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On the nonlinear dynamics of free bars in straight channels

Published online by Cambridge University Press:  26 April 2006

R. Schielen ,
A. Doelman  and
H. E. de Swart
R. Schielen
Affiliation:
Mathematical Institute, Utrecht University, Budapestlaan 6, 3584 CP Utrecht, The Netherlands Delft Hydraulics, P.O. box 152, 8300 AD Emmeloord, The Netherlands
A. Doelman
Affiliation:
Mathematical Institute, Utrecht University, Budapestlaan 6, 3584 CP Utrecht, The Netherlands
H. E. de Swart
Affiliation:
Institute for Marine and Atmospheric Research Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands
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Abstract

A simple morphological model is considered which describes the interaction between a unidirectional flow and an erodible bed in a straight channel. For sufficiently large values of the width-depth ratio of the channel the basic state, i.e. a uniform current over a flat bottom, is unstable. At near-critical conditions growing perturbations are confined to a narrow spectrum and the bed profile has an alternate bar structure propagating in the downstream direction. The timescale associated with the amplitude growth is large compared to the characteristic period of the bars. Based on these observations a weakly nonlinear analysis is presented which results in a Ginzburg-Landau equation. It describes the nonlinear evolution of the envelope amplitude of the group of marginally unstable alternate bars. Asymptotic results of its coefficients are presented as perturbation series in the small drag coefficient of the channel. In contrast to the Landau equation, described by Colombini et al. (1987), this amplitude equation also allows for spatial modulations due to the dispersive properties of the wave packet. It is demonstrated rigorously that the periodic bar pattern can become unstable through this effect, provided the bed is dune covered, and for realistic values of the other physical parameters. Otherwise, it is found that the periodic bar pattern found by Colombini et al. (1987) is stable. Assuming periodic behaviour of the envelope wave in a frame moving with the group velocity, simulations of the dynamics of the Ginzburg-Landau equation using spectral models are carried out, and it is shown that quasi-periodic behaviour of the bar pattern appears.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 325 - 356
Copyright
© 1993 Cambridge University Press

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References

Bagnold, R. A. 1956 The flow of cohesionless grains in fluids. Proc. R. Soc. Lond. A 249, 235–279.Google Scholar
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417–430.Google Scholar
Blennerhasset, P. J. 1980 On the generation of waves by wind. Phil. Trans. R. Soc. Lond. 298, 451–494.Google Scholar
Blondeaux, P. & Seminara, G. 1985 A unified bar bend theory of river meanders. J. Fluid Mech. 157, 449–470.Google Scholar
Callander, R. A. 1969 Instability and river channels. J. Fluid Mech. 36, 465–480.Google Scholar
Colombini, M., Seminara, G. & Tubino, M. 1987 Finite-amplitude alternate bars. J. Fluid Mech. 181, 213–232.Google Scholar
Crosato, A. 1989 Simulation of meandering river processes. Commun. Hydraul., Rep. 3–90, Delft Univ. of Technology.
Doelman, A. 1989 Slow time periodic solutions of the Ginzburg-Landau equation. Physica D 40, 156–172.Google Scholar
Doelman, A. 1991 Finite dimensional models of the Ginzburg-Landau equation. Nonlinearity 4, 231–250.Google Scholar
Doering, C. R., Gibbon, J. D., Holm, D. D. & Nicolaenco, B. 1988 Low dimensional behaviour in the complex Ginzburg-Landau equation. Nonlinearity 1, 279–309.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Einstein, H. A. 1950 The bedload function for sediment transport in open channel flow. US Dept. Agric. Tech. Bull. 1026.
Engelund, F. 1974 Flow and bed topography in channel bends. J. Hydr. Div., ASCE 100 (HY11)1 1631–1648.Google Scholar
Engelund, F. & Skovgaard, O. 1973 On the origin of meandering and braiding in alluvial streams. J. Fluid Mech. 57, 289–302.Google Scholar
Fredsoe, J. 1978 Meandering and braiding of rivers. J. Fluid Mech. 84, 609–624.Google Scholar
Fukuoka, S. 1989 Finite amplitude development of alternate bars. In River Meandering (ed. Ikeda., S & Parker., G. AGU Water Resources Monograph, vol. 12, pp. 237–266. Washington DC.
Keefe, L. R. 1985 Dynamics of perturbed wavetrain solutions to the Ginzburg-Landau equation. Stud. Appl. Maths 73, 91–153.Google Scholar
Kuramoto, Y. 1984 Chemical Oscillations, Waves and Turbulence. Springer.
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.
Matkowski, B. J. & Volpert, V. 1993 Stability of plane wave solutions of complex Ginzburg-Landau equation. Q. Appl. Maths (to appear).Google Scholar
Newell, A. C. 1974 Envelope equations. Lect. Appl. Maths 15, 157–163.Google Scholar
Newell, A. C. & Whitehead, J. A. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 28, 279–303.Google Scholar
Olesen, K. 1983 Alternate bars in and meandering of alluvial rivers. Commun. Hydraul., Rep. 7–83. Delft Univ. of Technology.
Parker, G. 1976 On the cause and characteristic scales of meandering and braiding in rivers. J. Fluid Mech. 76, 457–480.Google Scholar
Parker, G. & Johannesson, H. 1989 Observations of several recent theories of resonance and overdeepening in meandering channels. I River Meandering (ed. Ikeda, S. & Parker, G.). AGU Water Resources Monograph, vol. 12, pp. 379–415. Washington DC.
Rijn, L. C. van 1989 Handbook of Sediment Transport by Currents and Waves. Delft Hydraulics, Delft, The Netherlands.
Rozovskij, I. L. 1957 Flow of Water in Bends of Open Channels. Kiev: Acad. Sci. Ukranian SSR.Google Scholar
Schöpf, W. & Zimmermann, W. 1989 Multicritical behaviour in binary fluid convection. Europhys. Lett. 8, 41–46.Google Scholar
Sekine, M. & Parker, G. 1992 Bed-load transport on transverse slope. J. Hydraul. Engng ASCE 118, (4)1 513–535.Google Scholar
Stewartson, K. & Stuart, J. T. 1971 A nonlinear instability theory for a wave system in plane Poiseuille flow. J. Fluid Mech. 48, 529–545.Google Scholar
Stuart, J. T. & DiPrima, R. C. 1978 The Eckhaus and Benjamin-Feir resonance mechanisms. Proc. R. Soc. Lond. A 362, 27–41.Google Scholar