No CrossRef data available.
Article contents
Buoyancy oscillations
Published online by Cambridge University Press: 03 April 2024
Abstract
The oscillations of buoyant bodies in stratified fluids are deduced from the variations of their added mass. Three configurations are considered: a body displaced from its neutral level then released; a Cartesian diver set into oscillation by a modulation of the hydrostatic pressure, then released; and a body attached to a pendulum to which an impulse is applied. The first configuration is related to the dynamics of Lagrangian floats in the ocean. Two particular bodies are considered: an elliptic cylinder of horizontal axis, typical of two-dimensional bodies; and a spheroid of vertical axis, typical of three-dimensional bodies. The ultimate motion of the body consists of oscillations at the buoyancy frequency with an amplitude decaying algebraically with time. Before that, the resonant response of the system is observed, either aperiodic exponential decay when the system has no intrinsic dynamics, or exponentially damped oscillation otherwise. Comparison with available measurements demonstrates the need to include viscous dissipation in the analysis. At high Stokes number, dissipation comes from the Basset–Boussinesq memory force and is affected negligibly by the stratification; at low Stokes number, dissipation comes from Stokes resistance and exhibits a significant effect of the stratification.
- Type
- JFM Papers
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press
References
Aagaard, E.E. & Ewart, T.E. 1973 Characteristics of a deep-sea neutrally buoyant float. In Ocean 73 – IEEE International Conference on Engineering in the Ocean Environment (ed. E.W. Early & T.F. Hueter), pp. 362–368. IEEE.CrossRefGoogle Scholar
Abate, J., Choudhury, G.L. & Whitt, W. 2000 An introduction to numerical transform inversion and its application to probability models. In Computational Probability (ed. W.K. Grassmann), pp. 257–273. Springer.CrossRefGoogle Scholar
Akulenko, L.D. & Baidulov, V.G. 2019 Extreme properties of oscillations of an elliptical float. Dokl. Phys. 64, 297–300.CrossRefGoogle Scholar
Akulenko, L.D., Mikhailov, S.A. & Nesterov, S.V. 1990 Oscillations of a float in a heterogeneous fluid in relation to the shape of its surface. Mech. Solids 25 (5), 24–31.Google Scholar
Akulenko, L.D., Mikhailov, S.A., Nesterov, S.V. & Chaikovskii, A.A. 1988 Numerical–analytic investigation of oscillations of a rigid body at the interface between two liquids. Mech. Solids 23 (4), 54–60.Google Scholar
Akulenko, L.D. & Nesterov, S.V. 1987 Oscillations of a solid at the interface between two fluids. Mech. Solids 22 (5), 30–36.Google Scholar
Ardekani, A.M., Doostmohammadi, A. & Desai, N. 2017 Transport of particles, drops, and small organisms in density stratified fluids. Phys. Rev. Fluids 2, 100503.CrossRefGoogle Scholar
Baidulov, V.G. 2022 Parametric control of float oscillations. Mech. Solids 57, 562–569.CrossRefGoogle Scholar
Barenblatt, G.I. 1978 Dynamics of turbulent spots and intrusions in a stably stratified fluid. Izv. Atmos. Ocean. Phys. 14, 139–145.Google Scholar
Basset, A.B. 1888 On the motion of a sphere in a viscous liquid. Phil. Trans. R. Soc. Lond. A 179, 43–63.Google Scholar
