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Steady free-surface flow over spatially periodic topography

  • B. J. Binder (a1), M. G. Blyth (a2) and S. Balasuriya (a1)


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.


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Akylas, T. R. 1984 On the excitation of long nonlinear water waves by a moving pressure distribution. J. Fluid Mech. 141, 455466.
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, 10081034.
Balasuriya, S. 2011 A tangential displacement theory for locating perturbed saddles and their manifolds. SIAM J. Appl. Dyn. Sys. 10, 11001126.
Balasuriya, S. & Binder, B. J. 2014 Nonautonomous analysis of steady Korteweg–de Vries waves under nonlocalised forcing. Physica D 285, 2841.
Balasuriya, S. & Padberg-Gehle, K. 2013 Controlling the unsteady analogue of saddle stagnation points. SIAM J. Appl. Maths 73, 1038.
Balasuriya, S. & Padberg-Gehle, K. 2014 Accurate control of hyperbolic trajectories in any dimension. Phys. Rev. E 90, 032903.
Billingham, J. & King, A. C. 2000 Wave Motion. Cambridge University Press.
Binder, B. J., Blyth, M. G. & Balasuriya, S. 2014 Non-uniqueness of steady free-surface flow at critical Froude number. Europhys. Lett. 105, 44003.
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.
Binder, B. J., Dias, F. & Vanden-Broeck, J.-M. 2005 Forced solitary waves and fronts past submerged obstacles. Chaos 15, 037106.
Cole, S. L. 1985 Transient waves produced by flow past a bump. Wave Motion 7, 579587.
Cox, E. A. & Mortell, M. P. 1986 The evolution of resonant water-wave oscillations. J. Fluid Mech. 162, 99116.
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419443.
Dias, F. & Vanden-Broeck, J.-M. 2002 Generalised critical free-surface flows. J. Engng Maths 42, 291.
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, 121.
Grimshaw, R. H. J. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429464.
Guckenheimer, J. & Holmes, P. J. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer.
Lamb, H. 1879 Hydrodynamics. Cambridge University Press.
Lighthill, M. J. 1978 Waves in Fluids. Cambridge University Press.
Ockendon, H., Ockendon, J. R. & Johnson, A. D. 1986 Resonant sloshing in shallow water. J. Fluid Mech. 167, 465479.
Ockendon, J. R. & Ockendon, H. 1973 Resonant surface waves. J. Fluid Mech. 59, 397413.
Reynolds, A. J. 1965 Waves on the erodible bed of an open channel. J. Fluid Mech. 22, 113133.
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.
Shen, S. S.-P. 1993 A Course on Nonlinear Waves. Kluwer Academic.
Wu, T. 1987 Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184, 7599.
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Steady free-surface flow over spatially periodic topography

  • B. J. Binder (a1), M. G. Blyth (a2) and S. Balasuriya (a1)


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