The unstable surface wave on a liquid sheet produced by an air-blast atomizer during primary breakup process was investigated by numerical simulation. The results of simulation were verified by comparison of primary breakup time and breakup length with accessible experimental data reported in technical papers. The frequency characteristics of stream-wise unstable wave at different axial locations were investigated by applying Discrete Fourier Transform (OFT). It was found that when there is no disturbance induced by internal flow, there is no specific frequency which is favored by shear instability near the nozzle exit, and the characteristic frequency of the dominant wave decreases along stream-wise direction due to the decrease of relative velocity. By applying Discrete Particle Method (OPM), the motion of fluid particles inside the liquid sheet was able to be tracked, and the Lagrangian characteristics of fluid particles can be partially revealed. The growth of stream-wise unstable wave was found to possess strong spatial characteristics by investigating the pathlines and streaklines of fluid particles. A rough evaluation for the stream-wise speed of fluid particles and the propagation velocity of unstable wave showed that fluid particles move faster than unstable wave in stream-wise direction, thus, relative motion exists between fluid particles and stream-wise wave. This relative motion could lead to huge acceleration of fluid particles, which could trigger Rayleigh-Taylor (RT) instability to induce transverse disintegration. Some complex behaviors of fluid particles inside the liquid sheet were observed, e.g. eddy-like structures formed by fluid particles.