New Equations for Rate of Energy Dissipation of a Stepped Spillway with Slope less than Critical and Specific Step Height

—Stepped spillways disperse energy in floodwater by utilizing their stepping nature. Several researchers investigated the hydraulic and geometric relationships of the stepped spillway with a dam slope above 26.6 o , which resulted in energy dissipation. But a few studied stepped spillways with a dam slope of 26.6 o with a step height of 0.61 m. This move led to a lack of knowledge and rules for the designers of stepped spillways with slope of 26.6 o and a step height of 0.61 m. The authors reviewed the researchers' publications on horizontal stepped spillways with dam slopes of 26.6 o with step height of 0.61 m conducted in transition and skimming flows in large-size facilities with phase-detection intrusive probes. They retrieved data sets from them, which they took into account to create two novel equations for energy dissipation in stepped spillways: one for nappe flow and one for skimming flow. The results showed that the measured data sets compared well with the data sets calculated from the two empirical models.


INTRODUCTION
In high velocity free-surface turbulent flows, large quantities of air bubbles/pockets move across the air-water interface being entrapped (air entrainment) in the water current and then are carried away within the flowing fluid and, eventually, exchanged back to the air flowing above the freesurface (Halbronn et al., 1953;Jevdjevich and Levin, 1953) [1,2] . The resulting air-water flow, or self-aerated flow, is a mixture of air and water consisting of both air packets within water and water droplets surrounded by air (Chanson, 1997) [3]. Aerated flows are used in a variety of applications including chemical, civil, environmental, mechanical, mining, nuclear, and water engineering (Rao and Kobus, 1974) [4].
.Aeration is a major feature of stepped spillway flows that is important for re-oxygenation applications such as waste water treatment, chemical applications, and rivers under environmental stress. They range in size from extremely little to quite huge. Self-aerated flows are frequently observed in water and environmental systems, such as mountain streams, storm waterways, culverts, dropshafts, spillway chutes, tidal channels, and stilling basins, where aeration is largely uncontrolled (Wood, 1991) [5], and such flows are also relevant to water quality, sediment transport, ecological sustainability, and, ultimately, environmental integrated assessment within such systems. Physical modeling is used to examine aerated flows experimentally.
Physical models for such experimental studies must be created with accuracy. Alternatively, size effects may have an impact on extrapolating experimental results to full-scale prototype structures (Kobus, 1984) [6]. Aerated flows are usually investigated using a Froude similitude; however the dynamic similitude in a geometrically comparable model incorporates extra dimensionless parameters. Shear flows are governed by viscous effects, whereas surface tension forces drive the mechanisms of bubble breakdown and coalescence (Chanson and Gualtieri, 2008) [7]. As a result, dynamic similitude in aerated flow demands that the Froude, Reynolds, and Morton values in the prototype and its model be equal. Yet, unless working at full-scale, this is hard to execute using geometrically identical models.
A Froude and Morton similitude can be implemented in the prototype and in the laboratory using the same fluids, namely air and water, but the model Reynolds number cannot be as great as in the prototype, resulting in viscous-scale effects in small-size models (Rao and Kobus, 1974;Wood, 1985;Chanson, 2009;Ozueigbo and Agunwamba, 2022) [4, 8 -10].
. Turbulence near the air-water interface drives the exchange of air across the air-water contact. When the turbulent shear stress exceeds the surface tension force per unit area opposing interfacial breakdown, free-surface breakup and air entrainment occur (Hino, 1961;Ervine and Falvey, 1987) [11,12]. When some air is entrained inside the majority of the flow, air pockets break up when the tangential shear stress exceeds the capillary force per unit area (Chanson, 2009) [13].
As bubbles and droplets are advected by the flow, particle collisions may cause them to coalesce, and air detrainment due to buoyancy occurs. Bubble clustering may occur as a result of the intricate interactions between entrained air and turbulence.
A bubble cluster is described as a group of two or more bubbles separated from the other bubbles before and after the cluster. Previous research has shown that clustering analysis can provide some useful information about the interplay between turbulence and bubbly flow Chanson 2010, 2013;Wang et al., 2015a) [14 -16].
The use of typical mono-phase flow instrumentation is not feasible due to the three-dimensional character of highvelocity air-water flows with multiple air-water contacts.
In stepped spillways, phase-detection intrusive probes are frequently used, and trials have been effective with optical fiber probes [17], conductivity probes [18 -21], air probes, and a back flushing Pitot-Static tube [22].
Intrusive probes with phase detection can measure characteristic air-water flow properties such as void fraction, bubble count rate, and interfacial velocity, as well as characteristic aeration parameters such as the mean air concentration Cmean: where Y90 is the characteristic flow depth and C = 0.9. The depth averaged characteristic void fraction metric representing aeration inside a cross-section is the mean air concentration Cmean.
The basis of the conductivity probe is the difference in resistance between air and water, which generates an immediate voltage signal. The time averaged local air concentration or void fraction C, the number of air-to-water (or water-to-air) voltage shifts described as bubble count rate F, and the air bubble and water droplet chord sizes can all be calculated from a single sensor's signal using a threshold technique.
The cross-correlation analysis of the data from a double-tip conductivity probe with longitudinal spacing between the two probe sensors leads to the local time-averaged interfacial velocity V. [23] and [24] have more information on signal processing techniques.

