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Research Article | | Peer-Reviewed

New Energy Loss Expression for Stepped Spillways with Channel Slopes 26.6° and Below

Okechukwu Ozueigbo*,  Jonah Agunwamba
Published in Journal of Civil, Construction and Environmental Engineering (Volume 10, Issue 1)
Received: 14 October 2024     Accepted: 13 November 2024     Published: 22 January 2025
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Abstract

A stepped spillway is a hydraulic structure built at storage and detention dams to discharge flood water that cannot be safely kept in the reservoir. It was created to minimize the kinetic energy that would have otherwise produced dangerous scour at the natural river bed beneath the spillway. They discharge this energy in floodwater using their stepping nature. Several studies show the detrimental consequences of falling water's kinetic energy on the river bed underneath the structure. Only a handful of these studies, however, have evaluated the impact of energy losses caused by stepped spillways with channel slopes 26.6° or less. As a result, there are gaps in the rules and recommendations for designers of stepped spillways with channel slopes below 26.6°. Furthermore, the existing models for forecasting energy dissipation in stepped spillways for channels of all slopes include a parameter, the friction factor, f, which is difficult to estimate with precision, leading to its subjective estimation by those involved in the stepped spillway design. The goal of this study is not only to provide designers with design recommendations and information for stepped spillways with channel slopes below 26.6°, but also to eliminate the 'troublesome' frictional factor; f. Using phase-detection intrusive probes, air-water flow tests were carried out in nappe, transitional, and skimming flows on a stepped spillway with channel slope below 26.6° in a large facility. New expressions for evaluating energy losses in stepped spillways with slopes 26.6° or less are developed. Regarding energy dissipation, the data from the latest models compare well with the measured data, with high coefficients of correlation that range between 0.95 and 0.99. All of the measured data and the estimated data are in good agreement. The models are simple to use.

Published in Journal of Civil, Construction and Environmental Engineering (Volume 10, Issue 1)
DOI 10.11648/j.jccee.20251001.12
Page(s) 16-26
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

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Copyright © The Author(s), 2025. Published by Science Publishing Group
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Keywords

