Hostname: page-component-54dcc4c588-m259h Total loading time: 0 Render date: 2025-09-30T11:58:12.263Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear flexural-gravity waves for flows over bottom topography

Published online by Cambridge University Press:  22 September 2025

Chuang Liang ,
Baoyu Ni Open the ORCID record for Baoyu Ni [Opens in a new window]  and
Yuriy Semenov Open the ORCID record for Yuriy Semenov [Opens in a new window]
Chuang Liang
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, PR China
Baoyu Ni*
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, PR China
Yuriy Semenov*
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, PR China
*
Corresponding authors: Baoyu Ni, nibaoyu@hrbeu.edu.cn; Yuriy Semenov, yuriy.a.semenov@gmail.com
Corresponding authors: Baoyu Ni, nibaoyu@hrbeu.edu.cn; Yuriy Semenov, yuriy.a.semenov@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The interaction between the flow in a channel with multiple obstructions on the bottom and an elastic ice sheet covering the liquid is studied for the case of steady flow. The mathematical model employs velocity potential theory and fully accounts for the nonlinear boundary conditions at the ice/liquid interface and on the channel bottom. The integral hodograph method is used to derive the complex velocity potential of the flow, explicitly containing the velocity magnitude at the interface. This allows the boundary-value problem to be reduced to a system of nonlinear equations for the unknown velocity magnitude at the ice/liquid interface, which is solved using the collocation method. Case studies are carried out for a widened rectangular obstruction, whose width exceeds the wavelength of the interface, and for arrays of triangular ripples forming the undulating bottom shape. The influence of the bottom shape on the interface is investigated for three flow regimes: the subcritical regime, $F \lt F_{{cr}}$, for which the depth-based Froude number is less than the critical Froude number, and the interface perturbation decays upstream and downstream of the obstruction; the ice-supercritical and channel-subcritical regime, $F_{cr} \lt F \lt 1$, for which two waves of different wavelengths extend upstream and downstream to infinity; and the channel-supercritical regime, $F \gt 1$, for which the hydroelastic wave extends downstream to infinity. The results revealed a trapped interface wave above the rectangular obstruction and the ripple patch. The resonance behaviour of the interface over the undulating bottom occurs when the period of ripples approaches the wavelength of the ice/liquid interface.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Waves/Free-surface Flows: Channel flow Waves/Free-surface Flows: Wave-structure interactions

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1019 , 25 September 2025 , A40
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alam, M.-R., Liu, Y. & Yue, D.K.P. 2009 Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis. J. Fluid Mech. 624, 191–224.10.1017/S0022112008005478CrossRefGoogle Scholar
Barman, K.K., Chanda, A., Tsai, C.-C. & Hsu, T.-W. 2024 b Bragg scattering of gravity waves by a sea bed of varying depth in the presence of uniform current covered by a floating membrane. Phys. Fluids 36, 012118.10.1063/5.0183629CrossRefGoogle Scholar
Barman, K.K., Chanda, A., Tsai, C.-C. & Mondale, S. 2024 a Flexural-gravity wave interaction with undulating bottom topography in the presence of uniform current: an asymptotic approach. Appl. Math. Model. 133, 41–64.10.1016/j.apm.2024.05.012CrossRefGoogle Scholar
