Intermittency and velocity fluctuations in hopper flows prone to clogging

We study experimentally the dynamics of granular media in a discharging hopper. In such flows, there often appears to be a critical outlet size $D_c$ such that the flow never clogs for $D>D_c$. We report on the time-averaged velocity distributions, as well as temporal intermittency in the ensemble-averaged velocity of grains in a viewing window, for both $D<D_c$ and $D>D_c$, near and far from the outlet. We characterize the velocity distributions by the standard deviation and the skewness of the distribution of vertical velocities. We propose a measure for intermittency based on the two-sample Kolmogorov-Smirnov $D_{\text{KS}}$ statistic for the velocity distributions as a function of time. We find that there is no discontinuity or kink in these various measures as a function of hole size. This result supports the proposition that there is no well-defined $D_c$ and that clogging is always possible. Furthermore, the intermittency time scale of the flow is set by the speed of the grains at the hopper exit. This latter finding is consistent with a model of clogging as the independent sampling for stable configurations at the exit with a rate set by the exiting grain speed [C. C. Thomas and D. J. Durian, Phys. Rev. Lett. 114, 178001 (2015)].

Figure 1

Sample image of dry tapioca pearls in the quasi-2D hopper with back illumination. Only the grains nearest the camera, in sharp focus, are tracked and analyzed. For scale, the average grain diameter is $d=3.5\mathrm{mm}$.

Figure 2

Average mass $\langle m\rangle$ discharged before a clog occurs, as a function of slit width $D$. Fitting to the clogging transition form of Eq. (1), we find $\gamma =7.5\pm 2.5$ and $D_c=10.5\pm 0.5\text{mm}$, shown by the dashed curve. However, the approach of Ref. [24] suggests that $\langle m\rangle$ should follow the form of Eq. (2), shown by the solid curve.

Figure 3

Mean-squared displacement in the horizontal $x$ direction of grains near the exit. For all opening sizes, the motion is ballistic for short lag times $\mathrm{\Delta }t$. The thick black line indicates $\langle \mathrm{\Delta }{x}^2\rangle \sim \mathrm{\Delta }{t}^2$.

Figure 4

Velocity distribution of the grains in the vertical direction $y$ in a region near the exit. The distributions are broad and highly skewed in the downward direction.

Figure 5

Map of the hydrodynamic average vertical velocity $v_{h,y}(x,y)\equiv \langle v_y(x,y)\rangle$ for hoppers of varying slit width $D$. Scaled by the speed of the exiting grains $v_{\mathrm{exit}}$ and $D$, the shape of the velocity field is identical for all hoppers. It is also in agreement with the prediction of the kinematic model, given by Eq. (3) and shown on the far right with $b=2d$. The estimate for $D_c=10.5\pm 0.5\mathrm{mm}$ from Fig. 2 and Eq. (1) is indicated. Binned regions shown are typically $7\mathrm{mm}\times 5\mathrm{mm}$.

Figure 6

Exit velocity of the discharging grains, determined two ways. $W=33.4\mathrm{g}/\mathrm{s}$ is the mass discharge rate, ${\rho }_b=0.69\pm 0.01{\mathrm{g}/\mathrm{cm}}^3$ is the bulk mass density of the packing, $L=3.8\mathrm{cm}$ is the slit length (distance between the plates), $D=12.3\mathrm{mm}$ is the slit width, and $v_y(x,y)$ is the downward speed at height $y$. For large heights, $\int v(x,y)dx/D$ is constant and only slightly lower than $W/\left({\rho }_{b}LD\right)$, showing that the packing fraction is constant and wall friction is minimal.

Figure 7

Map of the individual particle velocity fluctuations within a region over all time. Note that $\delta v^2\equiv {\sigma }_{v_x}^2+{\sigma }_{v_y}^2$ and $v_h^2\equiv {\langle v_x\rangle }^2+{\langle v_y\rangle }^2$. White dashed lines indicate approximate locations of $\delta v=v_h/2$; dashed green (gray) lines indicate where $\delta v=v_h$. The relative velocity fluctuations increase in size for flows with smaller $D$, everywhere in the hopper. The estimate for $D_c=10.5\pm 0.5\mathrm{mm}$ from Fig. 2 and Eq. (1) is indicated. Binned regions shown are typically $7\mathrm{mm}\times 5\mathrm{mm}$.

