by a fixed geometry, which means that the functions x
_{b}(ξ) and z_{b}(ξ) in Equation (4) and all their derivatives are known prior to the computation of the flow.

#### 2.2. Dimensionless Quantities and Equations

#### 2.2.1. Velocity and Pressure Distribution Coefficients

_{b}/h tends toward infinity, the velocities tend to be constant across the flow (as in the shallow water equations), while as r

_{b}/h tends to 0, the difference between the upper and lower velocities increases.

_{b}/h with another dimensionless variable that is directly measurable on the velocity profile defined in Equation (16). This quantity is λ, the ratio of the lower and upper velocities, u

_{b}and u

_{s}, which, according to Equation (16), is:

_{b}, to the hydrostatic value, ρgh cosθ, that it would take if the flow had no curvature:

#### 2.2.2. Equations with λ and α as Unknowns

_{b}, becomes a function of λ:

_{b}or z

_{b}is known:

#### 2.2.3. Design Head, H_{d}, as a Reference for all Other Dimensionless Variables

_{b}). However, this quantity has two drawbacks. First, it is not constant over space (because of z

_{b}), which changes the behavior of the derivatives of the corresponding dimensionless quantities with respect to the original ones (e.g., they do not become 0 simultaneously). Second, it is not the same for all flows over a given structure, while studying different flows over the same structure is the aim of this study. For these reasons, it is convenient to use a characteristic length of the fixed shape of the structure to define dimensionless quantities. In the case of ogee crests, this quantity is the design head, H

_{d}:

_{d}, is given by:

#### 2.2.4. Set of Dimensionless Equations

#### 2.2.5. Working Hypothesis on r′

#### 2.3. Flow over a Sharp-Crested Weir

_{d}, the head ratio to consider here is $\overline{H}=1$. With these assumptions, the derivation of Equation (43) gives:

#### 2.4. Assessment of the Consistency of the Model

_{d}due to the approximations on the velocity distribution.

#### 2.4.1. Slope of the Upper Flow Boundary and Assumption w = 0

_{est.}and w

_{est.}be the velocity components in directions ξ and η if the velocity distribution derived by Peltier et al. [15] is applied to the absolute velocity. This would imply (after dividing by $\sqrt{2g{H}_{\mathrm{d}}}$):

_{est.}and w

_{est.}would be connected through the slopes of the streamlines crossing the η-axis. Assuming a linear variation in the inclination of the streamlines from the bottom “b” to the top “s”, as suggested by the numerical results by Castro-Orgaz [28], would give:

#### 2.4.2. Curvature of the Upper Flow Boundary and Assumption r′ = −2

_{est.}be the derivative of the radius of curvature of the streamlines crossing the η-axis that would give a distribution compatible with the radius of curvature of the upper flow boundary:

_{d}to its better estimate is:

#### 2.5. Flow over an Ogee Crest

#### 2.5.1. System of Equations

_{α}, B

_{λ}, and B

_{α}are never equal to zero. This is also the case for T: except for $\overline{H}=0$, it is never equal to zero on an ogee crest, as suggested by Equations (54), (57) and (60)

#### 2.5.2. Approximation at the Crest Apex

_{α}, B

_{λ}, B

_{α}, and T are never equal to zero, the left-hand side in Equation (91) is zero if A

_{λ}= 0, which implies:

_{0}:

_{0}= 1 is obvious), it has three real solutions. The only solution that verifies λ

_{0}> 1, in line with (55), is:

#### 2.5.3. Formulas for the Flow at the Crest Apex of an Ogee Crest

_{d}= 0.5 for the design head. For extremely small heads, C

_{d}approaches 0.3849 (but, in this case, the effects of surface tension and viscosity can no longer be neglected). For extremely large heads, C

_{d}approaches 0.7698. This value is also purely theoretical, as nappe separation would occur. However, this value suggests that, even in the absence of nappe separation, C

_{d}cannot be increased endlessly by increasing the head ratio.

_{0}is, with Equations (99) and (101):

## 3. Model Validation

#### 3.1. Ogee Crest

_{b}= –H

_{d}/2 is compared to the radius of curvature of several crest profiles designed for spillways with a vertical upstream face and negligible approach velocities.

