This study presents numerical investigation for flow around cylinder at Reynolds number = 10⁴ using different turbulent models. Numerical simulations have been conducted for fixed cylinder case at Reynolds number = 10⁴ and for cylinder free to oscillate in cross-flow direction, at Reynolds number O (10⁴), mass–damping ratio = 0.011 and range of frequency ratio w_t = 0.4–1.4 using two-dimensional Reynolds-averaged Navier–Stokes equations. In the literature, the study has been conducted using detached eddy simulation, large eddy simulation and direct numerical simulation which are comparatively expensive in terms of computational cost. This study utilizes the Reynolds-averaged Navier–Stokes shear stress transport k-ω and realizable k-ε models to investigate the flow around fixed cylinder and flow around cylinder constrained to oscillate in cross-flow direction only. Hydrodynamic coefficients, vortex mode shape and maximum amplitude ( Ay/ D) extracted from this study are compared with detached eddy simulation, large eddy simulation and direct numerical simulation results. Results obtained using two-dimensional Reynolds-averaged Navier–Stokes shear stress transport k-ω model are encouraging, while realizable k-ε model is unable to capture the entire response branches. In addition, broad range of “lock-in” region is observed due to delay in capturing the transition from upper to lower branch during two-dimensional realizable k-ε analyses.