Stability of viscous St. Venant roll-waves: from onset to infinite-Froude number limit

TL;DR

This paper analyzes the spectral stability of viscous St. Venant roll-waves across a range of Froude numbers, rigorously validating the weakly unstable limit and exploring stability transitions at high Froude numbers.

Contribution

It provides the first rigorous verification of stability for viscous St. Venant roll waves and classifies their stability behavior from onset to infinite Froude number.

Findings

Stability at F→2+ is equivalent to KdV-KS wave stability.

Validated the formal limit connecting shallow water equations to KdV-KS.

Identified a stability transition around F=2.3 with power law relations.

Abstract

We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude number or "weakly unstable" limit $F\to 2^+$ and for general values of the Froude number $F$, including the limit $F\to +\infty$. In the former, $F\to 2^+$, 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 this formal limit, showing that stability as $F\to 2^+$ is equivalent to stability of the corresponding KdV-KS waves 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 remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around $F=2.3$ from weakly unstable to different, large-$F$ behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $2.5\leq F\leq 6.0$.