With the growing importance of renewable energy, the interest in energy-harvesting technologies based on flow-induced vibrations of airfoils has been rekindled in the past few years. Compared to conventional turbines, these devices are centrifugal stress-free and hence they are structurally more robust. They are also environmentally friendly due to the relatively low speeds, thus reducing the impact on flying animals. In order to numerically investigate these types of devices, mesh motion techniques must be used so that the computational grid can adapt to the time-varying shape of the domain and boundaries, thus preserving its robustness and quality. During the last years, many different dynamic mesh approaches have been developed to better predict the behaviour of a flow interacting with solid moving bodies. The aim of the present work is to apply and validate the overset grid method offered by the OpenFOAM software. In this type of mesh, one or more grid blocks are allowed to overlap with other sets of cells to solve the fluid domain with moving bodies, thus showing many potential advantages compared to the other existing dynamic mesh approaches. In the first part of this work, a numerical investigation of the flow over a stationary SD 7003 airfoil at low Reynolds number has been performed using an unsteady RANS approach. The computed results have then been compared to experimental and high-fidelity computational data available from the literature in order to validate the presented model in terms of numerical domain configuration, mesh refinement, and turbulence modelling. Subsequently, unsteady RANS simulations have been performed on the SD 7003 and NACA 0012 airfoils for two different amplitudes of flapping motion, i.e., 30° and 45°. Finally, the results obtained with the OpenFOAM overset grid solver have been compared with the numerical and experimental benchmarks presented by Kurtulus, showing an excellent agreement with both computed and measured data.