Critical Transition Lab
Shape a smooth transition in bed slope for one discharge, then move away from the design point.
About this phenomenon
A supercritical flow normally becomes subcritical through a hydraulic jump: an abrupt, turbulent rise in depth. At critical flow, the gradually varied flow equation becomes singular because the denominator approaches zero.
But there is a way through the singularity. If the numerator also approaches zero at the same point, the ratio can retain a finite slope. That 0/0 balance can look like a mathematical trick, but suitably shaped channels can realize it physically.
This lab shapes the bed for one design discharge and then keeps the geometry fixed while the operating discharge changes. The smooth transition therefore exists over a finite operating envelope, not only at one exact discharge.
Channel width can provide another geometric compatibility term. Coordinating bed and width transitions can extend the smooth operating envelope further, but the bed transition alone is enough to reveal the mechanism.
Scientific basis
With fixed channel width, the steady gradually varied flow equation is
A smooth target depth is prescribed at the design discharge and the equation is rearranged to obtain the required bed slope:
At a finite-slope critical crossing, both numerator and denominator vanish:
L’Hôpital’s rule gives the local slopes through the singular point. If those real slopes disappear, the gradually varied branches can no longer meet smoothly. In a non-prismatic channel, a width gradient adds another compatibility term and can broaden the smooth operating range.
References.
Chen, J. & Dracos, T. (1996), Journal of Hydraulic Research 34(4), 517–536.
Kabiri-Samani, A. et al. (2014), Journal of Hydraulic Research 52(1), 129–136.
Rezashahreza, M. et al. (2025), Flow Measurement and Instrumentation 104, 102889.