- Type 1: a WES profile with an upstream quadrant composed of three circles of radii 0.04 H
_{d}, 0.2 H_{d}, and 0.5 H_{d}and a downstream quadrant given by Equation (105); - Type 2: an upstream quadrant described by Equation (103) and a downstream quadrant described by Equation (105);$${\overline{z}}_{\mathrm{b}}=-0.724{\left({\overline{x}}_{\mathrm{b}}+0.270\right)}^{1.85}-0.126+0.4315{\left({\overline{x}}_{\mathrm{b}}+0.270\right)}^{0.625}$$
- Type 3: an elliptical upstream quadrant given by Equation (104) and a downstream quadrant given by Equation (105);$${\left(\frac{{\overline{x}}_{\mathrm{b}}}{0.280}\right)}^{2}+{\left(\frac{{\overline{z}}_{\mathrm{b}}}{0.167}+1\right)}^{2}=1$$
- Type 4: a USBR with an upstream quadrant composed of two circles of radii 0.235 H
_{d}and 0.530 H_{d}and a downstream quadrant given by Equation (106).

_{b}> 0) of all four types are described by a power law. Types 1–3 follow Equation (105), while type 4 follows Equation (106). Note that the radius of curvature at the crest apex is zero for both formulas, which generates a discontinuity with the radius of the upstream quadrant (Figure 10). Nevertheless, the radius of curvature of both formulas increases very rapidly and reaches values in the order of the upstream quadrant values within a distance less than 0.002 H

_{d}, so it seems unlikely that this discontinuity has a real impact on the physical models or prototypes.

_{b}= –0.5 H

_{d}at the crest apex on the upstream quadrant, which is exactly the value of the theoretical model; type 3 has r

_{b}≈ –0.47 H

_{d}(difference of 6%); and type 4 has r

_{b}= –0.53 H

_{d}(difference of 6%). Locally, the values given by the theoretical model are very good. However, the spatial distribution of the radius of curvature shows that its absolute values are generally smaller than those of the practical profiles.

#### 3.2. Discharge Coefficient

_{d}= 0.502 for type 3 crests [39], while USBR gives C

_{d}= 0.492 for type 4 crests [33]. Both values have been converted to fit Definition (1). The value given by the present model (C

_{d}= 0.5) differs from these values by 0.002 and 0.008, respectively. Let ε

_{q}be that difference. According to Equation (107), where the subscript “ref” refers to a reference value (an empirical value, as opposed to the theoretical value), a difference of 0.008 means that the difference in discharge is equal to 0.8% of (2gH

^{3})

^{1/2}:

_{d}at head ratios larger than 1.

_{d}= 0.10 m) with a 7.8% difference between theory and experiments (Table 3)—this point is easily identifiable in Figure 11a—the 7.8% difference is smaller than the uncertainty on the measurement. If this point is taken out of the dataset, the maximum difference is less than 3.6% for all datasets, and the mean difference is less than 0.9%.

#### 3.3. Velocity Distribution

_{d}values above, as shown in Figure 12 in comparison to experimental data from the literature. Note that the experimental data in Figure 12a,b are not available for the whole cross section, only the lower part of it. Moreover, in contrast to [15], the measurements presented in Figure 12a,b were not taken along an isopotential line but along a vertical cross section so that they could be compared to the theoretical model. Finally, note that the velocities in Figure 12a,b are the horizontal velocity component, while the values in Figure 12c are absolute flow velocities.

#### 3.4. Water Depth at the Crest Apex

_{h}be the difference between a theoretical value, h/H, and a measured value, h

_{ref}/H:

_{d}= 0.153), which means that the difference on h is, on average, less than 0.5% of H. This very good agreement confirms that the slight underestimation of the theoretical discharge coefficient compared to the measurements is due to a slight underestimation of the velocities, rather than a difference in the water depth.