Beck, R.F. & Liapis, S. 1987 Transient motions of floating bodies at zero forward speed. J. Ship Res. 31, 164–176.CrossRefGoogle Scholar
Biró, I., Szabó, K.G., Gyüre, B., Jánosi, I.M. & Tél, T. 2008 Power-law decaying oscillations of neutrally buoyant spheres in continuously stratified fluid. Phys. Fluids 20, 051705.CrossRefGoogle Scholar
Boussinesq, J. 1885 Sur la résistance qu'oppose un liquide indéfini en repos, sans pesanteur, au mouvement varié d'une sphère solide qu'il mouille sur toute sa surface, quand les vitesses restent bien continues et assez faibles pour que leurs carrés et produits soient négligeables. C. R. Hebd. Séances Acad. Sci. 100, 935–937.Google Scholar
Brennen, C.E. 1982 A review of added mass and fluid inertial forces. Tech. Rep. CR 82.010. Naval Civil Engineering Laboratory. Available at: https://resolver.caltech.edu/CaltechAUTHORS:BREncel82.Google Scholar
Brouzet, C., Ermanyuk, E.V., Moulin, M., Pillet, G. & Dauxois, T. 2017 Added mass: a complex facet of tidal conversion at finite depth. J. Fluid Mech. 831, 101–127.CrossRefGoogle Scholar
Cairns, J., Munk, W. & Winant, C. 1979 On the dynamics of neutrally buoyant capsules; an experimental drop in Lake Tahoe. Deep-Sea Res. A 26, 369–381.CrossRefGoogle Scholar
Cairns, J.L. 1975 Internal wave measurements from a midwater float. J. Geophys. Res. 80, 299–306.CrossRefGoogle Scholar
Candelier, F., Mehaddi, R. & Vauquelin, O. 2014 The history force on a small particle in a linearly stratified fluid. J. Fluid Mech. 749, 184–200.CrossRefGoogle Scholar
Cerasoli, C.P. 1978 Experiments on buoyant-parcel motion and the generation of internal gravity waves. J. Fluid Mech. 86, 247–271.CrossRefGoogle Scholar
Chashechkin, Y.D. & Levitskii, V.V. 1999 Hydrodynamics of free oscillations of a sphere on the neutral-buoyancy horizon in continuously stratified fluid. Dokl. Phys. 44, 48–53.Google Scholar
Chashechkin, Y.D. & Levitskii, V.V. 2003 Pattern of flow around a sphere oscillating an neutrally buoyancy horizon in a continuously stratified fluid. J. Vis. 6, 59–65.CrossRefGoogle Scholar
Chashechkin, Y.D. & Prikhod'ko, Y.V. 2006 The structure of flows occurring under the free oscillations of a cylinder on the neutral-buoyancy horizon in a continuously stratified fluid. Dokl. Phys. 51, 215–218.CrossRefGoogle Scholar
Chashechkin, Y.D. & Prikhod'ko, Y.V. 2007 Regular and singular flow components for stimulated and free oscillations of a sphere in continuously stratified liquid. Dokl. Phys. 52, 261–265.CrossRefGoogle Scholar
D'Asaro, E. 2018 Oceanographic floats: principles of operation. In Observing the Oceans in Real Time (ed. R. Venkatesan, A. Tandon, E. D'Asaro & M.A. Atmanand), pp. 77–98. Springer.CrossRefGoogle Scholar
D'Asaro, E.A. 2003 Performance of autonomous Lagrangian floats. J. Atmos. Ocean. Technol. 20, 896–911.2.0.CO;2>CrossRefGoogle Scholar
D'Asaro, E.A., Farmer, D.M., Osse, J.T. & Dairiki, G.T. 1996 A Lagrangian float. J. Atmos. Ocean. Technol. 13, 1230–1246.2.0.CO;2>CrossRefGoogle Scholar
Damaren, C.J. 2000 Time-domain floating body dynamics by rational approximation of the radiation impedance and diffraction mapping. Ocean Engng 27, 687–705.CrossRefGoogle Scholar
Davies, B. & Martin, B. 1979 Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. 33, 1–32.CrossRefGoogle Scholar
