Fixed and forced moving circular cylinders in turbulent flows are studied by using the Large Eddy Simulation (LES) and two-equation based Detached Eddy Simulation (DES) turbulence models. The Cartesian cut cell approach is adopted to track the body surface across a stationary background grid covering the whole computational domain. A cell-centered finite volume method of second-order accuracy in both time and space is developed to solve the flow field in fluid cells, which is also modified accordingly in cut cells and merged cells. In order to compare different turbulence models, the current flow past a fixed circular cylinder at a moderate Reynolds number, Re = 3900, is tested first. The model is also applied to the simulation of a forced oscillating circular cylinder in the turbulent flow, and the influences of different oscillation amplitudes, frequencies and free stream velocities are discussed. The numerical results indicate that the present numerical model based on the Cartesian cut cell approach is capable of solving the turbulent flow around a body undergoing motions, which is a foundation for the possible future study on wake induced oscillation and vortex induced vibration.