Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit
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- 作者：Blake Barker ; Mathew A. Johnson ; Pascal Noble…
- 刊名：Journal of Nonlinear Science
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：27
- 期：1
- 页码：285-342
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Analysis; Theoretical, Mathematical and Computational Physics; Classical Mechanics; Appl.Mathematics/Computational Methods of Engineering; Economic Theory/Quantitative Economics/Mathematical Methods;
- 出版者：Springer US
- ISSN：1432-1467
- 卷排序：27
文摘
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 \(2.5\le F\le 6.0\).