The hydraulic bump: The surface signature of a plunging jet

Abstract

When a falling jet of fluid strikes a horizontal fluid layer, a hydraulic jump arises downstream of the point of impact provided a critical flow rate is exceeded. We here examine a phenomenon that arises below this jump threshold, a circular deflection of relatively small amplitude on the free surface, that we call the hydraulic bump. The form of the circular bump can be simply understood in terms of the underlying vortex structure and its height simply deduced with Bernoulli arguments. As the incoming flux increases, a breaking of axial symmetry leads to polygonal hydraulic bumps. The relation between this polygonal instability and that arising in the hydraulic jump is discussed. The coexistence of hydraulic jumps and bumps can give rise to striking nested structures with polygonal jumps bound within polygonal bumps. The absence of a pronounced surface signature on the hydraulic bump indicates the dominant influence of the subsurface vorticity on its instability.PACS numbers: 47.20.Ma, 47.32.cd,

Harmonic solutions for polygonal hydraulic jumps in thin fluid films

“…Higher-order equations are neglected, but nevertheless are related to the description of diverse patterns found by Labousse & Bush (2013), which could explain, for instance, different polygonal shapes with the same number of vertices. The wavelengths of the polygons are found to be similar to the variable r * 0 , which is supported by comparison with experimental measurements and a comparative study with Watson's (1964) theory.…”

“…We remark that in this work we neglect higher orders in α that provide more information about nonlinear corrections for the patterns of polygonal hydraulic jumps and bumps found recently (Labousse & Bush 2013). Replacing the harmonic solutions ξ j (θ) = ξ 0 + α(A ξ /m) sin mθ into (2.9) and using (2.16), we find…”

“…The polygonal hydraulic jump is composed by a separation vortex and a second upper roller within the angular cross-section θ = const., and it was first suggested as a Plateau-Rayleigh instability of the circular state by Bush et al (2006). Type II jumps have been classified into type IIa and IIb according to whether or not they exhibit a bump downstream of the jump (Bush et al 2006;Labousse & Bush 2013).…”

Polygonal instabilities on interfacial vorticities

“…We note that polygonal hydraulic bumps can also be observed in the presence of the hydraulic jump, presumably owing to the instability of the roller vortex downstream of the jump [4,25]. One may thus obtain polygonal jumps bound by polygonal bumps [5] (e.g. see the six-sided outer surface structure in Figure 3a).…”

“…Labousse and Bush [5] reported that below a critical incoming flow rate, a plunging jet can give rise to a surface deflection called the hydraulic bump. The flow is marked by a subsurface poloidal vortex that is circular at low flow rates, but may destabilize into a polygonal form (see Figure 4a).…”