https://orcid.org/0000-0002-3542-8416
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ABSTRACT
Although stepped chute flows have been studied for a variety of chute geometries and flow conditions, reports on air–water flow development in the fully developed aerated region are not as comprehensive as those in the nonaerated and developing aerated regions. In this paper, skimming air–water flows in a large stepped chute (1V:2.5H) were studied comparatively for two step heights, showing different stages for velocity and aeration development downstream the inception point. The mean air concentration reached a constant level while the bubble frequency kept increasing slowly in the fully developed region. The developed mean air concentration was related to the sloping angle, step height and flow discharge. The results provided possible prediction of maximum bubble frequency and its longitudinal evolution, and a glance at the scale effect on it. The free-surface fluctuations and splash height were presented, with particular attention to the effects of step height on the turbulence and aeration modification.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notation
| amean | = | mean specific air–water interfacial area (m−1) |
| C | = | air concentration (–) |
| Ce | = | mean air concentration in fully developed aerated region (–) |
| CFmax | = | air concentration corresponding to maximum bubble frequency (–) |
| Cmean | = | mean air concentration (–) |
| d | = | equivalent clear-water depth (m) |
| d1 | = | initial water depth at x = 0 (m) |
| dc | = | critical flow depth (m) |
| de | = | equivalent clear-water depth in fully developed aerated region (m) |
| di | = | flow depth at inception point (m) |
| dmax | = | maximum splash height (m) |
| fe | = | equivalent air–water friction factor (–) |
| F* | = | roughness Froude number (–) |
| F | = | bubble frequency (Hz) |
| Fmax | = | maximum bubble frequency (Hz) |
| g | = | gravity acceleration (m s−2) |
| h | = | step height (m) |
| Hres | = | residual energy (m) |
| ks | = | step roughness height (m) |
| Li | = | distance from x = 0 to the aeration inception point (m) |
| m1 | = | hydrostatic energy correction coefficient (–) |
| m2 | = | kinetic energy correction coefficient (–) |
| N | = | power law exponent (–) |
| qw | = | specific water discharger (m2 s−1) |
| Re | = | Reynolds number (–) |
| Re* | = | roughness Reynolds number (–) |
| ur | = | bubble rise velocity (m s−1) |
| V | = | flow velocity (m s−1) |
| Vc | = | critical flow velocity (m s−1) |
| Vmean | = | mean velocity (m s−1) |
| V90 | = | characteristic interfacial velocity at normal position where C = 0.9 (m s−1) |
| x | = | distance parallel to pseudo-bottom (m) |
| y | = | distance normal to pseudo-bottom (m) |
| y90 | = | characteristic flow depth for C = 0.9 (m) |
| y95 | = | characteristic flow depth for C = 0.95 (m) |
| y98 | = | characteristic flow depth for C = 0.98 (m) |
| y99 | = | characteristic flow depth for C = 0.99 (m) |
| α | = | chute slope (°) |
| η | = | free-surface elevation above pseudo-bottom (m) |
| η′ | = | characteristic free-surface fluctuation magnitude (m) |
| ν | = | kinematic viscosity of water (m2 s−1) |
Subscripts
| c | = | critical flow condition |
| e | = | quantity equilibrium fully developed aerated region |
| i | = | quantity at inception point |
| max | = | maximum quantity |
| mean | = | depth-averaged quantity |
| 90, 95, 98, 99 | = | quantity corresponding to C = 0.9, 0.95, 0.98, 0.99 |
Acknowledgements
The authors thank Yongcheng Water Conservancy Bureau (Henan, China) for the strong support during the model construction and experiments. They thank Mr Wei Sang and Mr Wei Liu for their help with the field experiment coordination.