#### 3.5. Crest Pressure at the Crest Apex

_{p}be the difference between a theoretical value, p

_{b}/ρgH, and a measured value, p

_{b,ref}/ρgH:

_{b}is, on average, equal to 8% of ρgH), with a maximum of 17% at H/H

_{d}= 5. For the datasets of Figure 14b, the mean differences are less than 3.8%, and the maximum differences are less than 5.6%.

#### 3.6. Summary

## 4. Discussion

_{d}. For one of the experimental datasets ([5]—H

_{d}= 0.15 m) it is even more accurate than the present theory. For the other datasets, the new model is more accurate. The fact that both models give similar results underlines the stability that comes from computing q by equating a derivative of it to 0: changes in the variables cannot have large impacts on the discharge.

_{d}due to underestimated water depths and crest pressures, i.e., overestimated bottom velocities. The present theory predicts these quantities more accurately and, therefore, is more consistent.

## 5. Conclusions

_{b}| = H

_{d}/2, consistent with common ogee crest profiles, despite being slightly smaller.

^{3})

^{1/2}. At the crest apex, it also predicts the water depth, with a mean difference below 0.5% of H (data available for H/H

_{d}< 2 only), and the crest pressure, with a mean difference below 8% of ρgH for the data points related to the largest weir tested. The discharge and water depth are therefore well-reproduced by the new model, while the crest pressure and the radius of curvature are more sensitive to the approximations of the present theory.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Comparison of flows over sharp-crested weir and ogee crest for two heads (heads are measured from point O): (

**a**) design head; (

**b**) effective head larger than the design head.

**Figure 2.**(

**a**) Possible definitions of abscissa ξ and water depth h; (

**b**) Contour plot showing the values of function f (H, h, ξ) with their subcritical (in blue) and supercritical branches (in orange); the transcritical solution shown in (

**a**) is also plotted in (

**b**) (black dotted line); point (ξ

_{c}, h

_{c}) is a critical point of function f and a singular point of the corresponding subcritical and supercritical branches.

**Figure 3.**Outline of the development of the model, with reference to the Section 2.1, Section 2.2, Section 2.3, Section 2.4 and Section 2.5.

**Figure 5.**(

**a**) Velocity distribution used by Peltier et al. [15], where v is the absolute velocity across an isopotential line, and r is the radius of curvature of the streamline; (

**b**) Velocity distribution along the η-axis, where u is the velocity in direction ξ, and w is the velocity in direction η (neglected here).

**Figure 6.**Definition of (

**a**) velocity distribution coefficient λ and (

**b**) pressure distribution coefficient α with reference to parallel flows.

**Figure 8.**Comparison of a fully ventilated flow over a sharp-crested weir (free jet flow) and the flow over the corresponding ogee crest for the same head (results of a numerical implementation of the equations presented above). The only difference is the position of the critical section.

**Figure 10.**Comparison of the radius of curvature of the theoretical model (Taylor series based on r

_{b}and its two first derivatives at the crest apex, i.e., Equations (54), (57) and (60)) with common crest shapes.

Lower BC: Atmospheric Pressure | Lower BC: Prescribed Shape | ||||
---|---|---|---|---|---|

Known Values | Unknown Values | Equations | Known Values | Unknown Values | Equations |