Figure 8

Skewness of $v_y$ at different locations within hoppers of varying slit width $D$. The fluctuations in the velocity are systematically more asymmetric for smaller $D$ and closer to the exit. The estimate for $D_c=10.5\pm 0.5\mathrm{mm}$ from Fig. 2 and Eq. (1) is indicated. Black dashed lines indicate where the skewness $=-2$; blue (gray) dashed lines where it is equal to $-4$. Binned regions shown are typically $7\mathrm{mm}\times 5\mathrm{mm}$.

Figure 9

Vertical position of grains in the hopper near the exit over time $t$, where $t$ is scaled by the typical time $d/v_h$ for a grain to move a distance of its diameter $d$ downward. All of the grains shown are within a square region of dimensions $\left(D+2d\right)\left(D+2d\right)$. The bottom of this region is at a vertical distance $D/2$ from the slit and is centered about the center of the slit, $x=0$. The dynamics of the grains vary substantially with slit width $D$: for small widths, the grains will often come to a rest for a period of time before flowing again. For large $D$, the ensemble speeds do not fluctuate as much and are never observed to approach zero.

Figure 10

Behavior near the exit region for two hoppers with different slits widths of $D=8.6\mathrm{mm}$ (left) and $D=18.2\mathrm{mm}$ (right). Particles are in a square region near the exit with dimensions $\left(D+2d\right)\left(D+2d\right)$. (a),(b) Vertical position $y$ of grains over time, as shown in Fig. 9. (c),(d) The component $v_L$ of the individual grain velocities ${\mathit{v}}_i\left(t\right)$ over time. The ensemble velocity $v_{E,L}\equiv {\langle v_{i,L}\left(t\right)\rangle }_i$ is overlaid as a thick black curve. The horizontal dashed line indicates the average $v_L$ over all time. Note that the $y$ ranges in (c) and (d) are such that the zero and average values align, in order to make very apparent that the fluctuations are much larger in (c). (e),(f) Signed Kolmogorov-Smirnov statistic $D_{\text{KS}}\left(t\right)$. The statistic characterizes the deviation of the velocity distributions at time $t$ with the velocity distribution for all time. When $D_{\text{KS}}\left(t\right)\approx 0$, the velocity distribution at time $t$ is very similar to the distribution over all time. When $D_{\text{KS}}\left(t\right)>0$, the grains are moving slower than usual, and when $D_{\text{KS}}\left(t\right)<0$, the grains are moving faster than usual. We therefore classify the behavior at time $t$ as “fast” or “slow” by the sign of $D_{\text{KS}}\left(t\right)$.

Figure 11

Probability distribution function of the signed Kolmogorov-Smirnov statistic $D_{\text{KS}}\left(t\right)$, for various slit widths $D$. Data are for the square region near the exit, as shown in Figs. 9 and 10, with dimensions $\left(D+2d\right)\left(D+2d\right)$. Times where $D_{\text{KS}}\left(t\right)<0$ or $>0$ can be classified as “fast” or “slow,” respectively. The curve with $D⪆D_c$ is indicated as a black lines with heavier weight. The growing magnitude of $D_{\text{KS}}\left(t\right)$ for small $D$ indicates that the velocity distributions at any given time are less similar to the global velocity distribution, a signal of growing intermittency.

