Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit - Institut Camille Jordan Accéder directement au contenu
Article Dans Une Revue Journal of Nonlinear Science Année : 2017

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

Résumé

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 → 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 this formal limit, showing that stability as F → 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 ≤ F ≤ 6.0.
Fichier principal
Vignette du fichier
SV_Froude-inspection.pdf (1.01 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01135510 , version 1 (25-03-2015)

Identifiants

  • HAL Id : hal-01135510 , version 1

Citer

Blake Barker, Mathew A. Johnson, Pascal Noble, Luis Miguel Miguel Rodrigues, Kevin Zumbrun. Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit. Journal of Nonlinear Science, 2017, 27 (1), pp.285--342. ⟨hal-01135510⟩
312 Consultations
143 Téléchargements

Partager

Gmail Facebook X LinkedIn More