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Nonlinear hydroelastic waves on a linear shear current at finite depth

  • T. Gao (a1), Z. Wang (a2) (a3) and P. A. Milewski (a1)


This work is concerned with waves propagating on water of finite depth with a constant-vorticity current under a deformable flexible sheet. The pressure exerted by the sheet is modelled by using the Cosserat thin shell theory. By means of multi-scale analysis, small amplitude nonlinear modulation equations in several regimes are considered, including the nonlinear Schrödinger equation (NLS) which is used to predict the existence of small-amplitude wavepacket solitary waves in the full Euler equations and to study the modulational instability of quasi-monochromatic wavetrains. Guided by these weakly nonlinear results, fully nonlinear steady and time-dependent computations are performed by employing a conformal mapping technique. Bifurcation mechanisms and typical profiles of solitary waves for different underlying shear currents are presented in detail. It is shown that even when small-amplitude solitary waves are not predicted by the weakly nonlinear theory, we can numerically find large-amplitude solitary waves in the fully nonlinear equations. Time-dependent simulations are carried out to confirm the modulational stability results and illustrate possible outcomes of the nonlinear evolution in unstable cases.


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Akers, B. & Milewski, P. A. 2009 A model equation for wavepacket solitary waves arising from capillary–gravity Flows. Stud. Appl. Maths 122 (3), 249274.
Ashton, G. D. 1986 River and Lake Ice Engineering. Water Resources Publication.
Balint, T. S. & Lucey, A. D. 2005 Instability of a cantilevered flexible plate in viscous channel flow. J. Fluid Struct. 20 (7), 893912.
Benney, D. J. 1977 A general theory for interactions between short and long waves. Stud. Appl. Maths 56 (1), 8194.
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water Part 1. Theory. J. Fluid Mech. 27 (3), 417430.
Bhattacharjee, J. & Sahoo, T. 2009 Interaction of flexural gravity waves with shear current in shallow water. Ocean Engng 36, 831841.
Choi, W. 2009 Nonlinear surface waves interacting with a linear shear current. Maths Comput. Simul. 80 (1), 2936.
Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. ASCE 125, 756760.
Craik, A. D. 1988 Wave Interactions and Fluid Flows. Cambridge University Press.
Davey, A. & Stewartson, K. On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338 (1613), 101110.
Djordjevic, V. D. & Redekopp, L. G. 1977 On two-dimensional packets of capillary-gravity waves. J. Fluid Mech. 79 (4), 703714.
Dyachenko, A. I., Kuznetsov, E. A., Spectorm, M. D. & Zakharov, V. E. 1996a Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 7379.
Dyachenko, A. I., Zakharov, V. E. & Kuznetsov, E. A. 1996b Nonlinear dynamics on the free surface of an ideal fluid. Plasma Phys. Rep. 22, 916928.
Forbes, L. K. 1986 Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution. J. Fluid Mech. 169, 409428.
Francius, M. & Kharif, C. 2017 Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity. J. Fluid Mech. 830, 631659.
Gao, T., Milewski, P. A. & Vanden-Broeck, J.-M. 2019 Hydroelastic solitary waves with constant vorticity. Wave Motion 85, 8497.
Gao, T. & Vanden-Broeck, J.-M. 2014 Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves. Phys. Fluids 26 (8), 087101.
Gao, T., Vanden-Broeck, J.-M. & Wang, Z. 2018 Numerical computations of two-dimensional flexural-gravity solitary waves on water of arbitrary depth. IMA J. Appl. Maths 83 (3), 436450.
Gao, T., Wang, Z. & Vanden-Broeck, J.-M. 2016 New hydroelastic solitary waves in deep water and their dynamics. J. Fluid Mech. 788, 469491.
Goodman, D. J., Wadhams, P. & Squire, V. A. 1980 The flexural response of a tabular ice island to ocean swell. Ann. Glaciol. 1, 2327.
Guyenne, P. 2017 A high-order spectral method for nonlinear water waves in the presence of a linear shear current. Comput. Fluids 154, 224235.
Guyenne, P. & Părău, E. I. 2012 Computations of fully nonlinear hydroelastic solitary waves on deep water. J. Fluid Mech. 713, 307329.
Guyenne, P. & Părău, E. I. 2014 Finite-depth effects on solitary waves in a floating ice sheet. J. Fluid Struct. 49, 242262.
Hsu, H., Kharif, C., Abid, M. & Chen, Y. 2018 A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1. J. Fluid Mech. 854, 146163.
Jaiman, R. K., Parmar, M. K. & Gurugubelli, P. S. 2014 Added mass and aeroelastic stability of a flexible plate interacting with mean flow in a confined channel. J. Appl. Mech. 81 (4), 041006.