Davis, A.M.J. & Llewellyn Smith, S.G. 2010 Tangential oscillations of a circular disk in a viscous stratified fluid. J. Fluid Mech. 656, 342–359.CrossRefGoogle Scholar
Ermanyuk, E.V. 2000 The use of impulse response functions for evaluation of added mass and damping coefficient of a circular cylinder oscillating in linearly stratified fluid. Exp. Fluids 28, 152–159.CrossRefGoogle Scholar
Ermanyuk, E.V. 2002 The rule of affine similitude for the force coefficients of a body oscillating in a uniformly stratified fluid. Exp. Fluids 32, 242–251.CrossRefGoogle Scholar
Ermanyuk, E.V. & Gavrilov, N.V. 2002 a Force on a body in a continuously stratified fluid. Part 1. Circular cylinder. J. Fluid Mech. 451, 421–443.CrossRefGoogle Scholar
Ermanyuk, E.V. & Gavrilov, N.V. 2002 b Oscillations of cylinders in a linearly stratified fluid. J. Appl. Mech. Tech. Phys. 43, 503–511.CrossRefGoogle Scholar
Ermanyuk, E.V. & Gavrilov, N.V. 2003 Force on a body in a continuously stratified fluid. Part 2. Sphere. J. Fluid Mech. 494, 33–50.CrossRefGoogle Scholar
Fitzgerald, C.J. & Meylan, M.H. 2011 Generalized eigenfunction method for floating bodies. J. Fluid Mech. 667, 544–554.CrossRefGoogle Scholar
Goodman, L. & Levine, E.R. 1990 Vertical motion of neutrally buoyant floats. J. Atmos. Ocean. Technol. 7, 38–49.2.0.CO;2>CrossRefGoogle Scholar
Gorodtsov, V.A. 1991 Collapse of asymmetric perturbations in a stratified fluid. Fluid Dyn. 26, 834–840.CrossRefGoogle Scholar
Gorodtsov, V.A. 1992 Behavior of a sphere in an ideal, uniformly stratified medium. Fluid Mech. Res. 21 (6), 100–106.Google Scholar
Gould, W.J. 2005 From Swallow floats to Argo – the development of neutrally buoyant floats. Deep-Sea Res. II 52, 529–543.Google Scholar
Güémez, J., Fiolhais, C. & Fiolhais, M. 2002 The Cartesian diver and the fold catastrophe. Am. J. Phys. 70, 710–714.CrossRefGoogle Scholar
Hanazaki, H., Nakamura, S. & Yoshikawa, H. 2015 Numerical simulation of jets generated by a sphere moving vertically in a stratified fluid. J. Fluid Mech. 765, 424–451.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics, 2nd edn. Springer.CrossRefGoogle Scholar
Harris, J.B. & Pittman, J.F.T. 1975 Equivalent ellipsoidal axis ratios of slender rod-like particles. J. Colloid Interface Sci. 50, 280–282.CrossRefGoogle Scholar
Hartman, R.J. & Lewis, H.W. 1972 Wake collapse in a stratified fluid: linear treatment. J. Fluid Mech. 51, 613–618.CrossRefGoogle Scholar
Holdsworth, A.M., Barrett, K.J. & Sutherland, B.R. 2012 Axisymmetric intrusions in two-layer and uniformly stratified environments with and without rotation. Phys. Fluids 24, 036603.CrossRefGoogle Scholar
Holdsworth, A.M., Décamp, S. & Sutherland, B.R. 2010 The axisymmetric collapse of a mixed patch and internal wave generation in uniformly stratified fluid. Phys. Fluids 22, 106602.CrossRefGoogle Scholar
Hurlen, E.C. 2006 The motions and wave fields produced by an ellipse moving through a stratified fluid. PhD thesis, University of California at San Diego. Available at: https://escholarship.org/uc/item/40m4494n.Google Scholar
Hurlen, E.C. & Llewellyn Smith, S.G. 2024 The fall of an ellipse in a stratified fluid. Fluid Dyn. Res. (in preparation).Google Scholar
Hurley, D.G. & Hood, M.J. 2001 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
den Iseger, P. 2006 Numerical transform inversion using Gaussian quadrature. Probab. Engng Inf. Sci. 20, 1–44.CrossRefGoogle Scholar
