Insects rely on their olfactory system to forage, prey, and mate. They can sense odorant plumes emitted from sources of their interests with their bilateral odorant antennae, and track down odor sources using their highly efficient flapping-wing mechanism. The odor-tracking process typically consists of two distinct behaviors: surging upwind and zigzagging crosswind. Despite the extensive numerical and experimental studies on the flying trajectories and wing flapping kinematics during odor tracking flight, we have limited understanding of how the flying trajectories and flapping wings modulate odor plume structures. In this study, a fully coupled three-way numerical solver is developed, which solves the 3D Navier-Stokes equations coupled with equations of motion for the passive flapping wings, and the odorant convection-diffusion equation. This numerical solver is applied to investigate the unsteady flow field and the odorant transport phenomena of a fruit fly model in both surging upwind and zigzagging crosswind cases. The unsteady flow generated by flapping wings perturbs the odor plume structure and significantly impacts the odor intensity at the olfactory receptors (i.e., antennae). During zigzagging crosswind flight, the differences in odor perception time and peak odor intensity at the receptors potentially help create stereo odorant mapping to track odor source. Our simulation results will provide new insights into the mechanism of how fruit flies perceive odor landscape and inspire the future design of odor-guided micro aerial vehicles (MAVs) for surveillance and detection missions.