Marine risers used to convey oil from the sea bed to the sea level, marine cables, heat exchanger pipes, civil engineering structures and aircraft wings vibrate due to the formation of the vortex streets behind these structures. The interactions between the flow oscillations and the structure give rise to complicated vibrations of the structure which could cause structural damage due to the fatigue. Numerical simulation of the vortex-induced vibrations on circular cylinders is used to investigate the possibility of suppressing these vibrations for different engineering applications. The unsteady, incompressible, two-dimensional Navier-Stokes equations are solved numerically on a structured grid using the finite difference method. The effect of flow control using the boundary layer suction for fixed or moving cylinder is investigated by applying the appropriate boundary conditions on the cylinder surface. In the current study, natural motion of the cylinder is not considered. Flow oscillations are investigated only for fixed cylinder and for forced motion of the cylinder. The results indicate that the flow oscillations are completely damped for a fixed cylinder using suction on the cylinder surface. For forced motion of the cylinder, the vortex shedding from the surface is eliminated using the boundary layer suction. The flow oscillates only due to the forced motion of the body vibrations which means that the flow oscillations can be damped completely if the cylinder is left to oscillate naturally.