Stability of a Liquid Jet Impinging on Confined Saturated Sand

This Letter focuses on the dynamics of a liquid jet impacting the surface of a confined, immersed granular bed. Although previous works have considered the erosion process and surface morphology, less attention has been given to the jet hydrodynamics. Based on laboratory experiments, we show that when the liquid jet forms a crater, two situations arise. For weak or no erosion and for open craters, the jet is stationary. For vertical or overhanging crater walls, the jet displays a wide range of behaviors, from quasiperiodic oscillations to symmetry breaking and exploration of different states in time. An analysis of the different system states leads to the emergence of a bifurcation diagram depending on a dimensionless parameter, $J$, comparing the jet impact force to the force necessary to eject a grain. The frequency of the jet oscillations depends on the inertial velocity, the jet dispersion and the ratio between the injector cross section and the confinement length.

Figure 1

Crater morphology for different liquid flow rates $Q$ [$h=3\mathrm{cm}$]. The upper panels (a)–(d) show a snapshot and the lower panels (e)–(h) display the average over the last 1000 images. Panel (c) shows an example of PIV computation. The red dashed lines in (c),(d) display the section upon which the incident (downward) flux $Q_w$ and the upward side fluxes $Q_1$ and $Q_2$ are computed.

Figure 2

Crater morphology and jet stability in the phase space $(Q,h)$. (Cross) no erosion; (Square) weak erosion; (Plus) jet indentation; (Inverted triangle) open crater; (Circle) close crater with stationary (black and white) or nonstationary (black) jet. Solid black line: $J=0.6$ [Eq. (1)] for the jet angle determined experimentally, with error estimation (dashed gray lines). The gray zone (guide for the eye) shows the existence of jet oscillations.

Figure 3

(a)–(d) Different flow regimes [$Q_1$, orange; $Q_2$, black; $⟨Q_w⟩$, dashed where $⟨.⟩$ denotes the time average ($\mathrm{error}\mathrm{bar}=\mathrm{standard}\mathrm{deviation}$ of $⟨Q_w⟩$ fluctuations); $⟨Q_w/2⟩$, dash-dotted] ($h[\mathrm{cm}]$, $Q[\mathrm{mL}/\mathrm{s}]$, $J$). (a) Symmetric (3,1,0.95); (b) asymmetric (1,2,4.18); (c) different states (1,1,2.86), the gray parts of the signals indicate the transition regions; (d) quasiperiodic oscillations (3,4,2.78). (e) System asymmetry $\xi$ as a function of $J$. Hatched region: no crater formation, inverted triangles = open craters, circles = close craters. Gray zone: nonstationary jet that explores different states. The colorscale gives the probability $p$ for the system to be in a state $\{\xi \}$ for a given $J$.

Figure 4

(a) Strouhal number $\text{St}$ as a function of $J$ [(crosses) from fluxes cross-correlation, (stars) data from fluxes averaged for a given $J$, (circles) from spatiotemporal]. The error bar is smaller than each symbol size. Dashed line: fit of the average values, $\text{St}=\beta J^{-1}$ with $\beta \simeq 0.09$. Inset: jet oscillation frequency $f$. Hatched region: no or weak erosion; gray area: oscillations. Dashed line: frequency given by Eq. (2) for $\alpha ={\alpha }^{*}$. (b) Spatiotemporal evolution of the images intensity along a horizontal region encompassing the crater aperture. The crater aperture width $W$ is visible as the clear, oscillating region [$h=3\mathrm{cm}$, $Q=4\mathrm{mL}/\mathrm{s}$]. Orange and black lines: data $Q_1$ and $Q_2$ from Fig. 3, vertically shifted for clarity. White dashed lines: harmonic oscillations at 2.5 Hz (guides for the eye).