Non-Linear Evolution of Steady and Migrating Alternate Bars in a Straight
Channel
H. N. Southgate1 and A. Crosato2,3
1Section of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology,
Stevinweg 1, 2628 CN Delft, The Netherlands
2Section of Environmental Fluid Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of
Technology, Stevinweg 1, 2628 CN Delft, The Netherlands
3UNESCO-IHE, Westvest 7, 2601 DA Delft, The Netherlands
1. Introduction
This paper contains an analysis of a long-duration experiment that shows the evolution of alternate bars in a straight channel. The theoretical predictions are based on a weakly non-linear theory of the morphological development. Both the experiment and theory have several innovative features.
2. Summary of experiment
Between June and September 2009, an experiment was conducted at Delft University of Technology to investigate the long-term evolution of alternate bars in a straight channel with constant discharge and uniform flow at the upstream boundary (Crosato et al, 2011 and 2012). The duration of this experiment, about three months, was much longer than previous experiments of this type, and allowed the observation of long-term behaviour of migrating and steady (nonmigrating) alternate bars.
One or two steady bars were seen at the upstream end of the channel, while migrating bars developed in the rest of the channel. Migrating bars were seen to evolve over a few days but steady bars evolved much more slowly, and they were not fully developed when the experiment ended. An unexpected phenomenon was the occasional near-complete disappearance of the migrating bars after they had achieved full development. This occurred quite rapidly, over a few hours, and they subsequently reformed more slowly over one or two days.
3. Theoretical development
The theoretical analysis is based on the complex Ginzburg-Landau (CGL) equation, which is a weakly nonlinear equation that describes morphological development for channels whose width-to-depth ratio is greater than but close to the critical value for unstable behaviour. The analysis uses the framework of Schielen et al (1993) for the underlying physical equations, the derivation of the CGL equation and the complex constants that appear in this equation.
A new feature is the treatment of advection terms involving the linear group velocity in the CGL equation. Our work shows that these do not describe the dynamic evolution of bedforms, but represent approximations of the differences of values of the linear wavenumber and frequency at the linear maximum growth state from their values at the critical state.
A second innovation is the use of a class of solutions for migrating bars that allow for full variations of the wavenumber and amplitude of these bedforms in space and time, and which commonly reduce to approximate
plane wave forms. These solutions can describe the evolution of quasi-plane waves from an initial spatially uniform state to a final equilibrium state during which both the amplitude and wavenumber change significantly. In contrast, the conventional plane wave solution (which in other work is often used to investigate modulational instabilities) has fixed values of these quantities and therefore does not allow them to evolve in time.
The method is extended to describe the evolution of steady bars at the upstream boundary. The same approximate plane wave solutions are used, but with additional terms from the CGL equation that describe spatial variations near the boundary.
4. Comparison with measurements
The new theory provides quantitative agreement in predictions of the amplitude, wavelength and celerity of the migrating bars with errors of the order of the scatter of the data. Predictions of the equilibrium amplitude and wavelength of the steady bars are made although these are not reached in the experiment. Finally, a suggestion is put forward to explain the puzzling occasional near-complete collapse of the migrating bars in the experiment, based on a boundary-triggered effect. Acknowledgments
The authors would like to thank Ralph Schielen and Arjen Doelman for helpful discussions during this work. References
Crosato A., Mosselman E., Desta F.B., and Uijttewaal W.S J. (2011). Experimental and numerical evidence for intrinsic nonmigrating bars in alluvial channels.
Water Resources Research, 47(3). W03511,
doi:10.1029/2010WR009714.
Crosato A., Desta F.B., Cornelisse J., Schuurman F. and Uijttewaal W.S.J. (2012). Experimental and numerical findings on the long-term evolution of migrating alternate bars in alluvial channels. Water
Resources Research, 48(6), W06524,
doi:10.1029/2011WR011320.
Schielen R., Doelman A. and De Swart H.E. (1993). On the nonlinear dynamics of free bars in straight channels, J. Fluid Mech., 252: 325-356.