Jones, T.B. 1982 Generation and propagation of acoustic gravity waves. Nature 299, 488–489.CrossRefGoogle Scholar
Kabarowski, J.K. & Khair, A.S. 2020 The force on a slender particle under oscillatory translational motion in unsteady Stokes flow. J. Fluid Mech. 884, A44.CrossRefGoogle Scholar
Kao, T.W. 1976 Principal stage of wake collapse in a stratified fluid: two-dimensional theory. Phys. Fluids 19, 1071–1074.CrossRefGoogle Scholar
Kotik, J. & Lurye, J. 1964 Some topics in the theory of coupled ship motions. In Proceedings of the Fifth Symposium on Naval Hydrodynamics (ed. J.K. Lunde & S.W. Doroff), pp. 407–424. US Government Printing Office. Available at: http://resolver.tudelft.nl/uuid:73776ccf-258f-4d5d-ab6c-88c95a002091.Google Scholar
Kotik, J. & Lurye, J. 1968 Heave oscillations of a floating cylinder or sphere. Schiffstechnik 15, 37–38.Google Scholar
Kumar, K.K. 2007 VHF radar investigations on the role of mechanical oscillator effect in exciting convectively generated gravity waves. Geophys. Res. Lett. 34, L01803.CrossRefGoogle Scholar
Lane, T.P. 2008 The vortical response to penetrative convection and the associated gravity-wave generation. Atmos. Sci. Lett. 9, 103–110.CrossRefGoogle Scholar
Larsen, L.H. 1969 Oscillations of a neutrally buoyant sphere in a stratified fluid. Deep-Sea Res. 16, 587–603.Google Scholar
Lavrentiev, L. & Chabat, B. 1972 Méthodes de la théorie des fonctions d'une variable complexe. Mir.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.CrossRefGoogle 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.CrossRefGoogle Scholar
Le Dizès, S. & Le Bars, M. 2017 Internal shear layers from librating objects. J. Fluid Mech. 826, 653–675.CrossRefGoogle Scholar
Le Gal, P., Castillo Morales, B., Hernandez-Zapata, S. & Ruiz Chavarria, G. 2022 Swimming of a ludion in a stratified sea. J. Fluid Mech. 931, A14.CrossRefGoogle Scholar
Levitskii, V.V. & Chashechkin, Y.D. 1999 Natural oscillations of a neutrally buoyant body in a continuously stratified fluid. Fluid Dyn. 34, 641–651.Google Scholar
Loewenberg, M. 1993 a Stokes resistance, added mass, and Basset force for arbitrarily oriented, finite-length cylinders. Phys. Fluids A 5, 765–767.CrossRefGoogle Scholar
Loewenberg, M. 1993 b The unsteady Stokes resistance of arbitrarily oriented, finite-length cylinders. Phys. Fluids A 5, 3004–3006.CrossRefGoogle Scholar
Loewenberg, M. 1994 a Axisymmetric unsteady Stokes flow past an oscillating finite-length cylinder. J. Fluid Mech. 265, 265–288.CrossRefGoogle Scholar
Loewenberg, M. 1994 b Asymmetric, oscillatory motion of a finite-length cylinder: the macroscopic effect of particle edges. Phys. Fluids 6, 1095–1107.CrossRefGoogle Scholar
Maas, L.R.M. 2011 Topographies lacking tidal conversion. J. Fluid Mech. 684, 5–24.CrossRefGoogle Scholar
Magnaudet, J. & Mercier, M.J. 2020 Particles, drops, and bubbles moving across sharp interfaces and stratified layers. Annu. Rev. Fluid Mech. 52, 61–91.CrossRefGoogle Scholar
Maskell, S.J. & Ursell, F. 1970 The transient motion of a floating body. J. Fluid Mech. 44, 303–313.CrossRefGoogle Scholar
McIver, M. & McIver, P. 2011 Water waves in the time domain. J. Engng Maths 70, 111–128.CrossRefGoogle Scholar
