Research on vortex-induced vibrations (VIV) mainly involves experimental science but building laboratory setups to investigate the flow are "expensive and time consuming. Computational fluid dynamics (CFD) methods may offer a faster and a cheaper way to understand this phenomenon depending on the solution approach to the problem. The context of this paper is to present the author's computational approach to solve for vortex -induced vibrations which cover extensive explanations on the mathematical background, the grid structure and the turbulence models implemented. Current computational research on VIV for smooth cylinders is currently restricted to flows that have Reynolds numbers below 10,000. This paper describes the method to approach the problem with URANS and achieves to return satisfactory results for higher Reynolds numbers. The computational approach is first validated with a benchmark experimental study for rather low Reynolds number which falls into TrSL2 flow regime. Then, some numerical results up to Re = 130,000, which falls into TrSL3 flow regime,are given at the end of the paper to reveal the validity of the approach for even higher Reynolds numbers.