Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-10-31T23:00:51.143Z Has data issue: false hasContentIssue false

Subcritical, transcritical and supercritical flows over a step

Published online by Cambridge University Press:  25 February 1997

YINGLONG ZHANG
Affiliation:
Department of Mathematics, The University of Wollongong, Wollongong, NSW 2522, Australia
SONGPING ZHU
Affiliation:
Department of Mathematics, The University of Wollongong, Wollongong, NSW 2522, Australia

Abstract

Free-surface flow over a bottom topography with an asymptotic depth change (a ‘step’) is considered for different ranges of Froude numbers varying from subcritical, transcritical, to supercritical. For the subcritical case, a linear model indicates that a train of transient waves propagates upstream and eventually alters the conditions there. This leading-order upstream influence is shown to have profound effects on higher-order perturbation models as well as on the Froude number which has been conventionally defined in terms of the steady-state upstream depth. For the transcritical case, a forced Korteweg–de Vries (fKdV) equation is derived, and the numerical solution of this equation reveals a surprisingly conspicuous distinction between positive and negative forcings. It is shown that for a negative forcing, there exists a physically realistic nonlinear steady state and our preliminary results indicate that this steady state is very likely to be stable. Clearly in contrast to previous findings associated with other types of forcings, such a steady state in the transcritical regime has never been reported before. For transcritical flows with Froude number less than one, the upstream influence discovered for the subcritical case reappears.

Type
Research Article
Copyright
© 1997 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)