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The hydraulic bump: The surface signature of a plunging jet

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.


I. INTRODUCTION
When a falling jet of fluid strikes a horizontal fluid layer, several flow regimes may arise.
The most distinctive phenomenon, the hydraulic jump, arises above a critical flow rate, and consists of a large-amplitude increase in fluid depth at a critical distance from the site of jet impact (Figure 1 a). The circular hydraulic jump was first reported by Bélanger 1 and Rayleigh 2 , and subsequently studied theoretically and experimentally by a number of investigators (see [3][4][5][6][7][8][9] and references therein).
Bohr et al. 6 and Watanabe et al. 10  Bush et al. 14 also reported the emergence of double jump structures in certain parameters regimes, wherein the free surface is marked by two discrete changes in depth.
Remarkably, in certain parameter regimes, the circular hydraulic jump becomes unstable to polygons (Figure 1b), a phenomenon first reported by Ellegaard 15,16 , and subsequently examined by Bohr   We here report a phenomenon that occurs well below the hydraulic jump threshold, when the free surface is only weakly perturbed by the plunging jet. When the fluid layer is sufficiently deep, a small-amplitude circular deflection arises at the free surface, a phenomenon that we christen the hydraulic bump (Figure 1 g). As is the case for the hydraulic jump, as the incoming flux increases, the bump radius expands until a breaking of axial symmetry results in polygonal forms (Figure 1 h). In §II, we report the results of our experimental investigation, and describe the flows observed. We rationalize the radius of the bump via simple scaling laws. Finally, in §III we explore the connection between the polygonal instabilities on the hydraulic bump and their counterparts on the hydraulic jump.

II. EXPERIMENTS
The experimental apparatus is shown in figure 2. A glycerine-water solution with density ρ, kinematic viscosity ν, and a surface tension γ is pumped from the tank through a flow meter and a source nozzle of radius R n = 2.5 mm. The resulting jet has a flux Q and a radius at impact r j that differs from R n , and varies weakly with flow rate and height in a manner detailed by Bush and Aristoff 8 . The jet impacts the center of a flat plate of radius R p = 16.8 cm surrounded by an outer wall whose height can be adjusted in order to control the outer fluid depth H. Radial gradations on the base plate indicate 0.5 cm increments. Special care is taken to level the plate by adjusting its three supports and measuring the level along two perpendicular directions. The plate is horizontal to within ± 0.1 • . We note that the flow structure is extremely sensitive to the levelling of the plate; indeed, an inclination of 1 − 2 • completely destroys the polygonal bump and jump forms.
The working fluid is a glycerine-water solution with viscosity ranging from 58 to 96 cS.
During the course of the experiments, water was added to compensate for evaporative losses.
For the fluids considered, surface tension is roughly constant and equal to 68 mN.m −1 . The average depth is determined by measuring the volume V t above the impact plate, which is known with a precision ± δV t = 2 ml. Typically, V t 500 ml and H 5 mm , so the error in depth (δV t H)/V t 0.01 mm, is sufficient for our experiments and smaller than would arise from a direct measurement. We visualize the flow structure by injecting submillimetric bubbles into the jet inlet with a syringe and taking photos that yield streak images of the bubble circulation. In passing through the pump and the flowmeter, these bubbles are generally fractured into microbubbles that do not appreciably perturb the flow. We denote by R bump the bump radius, δH its height and H int the height just upstream of the bump.
We denote the fluid velocity by v and its speed by v. This suggests the introduction of the Reynolds number Re = v j H/ν, with the jet speed v j = Q/(πR 2 n ) being evaluated at the nozzle output, and a local Weber number W e = ρQ 2 /(γπ 2 HR 2 ), with R being the radius of the jump or the bump. signature, much less than the jump, we infer that the subsurface vortical structure is critical in its instability.
The height and radius of the circular bump are readily rationalized via scaling arguments.
We consider a point A at the surface near the plunging jet and a point B on the bump (see Figure 3b). We denote by δH the amplitude of the bump. Since B can be considered as a 6 stagnation point and since curvature pressures are expected to be negligible with respect to the hydrostatic pressure within the bump, Bernoulli's Theorem dictates that v 2 A /2 − gδH = const., so we expect that δH = c 1 v 2 A /2g with v A Q/(2πR n H). Figure 4a illustrates the dependence of 2gδH on Q 2 /(4π 2 R 2 n H 2 ) over the parameter range in which circular bumps arise. This simple scaling is roughly validated, and a proportionality constant of c 1 = 0.41 is indicated.  We have also reported that, as the flux increases, the circular bump goes unstable to a polygonal form reminiscent of that arising in the hydraulic jump 14 . We note that the 9 polygonal instabilities of both the jump and bump are associated with a toroidal vortex with a surface signature. Since the bump has a relatively small surface signature, we expect its accompanying subsurface vorticity to provide the dominant mechanism for its instability. This surface vortex instability mechanism, and its relation to the hydraulic bump, the hydraulic jump, and the toroidal Leidenfrost vortex 29 , will be the subject of a theoretical investigation to be reported elsewhere.