Stepped Spillway, Energy Dissipation, Nappe Flow, Skimming Flow

1. Introduction
Many scholars have conducted investigations and developed equations that govern Nappe flow, Transition, and Skimming flow regimes in stepped spillways, including
[9, 18, 20, 34]
.
Many of these equations still need to be verified by researchers, while some have already been confirmed. These researchers looked into and tried to find the ideal design and configuration of steps to maximize energy dissipation and decrease basin sizes.
The effect of step geometry on energy dissipation in the stepped spillway has not been extensively studied by scholars. In this field, there is a knowledge deficit.
Therefore, this study is conducted in an effort to offer some knowledge in that field.
If floodwater's energy, particularly its kinetic energy, is not securely dispersed, significant harm could result. One kind of device used to release floodwaters is the stepped spillway. It is made up of an open channel and several drips that are inverted. These drops present high flow impediment, which causes energy loss. Due to this energy loss, smaller and more cost-effective dissipation structures are designed downstream of the chute.
A critical point at which air would be entrained in the flow would be achieved as the discharge down the spillway was raised.
Two-phase flow or air-water flow is the term used to describe this phenomenon.
Chanson and Stephenson [18, 34] classified the flow over the stepped spillway based on experimental observations into three different flow regimes:
a) Nappe flow regime or jet flow regime for lower discharges;
b) Transition flow for medium discharges; and
c) Skimming flow regime for higher discharges.
1.1. Nappe Flow
Similarly, Peyras et al
[30]
identified two types of nappe flow: flow with a fully developed hydraulic jump for low discharge and small flow depth, and flow with a partially developed hydraulic jump. In the nappe flow regime, a sequence of freely falling water flows down from one step to the next lower step with the development of a hydraulic jump at each step. This type of flow can be likened to a sequence of separate drop structures linked together
[9, 16]
. It flows from each step and passes through critical depth at the brink of the step forming a supercritical free-falling jet and returning to subcritical flow downstream of the jump. The flow hits the step below as a freely falling jet, with the energy dissipation occurring by a jet breakup in air, by jet mixing on the step, and by the formation of a fully developed or partial hydraulic jump on the step.
For nappe flow to occur, the step horizontal tread needs to be greater than the water depth
[34]
. In dam design, this would result in a relatively flat slope. The step height, h, must be large for nappe flow to occur. This situation may apply to relatively flat stepped channels or at low flow rates.
According to Peyras et al
[30]
the step height needs to be at least two or three times the critical flow depth and the tread of the steps needs to be two or three times the step height.
The slopes of these reaches were such that a hydraulic jump occurred at the base of each drop.
If the length of the drop added to the length of the roller, Lr, is less than the length of the horizontal step, l, a fully developed hydraulic jump can take place (Figure 1). Chanson
[10] 
states that a fully developed nappe flow with a fully developed hydraulic jump occurs for discharges smaller than a critical value defined by:
dchchar= 0.0916hl-1.276(1)
Where the correlation - Eq (1) - is valid for 0.2 ≤ h/l ≤ 6
Where l is the step length.
Nappe flow for a fully developed hydraulic jump occurs for
tanθ = hl < 0.2(2)
Several dams in South Africa were built with stepped spillways. Stephenson
[34]
, from his experience, suggested that the most suitable conditions for nappe flow situations are:
tan∝ = hl < 0.2
Figure 1. Nappe flow regime (Flow at a drop structure).
Energy Dissipation
In a nappe flow regime with a fully developed hydraulic jump, the head loss at any intermediate step equates the energy loss. The total head loss, ∆H, along the spillway equals the maximum head available, Hmax, and the residual heat, H1, at the bottom of the spillway
[11]
.
∆HHmax = 1 - d1dc +12dcd1232 + Hdamdc ungated spillway(3)
∆HHmax = 1 - d1dc +12dcd12Hmax +Hodc gated spillway(4)
Where Hdam is the dam height and H0 is the reservoir free-surface elevation above the spillway crest. The residual energy is dissipated at the toe of the spillway by a hydraulic jump in the dissipation basin.
1.2. Partial Nappe Flow
In this type of flow, the nappe does not fully impinge on the step surface and it disperses with considerable turbulence. The flow is supercritical down the length of the spillway. For a given step geometry, an increase in flow rate may lead to an intermediate flow pattern between nappe and skimming flow - the transition flow regime is also called a partial nappe flow. The transition flow is characterized by a pool of circulating water and is often accompanied by a small air bubble (cavity), significant water spray, and the deflection of the water jet immediately downstream of the stagnation point (Figure 2). The transition flow pattern exhibits significant longitudinal variations of the flow properties on each step. It does not present the coherent appearance of skimming flows. The transition from one type of flow to another is gradual and continuous. As a result, both the nappe flow and the skimming flow appear simultaneously in a certain range, one of them on some steps and the other on the remaining, both changing spatially and temporarily.
