Hydraulic jumps in a shallow flow down a slightly inclined substrate

Abstract

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 → 0 and Re → ∞, 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 (η, s) parameter space, where η 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, η 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.

Oblique liquid curtains with a large Froude number

“…They were put forward by Odulo (1979) in his study of shallow-water equations with rotation, and later used for various shallow-water/slender-flow problems (e.g. Zakharov 1981;Benilov 1995Benilov , 2015. These variables are called 'semi' because only the cross-stream Eulerian coordinate is replaced with a Lagrangian marker, whereas the streamwise coordinate remains Eulerian.…”

Stable Vortices in a Continuously Stratified Ocean with Thin Active Layer

“…Such coordinates have been used previously in [20][21][22] for shallow-water models similar to ours, and in [23], for a model comprising ours as a particular case. Similar coordinates are often used for general oceanographic modeling (e.g., [24]), where they are referred to as "isopycnic".…”