Standing and travelling waves in the shallow-water circular hydraulic jump

TL;DR

This paper models shallow-water hydraulic jumps using wave equations that emulate acoustic white holes, revealing stability and instability behaviors of standing and traveling waves, with experimental validation of flow destabilization near the jump.

Contribution

It introduces a wave equation framework that captures wave behavior in hydraulic jumps, including stability analysis and experimental support for flow destabilization.

Findings

Standing waves in sub-critical flow are stabilized by viscosity.

Standing waves in super-critical flow are inherently unstable.

Traveling upstream waves destabilize flow near the jump, supported by experiments.

Abstract

A wave equation for a time-dependent perturbation about the steady shallow-water solution emulates the metric an acoustic white hole, even upon the incorporation of nonlinearity in the lowest order. A standing wave in the sub-critical region of the flow is stabilised by viscosity, and the resulting time scale for the amplitude decay helps in providing a scaling argument for the formation of the hydraulic jump. A standing wave in the super-critical region, on the other hand, displays an unstable character, which, although somewhat mitigated by viscosity, needs nonlinear effects to be saturated. A travelling wave moving upstream from the sub-critical region, destabilises the flow in the vicinity of the jump, for which experimental support has been given.