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Resonant triad interactions of gravity waves in cylindrical basins

Published online by Cambridge University Press:  03 July 2023

Matthew Durey Open the ORCID record for Matthew Durey [Opens in a new window]  and
Paul A. Milewski Open the ORCID record for Paul A. Milewski [Opens in a new window]
Matthew Durey*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
Paul A. Milewski
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
*
†Email address for correspondence: matthew.durey@glasgow.ac.uk
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Abstract

We present the results of a theoretical investigation into the existence, evolution and excitation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of finite depth. It is well known that resonant triads are impossible for gravity waves in laterally unbounded domains; we demonstrate, however, that horizontal confinement of the fluid may induce resonant triads for particular fluid depths. For any three correlated wave modes arising in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the existence of a depth at which nonlinear resonance may arise, and show that the resultant critical depth is unique. We enumerate the low-frequency triads for circular cylinders, including a new class of resonances between standing and counter-propagating waves, and also briefly discuss annular and rectangular cylinders. Upon deriving the triad amplitude equations for a finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always periodic, and determine parameters controlling the efficiency of energy exchange. In order to excite a particular triad, we explore the influence of external forcing; in this case, the triad evolution may be periodic, quasi-periodic or chaotic. Finally, our results have potential implications on resonant water waves in man-made and natural basins, such as industrial-scale fluid tanks, harbours and bays.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 966 , 10 July 2023 , A25
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Annenkov, S.Y. & Shrira, V.I. 2006 Direct numerical simulation of downshift and inverse cascade for water wave turbulence. Phys. Rev. Lett. 96 (20), 204501.CrossRefGoogle ScholarPubMed
Ball, F.K. 1964 Energy transfer between external and internal gravity waves. J. Fluid Mech. 19 (3), 465–478.CrossRefGoogle Scholar
Benney, D.J. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14 (4), 577–584.CrossRefGoogle Scholar
Benney, D.J. & Luke, J.C. 1964 On the interactions of permanent waves of finite amplitude. J. Math. Phys. 43, 309–313.CrossRefGoogle Scholar
Berger, K.M. & Milewski, P.A. 2003 Simulation of wave interactions and turbulence in one-dimensional water waves. SIAM J. Appl. Maths 63 (4), 1121–1140.CrossRefGoogle Scholar
Bretherton, F. 1964 Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20 (3), 457–479.CrossRefGoogle Scholar
Bridges, T.J. & Dias, F. 1990 An analysis of two-dimensional water waves based on $O(2)$ symmetry. Nonlinear Anal. Theory Meth. Applics. 14 (9), 733–764.CrossRefGoogle Scholar
Bryant, P.J. 1973 Periodic waves in shallow water. J. Fluid Mech. 59 (4), 625–644.CrossRefGoogle Scholar
Bryant, P.J. 1989 Nonlinear progressive free waves in a circular basin. J. Fluid Mech. 205, 453–467.CrossRefGoogle Scholar
Case, K.M. & Chiu, S.C. 1977 Three-wave resonant interactions of gravity-capillary waves. Phys. Fluids 20, 742.CrossRefGoogle Scholar
Chabane, M. & Choi, W. 2019 On resonant interactions of gravity-capillary waves without energy exchange. Stud. Appl. Maths 142, 528–550.CrossRefGoogle Scholar
Choi, W., Chabane, M. & Taklo, T.M.A. 2021 Two-dimensional resonant triad interactions in a two-layer system. J. Fluid Mech. 907, A5.CrossRefGoogle Scholar
Chossat, P. & Dias, F. 1995 The 1:2 resonance with $O(2)$ symmetry and its applications in hydrodynamics. J. Nonlinear Sci. 5, 105–129.CrossRefGoogle Scholar
Chow, C., Henderson, D. & Segur, H. 1996 A generalized stability criterion for resonant triad interactions. J. Fluid Mech. 319, 67–76.CrossRefGoogle Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108 (1), 73–83.CrossRefGoogle Scholar
Craik, A.D.D. 1986 Wave Interactions and Fluid Flows. Cambridge University Press.CrossRefGoogle Scholar
Davis, R.E. & Acrivos, A. 1967 The stability of oscillatory internal waves. J. Fluid Mech. 30 (4), 723–736.CrossRefGoogle Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. 121, 299–340.Google Scholar
Guza, R.T. & Davis, R.E. 1974 Excitation of edge waves by waves incident on a beach. J. Geophys. Res. 79 (9), 1285–1291.CrossRefGoogle Scholar
Hammack, J.L. & Henderson, D.M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25 (1), 55–97.CrossRefGoogle Scholar
Harris, J., Bustamante, M.D. & Connaughton, C. 2012 Externally forced triads of resonantly interacting waves: boundedness and integrability properties. Commun. Nonlinear Sci. Numer. Simul. 17, 4988–5006.CrossRefGoogle Scholar
Hasselmann, K. 1961 On the non-linear energy transfer in a gravity-wave spectrum. Part 1. General theory. J. Fluid Mech. 12 (4), 481–500.CrossRefGoogle Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30 (4), 737–739.CrossRefGoogle Scholar