Figure 2. Nappe flow with partially developed hydraulic jump.
1.3. Skimming Regime
In skimming flow, the water flows down the stepped face as a coherent stream. It skims over the steps and is supported by the recirculating fluid trapped between them. The external edges of the steps form a pseudo bottom over which the flow skims. Beneath this bottom, recirculating vortices are formed, which are sustained by the transmission of shear stress from the water flowing past the edge of the steps. At the upstream end, the flow is transparent and has a glossy appearance and no air entrainment takes place. After a few steps, the flow is characterized by air entrainment similar to a self-aerated flow down a smooth inverted spillway. There is a continuous exchange of flow between the top layer and vortices formed on steps. The flow rotates in the vortex for a brief period and then returns to the main flow to proceed on down the spillway face. These vortices are supported through the transmission of shear stress from the fluid flowing past the edges of the steps (Figure 4). Chanson
[14]
states that in addition to this, small vortices are generated continuously at the corner of the steps. It occurs with the submergence of the steps with the development of a fully aerated uniform flow downstream of a long chute. Along the upstream steps, a non-aerated flow region exists in which a turbulent boundary layer develops. Air entrainment in the flow begins where the boundary layer intersects the free surface, referred to as the point of inception. Downstream of the point of inception, the flow continues to aerate and varies gradually in depth (Figure 3). The flow eventually becomes a fully aerated uniform flow in which the water depth, velocity, and air concentration become constant
[3, 10]
(Figure 4).
The condition is very difficult to establish analytically. Therefore, empirical equations have been proposed by many investigators for the delineation of the skimming flow from nappe flow over the stepped spillway. Essery and Horner
[20, 26]
reported that it is very difficult to distinguish between nappe and skimming flow for flatter slopes having <0.4Shl
It must be noted that the data of Peyras et al.
[27, 30]
were obtained on a gabion stepped spillway model. The infiltration through the gabion is likely to affect the flow conditions and may explain smaller values of (dc/h)onset than for
[32, 33]
.
Chanson
[18, 21, 24]
stated that energy dissipation occurs by momentum transfer, or the transmission of turbulent shear stress, between the skimming stream and the vortices.
Many of the researches on the stepped spillway skimming flow has concentrated on determining the profile of the turbulent boundary layer, the location of the point of inception, and the concentration of air in the flow. All of these factors influence the design of stepped spillways. Specifically, knowledge of the concentration and distribution of aeration is important in determining the water depth, velocity, and hence the amount of energy dissipation along the slope
[24]
.
Numerous researchers have observed and analyzed the flow regime change from Nappe to Skimming in a spillway and linked the parameters, which have been related to flow critical depth, dc, and the spillway shape, which are the non-dimensional ratios dc /h and h/ l
[5, 6, 25
].
Chanson, Rajaratnam
[7, 8, 31]
took into account these non-dimensional parameters dc /h and h/ l. He proposed that values of dc/ h > 0.8 will produce skimming flow in the range of 0.42 ≤ h/l ≤ 0.842 of the skimming, leaving the nappe flow regime when flow dc/ h < 0.8 whatever the geometric specifications of the structure might be.
Studies have, however, shown that the majority of the changes in flow regime occur for the flow rate, dc/h, less than this proposed 0.8.
For small discharges and flat slopes, the flow regime is the nappe flow regime. An increase in discharge or channel slope may induce the appearance of a skimming flow regime.
Chanson
[18] 
in his result, also took into account the analysis of Beitz and Lawless, Degoutte et al, Essery and Horner
[1, 19, 20]
proposed that the skimming flow regime occurs for discharges larger than a critical value defined as:
(dc)onset/h =1.06−0.465 h/l(5)
Where (dc)onset is the characteristic critical depth.
Chanson
[8]
stated that skimming flows occur when dc/h > (dc/h)onset
He noted that Eq (5) was deduced for h/l that ranges from 0.2 to 1.3 and that further experimental work is required to verify or modify the equation outside the range.
Several researchers have published works on the impact, of scale effects in modeling stepped spillways
[2, 4, 15, 23, 31]
.
The scale effect is the slight misrepresentations, which occur when secondary forces like viscous forces and surface tension forces in turbulent flow are ignored. It is overlooked in many open-channel flows. However, if they are ignored in highly air-entrained flows in stepped spillways, where they play significant roles, they could lead to scale effects and wrong interpretations of results
[2]