McLaren, T.I., Pierce, A.D., Fohl, T. & Murphy, B.L. 1973 An investigation of internal gravity waves generated by a buoyantly rising fluid in a stratified medium. J. Fluid Mech. 57, 229–240.CrossRefGoogle Scholar
Meng, J.C.S. & Rottman, J.W. 1988 Linear internal waves generated by density and velocity perturbations in a linearly stratified fluid. J. Fluid Mech. 186, 419–444.CrossRefGoogle Scholar
Meylan, M.H. 2014 The time-dependent motion of a floating elastic or rigid body in two dimensions. Appl. Ocean Res. 46, 54–61.CrossRefGoogle Scholar
More, R.V. & Ardekani, A.M. 2023 Motion in stratified fluids. Annu. Rev. Fluid Mech. 55, 157–192.CrossRefGoogle Scholar
Morton, B.R., Taylor, G. & Turner, J.S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Newman, J.N. 1985 Transient axisymmetric motion of a floating cylinder. J. Fluid Mech. 157, 17–33.CrossRefGoogle Scholar
Nuriev, A.N., Egorov, A.G. & Kamalutdinov, A.M. 2021 Hydrodynamic forces acting on the elliptic cylinder performing high-frequency low-amplitude multi-harmonic oscillations in a viscous fluid. J. Fluid Mech. 913, A40.CrossRefGoogle Scholar
Orlanski, I. & Ross, B.B. 1973 Numerical simulation of the generation and breaking of internal gravity waves. J. Geophys. Res. 78, 8808–8826.CrossRefGoogle Scholar
Pierce, A. & Coroniti, S. 1966 A mechanism for the generation of acoustic–gravity waves during thunderstorm formation. Nature 210, 1209–1210.CrossRefGoogle Scholar
Pot, G. & Jami, A. 1991 Some numerical results in 3-D transient linear naval hydrodynamics. J. Ship Res. 35, 295–303.CrossRefGoogle Scholar
Pozrikidis, C. 1989 A singularity method for unsteady linearized flow. Phys. Fluids A 1, 1508–1520.CrossRefGoogle Scholar
Prikhod'ko, Y.V. & Chashechkin, Y.D. 2006 Hydrodynamics of natural oscillations of neutrally buoyant bodies in a layer of continuously stratified fluid. Fluid Dyn. 41, 545–554.CrossRefGoogle Scholar
Pyl'nev, Y.V. & Razumeenko, Y.V. 1991 Damped oscillations of a float of special shape, deeply immersed in a homogeneous and stratified fluid. Mech. Solids 26 (4), 67–76.Google Scholar
Renaud, A. & Venaille, A. 2019 Boundary streaming by internal waves. J. Fluid Mech. 858, 71–90.CrossRefGoogle Scholar
Rossby, T. 2007 Evolution of Lagrangian methods in oceanography. In Lagrangian Analysis and Prediction of Coastal and Ocean Dynamics (ed. A. Griffa, A.D. Kirwan, Jr, A.J. Mariano, T. Özgökmen & H.T. Rossby), pp. 1–38. Cambridge University Press.CrossRefGoogle Scholar
Rossby, T., Dorson, D. & Fontaine, J. 1986 The RAFOS system. J. Atmos. Ocean. Technol. 3, 672–679.2.0.CO;2>CrossRefGoogle Scholar
Sharman, R.D. & Trier, S.B. 2019 Influences of gravity waves on convectively induced turbulence (CIT): a review. Pure Appl. Geophys. 176, 1923–1958.CrossRefGoogle Scholar
Sretenskii, L.N. 1937 On damping of the vertical oscillations of the centre of gravity of floating bodies. Trudy TsAGI 330, 1–12 (In Russian).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 (2), 8–106.Google Scholar
Stommel, H. 1955 Direct measurements of sub-surface currents. Deep-Sea Res. 2, 284–285.Google Scholar
Sutherland, B.R., Chow, A.N.F. & Pittman, T.P. 2007 The collapse of a mixed patch in stratified fluid. Phys. Fluids 19, 116602.CrossRefGoogle Scholar
Sutherland, B.R., Flynn, M.R. & Dohan, K. 2004 Internal wave excitation from a collapsing mixed region. Deep-Sea Res. II 51, 2889–2904.Google Scholar