Henderson, D.M. & Hammack, J.L. 1987 Experiments on ripple instabilities. Part 1. Resonant triads. J. Fluid Mech. 184, 15–41.CrossRefGoogle Scholar
Henderson, D.M. & Miles, J.W. 1991 Faraday waves in 2:1 internal resonance. J. Fluid Mech. 222, 449–470.CrossRefGoogle Scholar
Joyce, T.M. 1974 Nonlinear interactions among standing surface and internal gravity waves. J. Fluid Mech. 63 (4), 801–825.CrossRefGoogle Scholar
Kadri, U. & Akylas, T.R. 2016 On resonant triad interactions of acoustic-gravity waves. J. Fluid Mech. 788, R1.CrossRefGoogle Scholar
Kadri, U. & Stiassnie, M. 2013 Generation of an acoustic-gravity wave by two gravity waves, and their subsequent mutual interaction. J. Fluid Mech. 735, R6.CrossRefGoogle Scholar
Kevorkian, J.K. & Cole, J.D. 1996 Multiple Scale and Singular Perturbation Methods, Appl. Math. Sci., vol. 114. Springer.CrossRefGoogle Scholar
Kreyszig, E. 1989 Introductory Functional Analysis with Applications. John Wiley & Sons.Google Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. Lond. A 452, 1113–1126.Google Scholar
Kumar, K. & Tuckerman, L.S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Dover.Google Scholar
Longuet-Higgins, M.S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12 (3), 321–332.CrossRefGoogle Scholar
Mack, L.R. 1962 Periodic, finite-amplitude, axisymmetric gravity waves. J. Geophys. Res. 67 (2), 829–843.CrossRefGoogle Scholar
McEwan, A., Mander, D. & Smith, R. 1972 Forced resonant second-order interaction between damped internal waves. J. Fluid Mech. 55 (4), 589–608.CrossRefGoogle Scholar
McGoldrick, L.F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21 (2), 305–331.CrossRefGoogle Scholar
McGoldrick, L.F. 1970 a An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40 (2), 251–271.CrossRefGoogle Scholar
McGoldrick, L.F. 1970 b On Wilton's ripples: a special case of resonant interactions. J. Fluid Mech. 42 (1), 193–200.CrossRefGoogle Scholar
McGoldrick, L.F. 1972 On the rippling of small waves: a harmonic nonlinear nearly resonant interaction. J. Fluid Mech. 52 (4), 725–751.CrossRefGoogle Scholar
Michel, G. 2019 Three-wave interactions among surface gravity waves in a cylindrical container. Phys. Rev. Fluids 4, 012801(R).CrossRefGoogle Scholar
Miles, J.W. 1967 Surface-wave damping in closed basins. Proc. R. Soc. A 297 (1451), 459–475.Google Scholar
Miles, J.W. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75 (3), 419–448.CrossRefGoogle Scholar
Miles, J.W. 1984 a Internally resonant surface waves in a circular cylinder. J. Fluid Mech. 149, 1–14.CrossRefGoogle Scholar
Miles, J.W. 1984 b Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.CrossRefGoogle Scholar
Miles, J.W. 1984 c Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.CrossRefGoogle Scholar
Miles, J.W. & Henderson, D.M. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22, 143–165.CrossRefGoogle Scholar
Milewski, P.A. & Keller, J.B. 1996 Three-dimensional water waves. Stud. Appl. Maths 97, 149–166.CrossRefGoogle Scholar
Phillips, O.M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9 (2), 193–217.CrossRefGoogle Scholar
Qadeer, S. & Wilkening, J.A. 2019 Computing the Dirichlet–Neumann operator on a cylinder. SIAM J. Numer. Anal. 57 (3), 1183–1204.CrossRefGoogle Scholar
Raupp, C.F.M. & Silva Dias, P.L. 2009 Resonant wave interactions in the presence of a diurnally varying heat source. J. Atmos. Sci. 66 (10), 3165–3183.CrossRefGoogle Scholar
Raupp, C.F.M., Silva Dias, P.L., Tabak, E.G. & Milewski, P.A. 2008 Resonant wave interactions in the equatorial waveguide. J. Atmos. Sci. 65 (11), 3398–3418.CrossRefGoogle Scholar
Schwartz, L. & Vanden-Broeck, J.-M. 1979 Numerical solution of the exact equations for capillary–gravity waves. J. Fluid Mech. 95 (1), 119–139.CrossRefGoogle Scholar
Segur, H. 1980 Resonant interactions of surface and internal gravity waves. Phys. Fluids 23, 2556.CrossRefGoogle Scholar
Simmons, W.F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309 (1499), 551–577.Google Scholar
Strogatz, S.H. 2015 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering, 2nd edn. CRC Press.Google Scholar
Taklo, T.M.A. & Choi, W. 2020 Group resonant interactions between surface and internal gravity waves in a two-layer system. J. Fluid Mech. 892, A14.CrossRefGoogle Scholar
Wang, Z. & Milewski, P.A. 2012 Dynamics of gravity-capillary solitary waves in deep water. J. Fluid Mech. 708, 480–501.CrossRefGoogle Scholar
Wang, Z., Vanden-Broeck, J.-M. & Milewski, P.A. 2013 Two-dimensional flexural–gravity waves of finite amplitude in deep water. IMA J. Appl. Maths 78 (4), 750–761.CrossRefGoogle Scholar
Whitham, G.B. 1965 a A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22 (2), 273–283.CrossRefGoogle Scholar
Whitham, G.B. 1965 b Non-linear dispersive waves. Proc. R. Soc. Lond. A 283 (1393), 238–261.Google Scholar
Whitham, G.B. 1967 a Non-linear dispersion of water waves. J. Fluid Mech. 27 (2), 399–412.CrossRefGoogle Scholar
Whitham, G.B. 1967 b Variational methods and applications to water waves. Proc. R. Soc. Lond. A 299, 6–25.Google Scholar
Wilton, J.R. 1915 On ripples. Phil. Mag. 29 (173), 688–700.CrossRefGoogle Scholar
Yang, X., Dias, F., Liu, Z. & Liao, S. 2021 Finite-amplitude steady-state resonant waves in a circular basin. J. Fluid Mech. 915, A136.CrossRefGoogle Scholar
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