Experiments and characterization of low-frequency oscillations in a granular column

The behavior of a vertically vibrated granular bed is reminiscent of a liquid in that it exhibits many phenomena such as convection and Faraday-like surface waves. However, when the lateral dimensions of the bed are confined such that a quasi-one-dimensional geometry is formed, the only phenomena that remain are bouncing bed and the granular Leidenfrost effect. This permits the observation of the granular Leidenfrost state for a wide range of energy injection parameters and more specifically allows for a thorough characterization of the low-frequency oscillation (LFO) that is present in this state. In both experiments and particle simulations we determine the LFO frequency from the power spectral density of the center-of-mass signal of the grains, varying the amplitude and frequency of the driving, the particle diameter, and the number of layers in the system. We thus find that the LFO frequency (i) is inversely proportional to the fast inertial timescale and (ii) decorrelates with a typical decay time proportional to the slow dissipative timescale in the system. The latter is consistent with the view that the LFO is driven by the inherent noise that is present in the granular Leidenfrost state with a low number of particles.

Figure 1

Schematic phase diagram of a vibrated quasi-two-dimensional granular bed as a function of the energy injection and the container's length. Here, B.B. indicates the bouncing bed state and the granular gas state is reached for very high energy injection which lies outside the range of the phase diagram.

Figure 2

(a) A full picture and a zoomed-in view of the experimental setup: The container is mounted on top of the mechanical shaker, with the bottom located at approximately $1\mathrm{cm}$ above the socket. Note that the granular material above it is at rest. (b) A schematic of the container, where the dimensions of the container and the filling height $H$, related to the filling factor $F$ and particle diameter $d$, are indicated. (c) Typical images of the partially filled container under shaking for the three different particle diameters, $d=0.5$, 1.0, 2.5 mm.

Figure 3

(a) A sequence of images of the system in different phases of the driving showing a dense cluster floating over a dilute, gaseous granular layer. This sequence corresponds to one period of oscillation of the shaker. (b) Average of the black and white pixels on every horizontal level for each frame as a function of time, referred to as density profile. (c) Time-averaged density profile of the system.

Figure 4

A comparison of the vertical position of the center of mass $z_{\mathrm{CM}}\left(t\right)$ computed directly from the position of the particles (blue line) and using the experimental method, i.e., from the density profile (red line); data obtained using numerical simulations. The frequency that dominates the main plot corresponds to that of the LFO, whereas in the region that is magnified in the inset the driving frequency $f_0$ can also be appreciated.

Figure 5

The PSD of the time evolution of the center of mass $z_{\mathrm{CM}}\left(t\right)$ for (a) different shaking strengths $S_d$ with constant particle diameter $d=1\mathrm{mm}$ and (b) different particle diameters $d$ with constant shaking strength $S_d=81$, obtained from experimental data for $F=12$. The LFO is clearly distinguished as a broad peak around the 10-Hz region and its frequency $f_{\mathrm{LFO}}$ can be determined from the location of the peak's maximum. The black dashed lines represent theoretical results calculated from the Langevin model [Eq. (11)].

Figure 6

The LFO frequency $f_{\mathrm{LFO}}$ is plotted versus the dimensionless shaking strength $S_d$ for (a) different numbers of layers $F$ and a constant particle diameter $d=1\mathrm{mm}$, and (b) different particle diameters and a constant number of layers $F=8$. The different symbols correspond to different amplitudes of the shaker, $\circ =1.0\mathrm{mm},\square =2.0\mathrm{mm},\diamond =3.0\mathrm{mm},▽=4$ mm. The solid symbols represent experimental data and the open symbols numerical simulation results.

Figure 7

(a) The granular temperature $T$ and (b) the ratio of the kinetic energy of the center of mass $z_{\mathrm{CM}}$ to the kinetic energy per particle, ${\overline{K}}_{\mathrm{CM}}/{\overline{K}}_{\text{pp}}$, are plotted versus ${\left(a_0{f}_0\right)}^2$ for different numbers of layers $F$ and a constant particle diameter $d=1\mathrm{mm}$, both correspond to numerical simulation results. The different symbols correspond to different amplitudes of the shaker, $\circ =1.0\mathrm{mm},\square =2.0\mathrm{mm},\diamond =3.0\mathrm{mm},▽=4$ mm.

Figure 8

The dimensionless frequency of the low-frequency oscillations ${\tilde{f}}_{\text{LFO}}$ as a function of the shaking parameter $B$ collapses all simulational data (as shown in Fig. 6) onto a single curve. Different symbols correspond to different numbers of layers $F$ and different colors to different particle diameters $d$. As in Fig. 6, the solid symbols represent experimental data and the open symbols numerical simulation results. The dashed-dotted line serves as a guide to the eye.

Figure 9

(a) Autocorrelation function of the original center-of-mass signal $z_{\mathrm{CM}}\left(t\right)$ (blue line) and of the low-pass-filtered center of mass (red line) for $a_0=4\mathrm{mm},f_0=55.7$ Hz, $S_d=200$, and $F=12$. (b) The autocorrelation function of the low-pass-filtered center-of-mass signal is plotted versus time for different number of layers and shaking strength $S_d=200$. The dashed line represents the fit of the maxima for $F=8$.

Figure 10

Decorrelation time $\tau$ determined from the autocorrelation function $\xi \left(t_0\right)$ of the center of mass signal $z_{\mathrm{CM}}\left(t\right)$. (a) Experimental data for $\tau$ as a function of the dimensionless shaking strength $S_d$ for different values of the number of layers $F$ and the particle diameter $d$ (see legend at the bottom). (b) Numerical data for $\tau$, again as function of $S_d$. (c) $\tau$ nondimensionalized with the dissipative timescale ${\tau }_d=a_0{f}_0/\left(gF\right)$ plotted versus the shaking parameter $B$. The horizontal lines correspond to the averaged value of $\tau /{\tau }_d$ for the numerical simulations and the experiments. As in previous plots, different symbols correspond to different numbers of layers $F$ and different colors to different particle diameters $d$, and the solid and open symbols represent experimental and numerical data, respectively.

Figure 11

Intensity $I_{\text{LFO}}$ obtained from the PSD of the center-of-mass signal $z_{\mathrm{CM}}\left(t\right)$. Top left: Experimental data for $I_{\text{LFO}}$ as a function of the dimensionless shaking strength $S_d$ for different values of the number of layers $F$ and the particle diameter $d$ (see legend at the bottom). Top right: Numerical data for $I_{\text{LFO}}$, again as function of $S_d$. Bottom: $I_{\text{LFO}}$ nondimensionalized by ${\left(a_0{f}_0\right)}^5{d}^2/\left(L^2{g}^{3}F\right)$ plotted versus the shaking parameter $B$. The dashed-dotted line serves as a guide to the eye. As in previous plots, different symbols correspond to different numbers of layers $F$, different colors to different particle diameters $d$, and the solid and open symbols represent experimental and numerical data, respectively.