. Scale effects in stepped spillway models are likely to occur with scales lower than 10: 1, Reynolds number smaller than 1 × 105, Weber number smaller than 100, and step heights lesser than 3 cm.
Energy Dissipation
The energy dissipated occurred to keep stable depression vortices. If uniform flow conditions are reached downstream of the spillway, the energy loss could be calculated as follows
[10]
(Figure 4):
∆HHmax=1- dwdccos⁡θ+12dcdw2Hdamdc+32(6)
Where dw is the clear water depth, Uavg is the average velocity, the total head loss may be rewritten in terms of the friction factor, f, the spillway slope, θ, in degree, the critical depth, dc, and the dam height, Hdam:
∆HHmax=1-f8sinθ1/3cosθ+E2f8sinθ-2/3Hdamdc+32(7)
Eq (6) computed for spillway slope with θ = 52 (degrees) and friction factor, f = 0.3 and f = 1.30 represented average flow resistance on smooth spillways and stepped spillways, respectively. where E is the kinetic energy correction coefficient, θ is the dam slope in degrees.
∆HHmax=0.049Nhdc0.353N0.06h0.124θ-0.157(8)
Eq (8) is developed for skimming flow when spillway slope is between 26.6° and 21.8°, h (cm) is not more than 20, N is not more than 20, and dc/h is between 1.0 and 3.7
[29]
.
∆HHmax=0.029Nhdc0.353N0.06h0.124θ-0.157(9)
Eq (9) is developed for skimming flow when h (cm) is not more than 20, N is not more than 20, θ (degrees) is between 21.8° and 3.4°, and dc/h is between 1.0 and 3.6
[29]
.
Figure 3. Skimming flow regime - Sorensen
[32]
.
Figure 4. Skimming flow regime with uniform flow conditions.
Figure 5. Arrangement of the spillway with the definition of the variables.
2. Material and Methods
The authors carefully selected eleven publications on horizontal stepped spillways with minimum step heights of 3 cm, undistorted Froude values, and large dimensionless discharges corresponding to Reynolds numbers between 1×105 and 1×106.
These researchers used experimental facilities on large scale to minimize scale effects that affect the air-water flow processes in high-velocity free-surface flows
[23]
. They measured air-water flow properties at all step edges downstream of the inception point of air-entrainment with conductivity phase-detection intrusive probes and optical fiber probes.
Use of a Dall Tube flow meter, or V-notch for flow rates, Prandtl-Pitot for flow velocities, or point gauge for clear water flow depth to obtain air-water flows properties is not practicable as large quantities of air are entrained at the air-water interface
[11, 12]
.
The principle of the conductivity probe is based on the difference between the resistivity of air and water, which provides an instantaneous voltage signal
[13, 17]
. The threshold technique analyzes signals - from a single sensor - used to calculate a) the time-averaged local air concentration or void fraction C, b) the number of air-to-water (or water-to-air) voltage changes expressed as bubble count rate F, and c) the air bubble and water droplet chord sizes. For a double-tip conductivity probe with longitudinal separation between the two probe sensors, the cross-correlation analysis of the signals leads to the local time-averaged interfacial velocity V
[10]
.
Details about the signal processing techniques could be found in
[16, 22]
.
The double-tip conductivity probes used had sensor sizes of Ø = 0.13 mm and 0.25 mm and were sampled for a period of 45 s with a frequency of 20 kHz per sensor.
The discharges comprised transition and skimming flow rates of 0.035 ≤ qw ≤ 0.234 m2/s for the spillways with θ = 8.9◦ and 0.02 ≤ qw ≤ 0.249 m2/s and for θ = 26.6◦, Reynolds numbers of 1.4 × 105 ≤ Re ≤ 9.3 × 105 and 8.1 × 104 ≤ Re ≤ 9.9 × 105
The National Water Institute's large-scale stepped spillway models with a slope of θ = 26.6°, 21.8°, 15.9°, 14.6°, and 8.9° were used for the experimental study.
The facility was 12 meters long and was made up of an input tank that continuously delivered an upstream water head. A 0.5 m wide uncontrolled broad-crested weir was used to let water into the experimental test area. The stepped spillway had the following number of steps: a) 10 No steps with a step height of 0.10 m and a step length of 0.20 m; b) 10 No steps with a step height of 0.10 m and a step length of 0.25 m; c) 9 No steps with a step height of 0.10 m and a step length of 0.35 m; d) 13 No steps with a step height of 0.10 m and a step W = 0.52 m was the width of the chute. Perspex was used for the channel walls and PVC for the steps.
A huge upstream intake basin with dimensions of 3.0 m 2.5 m and a depth of 1.6 m provided steady flow rates. A 1.0 m long smooth sidewall convergent with a 4: 1 contraction ratio provided a smooth inflow. A broad-crested weir with a height of 1 m, width of 0.52 m, crest length of 1.01 m 1.01 m, and an upstream rounded corner controlled the flow in the test portion.
An ultrasonic range finder and a sharp-crested weir were used to measure the discharge at the downstream channel's terminus.
The experimental facilities were large scale to minimize scale effects affecting the microscopic air–water flow processes in high-velocity free-surface flows
[15]
