Circular cylinders in cross-flow or the motion of circular cylinders in a fluid at rest are especially of interest in fields such as offshore and civil engineering or heat exchanger design in particular. A time-independent finite difference scheme, the basic equations are written in the form of the primitive-variable method, is developed to simulate the viscous flow across a streamwise oscillating circular cylinder. The mov-ing boundary of the oscillating cylinder is mapped to a fixed boundary and the boundary condition, therefore, becomes time independent. The finite difference ap-proximation and algorithm were first validated by the reported numerical simulation and flow visualization of the phenomenon £\ and phenomenon £] for a flow across a fixed circular cylinder. Detailed streamline patterns developed in the process are then described and discussed. Surface pressure distribution and position of separation point versus phases of various stationary and oscillating stages are discussed. The flow be-haviors of various amplitudes of exciting velocity and frequency of moving cylinder are simulated and compared. The relation between Keulegan-Carpenter and the drag force on cylinder during cylinder oscillation was also calculated under various Reynolds number.