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Optimal viscous damping of vibrating porous cylinders

Published online by Cambridge University Press:  09 July 2019

Saeed Jafari Kang ,
Esmaeil Dehdashti ,
Vahid Vandadi  and
Hassan Masoud Open the ORCID record for Hassan Masoud [Opens in a new window]
Saeed Jafari Kang
Affiliation:
Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA
Esmaeil Dehdashti
Affiliation:
Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA
Vahid Vandadi
Affiliation:
Polaris Industries Inc., Medina, MN 55340, USA
Hassan Masoud*
Affiliation:
Department of Mechanical Engineering–Engineering Mechanics, Michigan Technological University, Houghton, MI 49931, USA
*
†Email address for correspondence: hmasoud@mtu.edu
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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.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 339 - 358
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Copyright
© 2019 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Ahsan, S. N. & Aureli, M. 2015 Finite amplitude oscillations of flanged laminas in viscous flows: vortex–structure interactions for hydrodynamic damping control. J. Fluids Struct. 59, 297–315.Google Scholar
An, S. & Faltinsen, O. M. 2013 An experimental and numerical study of heave added mass and damping of horizontally submerged and perforated rectangular plates. J. Fluids Struct. 39, 87–101.Google Scholar
Avudainayagam, A. & Geetha, J. 1994 Oscillatory Stokes flow in two dimensions. Mech. Res. Commun. 21 (6), 617–628.Google Scholar
Barta, E. 2011 Motion of slender bodies in unsteady Stokes flow. J. Fluid Mech. 688, 66–87.Google Scholar
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 27–34.Google Scholar
Brinkman, H. C. 1948 On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. A 1, 81–86.Google Scholar
Chwang, A. T. & Wu, T. Y.-T. 1975 Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. Fluid Mech. 67 (4), 787–815.Google Scholar
Debye, P. & Bueche, A. M. 1948 Intrinsic viscosity, diffusion, and sedimentation rate of polymers in solution. J. Chem. Phys. 16, 573–579.Google Scholar
Felderhof, B. U. 2014 Velocity relaxation of a porous sphere immersed in a viscous incompressible fluid. J. Chem. Phys. 140 (13), 134901.Google Scholar
Graham, D. R. & Higdon, J. J. L. 2002 Oscillatory forcing of flow through porous media. Part 2. Unsteady flow. J. Fluid Mech. 465, 237–260.Google Scholar
Kanwal, R. P. 1955 Vibrations of an elliptic cylinder and a flat plate in a viscous fluid. Z. Angew. Math. Mech. 35 (1-2), 17–22.Google Scholar
Kanwal, R. P. 1964 Drag on an axially symmetric body vibrating slowly along its axis in a viscous fluid. J. Fluid Mech. 19 (4), 631–636.Google Scholar
Kanwal, R. P. 1970 Note on slow rotation or rotary oscillation of axisymmetric bodies in hydrodynamics and magnetohydrodynamics. J. Fluid Mech. 41 (4), 721–726.Google Scholar
Kolomenskiy, D. & Schneider, K. 2009 A Fourier spectral method for the Navier–Stokes equations with volume penalization for moving solid obstacles. J. Comput. Phys. 228 (16), 5687–5709.Google Scholar
Lai, R. Y. S. & Mockros, L. F. 1972 The Stokes-flow drag on prolate and oblate spheroids during axial translatory accelerations. J. Fluid Mech. 52 (1), 1–15.Google Scholar
Lawrence, C. J. & Weinbaum, S. 1986 The force on an axisymmetric body in linearized, time-dependent motion: a new memory term. J. Fluid Mech. 171, 209–218.Google Scholar
