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An elastic plate behind a rigid cylinder in viscous flow is modeled numerically by an immersed boundary-lattice Boltzmann method (IB-LBM). The hydrodynamic interactions of a upstream cylinder and a adjacent passively flapping plate are studied by varying the length and bending stiffness of the plate. Two patterns of the plate are found in the simulation results, if the plate is short enough, it would curl up because of the attraction of the suction zone behind the cylinder. With the increase of the plate's length, it will stretch and flap in a stable period. The cylinder has a strong influence on the dynamics of the plate and the plate also changes the vortex shedding of the cylinder and the shape of suction zone. With a downstream adjacent plate, the cylinder can always enjoys a drag reduction, longer the plate is, less drag the cylinder experiences. The vortex shedding frequency of the cylinder is larger than that of the single cylinder and equal to the plate's flapping frequency. This study may help to understand the energy saving of flexible bodies stay in the wake of a rigid structure and the hydrodynamic interactions between them.