Yes / 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. / This work was supported by National Natural Science Foundation of China [grant numbers 51939007, 51979183]; Sichuan Province Science and Technology support program [grant number 2019JDTD0007].