Optimal viscous damping of vibrating porous cylinders

Abstract

We theoretically study small-amplitude oscillations of permeable cylinders immersed in an unbounded fluid. Specifically, we examine the effects of oscillation frequency, permeability and shape on the effective mass and damping coefficients, the latter of which is proportional to the power required to sustain the vibrations. Cylinders of circular and elliptical cross-sections undergoing transverse and rotational vibrations are considered. The dynamics of the fluid flow through porous cylinders is assumed to obey the unsteady Brinkman–Debye–Bueche equations. We use a singularity method to analytically calculate the flow field within and around circular cylinders, whereas we introduce a Fourier-pseudospectral method to numerically solve the governing equations for elliptical cylinders. We find that, if rescaled properly, the analytical results for circular cylinders provide very good estimates for the behaviour of elliptical ones over a wide range of conditions. More importantly, our calculations indicate that, at sufficiently high frequencies, the damping coefficient of oscillations varies non-monotonically with the permeability, in which case it maximizes when the diffusion length scale for the vorticity is comparable to the penetration length scale for the flow within the porous material. Depending on the oscillation period, the maximum damping of a permeable cylinder can be many times greater than that of an otherwise impermeable one. This might seem counter-intuitive at first, since generally the power it takes to steadily drag a permeable object through a fluid is less than the power needed to drive the steady motion of the same, but impermeable, object. However, the driving power (or damping coefficient) for oscillating bodies is determined not only by the amplitude of the cyclic fluid load experienced by them but also by the phase shift between the load and their periodic motion. An increase in the latter is responsible for the excess damping coefficient of vibrating porous cylinders.