An experimental investigation of the stability of the circular hydraulic jump

Abstract

We present the results of an experimental investigation of the striking flow structures that may arise when a vertical jet of fluid impinges on a thin fluid layer overlying a horizontal boundary. Ellegaard et al. (Nature, vol. 392, 1998, p. 767; Nonlinearity, vol. 12, 1999, p. 1) demonstrated that the axial symmetry of the circular hydraulic jump may be broken, resulting in steady polygonal jumps. In addition to these polygonal forms, our experiments reveal a new class of steady asymmetric jump forms that include structures resembling cat's eyes, three- and four-leaf clovers, bowties and butterflies. An extensive parameter study reveals the dependence of the jump structure on the governing dimensionless groups. The symmetry-breaking responsible for the asymmetric jumps is interpreted as resulting from a capillary instability of the circular jump. For all steady non-axisymmetric forms observed, the wavelength of instability of the jump is related to the surface tension, $\sigma$, fluid density $\rho$ and speed $U_v$ of the radial outflow at the jump through $\lambda\,{=}\,(74\pm7)\sigma/(\rho U_v^2)$.

Experimental demonstration of the supersonic-subsonic bifurcation in the circular jump: A hydrodynamic white hole

“…Such effects might be interesting from a fluid mechanics' point of view, but are detrimental to the gravitational analogy, which assumes a smooth propagation of the surface waves. A low surface tension moreover guarantees that we avoid polygonal or more exotic jump shapes [5]. The jump that we obtain therefore corresponds to the most straightforward white hole analogy, namely the circularly symmetric (non-rotating) white hole.…”

Model for polygonal hydraulic jumps

“…(12) and it is likely that this behavior disappears by inclusion of some of the neglected terms. However, we note 036316-9 is given by δ(θ) (solid red line) and the radial flux by ξ r (θ) (blue dashed line), which is computed via the radial balance equation (12). Numbers refer to the symmetry N and increase down the columns.…”

“…This is the approach used in the present paper. It was recently pointed out [12], however, that the instability forming the polygonal jumps seems to be driven, at least in part, by surface tension (see also Ref. [8]).…”

A constrained variational model for radial symmetry breaking

“…Aristoff, Bush, Hosoi and coworkers [3,1,4,5] performed careful and thorough experiments, and interpreted the phenomenon of non-circular jumps to be due to surface tension. A model including the capillary effect was proposed and studied by Martens et al [11].…”