Anomalous diffusion in a monolayer of lightweight spheres fluidized in air flow

This paper presents statistical analyses of random motions in a single layer of fluidized lightweight spherical particles. Foam polystyrene spheres were driven by an upward airflow through the sieve mesh, and their two-dimensional motion was acquired using image analysis. In the bulk region, the particle velocity distributions changed from Gaussian to heavy-tailed distribution as the bulk packing fraction ${\varphi }_b$ was increased. The mean square displacement of the particles exhibited transition to subdiffusion at much lower ${\varphi }_b$ than observed in previous studies using similar setup but with heavier particles. A slight superdiffusion and significant growth of the correlation length in the two-body velocity correlation was observed at further large ${\varphi }_b$. The effect of the wall on the dynamics of the particles was also investigated, and the anisotropy of the granular temperature was found to be a useful index to discriminate between the wall region and the bulk. The turbulence statistics in the wake of a particle indicated a strong wall-normal asymmetry of aerodynamic forcing as the “thermal” agitation in the wall region.

Figure 1

Schematic of the experimental apparatus (not to scale).

Figure 2

Coordinate system used in the analysis. A representative image of particles ($N=80$) is displayed in background. Dotted circle shows the boundary between wall region and bulk region which will be defined in Sec. 3b, and particles in the bulk are shaded.

Figure 3

Spatial density distributions of particles for the various total particle numbers $N$, normalized by the mean number densities in the effective sieve radius $R_{\text{sieve}}-R_{\text{particle}}=140\mathrm{mm}$, at up-flow velocities (a) $U_a=0.42V_t$ and (b) $U_a=0.47V_t$. See Table 1 for ${\varphi }_b$ shown in the legend.

Figure 4

Spatial distribution of (total) granular temperature distribution $\langle v^2\left(r\right)\rangle$, for (a) $U_a=0.42V_t$ and (b) $U_a=0.47V_t$.

Figure 5

Spatial distribution of ratio of radial and azimuthal granular temperatures $\langle v_r^2\left(r\right)\rangle /\langle v_{\theta }^2\left(r\right)\rangle$, for (a) $U_a=0.42V_t$ and (b) $U_a=0.47V_t$.

Figure 6

(a, b) MSD of particles in the bulk region, vs time. Lines with slopes of 2 and 1 are shown as references. Saturation at around $6\times {10}^3{\mathrm{mm}}^2$ is due to the limited size ($r\le 90\mathrm{mm}$) of the observation domain (the bulk region). (c, d) Same data as in (a) and (b) but divided by $\tau$ to give diffusion coefficients. Airflow velocities are (a, c) $U_a=0.42V_t$ and (b, d) $U_a=0.47V_t$. The large spread at small ${\varphi }_b$ and large $\tau$ is caused by the decrease of tracable particles due to the increased escape from the bulk.

Figure 7

Velocity correlation functions for various ${\varphi }_b$ at (a) $U_a=0.42V_t$ and (b) $U_a=0.47V_t$. The entire particle velocity data were divided into 10 s segments, and the velocity autocorrelation for each segment was calculated and then averaged over 420 segments (with 50% overlap) and over particles that did not exit the bulk throughout each time segment. The large spreads in the correlations at ${\varphi }_b\le 0.14$ and large $\tau$ is caused by the decrease of correlatable particle trajectories due to the increased escape from the bulk.

Figure 8

Velocity distribution functions $P\left(v_r\right)$ for various ball numbers ${\varphi }_b$, at (a) $U_a=0.42V_t$ and (b) $U_a=0.47V_t$. Fitting the generalized exponential distribution $exp\left(\right|v_r{|}^{-\alpha })$ to the data points gives $\alpha$ as a function of $U_a$ and ${\varphi }_b$ as shown in Table 1. Curves are given only as guides for the eye: $\alpha =1.0$ (Cauchy, tent-shaped) and 2.0 (Gaussian, parabola).

Figure 9

Sample trajectories of a particle near the center of the sieve at $U_a=0.42V_t$. Sample trajectories of a particle in an interval of $100\mathrm{s}$ and for (a) ${\varphi }_b\approx 0.32$ and (b) ${\varphi }_b\approx 0.44$ are shown with time interval of $1/30\mathrm{s}$ between the dots. The arrows show the direction of particle movement. The circles designate the effective sieve boundary at radius $R_{\text{sieve}}-R_{\text{particle}}=140\mathrm{mm}$.

Figure 10

Two-body velocity correlation functions for various ${\varphi }_b$, at (a) $U_a=0.42V_t$ and (b) $U_a=0.47V_t$. The segmentation of the time series and the averaging were performed in the same way as Fig. 7.

Figure 11

(a) Histogram of aerodynamic force in the wake of single spherical particle rigidly fixed to the sieve surface at various radial positions in sieve $r_{\mathrm{particle}}$. The sieve-radial velocity $v_{\mathrm{wc}}$, which was measured at 5 mm downstream along wake centerline, is squared to be proportional to its contribution to aerodynamic force and is normalized by ${\overline{u}}^2$. Positive values on the horizontal axis are forces toward the wall and negative values are toward the center. (b) Autocorrelation of $v_{\mathrm{wc}}$. The lag is normalized by the characteristic timescale for the flow past a particle, $R_{\mathrm{particle}}/\overline{u}$.