This paper presents a numerical study on air-fuel mixing in a two-stroke direct injection spark ignition engine under homogeneous operation. The simulated engine is loop scavenged and uses an outwardly opening swirl injector. A generic mesh-snapping algorithm is developed to enable the moving piston to snap through transfer ports with complicated geometry. A spray model based on Linear Instability Sheet Atomization is used to describe the primary breakup of fuel sprays, and the initial rotational velocity of the conical sheet is determined from a CFD simulation of the nozzle internal flow. A wall film model accounting for the effect of contacting area is also developed to avoid the severe grid-dependence of the original film model in KIVA. Comparisons between simulations and experiments were made for sprays in quiescent ambient conditions, and a good agreement of the spray characteristics was obtained. The simulations were performed for four different injection timings. The model was shown to be capable to predict engine-out unburned hydrocarbon emission. It was also found that the mixture preparation around the spark plug is directly related to the engine-out emission, and leaner mixture tends to yield higher engine-out emission.