INSTRUMENTATION AND EXPERIMENT PROGRAM
The air-water flow experiments were carried out using air probes, backup flushing Pitotstatic tubes, and Point Gages at the step edges of Steps 1, 4, 8, 12, 16, and 20. VHS videotapes and 35 mm prints were used to obtain visual documentation.
The experimental program was designed to assess the hydraulic properties of flow through a simulated roller compacted concrete (RCC) stepped spillway under near-prototype circumstances.
The experimental facilities were built on a large scale to reduce viscous scale effects on microscopic air-water flow processes in high-velocity free-surface flows [25]. A water supply pipeline, baffled head box, entrance/transition, chute, stilling basin, and outlet works compose the test facility.
The concrete chute is roughly 34.0 m long, 3.0 m wide, and 1.5 m deep, with a slope of 1:2 (vertical: horizontal) and a total height of 15 m. The width of the chute was reduced to 1.2 m for the current study by employing a dividing wall ( Figure 1) In addition; flat-lashboards were placed on the walls of the test portion of the chute, extending the depth to 2.1 m, providing additional freeboard. Plexiglass windows, 1.2 m by 1.2 m were installed at five locations in the dividing wall to provide observation of flow in the chute [25].  The variations in flow regimes were consistent with the data in the literature (Chanson 2001, Chanson andToombes 2004) [26,27]. The developing flow measurements in this study were concentrated on the skimming flow regime (dc/h > 0.73). Table 1 summarizes the experimental flow conditions and compares them to previous extensive experimental research.

Flow Observations
Flow conditions were observed for each test series across a variety of discharges. The flow at the overtopping crest and along the spillway length was documented using videotape recordings and still pictures. Flow conditions were watched and recorded using five plexiglass windows installed along the spillway's sidewall.
It was observed in this study that Nappe flow regime occurred for unit discharges below 0.46 m 2 /s, Transition flow regime occurred for unit discharges between flows between 0.46 m 2 /s and 0.93 m 2 /s, and skimming flow regime for unit discharges above 0.93 m 2 /s

Formulation of the Model
The authors used Figure 3 along with Eq (7) to formulate Eq (8) and Eq (9).
Where ∆H/Hmax is the energy loss ratio, Hmax is the maximum available height, N is the number of spillway steps, h is the height of the spillway steps, θ is the spillway channel slope. ∆H is head loss, dc = critical water depth (m), αo, = constant, α1 and α2 = coefficients

Model Calibration
They divided the measured data sets into two parts. They used one portion and multiple regression analysis to calibrate expression (7). The calibration yielded the values of constant αo with the coefficients α1, and α2, which were then substituted in the expression to give the developed models in Eq (8) and Eq (9).

Model Verification
The authors used the remaining data sets, known as verification data sets, to evaluate the models' performance (interpolation).
If the model describes verification data well, then the model describes the real system and this is known the interpolation aspect [28].

Developed model for the nappe/transition flow regime
Eq 8 is valid for a stepped spillway with a channel slope, = 26.6 o , step height, h = 0.61 m, number of steps, N not more 20, dc/h not more than 0.73, and Nh/dc not more than 27.0.
The predicted data sets calculated with Eq (8) compared well with the measured data sets with the Pearson correlation of 0.99 (Figures 9 and 10).

Developed model for the skimming flow regime
The predicted data sets calculated with Eq (9) compared well with the measured data sets with the Pearson correlations of between 0.97 and 0.99 (Figures 11 to 15). Eq 9 is valid for a stepped spillway with a channel slope, = 26.6 o , step height, h = 0.61 m, number of steps, N not more 20, dc/h between 0.73 and 1.53, and Nh/dc not more than 20.0.
In order to explore the impacts of discharge on energy dissipation in the skimming flow regime, Nh/dc is given as a function of the dimensionless energy dissipation in Figures 9  through 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 compared well with Eq (8) for nappe flow and with Eq (9) for skimming flow.
In Figures 9 through 15, the measured data sets in the nappe flow and skimming flow compared well with developed data sets from Eq (8) and Eq (9) with Pearson Correlation of 0.99 for nappe flow and between 0.97 and 0.99 for skimming flow.

CONCLUSION
In both the nappe flow and skimming flow regimes, the measured data sets, in terms of energy dissipation, compared well with Eq (8) and Eq (9). The Pearson correlation for the nappe flow was 0.99 and for the skimming flow was between 0.97 and 0.99. For a given discharge, energy losses increase sharply with an increasing dam height, which is in line with [23,24,25,26].

ACKNOWLEDGEMENTS
We acknowledge the fruitful discussions and contributions of Prof Hubert Chanson of the Department of Civil Engineering, The University of Queensland, Brisbane QLD 4072, Australia.
We also acknowledge the kind permission of Ruff and Ward (2002) to reproduce the measured data sets used in this study.

LIST OF SYMBOLS
The following symbols are used in this report: C void fraction defined as the volume of air per unit volume of air and water; it is also called air concentration or local air content; DH hydraulic diameter (m); d equivalent clear water flow depth (m); dc critical flow depth (m): ; g gravity constant (m/s 2 ); H total head (m); Hdam dam height (m); Hmax maximum upstream head (m) above chute toe: