Characteristics of Skimming Flow over Stepped Spillways

This paper presents the results of a laboratory study on the characteristics of fully developed skimming flow in a large model of a stepped spillway for two slopes, for a range of discharges with yc∕h in the range of 0.7–4.4. Fully developed aerated flow on a stepped spillway can be divided into lower and upper regions, similar to those for self-aerated flow in steep chutes. The air concentration distributions in these two regions agree with the equations developed by Straub and Anderson for flow in steep chutes. It was found that the depth at which the air concentration is equal to 90% can be considered as the depth of aerated flow on stepped spillways. In the lower region, the velocity profiles were described by the Karman-Prandtl equation for rough turbulent flow when an equivalent bed roughness was used. A correlation was developed for the skin friction coefficient to predict the Reynolds shear stress at the virtual bed of the stepped spillway. It was found that the relative energy loss in the stepped spillway is in the range of 48–63%. It was also found that the mean air concentration on a stepped spillway is larger than that in a corresponding chute.     

Hydraulics of Skimming Flow on Modeled Stepped Spillways

Two sets of models of stepped spillways of slope 1 upon 0.6 with the same crest-shape and with a range of step sizes (0.25–2.0 m on the 1:10 scale and 0.5–2.0 m on the 1:20 scale models) were built and studied. In the range of prototype unit flows considered (0.8–3.8 m2∕s on the 1:10 scale models and 1.8–21.7 m2∕s on the 1:20 models), the residual specific energy is independent of these step sizes, at positions where measurements were made on the spillways (from 30 m below the crest) once fully aerated skimming flow is established. The residual specific energy at the toe of a 50-m-high (or higher) stepped spillway, within the range of step heights tested, is <60% of the residual specific energy at the same level on a similar smooth spillway experiencing flows up to 20 m2∕s; at these flows and at this height, the stepped spillway was found to be in equilibrium. The reduction of specific energy is lower than that expected from previous studies; these findings are likely to impact on the design of stepped spillways.     

Skimming Flow in the Nonaerated Region of Stepped Spillways over Embankment Dams

Traditionally, research on stepped spillway hydraulics has been focused on the air-water flow region but for the hydraulic design of small embankment dams experiencing relatively large overtopping flows, the nonaerated region can be very important. Empirical formulas are presented for predicting skimming flow properties upstream of the point of inception of air entrainment for 1V:2H sloping stepped spillways, and the location and flow depth at the point of inception. Particular emphasis is placed on the clear-water depth, velocity distribution, and the energy dissipation characteristics in the developing nonaerated flow region. The velocity distribution is well described by a power law. The normalized clear-water depth and the normalized specific energy varied with the relative distance along the spillway and the effect of the normalized critical depth was negligible. Finally, the rate of energy dissipation was small, which has direct implications for the design of the downstream energy dissipator.     

Two-Phase Flow Characteristics of Stepped Spillways

An experimental study on a large model flume with fiber-optical instrumentation indicated that minimum Reynolds and Weber numbers of about 105 and 100, respectively, are required for viscosity and surface tension effects to become negligible compared to gravitational and inertial forces expressed by Froude similitude. Both the location of and the flow depth at the inception point of air entrainment can be expressed as functions of a so-called roughness Froude number containing the unit discharge, step height and chute angle. The depth-averaged air concentration is found to depend only on a normalized vertical distance from the spillway crest and the chute angle for chute slopes ranging from embankment to gravity dam spillways. Air concentration profiles can be expressed by an air bubble diffusion model. The pseudobottom air concentration allows the assessment of the cavitation risk of stepped chutes and is approximated by a regression function. Finally, a new velocity distribution function is presented consisting of a power law up to 80% of the characteristic nondimensional mixture depth, and a constant value above.     

Flow Characteristics of Skimming Flows in Stepped Channels

Skimming flows in stepped channels are systematically investigated under a wide range of channel slopes (5.7°?θ?55°). The flow conditions of skimming flows are classified into two flow regimes, and the hydraulic conditions required to form a quasi-uniform flow are determined. An aerated flow depth of a skimming flow is estimated from the assumption that the residual energy at the end of a stepped channel coincides with the energy at the toe of the jump formed immediately downstream of the stepped channel. In a quasi-uniform flow region, the friction factor of skimming flows is represented by the relative step height and the channel slope. The friction factor for the channel slope of θ=19° appears to have a maximum. The residual energy of skimming flows is formulated for both nonuniform and quasi-uniform flow regions. Further, a hydraulic-design chart for a stepped channel is presented.     

Developing Flow Region and Pressure Fluctuations on Steeply Sloping Stepped Spillways

The hydrodynamic pressure field is important for the design and safety of steeply sloping stepped spillways, which are typically designed for considerably lower maximum specific discharges than smooth spillways. The hydraulic performance of stepped spillways at high velocities may compromise its use due to major concern with safety against cavitation damage. Hydraulic model investigations were conducted in different large-size stepped chutes to characterize the nonaerated flow region which is potentially prone to cavitation damage and the pressure field acting on the step faces. The clear water depths and energy dissipation in the developing flow region are described in terms of integral measures of the turbulent boundary layer. Expressions for the location of and the flow depth at the inception point of air entrainment are derived. Pressure distributions on the horizontal and vertical faces of the step along the spillway are presented. Measurements indicated a different behavior of the pressure field in the aerated and nonaerated flow region. The mean and fluctuating pressure coefficients along the spillway are approximated by a regression function. The vertical face near the outer step edge close to the inception point of air entrainment is identified as a critical region for predicting cavitation inception in flow over stepped spillways. From the analysis of the pressure fluctuations in that region a maximum velocity of 15 m/s is proposed as a criterion to avoid extreme negative pressures in typical prototype steeply sloping stepped spillways, eventually leading to the occurrence of cavitation in the nonaerated flow.     

Bottom Aeration of Stepped Spillways

The upstream reach of stepped spillway flows may become prone to cavitation damage for large specific discharges because of the absence of air close to its bottom, until the point of bottom self-aeration is reached. This study considers the effect of two aerator types located at the first vertical step face to add air to the chute bottom. Compared to standard stepped spillway flow, considerable differences may be observed closely downstream of the aerator, whereas no significant deviations occur in the far-downstream chute reach. The characteristics of bottom air concentration curves on stepped chutes are investigated with an experimental approach. The results are then compared with flows on both smooth chutes and standard stepped chutes. The data analysis results in design equations that may be applied to usual stepped spillways of chute angles around 50°. In addition, a sinusoidal variation of air concentration about the average value as a novel phenomenon is described relating to a local instability in the minimum bottom air concentration reach.     

Hydraulic Design of Stepped Spillways

An experimental study on a large model flume using fiber-optical instrumentation indicated that the onset of skimming flow is a function of critical depth, chute angle, and step height. Uniform mixture depths that determine the height of chute sidewalls and uniform equivalent clear water depths are described in terms of a roughness Froude number containing unit discharge, chute angle and step height. The spillway length needed to attain uniform flow is expressed as a function of critical depth and chute angle. The flow resistance of stepped spillways is significantly larger than for smooth chutes due to the macro roughness of the steps. The friction factor for uniform aerated flow is of the order of 0.1 for typical gravity dam and embankment dam slopes, whereas the effect of relative roughness is rather small. The energy dissipation characteristics of stepped spillways and the design of training walls are also discussed. The paper aims to focus on significant findings of a research program and develops design guidance to lessen the need for individual physical model studies. A design example is further presented.