Free surface axially symmetric flows and radial hydraulic jumps

TL;DR

This paper analytically investigates free surface axially symmetric shallow flows, including sub- and supercritical solutions, hydraulic jumps, and the effects of friction and bottom topography, providing explicit solutions and conditions for flow transitions.

Contribution

It offers new analytical solutions for steady axially symmetric flows with and without friction, including hydraulic jump characterization, based on specific energy ratios.

Findings

Analytical solutions for sub- and supercritical flows over flat plates and topography.

Explicit derivation of hydraulic jump position and downstream depths.

Flow solutions depend uniquely on the specific energy ratio.

Abstract

Free surface, axially symmetric shallow flow is analysed in both the centrifugal and centripetal directions. Referring to the inviscid steady flow over a flat plate characterised by a unique value of specific energy, the analytical sub- and supercritical solutions are determined. Furthermore, the sub- and supercritical steady solutions for the flow with friction over a flat plate are determined, provided that inertial terms are important compared with frictional terms. As the inviscid case, the sub- and supercritical solutions over a bottom topography are determined, provided that the bottom unevenness is compatible with a continuous solution. In the fundamental case of inviscid flow over a flat plate, the discontinuous solution is also analysed for a direct hydraulic jump imposed by proper boundary conditions. In the simple scheme of an inviscid shock of zero length, the jump position and the sequent depths are analytically derived, thus indicating that all results are uniquely functions of one dimensionless number, i. e. the specific energy ratio between the sub- and supercritical flows.