Model for polygonal hydraulic jumps

We propose a phenomenological model for the polygonal hydraulic jumps discovered by Ellegaard and co-workers [Nature (London) 392, 767 (1998);   Nonlinearity 12, 1 (1999); Physica B 228, 1 (1996)], based on the known flow structure for the type-II hydraulic jumps with a “roller” (separation eddy) near the free surface in the jump region. The model consists of mass conservation and radial force balance between hydrostatic pressure and viscous stresses on the roller surface. In addition, we consider the azimuthal force balance, primarily between pressure and viscosity, but also including nonhydrostatic pressure contributions from surface tension in light of recent observations by Bush and co-workers [J. Fluid Mech. 558, 33 (2006);   Phys. Fluids 16, S4 (2004)]. The model can be analyzed by linearization around the circular state, resulting in a parameter relationship for nearly circular polygonal states. A truncated but fully nonlinear version of the model can be solved analytically. This simpler model gives rise to polygonal shapes that are very similar to those observed in experiments, even though surface tension is neglected, and the condition for the existence of a polygon with $N$ corners depends only on a single dimensionless number $\phi$. Finally, we include time-dependent terms in the model and study linear stability of the circular state. Instability occurs for sufficiently small Bond number and the most unstable wavelength is expected to be roughly proportional to the width of the roller as in the Rayleigh-Plateau instability.

Figure 1

Circular hydraulic jump. A cylindrical jet of fluid impinges vertically on a horizontal glass plate and forms a circular hydraulic jump. The fluid used here is ethylene glycol (about 10 times the viscosity of water).

Figure 2

Polygonal jump with eight corners. (a) Top view from an oblique angle. (b) View from below through a glass plate. The jump line becomes visible by diffracting light shone on the jump from above. The “belly” between two corners (or “necks”) is where the roller is thickest.

Figure 3

Measured height profiles of a pentagonal polygonal hydraulic jump in ethylene glycol. The solid curve represents the profile measured along the radial direction through a corner. The dashed curve represents the profile along the radial direction midway between two corners (“belly”), where the edge of the jump is almost straight [see, e.g., Fig. 2b]. The height of the fluid layer was measured manually with a depth micrometer attached to a translation table. The flow rate is $Q=40{\mathrm{ml}\mathrm{s}}^{-1}$. (This figure has been reproduced with permission from Ref. [13].)

Figure 4

Flow visualization of the roller vortex in the case of a triangular jump in ethylene glycol. The jump is seen from below through a glass plate, where the impinging jet is visible as the black center region and the jump line as the black line surrounding it. Red dye is injected into the roller flow by letting droplets of the dye fall from above into the vortex. It is seen that the roller structure extends from the jump line to the outer radius. In the corner region, fluid is expelled in clearly visible radial jets. Note also the fine white line between the roller vortices, which indicate that the vortices are actually disconnected structures. For a visualization of the flow see also Ref. [8].

Figure 5

Measured phase diagram, reproduced with permission from Ref. [6]. Polygonal hydraulic jumps are observed using ethylene glycol, which has a viscosity about 10 times that of water. Polygons appear in the parameter regime between the circular type-I state (marked as $I$) and the closed state (i.e., no jump, marked as $C$) when the jet flux $Q$ and the outer height $h_o$ are varied. The height of the nozzle about the surface is set to 2 cm. The number of corners is found to be less sensitive to the nozzle height [6]. Polygonal states with the number of corners $N=1,2,\cdots ,8$ are found in this experiment.

Figure 6

Three-dimensional view of the infinitesimal wedge $[\theta ,\theta +d\theta ]$ for the hydraulic jump. The gray shaded region is a part of the roller eddy near the surface just after the jump and represents the control volume we consider for the mass and force balances. Areas of the side, bottom and rear face, as well as the free-surface portion of the volume, are labeled as $A_t\left(\theta \right)$, $d{A}_b\left(\theta \right)$, $d{A}_r$, and $d{A}_f\left(\theta \right)$, respectively. The bold blue arrows indicate the flow direction.

