Continuous hydraulic jumps in laminar channel flow

TL;DR

This paper models hydraulic jumps in laminar flow as continuous, stable shock structures derived from viscous Saint-Venant equations, providing analytical and numerical insights into their properties.

Contribution

It introduces a novel continuous shock model for hydraulic jumps in laminar flow based on viscous equations, avoiding classical discontinuous jump relations.

Findings

Hydraulic jumps are stable stationary solutions in laminar flow.

Derived an analytical expression for jump length involving Froude and Reynolds numbers.

Numerical experiments confirm the stability of the continuous jumps.

Abstract

On the basis of the viscous Saint-Venant equations, hydraulic jumps in laminar open channel flow are obtained as continuous shock structures. Thanks to the inclusion of viscosity, the jumps are not abrupt, rendering the classic patchwork via the Rankine-Hugoniot shock relations unnecessary. The jumps arise as stable stationary solutions of the governing equations and lend themselves excellently to a Dynamical Systems analysis, manifesting themselves as near-parabolic trajectories in phase space. Based on this, we derive an analytic expression for the jump length as a function of the Froude and Reynolds numbers, reflecting the fact that both gravity and viscosity contribute to the balance of forces that shape the jump. The paper concludes with a numerical experiment confirming the stability of the jumps.