Hydraulic jumps in a shallow flow down a slightly inclined substrate

• Published in: Journal of fluid mechanics 2015-11, Vol.782, p.5-24
• Main Author: Benilov, E. S.
• Format: Article
• Language: English
• Subjects: Criteria; Fluid dynamics; Fluid flow; Fluid mechanics; Free surfaces; Hydraulics; Mathematical analysis; Parameters; Reynolds number; Shallow water; Slopes; Substrates
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2015.521

This work examines free-surface flows down an inclined substrate. The slope of the free surface and that of the substrate are both assumed small, whereas the Reynolds number $Re$ remains unrestricted. A set of asymptotic equations is derived, which includes the lubrication and shallow-water approximations as limiting cases (as $Re\rightarrow 0$ and $Re\rightarrow \infty$ , respectively). The set is used to examine hydraulic jumps (bores) in a two-dimensional flow down an inclined substrate. An existence criterion for steadily propagating bores is obtained for the $({\it\eta},s)$ parameter space, where ${\it\eta}$ is the bore’s downstream-to-upstream depth ratio, and $s$ is a non-dimensional parameter characterising the substrate’s slope. The criterion reflects two different mechanisms restricting bores. If $s$ is sufficiently large, a ‘corner’ develops at the foot of the bore’s front – which, physically, causes overturning. If, in turn, ${\it\eta}$ is sufficiently small (i.e. the bore’s relative amplitude is sufficiently large), the non-existence of bores is caused by a stagnation point emerging in the flow.