Abstract
The performance and stability of a numerical model which consist of a shallow water flow model and an equilibrium bedload transport model for calculating bed evolution in rivers is discussed. Such system equations which are generally adopted for calculating reach-scale river bed evolution have a hyperbolic feature. This fact has implied that upwind schemes to the space derivative of bedload transport in Exner equation are useful numerical methods to avoid bed instabilities in calculating the propagation of a small disturbance. Whereas, a bed diffusion effect which is essentially included in bedload dominated bed evolution has been modelled in equilibrium bedload transport models by considering the local slope effect to the bedload transport. The relation between the numerical and physical diffusion effect to the bed evolution should be investigated to understand the performance and stability of the model. A hyperbolic differential equation controlling the bed evolution was derived from the linearized system equations to understand the numerical characteristic of the propagation of the bed disturbance. We defined a Péclet number, Pe, as the ratio between the advective effect which controls the propagation of the bed disturbance and the diffusive effect associated with the bed slope effect. The effect of physical diffusion of the bed exceeds the effect of numerical diffusion by the upwind scheme to the bedload transport in the condition which Pe < 2. The series of computations which focus on the propagation of hump type bed disturbance were conducted. The numerical results show that the calculation of propagating the bed disturbance can be physically stable without the upwind scheme when the grid size which satisfies Pe < 2 is used. The grid size which is required such physically stable computation depends on how the bed slope effect is modelled in the bedload transport model.
