A method is provided to measure a distribution of axial velocities and a flowrate in a pipe or a vessel. The method comprises selecting a single cross-section at a stable-flow segment in a pipe or a vessel, installing a plurality of acoustic wave sensors along a peripheral wall of the pipe or the vessel to form a plurality of effective sound wave paths; measuring sound wave travelling time on each sound wave path; substituting the measured sound wave travelling time into the model formula based on a sound path refraction principle for reconstruction calculation to obtain a distribution of axial velocity in the measured cross-section of the pipe or the vessel, u(x,y); and integrating the distribution of the axial velocity u(x,y) along the cross-section to obtain a flow rate. A system is also provided to measure an axial velocity distribution and a flow rate in a pipe or a vessel.