Subcritical, transcritical and supercritical flows over a step

Free-surface flow over a bottom topography with an asymptotic depth change (a `step') is considered for di erent 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 e ects on higher-order perturbation models as well as on the Froude number which has been conventionally de ned in terms of the steady-state upstream depth. For the transcritical case, a forced Korteweg{de Vries (f KdV) 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 ndings 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. Disciplines Physical Sciences and Mathematics Publication Details This article was originally published as Zhang, Y and Zhu, S, Subcritical, transcritical and supercritical flows over a step, Journal of Fluid Mechanics, 333, 1997, 257-271. Copyright Cambridge University Press. Original journal available here. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/429 J. Fluid Mech. (1997), vol. 333, pp. 257–271 Copyright c © 1997 Cambridge University Press 257 Subcritical, transcritical and supercritical flows over a step By Y I N G L O N G Z H A N G AND S O N G P I N G Z H U Department of Mathematics, The University of Wollongong, Wollongong, NSW 2522, Australia (Received 30 October 1995 and in revised form 19 August 1996) 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.