. Scale effect is a term used to describe slight distortions that are introduced by ignoring secondary forces such as viscous forces, surface tension in stepped spillway models. Viscous forces and surface tension in most open-channel applications are deemed negligible, but in highly air- entrained flows like those expected in stepped spillways, these forces are more dominating and cannot be simply ignored. Scale effects can develop if these factors are ignored, leading to data misinterpretation. Scale effects in modeling stepped spillways have been thoroughly established Boes and Hager
[4]
. In stepped spillway models, scale effects are most typically linked with scales less than 10: 1.
According to Chanson
[10]
, a model scale of 10: 1 or greater is recommended, and Boes and Hager
[3]
recommended a minimum Reynolds number of 105 and a minimum Weber number of 100. Takahashi et al.
[35]
urge that Froude, Reynolds, and Morton similarity be satisfied when simulating strongly air entrained flow, but they acknowledge that this can only be done at full scale. Although researchers have not established an agreement on the boundaries to reduce scale effects in physical models of stepped spillways, some information is available. The use of traditional mono-phase flow instrumentation in high-velocity air-water flows is not possible due to the three-dimensional air-water flow with enormous volumes of air-water.
Because substantial quantities of air are entrained at the air-water interface, using a Dall Tube flow meter, or V-notch for flow rates, Prandtl-Pitot for flow velocities, or point gauge for clear water flow depth to get air-water flow attributes is impractical
[11, 12]
.
In stepped spillways, detection invasive probes are widely utilized, and experimental tests with optical fiber probes
[1]
and conductivity probes
[2, 10, 13, 28]
have been effective.
Extensive studies were carried out with all of the stepped spillways for a wide variety of discharges 0.018 m3 /s < Q < 0.117 m3 /s. Air-water flow studies were carried out with conductivity phase-detection intrusive probes at all step edges downstream of the inception site of air entrainment for all stepped sizes. The sensor diameters of the double-tip conductivity probes were 0.13 mm and 0.25 mm, and they were sampled for 45 seconds at a frequency of 20 kHz per sensor. Typically, the probe was placed at step edges in the air-water flow zone.
The inner diameter of the probe was 0.13 mm for both points, which were separated in the streamwise direction x = 5.1 mm and the transverse direction z = 1 mm.
All measurements lasted 45 seconds at a sampling rate of 20 kHz per probe tip. The conductivity probe's basis is based on the differing resistance of air and water, which provides an immediate voltage signal. A single sensor's signal can be analyzed using a threshold technique to determine the time averaged local air concentration or void fraction C, the number of air-to-water voltage shifts expressed as bubble count rate F, and the air bubble and water droplet chord diameters. The cross-correlation analysis of the data for a double-tip conductivity probe with longitudinal spacing between the two probe sensors yields the local time-averaged interlayer velocity V. Chamani, Chanson
[8, 13]
provide more information on signal processing techniques
The trials were carried out for a wide variety of discharges at numerous step edges downstream of the free surface aeration inception point.
The discharges had transition and skimming flow rates of 0.035 ≤ qw ≤ 0.234 m2/s for the spillways with θ = 8.9o and 0.02 ≤ qw ≤ 0.249 m2/s for θ = 26.6°, with Reynolds numbers of 1.4 × 105 ≤ Re ≤ 9.3 × 105 and 8.1 × 104 ≤ Re ≤ 9.9 × 105, respectively
3. Results
The Developed Models for the Nappe/Transition/Skimming Flow Regime.
The discharges had transition and skimming flow rates of 0.035 ≤ qw ≤ 0.234 m2/s for the spillways with θ = 8.9° and 0.02 ≤ qw ≤ 0.249 m2/s for θ = 26.6°, with Reynolds numbers of 1.4 × 105 ≤ Re ≤ 9.3 × 105 and 8.1 × 104 ≤ Re ≤ 9.9 × 105, respectively.
The authors then used the developed models, Eq (10) through Eq (19), to predict the rates of energy dissipation and displayed them vis-à-vis the measured data sets in Figure 6 through Figure 15.
∆H/Hmax=(0.92 Nh/dc)0.29N0.20h--0.17 θ--0.32(10)
This equation is valid for 𝜽 of 26.6°, Nh/dc between 5.0 and 10.0, N between 5 and 20, dc/h between 0.80 and 3.30, h (cm) between 5 and 10. Spearman coefficient is 0.97.
∆H/Hmax=(0.28 Nh/dc)0.29N0.20h--0.17 θ--0.32(11)
This equation is valid for 𝜽 between 21.8° and 26.6°, Nh/dc between 5.0 and 25, N between 5 and 20, dc/h between 0.80 and 1.85, h (cm) between 5 and 10. Spearman's correlation coefficient is 0.94.
Figure 6. ∆H/Hmax as a function of Nh/dc for qw = (0.073 - 0.249 m2/s) & Re = (2.92 × 105 - 9.96 × 105), dc/h, of (0.80 - 1.85) ∆H/Hmax as a function of Nh/dc for qw = (0.073 - 0.249 m2/s) & Re = (2.92 × 105 - 9.96 × 105), dc/h, of (0.80 - 1.85)
Figure 7. ∆H/Hmax as a function of Nh/dc between 4.00 and 10.00, qw = (0.020 - 0.227) m2/s & Re = (8.0 × 104 - 9.08 × 105), flow rate dc/h, of (0.69 - 3.30). For 𝜽 = 26.6°, N = 20, h (cm) = 5. ∆H/Hmax as a function of Nh/dc between 4.00 and 10.00, qw = (0.020 - 0.227) m2/s & Re = (8.0 × 104 - 9.08 × 105), flow rate dc/h, of (0.69 - 3.30). For 𝜽 = 26.6°, N = 20, h (cm) = 5.
∆H/Hmax=(0.30 Nh/dc)0.29N0.20h--0.17 θ--0.32(12)
This equation is valid for 𝜽 of 21.8°, Nh/dc between 5.0 and 11.5, N between 5 and 10, dc/h between 1.00 and 3.16, h (cm) between 5.0 and 10.0. Spearman's rank correlation coefficient = 0.8.
∆H/Hmax=(0.25 Nh/dc)0.29N0.20h--0.17 θ--0.32(13)
