Oscillations of a particle-laden fountain

Different regimes are usually observed for fluid migration through an immersed granular layer. In this work, we report a puzzling behavior when injecting water at a constant flow rate through a nozzle at the bottom of an immersed granular layer in a Hele-Shaw cell. In a given range of parameters (granular layer height and fluid flow rate) the granular bed is not only fluidized, but the particle-laden jet also exhibits periodic oscillations. The frequency and amplitude of the oscillations are quantified. The Strouhal number displays a power-law behavior as a function of a nondimensional parameter, $J$, defined as the ratio between the jet velocity at the initial granular bed height and the inertial particle velocity. Fluid-particle coupling is responsible for the jet oscillations. This mechanism could be at the origin of the cyclic behavior of pockmarks and mud volcanoes in sedimentary basins.

Figure 1

(a)–(e) Snapshots of the experiment illustrating one period of oscillations of the particle-laden jet ($h_g=2\mathrm{cm}$, $Q=35$ mL/min). (a) $t=103.46$ s; (b) $t=103.9$ s; (c) $t=104.3$ s; (d) $t=104.6$ s; (e) $t=104.9\mathrm{s}$. Time $t=0$ corresponds to the valve opening and the fluid entrance from the nozzle (in black at the bottom center of the image). Image (e) indicates the $x$ and $z$ axes and the inner angle of the crater, $\theta$. (f) Space-time diagrams illustrating the jet oscillations along a horizontal profile [white dashed line in (a) located 2.4 cm above the nozzle, upper panel] and a vertical profile above the injection nozzle [black dashed line in (a), lower panel].

Figure 2

Different regimes for fluid migration through the granular layer ($d_p=106\text{–}212\mu \mathrm{m}$). $\square$: percolation; $▵$: fluidization; $◯$: fluidization with particle-laden jet oscillations (see text). Double symbols for $h_g=4\mathrm{cm}$, $Q=60\text{–}70$ mL/min indicate different regimes observed when repeating the experiment. The colored regions are guides for the eye. The dashed line indicates the limit between percolation and fluidization for $d_p=400\text{–}600\mu \mathrm{m}$ (see text). For these largest particles, no oscillations are observed. For $h_g=0$ (no grains), a stable jet is observed for all flow rates.

Figure 3

Temporal evolution of the jet maximum height (a) horizontal position $X_j$ and (b) vertical position $Z_j$ for different flow rates $(h_g=2\mathrm{cm})$. After a transient, the jet height $Z_j$ reaches a plateau $p_z^{*}$ superimposed with oscillations. Inset: zoom between $t=110$ and 120 s showing the oscillations for the lowest flow rates.

Figure 4

Normalized jet height $p_z^{*}/h_g$ in the stationary regime as a function by the normalized flow rate $Q/Q_c$, where $Q_c$ is the critical fluid flow rate above which oscillations appear, for different initial granular bed height $h_g$. Open symbols: no jet oscillations; filled symbols: jet oscillations (gray region). The error bars are smaller than the symbol size. The solid colored lines correspond to the prediction of Eq. (1) for each $h_g$. Inset: critical flow rate $Q_c$ as a function of $h_g$.

Figure 5

(a) Oscillation frequency along the $z$ axis, $f_z$, as a function of the oscillation frequency along the $x$ axis, $f_x$, for different initial granular height (symbols corresponding to the legend in (b). When not visible, the error bars are smaller than the symbol size. The color scale indicates the value of the injection flow rate $Q$. The dashed line indicates $f_z=2f_x$. Inset: $f_x$ as a function of the inner angle $\theta$ of the crater [see Fig. 1]. (b) Frequency of the jet horizontal oscillations $f_x$ as a function of the normalized flow rate $Q/Q_c$ for different initial granular height. The dashed line is a guide for the eye.

Figure 6

(a) Amplitude of the jet horizontal oscillations in the steady-state regime, $A_x^{*}$, as a function of the amplitude predicted by the metronome model, $A_x^{\mathrm{theo}}$ (inset). The dashed line indicates 1:1 relation. (b) Amplitude of the jet vertical oscillations in the steady-state regime, $A_z^{*}$, normalized by the initial granular bed height $h_g$, as a function of the normalized flow rate $Q/Q_c$. The colored lines correspond to Eq. (3) for each data set (see text). Inset: coefficient $\xi$ [Eq. (3)].

Figure 7

Strouhal number as a function of the $J$ parameter (see text). The dashed line indicates $\text{St}=\beta J^{\gamma }$ with $\beta \simeq 0.06$ and $\gamma \simeq -5/2$.