Figure 12

Distributions of the individual particle velocities $v_{i,L}\left(t\right)$ for several different slit widths $D$ in the region near the exit. The region has dimensions $\left(D+2d\right)\left(D+2d\right)$. Red solid curves indicate the distributions over all time: they are characteristically more skewed for slower flows more prone to clogging. The distributions of the individual velocities during “slow” times, when $D_{\text{KS}}\left(t\right)>0$, are shown as dashed blue curves, and for “fast” times, when $D_{\text{KS}}\left(t\right)<0$, as dotted green curves. For large opening sizes $D$, the velocity distributions are similarly Gaussian in character. However, for small $D$ the distributions during slow periods increasingly deviate from the fast velocity distributions, including larger probabilities of velocities that are zero or even opposite the direction of the mean flow ($v_L<0$).

Figure 13

Autocorrelation functions of (a) ensemble velocity $v_{E,L}\left(t\right)$ in the direction of the mean flow and (b) the signed Kolmogorov-Smirnov statistic $D_{\text{KS}}\left(t\right)$, where examples of $v_{E,L}\left(t\right)$ and $D_{\text{KS}}\left(t\right)$ are shown in Figs. 10 and 10 and Figs. 10 and 10, respectively. These are shown here only for the region near the exit, with dimensions $\left(D+2d\right)\left(D+2d\right)$. The autocorrelation function time scales grow slowly with decreasing $D$. The curve with $D⪆D_c$ is indicated by a black dashed line with heavier weight. The time scales and magnitude of the fluctuations are shown in more detail in Fig. 14.

Figure 14

Signatures of intermittency for the square region near the exit of dimensions $\left(D+2d\right)\left(D+2d\right)$. The value of $D_c$, as found in Fig. 2 and Eq. (1), is displayed as a vertical line with a gray band indicating the confidence intervals of $D_c$. (a) Excess kurtosis of the longitudinal velocity during slow (blue circles), fast (green triangles), and all (red squares) times, as a function of slit width $D$. Fast or slow times are classified according to the sign of the signed Kolmogorov-Smirnov statistic $D_{\text{KS}}$. For smaller $D$, the difference between the shape of the slow or fast velocity distributions is much larger, and there is a more sharply pronounced difference between the fast and the slow states. Error bars in $y$ are smaller than the displayed data points. (b) Time for the autocorrelation functions (Fig. 13) of the ensemble velocity $v_{E,L}\left(t\right)$ and the signed Kolmogorov-Smirnov statistic $D_{\text{KS}}\left(t\right)$ to decay $1/e$ of their initial values above the baseline, scaled by the time that exiting grains move by their own diameter. (c) Fluctuations of $v_{E,L}\left(t\right)$ and $D_{\text{KS}}\left(t\right)$, as determined by the variance of these quantities (alternatively, the difference between the $y$ intercept and baseline in Fig. 13). (d) Average value of $D_{\text{KS}}$ (black squares, left axis) and significance (red triangles, right axis). Larger values of $\langle \left|D_{\text{KS}}\right|\rangle$ indicate more intermittent flows. The significance of $D_{\text{KS}}$ is found as the $p$ value [46], with average significance plotted on the right. Even though the sample exit regions are smaller for smaller slit widths, the significance of $|D_{\text{KS}}|$ is greater (smaller $p$).

Figure 15

Plots of various quantities for regions within the hoppers as a function of vertical position $y$ and hopper opening width $D$. The curves with $D⪆D_c$ are indicated by black dashed lines with heavier weights. (a) Excess kurtosis of the longitudinal velocity $v_L$ distributions, as a function of opening size $D$ and position in the hopper $y$. (b) Autocorrelation time scale ${\tau }_e$ for $D_{\text{KS}}\left(t\right)$, scaled by the average time it takes for a grain to shift by its own diameter $d$ at location $y$. Relative to the time scales of the average flow speed $v_h$, the intermittency time scale grows for small $y$ and large $D$. (c) Fluctuations in the ensemble velocity $v_{E,L}$ scaled by the hydrodynamic velocity $v_h$, for all $y$. (d) Fluctuations in $D_{\text{KS}}$. The fluctuations in the signed Kolmogorov-Smirnov statistic follow a different pattern for small $D$ than for large $D$: for a hopper highly prone to clogging, the most intermittent flow is near the exit. For hoppers unlikely to clog, the most intermittent flow is far from the exit.