This equation is valid for 𝜽 between 18.4° and 21.8°, Nh/dc between 5.0 and 12.0, N between 5 and 40, dc/h and 0.80 and 1.85, h (cm) between 5.0 and 10.0. Spearman's rank correlation coefficient = 0.95.
Figure 8. ∆H/Hmax as a function of Nh/dc between 5.00 and 12.00, qw = (0.095 - 0.180) m2/s, Re = (3.80 × 105 - 7.20 × 105), flow rate, dc/h, of (1.00 - 1.57). For 𝜽 = 21.8°, N = 10, h (cm) = 10.
Figure 9. ∆H/Hmax as a function of Nh/dc between 5.50 and 11.00, qw = (0.059 - 0.158) m2/s, Re = (2.36 × 105 - 6.32 × 105), and flow rate, dc/h, of (0.80 - 1.85). For 𝜽 = 21.8°, N = 20, h (cm) = 5.
∆H/Hmax= (0.1 Nh/dc)0.29N0.20h--0.17 θ--0.32(14)
This equation is valid for 𝜽 between 15.9° and 18.4°, Nh/dc between 21.00 and 30.00, N between 5 and 40, dc/h between 0.6 and 3.58, h (cm) between 5.0 and 10.0. Spearman's rank correlation coefficient = 1.00.
Figure 10. ∆H/Hmax as a function of Nh/dc between 21.00 and 30.00, qw = (0.059 - 0.158) m2/s, Re = (2.36 × 105 - 6.32 × 105), flow rate, dc/h, of (0.80 - 1.85). For 𝜽 = 15.9°, N = 18, h (cm) = 6.
∆H/Hmax=(0.12 Nh/dc)0.29N0.20h--0.17 θ--0.32(15)
This equation is valid for 𝜽 of 15.9°, Nh/dc between 5.0 and 9.5, N between 5 and 18, dc/h between 0.6 and 3.2, h (cm) between 5.0 and 10.0. Spearman's rank correlation coefficient =.0.82.
Figure 11. ∆H/Hmax as a function of Nh/dc for 𝜽 = 15.9°, N = 9, h (cm) = 10, qw = (0.069 - 0.188) m2/s, Re = (2.76 × 105 - 7.52 × 105), dc/h = (0.78 - 1.53).
∆H/Hmax=(0.15 Nh/dc)0.29N0.20h--0.17 θ--0.32(16)
This equation is valid for 𝜽 between 14.6° and 15.9°, Nh/dc between 5.0 and 9.5, N between 5 and 18, dc/h between 0.60 and 3.20, h (cm) between 5.0 and 10.0. Spearman's rank correlation coefficient = 0.79.
Figure 12. ∆H/Hmax as a function of Nh/dc between 4.50 and 6.60, qw = (0.069 - 0.188) m2/s, Re = (2.76 × 105 - 7.52 × 105), flow rate, dc/h, of (0.78 - 1.53). For 𝜽 = 15.9°, N = 9, h (cm) = 1.
∆H/Hmax=(0.25 Nh/dc)0.29N0.20h--0.17 θ--0.32(17)
This equation is valid for 𝜽 of 14.6°, Nh/dc between 5.0 and 9.5, N between 5 and 26, dc/h between 1.27 and 3.55, h (cm) between 5.0 and 10.0. Spearman's rank correlation coefficient = 1.00.
Figure 13. ∆H/Hmax as a function of Nh/dc between 6.25 and 8.60, qw = (0.069 - 0.188) m2/s, Re = (2.76 × 105 - 7.52 × 105), flow rate, dc/h, of (0.78 - 1.53). For 𝜽 = 14.6°, N = 13, h (cm) = 10.
∆H/Hmax=(0.10 Nh/dc)0.29N0.20h--0.17 θ--0.32(18)
This equation is valid for 𝜽 between 14.6° and 15.9°, Nh/dc between 7.5 and 17.0, N between 5 and 18, dc/h between 0.60 and 3.20, h (cm) between 5.0 and 10.0. Spearman's rank correlation coefficient = 0.79.
Figure 14. ∆H/Hmax as a function of Nh/dc between 6.20 and 8.00, qw = (0.05 - 0.234m2/s), Re = (2.0 × 105 - 9.36 × 105), & flow rate, dc/h, of (1.27 - 3.55). For 𝜽 = 14.6°, N = 26, h (cm) = 5. ∆H/Hmax as a function of Nh/dc between 6.20 and 8.00, qw = (0.05 - 0.234m2/s), Re = (2.0 × 105 - 9.36 × 105), & flow rate, dc/h, of (1.27 - 3.55). For 𝜽 = 14.6°, N = 26, h (cm) = 5.
∆H/Hmax= 0.18Nh/dc0.29N0.20h-0.17θ-0.32(19)
This equation is valid for 𝜽 between 3.4° and 8.9°, Nh/dc between 8.5 and 14.0, N between 5 and 21, h (cm) between 3.0 and 14.3. Spearman's rank correlation coefficient = (1.00).
Figure 15. ∆H/Hmax as a function of Nh/dc between 8.50 and 14.00, qw = (0.035 - 0.234) m2/s, Re = (1.40 × 105 - 9.36 × 105), & flow rate, dc/h, of (1.0 - 3.55). For 𝜽 = 8.9°, N = 21, h (cm) = 3/6. ∆H/Hmax as a function of Nh/dc between 8.50 and 14.00, qw = (0.035 - 0.234) m2/s, Re = (1.40 × 105 - 9.36 × 105), & flow rate, dc/h, of (1.0 - 3.55). For 𝜽 = 8.9°, N = 21, h (cm) = 3/6.
4. Discussions
Figure 6 to Figure 15 depict the energy loss rates as a function of the expression of a dam height divided by the critical depth for the measured data, the developed analytical formulation Eq (10) to Eq (19).
The results from the developed models, Eq (10) to Eq (19), compare well with the measured data sets in terms of energy dissipation, with the Spearman coefficients that range between 0.79 and 1.0.
The models are simple and straightforward to use.
From Figures 6-15 energy losses for a given discharge rise progressively with an increasing dam height, which is consistent with
[27]
.
For all the measured data sets, the dimensionless energy dissipation rates distribution compare well with all the flow regime.
In Figures 9-15, the measured data sets in the nappe flow and skimming flow compare well with the developed data sets from Eq (10) through Eq (19) with the Spearman Correlations from 0.79 to 1.00.
5. Conclusion
The figures depict the energy loss rates as a function of the expression of a dam height divided by the critical depth for the measured data as well as the developed analytical formulation. Both the measured and the estimated data distribution also show same traditional concave shape, which is consistent with
[27]
. For all the measured data sets, the results from the proposed models, in terms of the dimensionless energy losses rates distribution compare well with the measured data for all the flow regimes with the Spearman Correlations Coefficients that range from 0.79 to 1.00. All the measured data and estimated data are in good agreement. The models are simple and straightforward to use, and they produce accurate results. The energy losses for given discharges rise progressively with increasing dam height, which is consistent with the current findings.
Abbreviations

