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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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