In this numeric study, the two-phase flow phenomenon is considered in which the high speed gas stream is blasting the low speed liquid jet coming out from the central nozzle in a coaxial setup. The primary focus of this research is to study the primary breakup region and the liquid jet unsteadiness during coaxial airblast atomization. In this study, the flow is numerically modeled using the ANSYS Fluent. The DES (detached eddy simulation) technique is adopted to compute the gas-liquid flow, while the Volume of Fluid (VoF) technique is used to capture the air-water interface. The results of this study are compared with the in-house experimental data. Jet instabilities are characterized at different axial locations downstream of the atomizer exit. The qualitative comparison of simulation and experiment is done for the structure of the liquid jet core for the given injector operating condition. The mean and fluctuations of the liquid jet breakup are computed and compared against the experimental data. The simulation targets to gain further understanding of the flapping instabilities of the liquid jet and its role on the unsteady jet breakup process.