With the help of computational results of incompressible flow past a circular cylinder at Reynolds numbers of Re = 150 and 1000, we explain two possible mechanisms for the experimentally observed drag reduction by rotary oscillation. Here, detailed computed results are compared with available experimental and computational results in Thiria et al. (J Fluid Mech 2006; 560:123-147) for Re = 150. The time-varying loads and moments for various cases have been analyzed first to study bluff body flow control at low Reynolds numbers. We specifically focus upon the effects of amplitude and frequency of the rotary oscillation. Furthermore, to study the effects of Reynolds number, we report another case for a higher Reynolds number of Re = 1000. Proper orthogonal decomposition of computational data for these two Re cases have been performed to explain physical mechanisms behind drag reduction by rotary oscillation and reduced order modeling for different parameter combinations. We show that the drag reduction at the lower Reynolds number (Re = 150) is related to organization of the larger shed vortices in the wake with the presence or absence of subharmonics. At the higher Reynolds number (Re = 1000), this is achieved by breaking the larger vortices into smaller ones by imposed surface motion that we term as aerodynamic tripping. This is related to the bypass transition-triggered by unsteady separation of the kind discussed in literature.