DH

Hydraulic Diameter (m)

dw

Equivalent Clear Water Flow Depth (m)

dc

Critical Flow Depth (m)

g

Gravity Constant (m/s2)

H

Total Head (m)

Hdam

Dam Height (m)

Hmax

Maximum Upstream Head (m) Above Chute Toe Hmax = Hdam + 3/2×dc

Hres

Residual Head (m)

h

Vertical Step Height (m)

I

Horizontal Step Length (m)

qw

Water Discharge Per Unit Width (m2/s)

Re

Reynolds Number Defined in Terms of the Hydraulic Diameter: Re = ρw×Uw×DH/μw

Uw

Mean Flow Velocity (m/s): Uw = qw/d

Uw

Average Mean Flow Velocity (m/s) for Stepped Spillway

W

Channel Width (m)

Y90

Characteristic Depth (m) Where the Void Fraction Is -90%

D

Distance (m) Measured Normal to the Invert (or Channel Bed)

ΔH

Total Head Loss (m): ∆H = Hmax – Hres

θ

Angle Between Pseudo-bottom Formed by the Step Edges and the Horizontal

c

Critical Flow Conditions

max

Maximum Value

mean

Mean Signal Component

w

Water Properties
Acknowledgments
I acknowledge the fruitful discussions and contributions of Prof Hubert Chanson of the Department of Civil Engineering, The University of Queensland, Brisbane QLD 4072, Australia.
Author Contributions
Okechukwu Ozueigbo: Conceptualization, Data curation, Formal Analysis, Investigation, Resources, Software
Jonah Agunwamba: Conceptualization, Investigation, Project administration, Resources, Supervision, Validation
Conflicts of Interest
The authors declare no conflicts of interest.
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    Ozueigbo, O., Agunwamba, J. (2025). New Energy Loss Expression for Stepped Spillways with Channel Slopes 26.6° and Below. Journal of Civil, Construction and Environmental Engineering, 10(1), 16-26. https://doi.org/10.11648/j.jccee.20251001.12

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    Ozueigbo, O.; Agunwamba, J. New Energy Loss Expression for Stepped Spillways with Channel Slopes 26.6° and Below. J. Civ. Constr. Environ. Eng. 2025, 10(1), 16-26. doi: 10.11648/j.jccee.20251001.12

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    Ozueigbo O, Agunwamba J. New Energy Loss Expression for Stepped Spillways with Channel Slopes 26.6° and Below. J Civ Constr Environ Eng. 2025;10(1):16-26. doi: 10.11648/j.jccee.20251001.12