Lawrence, C. J. & Weinbaum, S. 1988 The unsteady force on a body at low Reynolds number; the axisymmetric motion of a spheroid. J. Fluid Mech. 189, 463–489.Google Scholar
Liu, Y., Li, H.-J., Li, Y.-C. & He, S.-Y. 2011 A new approximate analytic solution for water wave scattering by a submerged horizontal porous disk. Appl. Ocean Res. 33 (4), 286–296.Google Scholar
Loewenberg, M. 1993 The unsteady Stokes resistance of arbitrarily oriented, finite-length cylinders. Phys. Fluids 5 (11), 3004–3006.Google Scholar
Looker, J. R. & Carnie, S. L. 2004 The hydrodynamics of an oscillating porous sphere. Phys. Fluids 16 (1), 62–72.Google Scholar
Masoud, H., Stone, H. A. & Shelley, M. J. 2013 On the rotation of porous ellipsoids in simple shear flows. J. Fluid Mech. 733, R6.Google Scholar
Molin, B. 2001 On the added mass and damping of periodic arrays of fully or partially porous disks. J. Fluids Struct. 15 (2), 275–290.Google Scholar
Molin, B. 2011 Hydrodynamic modeling of perforated structures. Appl. Ocean Res. 33 (1), 1–11.Google Scholar
Ollila, S. T. T., Ala-Nissila, T. & Denniston, C. 2012 Hydrodynamic forces on steady and oscillating porous particles. J. Fluid Mech. 709, 123–148.Google Scholar
Phan, C. N., Aureli, M. & Porfiri, M. 2013 Finite amplitude vibrations of cantilevers of rectangular cross sections in viscous fluids. J. Fluids Struct. 40, 52–69.Google Scholar
Pozrikidis, C. 1989 A singularity method for unsteady linearized flow. Phys. Fluids 1 (9), 1508–1520.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.Google Scholar
Pozrikidis, C. 2011 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.Google Scholar
Prakash, J., Raja Sekhar, G. P. & Kohr, M. 2012 Faxen’s law for arbitrary oscillatory Stokes flow past a porous sphere. Arch. Mech. 64 (1), 41–63.Google Scholar
Ray, M. 1936 Vibration of an infinite elliptic cylinder in a viscous liquid. Z. Angew. Math. Mech. 16 (2), 99–108.Google Scholar
Sader, J. E. 1998 Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. J. Appl. Phys. 84 (1), 64–76.Google Scholar
Santhanakrishnan, A., Robinson, A. K., Jones, S., Low, A. A., Gadi, S., Hedrick, T. L. & Miller, L. A. 2014 Clap and fling mechanism with interacting porous wings in tiny insect flight. J. Expl Biol. 217 (21), 3898–3909.Google Scholar
Shatz, L. F. 2004 Singularity method for oblate and prolate spheroids in Stokes and linearized oscillatory flow. Phys. Fluids 16 (3), 664–677.Google Scholar
Shatz, L. F. 2005 Slender body method for slender prolate spheroids and hemispheroids on planes in linearized oscillatory flow. Phys. Fluids 17 (11), 113603.Google Scholar
Shu, J.-J. & Chwang, A. T. 2001 Generalized fundamental solutions for unsteady viscous flows. Phys. Rev. E 63 (5), 051201.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
Tsai, C.-C. & Hsu, T.-W. 2010 The method of fundamental solutions for oscillatory and porous buoyant flows. Comput. Fluids 39 (4), 696–708.Google Scholar
Tuck, E. O. 1969 Calculation of unsteady flows due to small motions of cylinders in a viscous fluid. J. Engng Maths 3 (1), 29–44.Google Scholar
Vainshtein, P. & Shapiro, M. 2009 Forces on a porous particle in an oscillating flow. J. Colloid Interface Sci. 330 (1), 149–155.Google Scholar
Williams, W. E. 1966 A note on slow vibrations in a viscous fluid. J. Fluid Mech. 25 (3), 589–590.Google Scholar
Zhang, W. & Stone, H. A. 1998 Oscillatory motions of circular disks and nearly spherical particles in viscous flows. J. Fluid Mech. 367, 329–358.Google Scholar