https://orcid.org/0000-0002-9513-6802
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Abstract
Self-aeration in open channel flows occurs owing to free surface air entrainment. Self-aeration development and fully cross-sectional distribution of air concentration are not thoroughly understood. In the present study, an analytical solution for the averaged cross-sectional air concentration in the gradually varying region is established using a simplified mechanism of free surface air entrainment. For a fully cross-sectional distribution of air concentration affected by the channel bottom, a model of a diffusion region without wall restraint is proposed, and two situations are classified based on averaged cross-sectional air concentration. Good agreement between measured data and calculations is obtained, and the computational accuracy of the air concentration distribution near the wall is improved. The results reveal that the channel slope determines the air entrainment quantity, while water flow discharge determines the self-aeration evolution distance. The solutions for the averaged cross-sectional air concentration and the effect of the bottom wall on air diffusion promote air–water flow applications in hydraulic engineering practices.
Acknowledgements
The authors gratefully acknowledge the suggestions and technical comments provided by the reviewers. These are helpful to improve the manuscript. The paper was completed within the research projects funded by the National Natural Science Foundation of China [grant nos. 51979183; 51939007], and Sichuan Science and Technology Program [grant no. 2019JDTD0007].
Disclosure Statement
No potential conflict of interest was reported by the authors.
Notation
| a | = | constant coefficient (–) |
| b | = | constant coefficient (–) |
| C | = | local air concentration (–) |
| Cmean | = | cross-sectional mean air concentration (–) |
| CI | = | air concentration obtained by equation (–) |
| Cadd | = | air concentration in the diffusion region without wall restraint (–) |
| D | = | diameter of the turbulent eddy (m) |
| D* | = | diffusivity coefficient (–) |
| D′ | = | air turbulent diffusivity (–) |
| d | = | local mixture flow depth (m) |
| d0 | = | flow depth at the inception cross-section (m) |
| dab | = | air chord length (m) |
| d* | = | non-aerated water flow depth (m) |
| Ee | = | turbulent kinetic energy at the local free surface (J) |
| g | = | gravity acceleration (m s–2) |
| K′ | = | constant coefficient (–) |
| m | = | correction coefficient (–) |
| qa | = | air discharge per unit width (m2 s–1) |
| qw | = | water flow discharge per unit width (m2 s–1) |
| sC | = | critical length scale of the local surface deformation (m) |
| V | = | local mean flow velocity (m s–1) |
| V0 | = | flow velocity at the inception cross-section (m s–1) |
| VC | = | critical flow velocity of air entrainment (m s–1) |
| VC–init | = | critical flow velocity at the inception point (m s–1) |
| Ve | = | characteristic air entrainment velocity (m s–1) |
| Vr | = | characteristic air bubble rise velocity (m s–1) |
| vτ | = | friction velocity near the free surface area (m s–1) |
| v′ | = | turbulent eddy fluctuation velocity in the vertical direction (m s–1) |
| x | = | streamwise direction along the channel bottom (m) |
| y | = | vertical direction to the channel bottom (m) |
| y90 | = | flow depth where local air concentration is 0.9 (–) |
| yD | = | diffusion region without the bottom wall restraint (m) |
| yI | = | initial aerated region (m) |
| α | = | channel bottom slope (°) |
| σ | = | surface tension coefficient of water (N m–1) |
| ζ | = | correction coefficient (–) |
| ρ | = | water density (kg m–3) |
| κ | = | von Karman constant (–) |
| ΔE | = | residual kinetic energy (J) |
| (Eσ)C | = | critical limitation of surface tension energy (J) |