Stability, resonance and role of turbulent stresses in 1D alluvial flows

Linear stability analysis is used to investigate the behavior of small perturbations of a uniform flow in a straight channel with an erodible bed composed by a unisize sediment. A shallow-water flow model is employed and bedload sediment transport is assumed. The mathematical structure of the linear problem, in terms of the eigenvalues and their associated eigenvectors is explored in detail and information is gathered on the wavespeed and growth rate of the perturbations as a function of their wavelength and of the relevant flow and sediment parameters. Several aspects of the solution are discussed, with particular focus on the behaviour in the transcritical region where the Froude number approaches unity. An approximate solution for the roots of the eigenrelationship is presented, which is not uniformly valid in the transcritical region, leading to the appearance of an unphysical instability. A regular perturbation expansion is then introduced that allows for the elimination of this singularity.