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

Content Provider	Springer Nature Link
Author	Barker, Blake Johnson, Mathew A. ble, Pascal Rodrigues, L. Miguel Zumbrun, Kevin
Copyright Year	2016
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 $$F\rightarrow 2^+$$ and for general values of the Froude number F, including the limit $$F\rightarrow +\infty $$ . In the former, $$F\rightarrow 2^+$$ , 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 $$F\rightarrow 2^+$$ 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 $$2.5\le F\le 6.0$$ .
Starting Page	285
Ending Page	342
Page Count	58
File Format	PDF
ISSN	09388974
Journal	Journal of Nonlinear Science
Volume Number	27
Issue Number	1
e-ISSN	14321467
Language	English
Publisher	Springer US
Publisher Date	2016-09-23
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Analysis Theoretical, Mathematical and Computational Physics Classical Mechanics ApplicationMathematics/Computational Methods of Engineering Economic Theory/Quantitative Economics/Mathematical Methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Engineering Modeling and Simulation

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