Introduction
Local geometrical perturbations in alluvial channels can generate a pattern of alternate bars called ‘overdeepening’. Previous linear analyses and laboratory experiments showed that these bars arise downstream of perturbations in the relatively narrow and deep channels corresponding to subresonant conditions, but both upstream and downstream of perturbations in the relatively wide and shallow channels corresponding to superresonant conditions. Previous numerical computations reproduced alternate bars under subresonant conditions, but failed to do so under superresonant conditions until the recent 2D depth-averaged computations by van der Meer et al. (2011) using Delft3D. They managed to simulate bar formation under superresonant conditions, but observed discrepancies between the numerical results and theoretical and experimental findings.
Approach
We extended the modelling work of van der Meer et al. (2011) to assess the agreement of numerical results with linear theory and experimental observations, by simulating more cases and by analyzing in detail the causes of the discrepancies.
The focus has been on the influence of numerical diffusion, introduced by the upwind scheme (see Box 2) for the morphological updating procedure and the influence of the horizontal eddy viscosity. To assess the influence of these parameters, the behaviour of small-amplitude free bars is tested and compared to linear theory (Box 1).
Conclusions
Upwinding in the morphological updating procedure has mainly two effects:
• Longer bars in the spectrum are damped • First order bars become more pronounced
than higher order bars (e.g. central bars)
In shallow flows (large width-to-depth ratios) this leads to an underprediction of the bar length. In deeper flows (low width-to-depth ratios) these effects lead to an overprediction of the bar
length. The significance of these effects depends on the ratio between the applied grid size and
the dominant alternate bar length.
The point of resonance is proportional to
the applied horizontal eddy viscosity. Due to
stability reasons, this horizontal eddy viscosity often needs to be increased, leading to an
overprediction of the point of resonance
Validation
Delft3D computations have been compared to experiments of Zolezzi et al. (2005), see Figure 1. The point of resonance is overpredicted in the Delft3D simulations.
Box 1: Linear theory
Alternate bar behaviour can be explained to some extent by a stability analysis of the mathematical equations for flow and sediment transport. In this research the analytical model of Colombini et al. (1987) has been applied, see Figure 2.
Box 2: Numerical diffusion
Numerical diffusion is introduced by the upwind scheme in the bed load transport procedure, because of a shift of the bed load transport distribution in downstream direction, see Figure 3.
Results
Bar length
Numerical diffusion (due to the upwind scheme) has two main effects:
• Longer bars in the spectrum are damped, see Figure 5 (and box 2)
• First order bars become more pronounced than higher order bars (e.g. central bars), see Figure 6. This leads to larger alternate bar heights.
Point of resonance
Figure 7 shows that that the point of resonance in
the numerical model is proportional to the horizontal eddy viscosity and that numerical diffusion has a
minor influence. Due to stability reasons, a horizontal eddy viscosity of 0.01 m2/s was applied, leading to an overprediction of the point of resonance.
for more information
1. Deltares, Delft, The Netherlands
2. Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands
Figure 2: Free bar diagram (linear theory), including modelled forced bar lengths
Figure 4: Free and forced bars
Figure 1: observed and modelled alternate bar patterns
Figure 5: Influence of numerical diffusion on amplification rate
Figure 6: Influence of numerical diffusion on bar spectrum
Figure 7: Influence of numerical diffusion and horizontal eddy viscosity on point of resonance
Figure 3: Effect of upwinding on the bed load transport distribution
www.deltares.nl
Numerical nonlinear analysis of alternate-bar
formation and overdeepening under superresonant
conditions
W.Verbruggen1, E. Mosselman1,2, G. Zolezzi3 and C.J. Sloff1,2
References
ZOLEZZI, G., GUALA, M., TERMINI, D. & Seminara, G. 2005. Experimental observations of upstream overdeepening. Journal of Fluid Mechanics, 531, 191-219
VAN DER MEER, C., MOSSELMAN, E., SLOFF, K., JAGER, B., ZOLEZZI, G. & TUBINO, M. 2011. Numerical simulations of upstream and downstream overdeepening. RCEM2011, Tsinghua University Press, Beijing, China
COLOMBINI, M., SEMINARA, G. & TUBINO, M. 1987. Finite-amplitude alternate bars. Journal of Fluid Mechanics, 181, 213-232
VANZO, D., SIVIGLIA, A., ZOLEZZI, G., STECCA, G. & TUBINO, M. 2011. Interaction between steady and migrating bars in straight channels. RCEM 2011, Tsinghua University Press, Beijing, China
3. Department of Civil and Environmental Engineering, University of Trento, Trento, Italy
