Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit

Abstract

International audienceWe study the spectral stability of roll-wave solutions of the viscous St. Venant equationsmodeling inclined shallow-water flow, both at onset in the small-Froude number or “weakly unstable”limit F → 2+ and for general values of the Froude number F , including the limit F → +∞. In the former,F → 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV)equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate thisformal limit, showing that stability as F → 2+ is equivalent to stability of the corresponding KdV-KSwaves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson–Noble–Rodrigues–Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St.Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainderof the paper, we investigate numerically and analytically the evolution of the stability diagram as Froudenumber increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable todifferent, large-F behavior, with stability determined by simple power law relations. The latter stabilitycriteria are potentially useful in hydraulic engineering applications, for which typically 2.5 ≤ F ≤ 6.0