Unsteady stokes flow for a vibrating cantilever under a free-surface

Vibration of a thin, rectangular-cross-section beam submerged in a viscous, quiescent fluid undergoing small amplitude oscillations is studied using a Boundary Element (BE) approach in which the free-surface is modeled through a stress-free boundary condition. The Stokes approximation is used where nonlinear convective terms are negligible and the problem is formulated in Fourier and Laplace transform space when appropriate. Results are expressed in terms of nondimensional hydrodynamic force and its components, namely added mass and damping coefficients. Several parametric studies are conducted to evaluate the effects of depth of submergence, frequency and the amplitude of oscillations on the hydrodynamic functions. The results are compared with the classical solution for a vibrating lamina in an infinite fluid as the limit case and with a recent study using Smoothed Particle Hydrodynamics (SPH) analysis in the presence of a freesurface.