A new vectorial bedload formulation and its application to the time evolution of straight river channels

The derivation of a new vectorial bedload formulation for the transport of coarse sediment by fluid flow is presented in the first part of the paper. This relation has been developed for slopes up to the angle of repose both in the streamwise and transverse directions. The pressure distribution is assumed to be hydrostatic. The bed shear stress for the onset of particle motion and mean particle velocity are obtained from the mean force balance on a particle. A new generalized Bagnold hypothesis is introduced to calculate the sediment content of the bedload layer. The new formulation possesses two innovative features. It is fully nonlinear and vectorial in nature, in addition, it behaves smoothly up to the angle of repose. A mathematical model of the time evolution of straight river channels is presented in the second half of the paper. This study focuses on the evolution process due to bank erosion in the presence of bedload only. The bed and bank material is taken to be coarse, non-cohesive and uniform in size. The sediment continuity and the fluid momentum conservation equations describe the time evolution of the bed topography and flow field. These equations are coupled through the fluid shear stress acting on the bed. This bed shear stress distribution is predicted with the aid of a simple algebraic turbulent closure model. As regards the computation of the sediment flux, the new fully nonlinear vectorial formulation is found to perform well and renders the evolution model fully mechanistic. The formation of an erosional front in the time development of straight river channels has been so far obscured in physical experiments. Herein, with the help of the new bedload formulation, the existence and migration speed of the front of erosion are inferred from the analysis of the sediment continuity equation. The model successfully describes the time relaxation of an initially trapezoidal channel toward an equilibrium cross-sectional shape, as evidenced by comparison with experimental data. This equilibrium is characterized by a constant width, vanishing sediment transport in the transverse direction, and a small but non-vanishing streamwise transport rate of bed sediment.

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