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On the convective nature of roll waves instability. (English) Zbl 1092.76028

Summary: A theoretical analysis of one-dimensional Saint-Venant flow model is performed in order to examine the nature of its instability. Following Brigg criterion, the investigation is carried out by examining the branch points singularities of dispersion relation in complex \(\omega\) and \(k\) planes, where \(\omega\) and \(k\) are complex pulsation and wave number of the disturbance, respectively. The nature of linearly unstable flow conditions is shown to be of convective type, independently of Froude number value. Starting from this result, a linear spatial stability analysis of one-dimensional flow model is performed, in terms of time-asymptotic response to a pointwise time-periodic disturbance. The study reveals the influence of disturbance frequency on the perturbation spatial growth rate, which constitutes the theoretical foundation of semiempirical criteria commonly employed for predicting roll wave occurrence.

MSC:

76E17 Interfacial stability and instability in hydrodynamic stability
76E15 Absolute and convective instability and stability in hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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