No CrossRef data available.
Article contents
Added mass of oscillating bodies in stratified fluids
Published online by Cambridge University Press: 20 May 2024
Abstract
The concept of added mass is generalized to stratified fluids, accounting for the presence of internal waves. Once the added mass of a moving body is known, so is the hydrodynamic force exerted on it by the fluid, and the energy imparted by it to the fluid. As a function of frequency, added mass is complex. Its real part is associated with inertia and its imaginary part, only present in the frequency range of propagating waves, with wave damping. Owing to causality, these two parts satisfy Kramers–Kronig relations. The added masses of an elliptic cylinder of horizontal axis, typical of two-dimensional bodies, and a spheroid of vertical axis, typical of three-dimensional bodies, are deduced from their dipole strengths, themselves deduced from their representations as single layers. The wave power is shown to be a maximum, for fixed oscillation amplitude, at approximately 
$0.8$ times the buoyancy frequency. In the temporal domain, added mass appears as a new memory force taking the form of a convolution integral. The kernel of this integral combines algebraically decaying oscillations at the buoyancy frequency on the one hand; and an exponentially damped oscillation for the horizontal motion of the spheroid, implying short-term memory, an aperiodic algebraic decay for its vertical motion, implying long-term memory, and a constant for the motion of the cylinder, implying everlasting memory, on the other hand. A limitation of the study is its restriction to translational motion.
- Type
- JFM Papers
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press
References
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
Baines, P.G. 1982 On internal tide generation models. Deep-Sea Res. A 29, 307–338.CrossRefGoogle Scholar
Balmforth, N.J. & Peacock, T. 2009 Tidal conversion by supercritical topography. J. Phys. Oceanogr. 39, 1965–1974.CrossRefGoogle 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
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
Cadby, J. & Linton, C. 2000 Three-dimensional water-wave scattering in two-layer fluids. J. Fluid Mech. 423, 155–173.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
Cummins, W.E. 1962 The impulse response function and ship motions. Schiffstechnik 9, 101–109.Google 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
Dickinson, R.E. 1969 Propagators of atmospheric motions. 1. Excitation by point impulses. Rev. Geophys. 7, 483–514.CrossRefGoogle Scholar
Eames, I. & Hunt, J.C.R. 1997 Inviscid flow around bodies moving in weak density gradients without buoyancy effects. J. Fluid Mech. 353, 331–355.CrossRefGoogle Scholar
Echeverri, P. & Peacock, T. 2010 Internal tide generation by arbitrary two-dimensional topography. J. Fluid Mech. 659, 247–266.CrossRefGoogle Scholar
Echeverri, P., Yokossi, T., Balmforth, N.J. & Peacock, T. 2011 Tidally generated internal-wave attractors between double ridges. J. Fluid Mech. 669, 354–374.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
Ferrari, R., Mashayek, A., McDougall, T.J., Nikurashin, M. & Campin, J. 2016 Turning ocean mixing upside down. J. Phys. Oceanogr. 46, 2239–2261.CrossRefGoogle Scholar
Ferrari, R. & Wunsch, C. 2009 Ocean circulation kinetic energy: reservoirs, sources, and sinks. Annu. Rev. Fluid Mech. 41, 253–282.CrossRefGoogle Scholar
Fox, D.W. 1976 Transient solutions for stratified fluid flows. J. Res. Nat. Bur. Stand. B 80, 79–88.CrossRefGoogle Scholar
Gabov, S.A. & Shevtsov, P.V. 1983 Basic boundary value problems for the equation of oscillations of a stratified fluid. Sov. Maths Dokl. 27, 238–241.Google Scholar
Gabov, S.A. & Shevtsov, P.V. 1986 The method of descent and singular solutions of the equation of dynamics of a stratified liquid. Differ. Equ. 22, 210–215.Google Scholar
Garrett, C. & Kunze, E. 2007 Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech. 39, 57–87.CrossRefGoogle Scholar
Gatignol, R. 1983 The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow. J. Méc. Théor. Appl. 2, 143–160.Google Scholar
