Axisymmetric sudden-expansion geometry of a co-flowing methane-air diffusion flame is considered to investigate the effect of air inlet conditions on $NO_x$ formation, flow field and temperature distribution using the k-${varepsilon}$ turbulence and ${eta}$-PDF combustion model. The predicted results are in acceptable agreement with the published experimental and numerical data. The obtained results show that increasing air turbulence intensity results in considerable decrease in NO formation. Increasing the inlet angle of the air causes the NO formation to decrease due to raising vorticity strength. As a new index, the mass-averaged integral of vorticity magnitude is introduced to investigate the effect of altering inlet angle of the air on the flow field.