Figure 7

(a) Schematic top view of a pentagonal jump. The gray shaded wedge represents the control volume with infinitesimal angle $d\theta$, also seen in Fig. 6. (b) Enlargement of the wedge-shaped control volume. Bold blue arrows represent the radial and tangential mass fluxes $q_r$ and $q_{\theta }$, i.e., the flux in and out of the control volume.

Figure 8

Side view of the wedge in Fig. 6. The heights inside and outside the jump are $h_i$ and $h_o$, respectively. We treat the roller eddy (shaded region) as a separate body of fluid from the main stream and assume the force balance of the hydrostatic force $d{F}_r^h$ and the shear force $d{F}_r^{\mu }$ in the radial direction (forces indicated by bold red arrows).

Figure 9

Potential function $V$ as a function of $x=\delta /{\delta }_0$ and $X=x^3/3$, respectively. A periodic solution, i.e. “bound” state, exists only when $-\frac{1}{6}<C<0$. The solution corresponds to a polygon with $N$ corners if the period is $2\pi /N$ in terms of $\theta$.

Figure 10

Polygonal jumps with $N$ corners exist in the parameter range shown in the truncated nonlinear model. The jumps bifurcate from the dashed line $\phi =1/N$ for the circular jumps $C$. When the parameters approach the line $\phi =\pi /\left(3N\right)$, the roller thickness approaches $\delta =0$ in the corners at some $\delta =0$, indicating rupture: solutions appear spiky near the solid line $S$. On the border ${\delta }_0=1/x_{\max }$ (gray line), the roller thickness reaches $\delta =1$ so the solution becomes unphysical.

Figure 11

Typical polygon solutions for the truncated model in Eqs. (56) and (57). The polygon shape (jump line) is given by $\delta \left(\theta \right)$ (solid red line) and the radial flux by ${\xi }_r\left(\theta \right)$ (blue dashed line), which is computed via the radial balance equation (12). Numbers refer to the symmetry $N$ and increase down the columns. From left to right, the transition from nearly circular to spiky is shown for each symmetry ($c$, $m$, and $h$ refer to center, middle, and homoclinic orbits in phase space (x,y), respectively), corresponding to the transition from the lines $C$ to $S$ in Fig. 10.

Figure 12

Bifurcation diagram in physical parameters ($h_o,Q$). The regions for polygonal solutions with each $N$ up to $N=8$ are shown. The $h_o$-$Q$ relationship for the border curves, especially in the enlarged inset, is qualitatively similar to the measured relationship in the phase diagram of Fig. 5 despite quantitative disagreement. The bands are predicted to be thinner and do not overlap in the model, unless $N⩾22$, unlike in experimental observations.

Figure 13

There is no maximally unstable wave number $k^{*}$ when $C_1>0$ [see Eq. (98)]. The stability curves $s_{+}\left(k\right)$ (blue dotted line) and $s_{-}\left(k\right)$ (red dashed line) from Eq. (95) are shown for $C_1=2$ (above) and $C_1=-2$ (below) with $C_2=2.5$, $C_3=0.1$, and $C_4=0.3$. The dashed lines show the curves for continuous $k$, but only integral values of $k$, corresponding to the crosses, are allowed due to $2\pi$ periodicity in $\theta$. The mode $k=0$ is treated separately due to the global conservation law (96).

Figure 14

Stability curve $s_{+}\left(k\right)$ (dashed line) from Eq. (95) for stability coefficients $C_1\approx -0.603$, $C_2\approx 0.415$, $C_3\approx 0.0156$, and $C_4\approx 0.374$, based on experimental parameters and the height profile of a pentagon (see the Appendix). Note that the constant branch $s(k>k_0)=(C_4-C_2)/2\approx -0.02$ is negative. We find that the maximally stable wave number (98) is $k^{*}\approx 4.4$; thus the most likely symmetry to occur has symmetry $k=4$ or 5, which matches well the expectation. The crosses mark the values of $s$ for integer $k$, respecting the $2\pi$ periodicity. The mode $k=0$ is treated separately according to the global conservation law (96).