Stability of a sand bed subjected to a shear flow of low Froude number

The problem of sand bed stability is formulated and solved for a model in which the local slope of the free surface is small relative to the slope of the perturbed lower boundary. The solution shows the bed to be unstable to perturbations of wavelength greater than the wavelengths for which the inertia of the sediment grains is important. Wave speeds and growth rates from numerical calculations are presented and compared with long- and short-wave asymptotic expressions. Although the fluctuating velocity field in the nonuniform flow is approximated by a constant eddy viscosity, a general picture of the relationship between the flow over a perturbed boundary and the observed geometrical and dynamical parameters of the resulting boundary waves can be developed. It is shown that the sand bed instability is due to local accelerations of a shear flow caused by the nonuniform boundary. If either the shear or the acceleration is eliminated from the analysis, the instability cannot be found. On the other hand, no wavelength is defined by such an analysis, and it is shown that the wavelength is determined by spatial adjustments in the turbulent velocity field.