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Upstream-propagating waves induced by steady current over a rippled bottom: theory and experimental observation

Published online by Cambridge University Press:  21 January 2021

Jun Fan Open the ORCID record for Jun Fan [Opens in a new window] ,
Jinhai Zheng ,
Aifeng Tao  and
Yuming Liu Open the ORCID record for Yuming Liu [Opens in a new window]
Jun Fan
Affiliation:
Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing210024, PR China College of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing210024, PR China
Jinhai Zheng*
Affiliation:
Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing210024, PR China College of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing210024, PR China
Aifeng Tao
Affiliation:
Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nanjing210024, PR China College of Harbour, Coastal and Offshore Engineering, Hohai University, Nanjing210024, PR China
Yuming Liu*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA02139, USA
*
†Email addresses for correspondence: jhzheng@hhu.edu.cn, yuming@mit.edu
†Email addresses for correspondence: jhzheng@hhu.edu.cn, yuming@mit.edu
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Abstract

We theoretically and experimentally investigate the mechanism underlying the generation of upstream-propagating waves induced by a steady current over a horizontal bottom with a patch of sinusoidal ripples. By considering the triad resonant wave–ripple interactions involving two unsteady wave components (which have the same frequency but different wavenumbers) and one bottom ripple component in the presence of a steady uniform current, we derive the general condition under which unsteady upstream- and/or downstream-propagating waves can be induced. The frequency and wavenumbers of the induced propagating waves are given by the triad resonance condition in terms of current speed, water depth and bottom ripple wavenumber. By means of a multiple-scale perturbation analysis, we obtain the nonlinear amplitude evolution equations governing the spatio-temporal evolution of resonance-generated waves. Based on these equations, we find that the amplitude of the generated upstream-propagating waves is dramatically amplified when the associated triad resonance occurs in the neighbourhood of the critical current speed/frequency (corresponding to zero group velocity of unsteady waves in the presence of a current). A series of laboratory experiments in a long wave flume with wide ranges of current speeds and water depths are conducted to verify the theory. The experiments confirm the observation of the phenomenon of upstream-propagating wave generation in a steady flow over a rippled bottom. In particular, the experimental measurements of the kinematics of upstream-propagating waves as well as the critical flow condition for the observation of such wave generation compare well with the theoretical prediction.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave scattering Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A49
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Binnie, A.M. 1960 Self-induced waves in a conduit with corrugated walls. I. Experiments with water in an open horizontal channel with vertically corrugated sides. Proc. R. Soc. Lond. A 259, 18–27.Google Scholar
Buckles, J., Hanratty, T.J. & Adrian, R.J. 1984 Turbulent flow over large-amplitude wavy surfaces. J. Fluid Mech. 140, 27–44.CrossRefGoogle Scholar
Fan, J., Zheng, J.-H., Tao, A.-F., Yu, H.-F. & Wang, Y. 2016 Experimental study on upstream-advancing waves induced by currents. J. Coast. Res. 75 (sp1), 846–850.CrossRefGoogle Scholar
Garabedian, P.R. 1964 Partial Differential Equations. John Willey and Sons.Google Scholar
Hasselmann, K. 1967 A criterion for nonlinear wave stability. J. Fluid Mech. 30 (4), 737–739.CrossRefGoogle Scholar
Hudson, J.D., Dykhno, L. & Hanratty, T.J. 1996 Turbulence production in flow over a wavy wall. Exp. Fluids 20 (4), 257–265.CrossRefGoogle Scholar
Kennedy, J.F. 1963 The mechanics of dunes and antidunes in erodible-bed channels. J. Fluid Mech. 16 (4), 521–544.CrossRefGoogle Scholar
Kirby, J.T. 1988 Current effects on resonant reflection of surface water waves by sand bars. J. Fluid Mech. 186, 501–520.CrossRefGoogle Scholar
Kuzan, J.D., Hanratty, T.J. & Adrian, R.J. 1989 Turbulent flows with incipient separation over solid waves. Exp. Fluids 7 (2), 88–98.CrossRefGoogle Scholar
Kyotoh, H. & Fukushima, M. 1997 Upstream-advancing waves generated by a current over a sinusoidal bed. Fluid Dyn. Res. 21 (1), 1–28.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
McHugh, J.P. 1986 The stability of stationary waves produced by flow through a channel with wavy sidewalls. PhD thesis, The University of Michigan.Google Scholar
McHugh, J.P. 1988 The stability of stationary waves in a wavy-walled channel. J. Fluid Mech. 189, 491–508.CrossRefGoogle Scholar
McHugh, J.P. 1992 The stability of capillary-gravity waves on flow over a wavy bottom. Wave Motion 16 (1), 23–31.CrossRefGoogle Scholar
Mei, C.C. 1969 Steady free surface flow over wavy bed. J. Engng Mech. Div. 95 (6), 1393–1402.Google Scholar
Mei, C.C. 1985 Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335.CrossRefGoogle Scholar
Mei, C.C., Stiassnie, M. & Yue, D.K.-P. 2005 Theory and Applications of Ocean Surface Waves. World Scientific.Google Scholar
Phillips, O.M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Raj, R. & Guha, A. 2019 On Bragg resonances and wave triad interactions in two-layered shear flows. J. Fluid Mech. 867, 482–515.CrossRefGoogle Scholar
Richardson, A.R. 1920 VIII. Stationary waves in water. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 40 (235), 97–110.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2017 Boundary-Layer Theory. Springer.CrossRefGoogle Scholar
Shen, L., Zhang, X., Yue, D.K.P. & Triantafyllou, M.S. 2003 Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J. Fluid Mech. 484, 197–221.CrossRefGoogle Scholar
White, F.M. 2006 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Yih, C.-S. 1976 Instability of surface and internal waves. In Advances in Applied Mechanics, vol. 16, pp. 369–419. Elsevier.CrossRefGoogle Scholar
Zilker, D.P. & Hanratty, T.J. 1979 Influence of the amplitude of a solid wavy wall on a turbulent flow. Part 2. Separated flows. J. Fluid Mech. 90 (2), 257–271.CrossRefGoogle Scholar