On the convective nature of roll waves instability

A theoretical analysis of the Saint-Venantone-dimensional flow model is performed in order to define thenature of its instability. Following the Brigg criterion, theinvestigation is carried out by examining the branch pointssingularities of dispersion relation in the complex ω andk planes, where ω and k are the complex pulsationand wave number of the disturbance, respectively. The nature ofthe linearly unstable conditions of flow is shown to be ofconvective type, independently of the Froude number value.Starting from this result a linear spatial stability analysis ofthe one-dimensional flow model is performed, in terms of timeasymptotic response to a pointwise time periodic disturbance.The study reveals an influence of the disturbance frequency onthe perturbation spatial growth rate, which constitutes thetheoretical foundation of semiempirical criteria commonlyemployed for predicting roll waves occurrence