$\mathrm{IV}:\left\{\begin{array}{l}\overline{q}\\ \overline{H}\\ {\overline{x}}_{\mathrm{b},0}\\ {\overline{z}}_{\mathrm{b},0}\\ {\theta}_{0}\end{array}\right.$$\phantom{\rule{0ex}{0ex}}\mathrm{BC}:\alpha =0$$\phantom{\rule{0ex}{0ex}}\mathrm{Other}:r\u2019$ | $\left\{\begin{array}{l}\lambda \left(\overline{\xi}\right)\\ {\overline{z}}_{\mathrm{b}}\left(\overline{\xi}\right)\\ \theta \left(\overline{\xi}\right)\end{array}\right.$$\phantom{\rule{0ex}{0ex}}{\overline{x}}_{\mathrm{b}}\left(\overline{\xi}\right)$ $\overline{h}\left(\overline{\xi}\right)$ | $\left\{\begin{array}{l}\left(33\right)\\ \left(40\right)\\ \left(41\right)\end{array}\right.$ (39) (34) | $\mathrm{IV}:\left\{\begin{array}{l}\overline{q}\\ \overline{H}\end{array}\right.$$\phantom{\rule{0ex}{0ex}}\mathrm{BC}:\left\{\begin{array}{l}{\overline{x}}_{\mathrm{b}}\left(\overline{\xi}\right)\\ {\overline{z}}_{\mathrm{b}}\left(\overline{\xi}\right)\\ \theta \left(\overline{\xi}\right)\\ {\overline{r}}_{\mathrm{b}}\left(\overline{\xi}\right)\end{array}\right.$$\phantom{\rule{0ex}{0ex}}\mathrm{Other}:r\u2019$ | $\left\{\begin{array}{l}\lambda \left(\overline{\xi}\right)\\ \alpha \left(\overline{\xi}\right)\end{array}\right.$$\phantom{\rule{0ex}{0ex}}\overline{h}\left(\overline{\xi}\right)$ | $\left\{\begin{array}{l}\left(33\right)\\ \left(35\right)\end{array}\right.$ (34) |

**Table 2.**Values of dimensionless quantities when the head ratio approaches 0, 1, and +∞ (both extremes are hypothetical).

$\overline{\mathit{H}}$ | 0 | 1 | +∞ |
---|---|---|---|

δ | 0 | 1 | +∞ |

λ_{0} | 1 | 2 | +∞ |

α_{0} | 1 | 0 | –∞ |

C_{d} | 2/3^{3/2} = 0.3849 | 0.5 | 4/3^{3/2} = 0.7698 |

p_{b,0}/ρgH | 2/3 | 0 | –∞ |

h_{0}/H | 2/3 | 3/4 | 2/3 |

**Table 3.**Comparison of the present theory (and Jaeger’s theory) with experimental data. Differences are computed for H/H

_{d}ranging from 0 to 5 (i.e., no flow separation). For Hager’s data, the point H/H

_{d}= 3.71 is excluded from the error computation because it seems to be influenced by flow separation.

Reference | Present Theory | Jaeger’s Theory | ||
---|---|---|---|---|

Mean ε_{q} | Maximum ε_{q} | Mean ε_{q} | Maximum ε_{q} | |

Erpicum et al. [5] (H_{d} = 0.15 m) | 0.8% | 3.1% (H/H_{d} = 0.459) | 0.5% | 2.6% (H/H_{d} = 0.459) |

Erpicum et al. [5] (H_{d} = 0.10 m) | 0.9% | 7.8% (H/H_{d} = 0.433) | 1.0% | 8.3% (H/H_{d} = 0.433) |

Hager [11] | 0.7% | 1.5% (H/H_{d} = 0.510) | 1.1% | 2.0% (H/H_{d} = 0.510) |

Rouse et al. [6] | 0.4% | 1.7% (H/H_{d} = 0.253) | 0.6% | 2.0% (H/H_{d} = 0.253) |

Melsheimer et al. [13] | 0.5% | 0.9% (H/H_{d} = 0.715) | 0.8% | 1.5% (H/H_{d} = 0.715) |

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**MDPI and ACS Style**

Stilmant, F.; Erpicum, S.; Peltier, Y.; Archambeau, P.; Dewals, B.; Pirotton, M. Flow at an Ogee Crest Axis for a Wide Range of Head Ratios: Theoretical Model. *Water* **2022**, *14*, 2337.
https://doi.org/10.3390/w14152337

**AMA Style**

Stilmant F, Erpicum S, Peltier Y, Archambeau P, Dewals B, Pirotton M. Flow at an Ogee Crest Axis for a Wide Range of Head Ratios: Theoretical Model. *Water*. 2022; 14(15):2337.
https://doi.org/10.3390/w14152337

**Chicago/Turabian Style**

Stilmant, Frédéric, Sebastien Erpicum, Yann Peltier, Pierre Archambeau, Benjamin Dewals, and Michel Pirotton. 2022. "Flow at an Ogee Crest Axis for a Wide Range of Head Ratios: Theoretical Model" *Water* 14, no. 15: 2337.
https://doi.org/10.3390/w14152337