Note on the stability of viscous roll-waves

TL;DR

This paper provides a comprehensive stability classification of viscous roll-wave solutions in shallow-water equations across all Froude numbers, offering both numerical and analytical insights relevant to hydraulic engineering.

Contribution

It offers the first complete stability classification of periodic roll-waves from onset to infinite Froude number, including a simple power-law relation and analytical stability boundaries.

Findings

Stable regions are characterized by a power-law relation between Froude number and wave period.

Roll-waves are unstable in the infinite-Froude limit.

Analytical expressions describe stability boundaries near Froude number 2.

Abstract

In this note, we announce a complete classification of stability of periodic roll-wave solutions of the viscous shallow-water equations, from their onset at Froude number $F\approx 2$ up to the infinite-Froude limit. For intermediate Froude numbers, we obtain numerically a particularly simple power-law relation between $F$ and the boundaries of the region of stable periods, that appears potentially useful in hydraulic engineering applications. In the asymptotic regime $F\to 2$ (onset), we provide an analytic expression of the stability boundaries whereas in the limit $F\to\infty$, we show that roll-waves are always unstable.