Gordeichik, B.N. & Ter-Krikorov, A.M. 1996 Uniform approximations of the fundamental solution of the equation of internal waves. J. Appl. Maths Mech. 60, 439–447.CrossRefGoogle Scholar
Gorgui, M.A., Faltas, M.S. & Ahmed, A.Z. 1995 Motion generated by a porous wave-maker in a continuously stratified fluid. Intl J. Engng Sci. 33, 757–771.CrossRefGoogle Scholar
Gorodtsov, V.A. & Teodorovich, E.V. 1980 On the generation of internal waves in the presence of uniform straight-line motion of local and nonlocal sources. Izv. Atmos. Ocean. Phys. 16, 699–704.Google Scholar
Gorodtsov, V.A. & Teodorovich, E.V. 1981 Cherenkov radiation of internal waves by a source in uniform motion. Preprint 183. Institute for Problems in Mechanics, Academy of Sciences of the USSR. In Russian.Google Scholar
Greenhow, M. 1984 A note on the high-frequency limits of a floating body. J. Ship Res. 28, 226–228.CrossRefGoogle Scholar
Greenhow, M. 1986 High- and low-frequency asymptotic consequences of the Kramers–Kronig relations. J. Engng Maths 20, 293–306.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
Hurley, D.G. 1997 The generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution. J. Fluid Mech. 351, 105–118.CrossRefGoogle 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
Kapitonov, B.V. 1980 Potential theory for the equation of small oscillations of a rotating fluid. Maths USSR Sbornik. 37, 559–579.CrossRefGoogle Scholar
Kennard, E.H. 1967 Irrotational flow of frictionless fluids, mostly of invariable density. Tech. Rep. 2299. David Taylor Model Basin. Available at: https://apps.dtic.mil/sti/citations/AD0653463.Google Scholar
King, B., Zhang, H.P. & Swinney, H.L. 2009 Tidal flow over three-dimensional topography in a stratified fluid. Phys. Fluids 21, 116601.CrossRefGoogle Scholar
Kotik, J. & Mangulis, V. 1962 On the Kramers–Kronig relations for ship motions. Intl Shipbuild. Prog. 9, 361–368.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
Lam, T., Vincent, L. & Kanso, E. 2019 Passive flight in density-stratified fluids. J. Fluid Mech. 860, 200–223.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1984 Electrodynamics of Continuous Media, 2nd edn. Pergamon.Google Scholar
Larsen, L.H. 1969 Oscillations of a neutrally buoyant sphere in a stratified fluid. Deep-Sea Res. 16, 587–603.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
Legg, S. 2021 Mixing by oceanic lee waves. Annu. Rev. Fluid Mech. 53, 173–201.CrossRefGoogle Scholar
Lighthill, J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.CrossRefGoogle Scholar
Lighthill, J. 1986 An Informal Introduction to Theoretical Fluid Mechanics. Oxford University Press.Google Scholar
Linton, C. & McIver, M. 1995 The interaction of waves with horizontal cylinders in two-layer fluids. J. Fluid Mech. 304, 213–229.CrossRefGoogle Scholar
Llewellyn Smith, S.G. & Young, W.R. 2002 Conversion of the barotropic tide. J. Phys. Oceanogr. 32, 1554–1566.2.0.CO;2>CrossRefGoogle Scholar
Machicoane, N., Cortet, P.-P., Voisin, B. & Moisy, F. 2015 Influence of the multipole order of the source on the decay of an inertial wave beam in a rotating fluid. Phys. Fluids 27, 066602.CrossRefGoogle Scholar
MacKinnon, J.A., et al. 2017 Climate process team on internal wave–driven ocean mixing. Bull. Am. Meteorol. Soc. 98, 2429–2454.CrossRefGoogle Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.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
Martin, P.A. & Llewellyn Smith, S.G. 2011 Generation of internal gravity waves by an oscillating horizontal disc. Proc. R. Soc. Lond. A 467, 3406–3423.Google Scholar
Martin, P.A. & Llewellyn Smith, S.G. 2012 Generation of internal gravity waves by an oscillating horizontal elliptical plate. SIAM J. Appl. Maths 72, 725–739.CrossRefGoogle Scholar
Maslennikova, V.N. 1968 
$L_p$-estimates and the asymptotic behavior as $t \to \infty$ of a solution of the Cauchy problem for a Sobolev system. Proc. Steklov Inst. Maths 103, 123–150.Google Scholar
$L_p$-estimates and the asymptotic behavior as 
