In the present study, circular cylinders in the cross-flow or the motions of circular cylinders in a fluid at rest are especially of interest in fields, such as offshore and civil engineering or heat exchanger. For last two decades, the researches of the force caused by the fluid on the cylinder surface are mainly studied by the ways of experiment and numerical methods. A time-independent finite different method is developed to solve the two-dimensional fixed or transversely oscillating cylinder passing by a cross flow. The present study focuses on the cylinder under a cross flow with only two kinds of conditions, which are Re = 100, KC = 5 and Re=200, KC=4. The benchmark tests of the present numerical results are made and validated by the reported numerical simulation and experimental results, for instant, the flow visualization of the vorticity contours and the in-line force for a flow across a moving circular cylinder. The developed numerical method can easily apply on the analyses of interactions between viscous flow and motion cylinder. Besides, we also consider the oscillatory flow passes a circular cylinder connecting with a spring. The spring -linking cylinder is released in the beginning on the position of zero deflection of the spring and stares moving due to the influence of the in-line force acting on the cylinder. We can find that the spring-linking cylinder under a oscillating flow produces restoring force and drag force due to considering the influence of the spring and damping effect, the developed numerical method can easily apply on the analyses of interactions between viscous flow and oscillating cylinder.