The non-cavitating and cavitating flow over a crossflow water turbine is simulated by using an unsteady Navier-Stokes formulation. For the cavitating flow case, a homogeneous mixture with a varying density is considered and one additional transport equation is explicitly solved in time for the liquid volume fraction. The instantaneous rate of vapor production and absorption appearing as a source term is governed by a hydrodynamic model based on a simplified bubble dynamic equation. The spatial discretization is achieved by a 2D multi block technique consisting in fixed and rotating blocks which were especially adapted for Darrieus geometry. A calculation of a mono-bladed non-cavitating crossflow turbine corresponding to the experiment of Oler is presented. Then a turbine, planned to be tested in the LEGI tunnel, is considered. The effect of cavitation on the dynamic stall phenomenon and on the turbine performance is analyzed.