Air-entrainment in hydraulic intakes with a vertical pipe: The mechanism and influence of pipe offset

•The air-entrained vortex formation procedures are consistent despite of the different pipe positions.•The vortex movement is dominated the vorticity tilting effect.•For the pipe-offset cases, the natural asymmetry characteristics makes the entrainment formation more easily and quickly.•For the pipe-centered cases, the asymmetric flow arises due to the asymmetric flow separation near pipe surface.