Jones, M. C. W. 1992 Nonlinear stability of resonant capillary-gravity waves. Wave Motion 15 (3), 267283.
Kawahara, T., Sugimoto, N. & Kabutani, T. 1975 Nonlinear interaction between short and long capillary–gravity waves. J. Phys. Soc. Japan 39 (5), 13791386.
Kharif, C. & Pelinovsky, E. 2003 Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. (B/Fluids) 22 (6), 603634.
Korobkin, A., Părău, E. I. & Vanden-Broeck, J.-M. 2011 The mathematical challenges and modelling of hydroelasticity. Phil. Trans. R. Soc. Lond. A 369 (1947), 28032812.
Marko, J. R. 2003 Observations and analyses of an intense waves-in-ice event in the Sea of Okhotsk. J. Geophys. Res. 108, 3296.
Massom, R. A., Scambos, T. A., Bennetts, L. G., Reid, P., Squire, V. A. & Stammerjohn, S. E. 2018 Antarctic ice shelf disintegration triggered by sea ice loss and ocean swell. Nature 558, 383389.
McGoldrick, L. F. 1970 On Wilton’s ripples: a special case of resonant interactions. J. Fluid Mech. 42 (1), 193200.
Milewski, P. A. & Wang, Z. 2013 Three dimensional flexural-gravity waves. Stud. Appl. Maths 131 (2), 135148.
Milewski, P. A., Vanden-Broeck, J.-M. & Wang, Z. 2011 Hydroelastic solitary waves in deep water. J. Fluid Mech. 679, 628640.
Milewski, P. A., Vanden-Broeck, J. M. & Wang, Z. 2013 Steady dark solitary flexural-gravity waves. Proc. R. Soc. A 469, 20120485.
Milinazzo, F. A. & Saffman, P. G. 1990 Effect of a surface shear layer on gravity and gravity-capillary waves of permanent form. J. Fluid Mech. 216, 93101.
Părău, E. I. & Dias, F. 2002 Nonlinear effects in the response of a floating ice plate to a moving load. J. Fluid Mech. 460, 281305.
Peake, N. 2001 Nonlinear stability of a fluid-loaded elastic plate with mean flow. J. Fluid Mech. 434, 101118.
Peake, N. 2004 On the unsteady motion of a long fluid-loaded elastic plate with mean flow. J. Fluid Mech. 507, 335366.
Ribeiro, R., Milewski, P. A. & Nachbin, A. 2017 Flow structure beneath rotational water waves with stagnation points. J. Fluid Mech. 812, 792814.
Simmen, J. A. & Saffman, P. G. 1985 Steady deep water waves on a linear shear current. Stud. Appl. Maths 73, 3557.
Simmons, W. F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309, 551579.
Shoele, K. & Mittal, R. 2016 Flutter instability of a thin flexible plate in a channel. J. Fluid Mech. 786, 2946.
Squire, V., Hosking, R. J., Kerr, A. D. & Langhorne, P. J. 1996 Moving Loads on Ice Plates, Solid Mechanics and its Applications. Kluwer.
Squire, V., Robinson, W., Langhorne, P. & Haskell, T. 1988 Vehicles and aircraft on floating ice. Nature 333 (6169), 159161.
Stiassnie, M. & Kroszynski, U. 1982 Long-time evolution of an unstable water-wave train. J. Fluid Mech. 116, 207225.
Takizawa, T. 1985 Deflection of a floating sea ice sheet induced by a moving load. Cold Regions Sci. Tech. 11, 171180.
Takizawa, T. 1988 Response of a floating sea ice sheet to a steadily moving load. J. Geophys. Res. 93, 51005112.
Tanaka, M. 1990 Maximum amplitude of modulated wavetrain. Wave Motion 2 (6), 559568.
Teles Da Silva, A. F. & Peregrine, D. H. 1988 Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281302.
Thomas, R., Kharif, C. & Manna, M. 2012 A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity. Phys. Fluids 24, 127102.
Toland, J. F. 2007 Heavy hydroelastic travelling waves. Proc. R. Soc. Lond. A 463, 23712397.
Trichtchenko, O., Milewski, P. A., Părău, E. I. & Vanden-Broeck, J.-M. 2019 Stability of periodic travelling flexural-gravity waves in two dimensions. Stud. Appl. Maths 142, 6590.
Trichtchenko, O., Părău, E. I., Vanden-Broeck, J.-M. & Milewski, P. A. 2018 Solitary flexural-gravity waves in three dimensions. Phil. Trans. R. Soc. Lond. A 376 (2129), 20170345.
Vanden-Broeck, J.-M. 1994 Steep solitary waves in water of finite depth with constant vorticity. J. Fluid Mech. 274, 339348.
Vanden-Broeck, J.-M. & Părău, E. I. 2011 Two-dimensional generalized solitary waves and periodic waves under an ice sheet. Phil. Trans. R. Soc. Lond. A 369, 29572972.
Wang, Z. & Milewski, P. A. 2012 Dynamics of gravity-capillary solitary waves in deep water. J. Fluid Mech. 708, 480501.
Wang, Z., Părău, E. I., Milewski, P. A. & Vanden-Broeck, J.-M. 2014 Numerical study of interfacial solitary waves propagating under an elastic sheet. Proc. R. Soc. Lond. A 470, 20140111.
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, 750761.
Wilton, J. R. On ripples. Phil. Mag. 29 (173), 688700.
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