A finite-volume numerical method is presented for the solution of the two-dimensional incompressible, unsteady Navier-Stokes equations in general moving curvilinear co-ordinates. The method employs the semi-strong conservation form of the governing equations with pressure and physical contravariant velocity components as dependent variables. The k-ε model is utilised to describe the turbulent flow process. The SIMPLE algorithm is used to handle the pressure-velocity coupling. Results for the calculation of the fluctuating flow past an airfoil are compared with the available experimental data. As a simulation of fluid-structure interaction, the calculation of the unsteady flow-field around a stay-vane hydrofoil in forced and free vibration is presented. The numerical results obtained in this work show very good agreement with experimental observations.