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Tangential oscillations of a circular disk in a viscous stratified fluid

Published online by Cambridge University Press:  21 May 2010

ANTHONY M. J. DAVIS  and
STEFAN G. LLEWELLYN SMITH
ANTHONY M. J. DAVIS
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
STEFAN G. LLEWELLYN SMITH*
Affiliation:
Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA
*
†Email address for correspondence: sgls@ucsd.edu
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Abstract

A complete solution is obtained for the wave field generated by the time-harmonic edgewise oscillations of a horizontal circular disk in an incompressible stratified viscous fluid. The linearized equations of viscous internal waves and the no-slip condition on the rigid disk are used to derive sets of dual integral equations for the fluid velocity and vorticity. The dual integral equations are solved by analytic reduction to sets of linear algebraic equations. Asymptotic results confirm that this edgewise motion no longer excites waves in the small-viscosity limit. Broadside oscillations and the effect of density diffusion are also considered.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 342 - 359
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Bardakov, R. N., Vasil'ev, A. Y. & Chashechkin, Y. D. 2007 Calculation and measurement of conical beams of three-dimensional periodic internal waves excited by a vertically oscillating piston. Fluid Dyn. 42, 612–626.CrossRefGoogle Scholar
Bleistein, N. & Handelsman, R. A. 1986 Asymptotic Expansions of Integrals. Dover.Google Scholar
Davis, A. M. J. 1993 Some asymmetric Stokes flows that are structurally similar. Phys. Fluids A 5, 2086–2094.CrossRefGoogle Scholar
Davis, A. M. J. & Nagem, R. J. 2004 Effect of viscosity on acoustic diffraction by a circular disk. J. Acoust. Soc. Am. 115, 2738–2748.CrossRefGoogle Scholar
Ermanyuk, E. V. & Gavrilov, N. V. 2008 On internal waves generated by large-amplitude circular and rectilinear oscillations of a circular cylinder in a uniformly stratified fluid. J. Fluid Mech. 613, 329–356.CrossRefGoogle Scholar
Gabov, S. A. & Pletner, Y. D. 1988 The problem of the oscillations of a flat disk in a stratified liquid. Comput. Math. Math. Phys. 28, 41–47.CrossRefGoogle Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Gordon, D. & Stevenson, T. N. 1972 Viscous effects in a vertically propagating internal wave. J. Fluid Mech. 56, 629–639.CrossRefGoogle Scholar
Görtler, H. 1943 Über eine Schwingungserscheinung in Flüssigkeiten mit stabiler Dichteschichtung. Z. Angew. Math. Mech. 23, 65–71.CrossRefGoogle Scholar
Hurley, D. G. & Hood, M. J. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 3. Angular oscillations and comparison of theory with recent experimental observations. J. Fluid Mech. 433, 61–75.CrossRefGoogle Scholar
Hurley, D. G. & Keady, G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution. J. Fluid Mech. 351, 119–138.CrossRefGoogle Scholar
Kistovich, Y. & Chashechkin, Y. D. 1995 The reflection of beams of internal gravity waves at a flat rigid surface. J. Appl. Maths Mech. 59, 579–585.CrossRefGoogle Scholar
Kistovich, Y. V. & Chashechkin, Y. D. 2001 Some exactly solvable problems of the radiation of three-dimensional periodic internal waves. J. Appl. Mech. Tech. Phys. 42, 228–236.CrossRefGoogle Scholar
Lai, R. Y. S. & Lee, C.-M. 1981 Added mass of a spheroid oscillating in a linearly stratified fluid. Intl J. Engng Sci. 19, 1411–1420.CrossRefGoogle Scholar
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Mowbray, D. E. & Rarity, B. S. H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28, 1–16.CrossRefGoogle Scholar
Munk, W. & Wunsch, C. 1998 Abyssal recipes. II. Energetics of tidal and wind mixing. Deep Sea Res. I 45, 1977–2010.CrossRefGoogle Scholar
Piessens, R., de Doncker-Kapenga, E., Überhuber, C. W. & Kahaner, D. 1983 Quadpack: A Subroutine Package for Automatic Integration. Springer.CrossRefGoogle Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid. Proc. Lond. Math. Soc. 9, 57–70 (also in Scientific Papers, 2, 200–207, Cambridge University Press, 1900).Google Scholar
Sarma, L. V. K. & Krishna, D. V. 1972 Oscillation of axisymmetric bodies in a stratified fluid. Zastosow. Matem. 13, 109–121.Google Scholar
Sneddon, I. N. 1966 Mixed Boundary Problems in Potential Theory. North-Holland.Google Scholar
Sutherland, B. R. & Linden, P. F. 2002 Internal wave excitation by a vertically oscillating elliptical cylinder. Phys. Fluids 14, 721–731.CrossRefGoogle Scholar
Tanzosh, J. P. & Stone, H. A. 1995 Transverse motion of a disk through a rotating fluid. J. Fluid Mech. 301, 295–324.CrossRefGoogle Scholar
Thomas, D. N. & Stevenson, T. N. 1972 A similarity solution for viscous internal waves. J. Fluid Mech. 54, 495–506.CrossRefGoogle Scholar
Thomas, N. H. & Stevenson, T. N. 1973 An internal wave in a viscous ocean stratified by both salt and heat. J. Fluid Mech. 61, 301–304.CrossRefGoogle Scholar
Tranter, C. J. 1966 Integral Transforms in Mathematical Physics. Wiley.Google Scholar
Vasil'ev, A. Y. & Chashechkin, Y. D. 2006 Generation of beams of three-dimensional periodic internal waves by sources of various types. J. Appl. Mech. Tech. Phys. 47, 314–323.CrossRefGoogle Scholar
Voisin, B. 2003 Limit states of internal wave beams. J. Fluid Mech. 496, 243–293.CrossRefGoogle Scholar