Blyth, M.G., Părău, E.I. & Vanden-Broeck, J.-M. 2011 Hydroelastic waves on fluid sheets. J. Fluid Mech. 689, 541–551.10.1017/jfm.2011.451CrossRefGoogle Scholar
Bonnefoy, F., Meylan, M.H. & Ferrant, P. 2009 Nonlinear wave generation and absorption in a 2D hydrodynamic flume. J. Fluid Mech. 621, 215–242.10.1017/S0022112008004849CrossRefGoogle Scholar
Boral, S., Meylan, M.H., Sahoo, T. & Ni, B.-Y. 2023 Time-dependent flexural gravity wave scattering due to uneven bottomin the paradigm of blocking dynamics. Phys. Fluids 35, 116603.10.1063/5.0173542CrossRefGoogle Scholar
Chamberlain, P.G. & Porter, D. 1995 The modified mild-slope equation. J. Fluid Mech. 291, 393–407.10.1017/S0022112095002758CrossRefGoogle Scholar
Davies, A.G. 1982 The reflection of wave energy by undulations on the seabed. Dyn. Atmos. Oceans 6 (4), 207–232.10.1016/0377-0265(82)90029-XCrossRefGoogle Scholar
Dias, F. & Vanden-Broeck, J.-M. 1989 Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170.10.1017/S0022112089002260CrossRefGoogle Scholar
Forbes, L.K. 1986 Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution. J. Fluid Mech. 169, 409–428.10.1017/S0022112086000708CrossRefGoogle Scholar
Forbes, L.K. 1988 Surface waves of large amplitude beneath an elastic sheet. Part 2. Galerkin solution. J. Fluid Mech. 188, 491–508.10.1017/S0022112088000813CrossRefGoogle Scholar
Gao, W., Wang, Z. & Milewski, P.A. 2019 Nonlinear hydroelastic waves on a linear shear current at finite depth. J. Fluid Mech. 876, 55–86.10.1017/jfm.2019.528CrossRefGoogle Scholar
Greenhill, A.G. 1887 Wave motion in hydrodynamics. Am. J. Math. 9, 62–112.10.2307/2369499CrossRefGoogle Scholar
Gurevich, M.I. 1965 Theory of Jets in Ideal Fluids. Academic Press.Google Scholar
Guyenne, P. & Părău, E.I. 2012 Computations of fully nonlinear hydroelastic solitary waves on deep water. J. Fluid Mech. 713, 307–329.10.1017/jfm.2012.458CrossRefGoogle Scholar
Joukowskii, N.E. 1890 Modification of Kirchhoff’s method for determination of a fluid motion in two directions at a fixed velocity given on the unknown streamline. Math. Sbornik. 15 (1), 121–278.Google Scholar
Karmakar, D., Bhattacharjee, J. & Sahoo, T. 2010 Oblique flexural gravity-wave scattering due to changes in bottom topography. J. Engng Maths 66, 325–341.10.1007/s10665-009-9297-8CrossRefGoogle Scholar
Kheisin, D.E. 1963 Moving load on an elastic plate which floats on the surface of an ideal fluid (in Russian). Izv. Akad. Nauk SSSR Otd. Tekh. Nauk. Mekh. i Mashinostroenie 1, 178–180.Google Scholar
Kheisin, D.E. 1967 Dynamics of ice cover. Tech. Rep. FSTC-HT-23-485-69. U.S. Army Foreign Science and Technology Center.Google Scholar
Kochin, N.E., Kibel, I.A. & Roze, N.V. 1964 Theoretical Hydromechanics, vol. 1964. Wiley Interscience.Google Scholar
Korobkin, A.A., Malenica, S. & Khabakhpasheva, T.I. 2018 Interaction of flexural-gravity waves in ice cover with vertical walls. Phil. Trans. R. Soc. A Math. Phys. Engng Sci. 376 (2129), 20170347.10.1098/rsta.2017.0347CrossRefGoogle ScholarPubMed
Korobkin, A.A., Părău, E.I. & Vanden-Broeck, J.-M. 2011 The mathematical challenges and modelling of hydroelasticity. Phil. Trans. R. Soc. Lond. A 369, 2803–2812.Google ScholarPubMed
Mei, C.C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.10.1017/S0022112085000714CrossRefGoogle Scholar
Michell, J.H. 1890 On the theory of free streamlines. Phil. Trans. R. Soc. Lond. A 181, 389–431 Google Scholar