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  • @article{10.11648/j.jccee.20251001.12,
  author = {Okechukwu Ozueigbo and Jonah Agunwamba},
  title = {New Energy Loss Expression for Stepped Spillways with Channel Slopes 26.6° and Below},
  journal = {Journal of Civil, Construction and Environmental Engineering},
  volume = {10},
  number = {1},
  pages = {16-26},
  doi = {10.11648/j.jccee.20251001.12},
  url = {https://doi.org/10.11648/j.jccee.20251001.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.jccee.20251001.12},
  abstract = {A stepped spillway is a hydraulic structure built at storage and detention dams to discharge flood water that cannot be safely kept in the reservoir. It was created to minimize the kinetic energy that would have otherwise produced dangerous scour at the natural river bed beneath the spillway. They discharge this energy in floodwater using their stepping nature. Several studies show the detrimental consequences of falling water's kinetic energy on the river bed underneath the structure. Only a handful of these studies, however, have evaluated the impact of energy losses caused by stepped spillways with channel slopes 26.6° or less. As a result, there are gaps in the rules and recommendations for designers of stepped spillways with channel slopes below 26.6°. Furthermore, the existing models for forecasting energy dissipation in stepped spillways for channels of all slopes include a parameter, the friction factor, f, which is difficult to estimate with precision, leading to its subjective estimation by those involved in the stepped spillway design. The goal of this study is not only to provide designers with design recommendations and information for stepped spillways with channel slopes below 26.6°, but also to eliminate the 'troublesome' frictional factor; f. Using phase-detection intrusive probes, air-water flow tests were carried out in nappe, transitional, and skimming flows on a stepped spillway with channel slope below 26.6° in a large facility. New expressions for evaluating energy losses in stepped spillways with slopes 26.6° or less are developed. Regarding energy dissipation, the data from the latest models compare well with the measured data, with high coefficients of correlation that range between 0.95 and 0.99. All of the measured data and the estimated data are in good agreement. The models are simple to use.},
 year = {2025}
}

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  • TY  - JOUR
T1  - New Energy Loss Expression for Stepped Spillways with Channel Slopes 26.6° and Below
AU  - Okechukwu Ozueigbo
AU  - Jonah Agunwamba
Y1  - 2025/01/22
PY  - 2025
N1  - https://doi.org/10.11648/j.jccee.20251001.12
DO  - 10.11648/j.jccee.20251001.12
T2  - Journal of Civil, Construction and Environmental Engineering
JF  - Journal of Civil, Construction and Environmental Engineering
JO  - Journal of Civil, Construction and Environmental Engineering
SP  - 16
EP  - 26
PB  - Science Publishing Group
SN  - 2637-3890
UR  - https://doi.org/10.11648/j.jccee.20251001.12
AB  - A stepped spillway is a hydraulic structure built at storage and detention dams to discharge flood water that cannot be safely kept in the reservoir. It was created to minimize the kinetic energy that would have otherwise produced dangerous scour at the natural river bed beneath the spillway. They discharge this energy in floodwater using their stepping nature. Several studies show the detrimental consequences of falling water's kinetic energy on the river bed underneath the structure. Only a handful of these studies, however, have evaluated the impact of energy losses caused by stepped spillways with channel slopes 26.6° or less. As a result, there are gaps in the rules and recommendations for designers of stepped spillways with channel slopes below 26.6°. Furthermore, the existing models for forecasting energy dissipation in stepped spillways for channels of all slopes include a parameter, the friction factor, f, which is difficult to estimate with precision, leading to its subjective estimation by those involved in the stepped spillway design. The goal of this study is not only to provide designers with design recommendations and information for stepped spillways with channel slopes below 26.6°, but also to eliminate the 'troublesome' frictional factor; f. Using phase-detection intrusive probes, air-water flow tests were carried out in nappe, transitional, and skimming flows on a stepped spillway with channel slope below 26.6° in a large facility. New expressions for evaluating energy losses in stepped spillways with slopes 26.6° or less are developed. Regarding energy dissipation, the data from the latest models compare well with the measured data, with high coefficients of correlation that range between 0.95 and 0.99. All of the measured data and the estimated data are in good agreement. The models are simple to use.
VL  - 10
IS  - 1
ER  -

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Author Information

  • Okechukwu Ozueigbo

    Department of Civil Engineering, University of Nigeria, Nsukka, Nigeria

    Research Fields: Civil Engineering, Water Resources, Environmental Analysis, Soil Erosion

    Contact Email

    http://orcid.org/0000-0003-1249-5648

  • Jonah Agunwamba

    Department of Civil Engineering, University of Nigeria, Nsukka, Nigeria

    Research Fields: Civil Engineering, Water Resource, Environmental Analysis, Soil Erosion

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