$t \to \infty$ of a solution of the Cauchy problem for a Sobolev system. Proc. Steklov Inst. Maths 103, 123–150.Google ScholarMathur, M., Carter, G.S. & Peacock, T. 2014 Topographic scattering of the low-mode internal tide in the deep ocean. J. Geophys. Res. Oceans 119, 2165–2182.CrossRefGoogle Scholar
Maxey, M.R. & Riley, J.J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889.CrossRefGoogle Scholar
Mercier, M.J., Ghaemsaidi, S.J., Echeverri, P., Mathur, M. & Peacock, T. 2012 iTides. Available at: https://doi.org/10.5281/zenodo.4421548.CrossRefGoogle Scholar
Mercier, M.J., Wang, S., Péméja, J., Ern, P. & Ardekani, A.M. 2020 Settling disks in a linearly stratified fluid. J. Fluid Mech. 885, A2.CrossRefGoogle Scholar
Miropol'skii, Y.Z. 1978 Self-similar solutions of the Cauchy problem for internal waves in an unbounded fluid. Izv. Atmos. Ocean. Phys. 14, 673–679.Google Scholar
More, R.V. & Ardekani, A.M. 2023 Motion in stratified fluids. Annu. Rev. Fluid Mech. 55, 157–192.CrossRefGoogle Scholar
More, R.V., Ardekani, M.N., Brandt, L. & Ardekani, A.M. 2021 Orientation instability of settling spheroids in a linearly density-stratified fluid. J. Fluid Mech. 929, A7.CrossRefGoogle Scholar
Morse, P.M. & Feshbach, H. 1953 Methods of Theoretical Physics. Part I. Feshbach Publishing.Google Scholar
Motygin, O.V. & Sturova, I.V. 2002 Wave motions in a two-layer fluid driven by small oscillations of a cylinder intersecting the interface. Fluid Dyn. 37, 600–613.CrossRefGoogle Scholar
Ogilvie, T.F. 1964 Recent progress toward the understanding and prediction of ship motions. In Proceedings of the 5th Symposium on Naval Hydrodynamics (ed. J.K. Lunde & S.W. Doroff), pp. 3–80. U.S. Government Printing Office. Available at: http://resolver.tudelft.nl/uuid:73776ccf-258f-4d5d-ab6c-88c95a002091.Google Scholar
Olver, F.W.J., Lozier, D.W., Boisvert, R.F. & Clark, C.W. 2010 NIST Handbook of Mathematical Functions. NIST/Cambridge University Press.Google Scholar
Palierne, J.F. 1999 On the motion of rigid bodies in incompressible inviscid fluids of inhomogeneous density. J. Fluid Mech. 393, 89–98.CrossRefGoogle Scholar
Papoutsellis, C.E., Mercier, M.J. & Grisouard, N. 2023 Internal tide generation from non-uniform barotropic body forcing. J. Fluid Mech. 964, A20.CrossRefGoogle Scholar
Pétrélis, F., Llewellyn Smith, S. & Young, W.R. 2006 Tidal conversion at a submarine ridge. J. Phys. Oceanogr. 36, 1053–1071.CrossRefGoogle Scholar
Pletner, Y.D. 1991 The fundamental solution of the equation of internal waves and some initial–boundary value problems. Comput. Maths Math. Phys. 31 (4), 79–88.Google Scholar
Riley, J.J., Metcalfe, R.W. & Weissman, M.A. 1981 Direct numerical simulations of homogeneous turbulence in density-stratified fluids. AIP Conf. Proc. 76, 79–112.CrossRefGoogle Scholar
Sarkar, S. & Scotti, A. 2017 From topographic internal gravity waves to turbulence. Annu. Rev. Fluid Mech. 49, 195–220.CrossRefGoogle Scholar
Sekerzh-Zen'kovich, S.Y. 1979 A fundamental solution of the internal-wave operator. Sov. Phys. Dokl. 24, 347–349.Google Scholar
Sekerzh-Zen'kovich, S.Y. 1981 a A uniqueness theorem and an explicit representation of the solution of the Cauchy problem for the equation of internal waves. Sov. Phys. Dokl. 26, 21–23.Google Scholar
Sekerzh-Zen'kovich, S.Y. 1981 b Construction of the fundamental solution for the operator of internal waves. J. Appl. Maths Mech. 45, 192–198.CrossRefGoogle Scholar
Sekerzh-Zen'kovich, S.Y. 1982 Cauchy problem for equations of internal waves. J. Appl. Maths Mech. 46, 758–764.CrossRefGoogle Scholar
Shmakova, N., Ermanyuk, E. & Flór, J.-B. 2017 Generation of higher harmonic internal waves by oscillating spheroids. Phys. Rev. Fluids 2, 114801.CrossRefGoogle Scholar
Simakov, S.L. 1993 Initial and boundary value problems of internal gravity waves. J. Fluid Mech. 248, 55–65.CrossRefGoogle Scholar
Sobolev, S.L. 1954 On a new problem of mathematical physics. Izv. Akad. Nauk SSSR Ser. Mat. 18, 3–50. In Russian; English transl. in Selected Works of S.L. Sobolev (ed. G.V. Demidenko & V.L. Vaskevich), pp. 279–332. Springer (2006).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
Sturova, I.V. 1994 Plane problem of hydrodynamic rocking of a body submerged in a two-layer fluid without forward speed. Fluid Dyn. 29, 414–423.CrossRefGoogle Scholar
