Aerodynamics of the Darrieus wind turbine is an extremely complex issue requiring the use of very advanced numerical methods. Additional structural components of this device, such as, for example, a rotating shaft disturb the flow through the rotor significantly impairing its aerodynamic characteristics. The main purpose of the presented research is to validate the commonly-used unsteady Reynolds averaged Navier–Stokes (URANS) approach with the shear stress transport (SST) k-ω turbulence model based on the particle image velocimetry (PIV) studies of a two-bladed rotor operating at the moderate tip speed ratio of 4.5. In the present numerical studies, a two-dimensional turbine rotor with a diameter of 1 m was considered. The following parameters were evaluated: instantaneous velocity fields; velocity profiles in the rotor shadow and aerodynamic blade loads. The obtained numerical results are comparable with the reference experimental results taken from the literature. The second purpose of this work was to examine the influence of the rotating rotor shaft/tower on the wind turbine performance. It has been proven that the cylindrical shaft reduces the power of the device by 2.5% in comparison with the non-shaft configuration.