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Steady free-surface flow over spatially periodic topography

Published online by Cambridge University Press:  16 September 2015

B. J. Binder ,
M. G. Blyth  and
S. Balasuriya
B. J. Binder*
Affiliation:
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
M. G. Blyth
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
S. Balasuriya
Affiliation:
School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
*
†Email address for correspondence: benjamin.binder@adelaide.edu.au
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Abstract

Two-dimensional free-surface flow over a spatially periodic channel bed topography is examined using a steady periodically forced Korteweg–de Vries equation. The existence of new forced solitary-type waves with periodic tails is demonstrated using recently developed non-autonomous dynamical-systems theory. Bound states with two or more co-existing solitary waves are also identified. The solution space for varying amplitude of forcing is explored using a numerical method. A rich bifurcation structure is uncovered and shown to be consistent with an asymptotic theory based on small forcing amplitude.

JFM classification

Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 781 , 25 October 2015 , R3
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Copyright
© 2015 Cambridge University Press 

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References

Akylas, T. R. 1984 On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141, 455–466.Google Scholar
Amundsen, D. E., Cox, E. A. & Mortell, M. P. 2007 Asymptotic analysis of steady solutions of the KdVB equation with application to resonant sloshing. Z. Angew. Math. Phys. 58, 1008–1034.Google Scholar
Balasuriya, S. 2011 A tangential displacement theory for locating perturbed saddles and their manifolds. SIAM J. Appl. Dyn. Sys. 10, 1100–1126.Google Scholar
Balasuriya, S. & Binder, B. J. 2014 Nonautonomous analysis of steady Korteweg–de Vries waves under nonlocalised forcing. Physica D 285, 28–41.Google Scholar
Balasuriya, S. & Padberg-Gehle, K. 2013 Controlling the unsteady analogue of saddle stagnation points. SIAM J. Appl. Maths 73, 1038.Google Scholar
Balasuriya, S. & Padberg-Gehle, K. 2014 Accurate control of hyperbolic trajectories in any dimension. Phys. Rev. E 90, 032903.Google Scholar
Billingham, J. & King, A. C. 2000 Wave Motion. Cambridge University Press.Google Scholar
Binder, B. J., Blyth, M. G. & Balasuriya, S. 2014 Non-uniqueness of steady free-surface flow at critical Froude number. Europhys. Lett. 105, 44003.Google Scholar
Binder, B. J., Blyth, M. G. & McCue, S. W. 2013 Free-surface flow past arbitrary topography and an inverse approach for wave-free solutions. IMA J. Appl. Maths 78, 685.Google Scholar
Binder, B. J., Dias, F. & Vanden-Broeck, J.-M. 2005 Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106.CrossRefGoogle ScholarPubMed
Cole, S. L. 1985 Transient waves produced by flow past a bump. Wave Motion 7, 579–587.Google Scholar
Cox, E. A. & Mortell, M. P. 1986 The evolution of resonant water-wave oscillations. J. Fluid Mech. 162, 99–116.CrossRefGoogle Scholar
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419–443.Google Scholar
Dias, F. & Vanden-Broeck, J.-M. 2002 Generalised critical free-surface flows. J. Engng Maths 42, 291.CrossRefGoogle Scholar
Grimshaw, R. & Tian, X. 1994 Periodic and chaotic behaviour in a reduction of the perturbed Korteweg–de Vries equation. Proc. R. Soc. Lond. A 445, 1–21.Google Scholar
Grimshaw, R. H. J. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.Google Scholar
Guckenheimer, J. & Holmes, P. J. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer.Google Scholar
Lamb, H. 1879 Hydrodynamics. Cambridge University Press.Google Scholar
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Ockendon, H., Ockendon, J. R. & Johnson, A. D. 1986 Resonant sloshing in shallow water. J. Fluid Mech. 167, 465–479.Google Scholar
Ockendon, J. R. & Ockendon, H. 1973 Resonant surface waves. J. Fluid Mech. 59, 397–413.Google Scholar
Reynolds, A. J. 1965 Waves on the erodible bed of an open channel. J. Fluid Mech. 22, 113–133.Google Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347.Google Scholar
Shen, S. S.-P. 1993 A Course on Nonlinear Waves. Kluwer Academic.Google Scholar
Wu, T. 1987 Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184, 75–99.Google Scholar
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Appendix A-C

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