Sturova, I.V. 1999 Problems of radiation and diffraction for a circular cylinder in a stratified fluid. Fluid Dyn. 34, 521–533.Google Scholar
Sturova, I.V. 2001 Oscillations of a circular cylinder in a linearly stratified fluid. Fluid Dyn. 36, 478–488.CrossRefGoogle Scholar
Sturova, I.V. 2003 Added masses of a cylinder intersecting the interface of a two-layer weightless fluid of finite depth. J. Appl. Mech. Tech. Phys. 44, 516–521.CrossRefGoogle Scholar
Sturova, I.V. 2006 Oscillations of a cylinder piercing a linearly stratified fluid layer. Fluid Dyn. 41, 619–628.CrossRefGoogle Scholar
Sturova, I.V. 2011 Hydrodynamic loads acting on an oscillating cylinder submerged in a stratified fluid with ice cover. J. Appl. Mech. Tech. Phys. 52, 415–426.CrossRefGoogle Scholar
Sturova, I.V. & Syui, C. 2005 Hydrodynamic load associated with the oscillations of a cylinder at the interface in a two-layer fluid of finite depth. Fluid Dyn. 40, 273–281.CrossRefGoogle Scholar
Sundukova, A.V. 1991 Fundamental solution of the gravitational–gyroscopic wave equation and the solvability of the internal and external Dirichlet problems. Comput. Maths Math. Phys. 31 (10), 87–93.Google Scholar
Ten, I. & Kashiwagi, M. 2004 Hydrodynamics of a body floating in a two-layer fluid of finite depth. Part 1. Radiation problem. J. Mar. Sci. Technol. 9, 127–141.CrossRefGoogle Scholar
Teodorovich, E.V. & Gorodtsov, V.A. 1980 On some singular solutions of internal wave equations. Izv. Atmos. Ocean. Phys. 16, 551–553.Google Scholar
Varanasi, A.K., Marath, N.K. & Subramanian, G. 2022 The rotation of a sedimenting spheroidal particle in a linearly stratified fluid. J. Fluid Mech. 933, A17.CrossRefGoogle Scholar
Voisin, B. 1991 Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources. J. Fluid Mech. 231, 439–480.CrossRefGoogle Scholar
Voisin, B. 2003 Limit states of internal wave beams. J. Fluid Mech. 496, 243–293.CrossRefGoogle Scholar
Voisin, B. 2007 Added mass effects on internal wave generation. In Proceedings of the 5th International Symposium on Environmental Hydraulics (ed. D.L. Boyer & O. Alexandrova). Available at: https://hal.archives-ouvertes.fr/hal-00268817.Google Scholar
Voisin, B. 2009 Added mass in density-stratified fluids. In Actes du 19ème Congrès Français de Mécanique (ed. C. Rey, P. Bontoux & A. Chrisochoos). Available at: https://hal.archives-ouvertes.fr/hal-00583091.Google Scholar
Voisin, B. 2021 Boundary integrals for oscillating bodies in stratified fluids. J. Fluid Mech. 927, A3.CrossRefGoogle Scholar
Voisin, B., Ermanyuk, E.V. & Flór, J.-B. 2011 Internal wave generation by oscillation of a sphere, with application to internal tides. J. Fluid Mech. 666, 308–357.CrossRefGoogle Scholar
Wehausen, J.V. 1971 The motion of floating bodies. Annu. Rev. Fluid Mech. 3, 237–268.CrossRefGoogle Scholar
Whalen, C.B., de Lavergne, C., Naveira Garabato, A.C., Klymak, J.M., MacKinnon, J.A. & Sheen, K.L. 2020 Internal wave-driven mixing: governing processes and consequences for climate. Nat. Rev. Earth Environ. 1, 606–621.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. & Nguyen, T. 1999 Radiation and diffraction of waves in a two-layer fluid. In Proceedings of the 22nd Symposium on Naval Hydrodynamics (ed. Nat. Res. Counc.), pp. 875–891. National Academy Press. Available at: https://doi.org/10.17226/9771.CrossRefGoogle Scholar
You, Y.-X., Shi, Q. & Miao, G.-P. 2007 The radiation and diffraction of water waves by a bottom-mounted circular cylinder in a two-layer fluid. J. Hydrodyn. B 19, 1–8.CrossRefGoogle Scholar
Zavol'skii, N.A. & Zaitsev, A.A. 1984 Development of internal waves generated by a concentrated pulse source in an infinite uniformly stratified fluid. J. Appl. Mech. Tech. Phys. 25, 862–867.CrossRefGoogle Scholar
Zhang, H.P., King, B. & Swinney, H.L. 2007 Experimental study of internal gravity waves generated by supercritical topography. Phys. Fluids 19, 096602.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
Zilman, G., Kagan, L. & Miloh, T. 1996 Hydrodynamics of a body moving over a mud layer. Part II. Added-mass and damping coefficients. J. Ship Res. 40, 39–45.CrossRefGoogle Scholar