Hydraulics of the Developing Flow Region of Stepped Spillways. II: Pressure and Velocity Fields

Abstract

In skimming flow on a stepped spillway, the upstream flow motion is non-aerated and a turbulent boundary layer develops until the outer edge of the boundary layer interacts with the free-surface: that is, at the inception point of air entrainment. Herein new experiments were performed in the developing flow region on a large size 1V:1H stepped spillway model with step height h = 0.10 m. The flow properties in the developing flow region were carefully documented. In the developing boundary layer, the velocity distributions followed a 1/4.5 th power law at step edges. Detailed velocity and pressure measurements showed some rapid flow redistribution between step edges and above step cavities. The application of the momentum integral equation indicated an average friction factor of 0.18, close to the observed air-water flow friction factor of 0.23, suggesting that the spatially-averaged dimensionless shear stress was comparable in the developing flow and fully-aerated flow regions.

Case Study on Application of the Step with Non-Uniform Heights at the Bottom Using a Numerical and Experimental Model

“…The results indicated that the energy dissipation rate of uniform and non-uniform step height was basically the same [7]. Further studies of air entrainment and pressure and velocity fields were conducted by Pegram et al [8], Ostad et al [9], Meireles et al [10], and Zhang G et al [11].…”

“…The negative pressure zone was more likely to induce cavitation according to Equations (11) and (12). Therefore, after the non-uniform-height steps are arranged on the curved spillway, an aeration device should be set in front of the steps to avoid cavitation.…”

Hydraulic Jump on a Partially Vegetated Bed

“…With the flume bed fully covered by the three roughness configurations (i.e., bare grate mat, grate mat with low‐density vegetation cover, and grate mat with high‐density vegetation cover), the flow velocity profiles were measured at two vertical cross sections in the developing flow region using a Prandtl‐Pitot tube. Applying the momentum integral equation to the section between the two velocity profiles yields (Zhang & Chanson, 2016): $$$\frac{\partial }{\partial x}\left({{V}_{0}}^{2}{\delta }_{2}\right)+{V}_{0}{\delta }_{1}\frac{\partial {V}_{0}}{\partial x}=\frac{{f}_{M}}{8}{{V}_{0}}^{2}$$$ where f M = the friction factor representing the mean dimensionless boundary shear stress between the two cross sections, V 0 = the free‐stream velocity, δ 1 = the displacement thicknesses, and δ 2 = the momentum thicknesses: $$${\delta }_{1}=\underset{0{}}{\overset{\delta }{\int }}\left(1-\frac{V}{{V}_{0}}\right)\mathrm{d}y$$$ $$${\delta }_{2}=\underset{0{}}{\overset{\delta }{\int }}\frac{V}{{V}_{0}}\left(1-\frac{V}{{V}_{0}}\right)\mathrm{d}y$$$ where V = the longitudinal velocity measured using the Prandtl‐Pitot tube, and δ = the boundary layer thickness from y = 0 to the upper position where V = 0.99 V 0 . Equations allow a calculation of f M from the velocity distributions.…”

Prioritizing Design Parameters for Stepped Chutes and Shear Stress Distribution

“…T analyzed data of stepped chutes with slopes of 30° and 26.6° showed similar beh of the pools on the chute performance. Other publications that have presented de hydraulic analysis on stepped chutes include Gonzalez [10], Meireles and Mato Bung [12], and Zhang and Chanson [13,14]. The focus of these studies varies in ter the effects produced from changes to some geometric parameters or to changes in A comprehensive analysis of the effects produced from changes in multiple geom parameters or the influence that parameters have on each other as well as overall p mance has not been well addressed.…”

“…The re-analyzed data of stepped chutes with slopes of 30 • and 26.6 • showed similar behavior of the pools on the chute performance. Other publications that have presented detailed hydraulic analysis on stepped chutes include Gonzalez [10], Meireles and Matos [11], Bung [12], and Zhang and Chanson [13,14]. The focus of these studies varies in terms of the effects produced from changes to some geometric parameters or to changes in slope.…”