Small-amplitude heave oscillations of an annular disk

We study small-amplitude broadside oscillations of an annular disk in an unbounded domain of fluid. Specifically, we formulate a semi-analytical framework to examine the effects of the oscillation frequency and pore radius on the added mass and damping coefficients of the disk. We break down the original problem into two simpler ones. The sub-problems are then bridged together by the reciprocal theorem and simplified further via a perturbation expansion in terms of the pore radius to arrive at dual integral equations. These equations are eventually reduced to two systems of algebraic equations and solved numerically. Our analysis reveals that the annular (porous) disk behaves nearly as a solid (impermeable) one in the Stokes regime, with the change in the force coefficients scaling with the cube of the pore radius. Remarkably, as the inertial effects become more pronounced, the damping coefficient initially increases with increasing the pore radius, reaches a maximum, and then decays as the inner hole of the disk becomes larger. The rate of decay of both the damping and added-mass coefficients scale with the pore radius in the asymptotic limit of high oscillation frequency. The non-monotonic damping behavior of annular disks can be harnessed in engineering applications.