In this work, we perform a numerical study on the flow induced by the motion of a rigid cantilever beam undergoing finite amplitude oscillations, in a viscous fluid, under a free surface. To this aim, we use a lattice Boltzmann volume of fluid (LB-VOF) integrated method, which includes the tracking of the fluid surface. The adopted approach couples the simplicity of the LB method with the possibility to track the free surface by means of a VOF strategy. Through a parametric analysis, we study the effects related to the depth of submergence, for several values of the oscillation frequency and amplitude. Results are provided in terms of a complex hydrodynamic function, whose real and imaginary parts are the added mass and the viscous damping, respectively, acting on the lamina. Validation of the results is carried out by comparing the solution, for the limit case of lamina submerged in an infinite fluid, with those from available literature studies. We find that the presence of the free surface strongly influences the flow physics around the lamina, especially at low values of the depth of submergence. In facts, when the lamina approaches to the free surface, the fluid waves, generated by the motion of the lamina, interact with the oscillating body itself, giving rise to additional effects, which we quantify in terms of added mass and viscous damping.

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