Swallow, J.C. 1955 A neutral-buoyancy float for measuring deep currents. Deep-Sea Res. 3, 74–81.Google Scholar
Swift, D.D. & Riser, S.C. 1994 RAFOS floats: defining and targeting surfaces of neutral buoyancy. J. Atmos. Ocean. Technol. 11, 1079–1092.2.0.CO;2>CrossRefGoogle Scholar
Torres, C.R., Hanazaki, H., Ochoa, J., Castillo, J. & Van Woert, M. 2000 Flow past a sphere moving vertically in a stratified diffusive fluid. J. Fluid Mech. 417, 211–236.CrossRefGoogle Scholar
Ursell, F. 1964 The decay of the free motion of a floating body. J. Fluid Mech. 19, 305–319.CrossRefGoogle Scholar
Vasil'ev, A.Y. & Chashechkin, Y.D. 2009 Damping of the free oscillations of a neutral buoyancy sphere in a viscous stratified fluid. J. Appl. Maths Mech. 73, 558–565.CrossRefGoogle Scholar
Vasil'ev, A.Y., Kistovich, A.V. & Chashechkin, Y.D. 2007 Free oscillations of a balanced ball on the horizon of neutral buoyancy in a continuously stratified fluid. Dokl. Phys. 52, 596–599.CrossRefGoogle Scholar
Voisin, B. 2007 Added mass effects on internal wave generation. In Proceedings of the Fifth International Symposium on Environmental Hydraulics (ed. D.L. Boyer & O. Alexandrova). Available at: https://hal.archives-ouvertes.fr/hal-00268817.Google Scholar
Voisin, B. 2021 Boundary integrals for oscillating bodies in stratified fluids. J. Fluid Mech. 927, A3.CrossRefGoogle Scholar
Voisin, B. 2024 Added mass of oscillating bodies in stratified fluids. J. Fluid Mech. (accepted).Google Scholar
Voorhis, A.D. 1971 Response characteristics of the neutrally buoyant float. Tech. Rep. 71-73. Woods Hole Oceanographic Institution. Available at: https://doi.org/10.1575/1912/24369.CrossRefGoogle Scholar
Voth, G.A. & Soldati, A. 2017 Anisotropic particles in turbulence. Annu. Rev. Fluid Mech. 49, 249–276.CrossRefGoogle Scholar
Warren, F.W.G. 1968 Gravity wave damping of hydrostatic oscillations for a buoyant disk. J. Fluid Mech. 31, 309–319.CrossRefGoogle Scholar
Wehausen, J.V. 1971 The motion of floating bodies. Annu. Rev. Fluid Mech. 3, 237–268.CrossRefGoogle Scholar
Wehausen, J.V. & Laitone, E.V. 1960 Surface waves. In Encyclopedia of Physics (ed. S. Flügge & C. Truesdell), vol. 9, pp. 446–778. Springer. Available at: http://surfacewaves.berkeley.edu.CrossRefGoogle Scholar
Weinheimer, A.J. 1987 Application of the Stokes drag on spheroids to the drag on disks and cylinders. J. Atmos. Sci. 44, 2674–2676.2.0.CO;2>CrossRefGoogle Scholar
Wolgamot, H.A., Meylan, M.H. & Reid, C.D. 2017 Multiply heaving bodies in the time-domain: symmetry and complex resonances. J. Fluids Struct. 69, 232–251.CrossRefGoogle Scholar
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 35, 531–544.CrossRefGoogle Scholar
Yeung, R.W. 1982 The transient heaving motion of floating cylinders. J. Engng Maths 16, 97–119.CrossRefGoogle Scholar
Zatsepin, A.G., Fedorov, K.N., Vorapayev, S.I. & Pavlov, A.M. 1978 Experimental study of the spreading of a mixed region in a stably stratified fluid. Izv. Atmos. Ocean. Phys. 14, 170–173.Google Scholar
Zhang, J., Mercier, M.J. & Magnaudet, J. 2019 Core mechanisms of drag enhancement on bodies settling in a stratified fluid. J. Fluid Mech. 875, 622–656.CrossRefGoogle 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.CrossRefGoogle Scholar