Milewski, P.A., Vanden-Broeck, J.-M. & Wang, Z. 2011 Hydroelastic solitary waves in deep water. J. Fluid Mech. 679, 628–640.10.1017/jfm.2011.163CrossRefGoogle Scholar
Milewski, P.A. & Wang, Z. 2013 Three dimensional flexural–gravity waves. Stud. Appl. Maths 131, 135–148.10.1111/sapm.12005CrossRefGoogle Scholar
Ni, B.-Y., Han, D.-F., Di, S.-C. & Xue, Y.-Z. 2020 On the development of ice-water-structure interaction. J. Hydrodyn. 32, 629–652.10.1007/s42241-020-0047-8CrossRefGoogle Scholar
Ni, B.-Y., Khabakhpasheva, T.I. & Semenov, Y.A. 2023 Nonlinear gravity waves in the channel covered by broken ice. Phys. Fluids 35, 102118.10.1063/5.0166567CrossRefGoogle Scholar
Ni, B.-Y., Semenov, Y.A., Khabakhpasheva, T.I., Părău, E.I. & Korobkin, A.A. 2024 Nonlinear ice sheet/liquid interaction in a channel with an obstruction. J. Fluid Mech. 983, A41.10.1017/jfm.2024.177CrossRefGoogle Scholar
Page, C. & Părău, E.I. 2014 Hydraulic falls under a floating ice plate due to submerged obstructions. J. Fluid Mech. 745, 208–222.10.1017/jfm.2014.92CrossRefGoogle Scholar
Părău, E.I. & Dias, F. 2002 Nonlinear effects in the response of a floating ice plate to a moving load. J. Fluid Mech. 460, 281–305.10.1017/S0022112002008236CrossRefGoogle Scholar
Plotnikov, P.I. & Toland, J.F. 2011 Modelling nonlinear hydroelastic waves. Phil. Trans. R. Soc. Lond. A 369, 2942–2956.Google ScholarPubMed
Porter, D. & Porter, R. 2004 Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography. J. Fluid Mech. 509, 145–179.10.1017/S0022112004009267CrossRefGoogle Scholar
Semenov, Y.A. 2021 Nonlinear flexural-gravity waves due to a body submerged in the uniform stream. Phys. Fluids 33 (5), 052115.10.1063/5.0048887CrossRefGoogle Scholar
Semenov, Y.A. & Cummings, L.J. 2007 Free boundary Darcy flows with surface tension: analytical and numerical study. Eur. J. Appl. Maths 17, 607–631.10.1017/S0956792506006759CrossRefGoogle Scholar
Semenov, Y.A. & Yoon, B.-S. 2009 Onset of flow separation at oblique water impact of a wedge. Phys. Fluids 21, 112103–112111.10.1063/1.3261805CrossRefGoogle Scholar
Shishmarev, K., Khabakhpasheva, T.I. & Korobkin, A.A. 2016 The response of ice cover to a load moving along a frozen channel. Appl. Ocean Res. 59, 313–326.10.1016/j.apor.2016.06.008CrossRefGoogle Scholar
Shishmarev, K., Khabakhpasheva, T.I. & Korobkin, A.A. 2019 Ice response to an underwater body moving in a frozen channel. Appl. Ocean Res. 91, 101877.10.1016/j.apor.2019.101877CrossRefGoogle Scholar
Squire, V.A. 2020 Ocean wave interactions with sea ice: a reappraisal. Annu. Rev. Fluid Mech. 52, 37–60.10.1146/annurev-fluid-010719-060301CrossRefGoogle Scholar
Squire, V.A., Dugan, J.P., Wadhams, P., Rottier, P.J. & Liu, A.K. 1995 Of ocean waves and sea ice. Annu. Rev. Fluid Mech. 27, 115–168.10.1146/annurev.fl.27.010195.000555CrossRefGoogle Scholar
Sturova, I.V. 2009 Time-dependent response of a heterogeneous elastic plate floating on shallow water of variable depth. J. Fluid Mech. 637, 305–325.10.1017/S0022112009990504CrossRefGoogle Scholar
Xiong, H., Ni, B., Semenov, Y. & Korobkin, A. 2025 Inertial motion of a circular cylinder approaching obliquely an ice cover. Sci Rep. 15, 9412.10.1038/s41598-025-93435-1CrossRefGoogle ScholarPubMed
Xue, Y.Z., Zeng, L.D., Ni, B.Y., Korobkin, A.A. & Khabakhpasheva, T. 2021 Hydroelastic waves in an ice-covered channel with leads. Phys. Fluids 33, 037109.10.1063/5.0037682CrossRefGoogle Scholar
Yoon, B.-S. & Semenov, Y.A. 2011 Separated inviscid sheet flows. J. Fluid Mech. 678, 511–534.10.1017/jfm.2011.123CrossRefGoogle Scholar