A mathematical model of river plan form movement

Meandering rivers are among the most dynamic features on earth's surface. A meandering river exhibits progressive change in position as it migrates across its floodplain. The planform dynamics of meandering rivers result from the interactions among flow, sediment transport, and evolving channel morphology that collectively scale with planform curvature. The view that planform curvature has a major influence on the rate and character of meander migration is supported by field observations and theoretical considerations. Morphological river changes (bed topography and/or plane-shape changes) occur as result of natural and/or of human interventions so that alluvial meandering streams strive to achieve an equilibrium configuration. As documented by many experimental and theoretical works the plane shape of a meander wave is the result of the complex interplay flow pattern-bed deformation bank erosion. The geometric shape of the meander wave gives rise to a complicated flow field which, in turn, modifies the bed topography and reshapes the meander wave through the bank erosion. In fact, in nature, different meandering streams exhibit different geometric characteristics so that the stream conditions may vary from one meander loop to another. Thus, many laboratory and field studies, apart numerical researches show that the evolution of a meander wave is mainly governed by the bed deformation that drives the erosion process at the channel banks. Particularly, outer-banks are considerably vulnerable to erosion processes. On the other side, the evaluation of the bank erosion and the consequent migration of meandering rivers are fundamental, both because of hazards associated with them and because of their effects in riparian ecosystem dynamics.
The modeling of meandering-river migration requires the simulation of the following processes: hydrodynamics, sediment transport, bed morphodynamics, and bank erosion. Furthermore, in this paper plane form of a meander wave was determined by the mathematical model. Under the conditions prevailing in natural streams, the channel centerlines follow sine-generated curves, with an assumed steady-state turbulent and subcritical flow, of large width-to-depth ratio and small Froude number. The plane deformation of the channel was caused by the action on the banks of the convective vertically-averaged meandering flow. It has to be pointed out and as it is well known that, the regime development of the flow width goes much faster than that of the channel slope. Therefore, in this model the regime development of slope (i.e., the loop expansion), a nearly constant flow width was assumed, which can be identified as the regime width. Accordingly, in this article, the flow width (with as the flow depth) was assumed to be constant along the channel length. Since the width-to-depth ratio was large, the effect of the cross-circulation was negligible. In overall, the channel deformation, as the result of both channel down valley migration and lateral expansion, is entirely due to the action of the convective vertically-averaged meandering flow. The growth (expansion and expansion) of meander loops is attributed to the regime-trend the loops continually grow in their amplitude (i.e., their length increases) and thus the channel slope progressively decreases to the regime state. This model was able to predict the migration and expansion of meander loops.
The following computational procedure was adopted to determine the migration and expansion speed of the meander loop under various values of the deflection angle along the channel centerline. (1) Set regime channel conditions, (2) Compute of meandering channel lengths, (3) Calculate regime slope, (4) Compute water depth, (5) Compute the average specific volumetric sediment transport rate (sediment transport is assumed to be the only bed-load in this article), (6) Compute the angle between the streamline and the coordinate line, (7) Compute the migration speed of meander loops and (8) Compute the expansion speed of meander Loops. The model was then applied to a sine-generated meandering channel and the migration and expansion component of bank speed deformation was calculated. Modeling results showed that, for small deflection angle migration was the main deformation in the meandering channels, on the other hand, for large deflection angles both migration and expansion became smaller in the meandering channels. Furthermore the effect of deflection angle along the channel centerline and channel-averaged specific volumetric sediment transport rate, have been proposed by model. The result showed that the specific volumetric sediment transport rate had significant effect on channel migration and expansion rates.