Clustering instability in a freely falling granular jet

This paper investigates a clustering instability of a freely falling granular jet composed of $100\mathrm{\mu }\mathrm{m}$ glass spheres. The granular flow out of a circular nozzle starts out spatially uniform and then, further downstream, breaks up into well-defined clusters. An optical method is used that measures inhomogeneities in the flow in order to quantify the growth of the clusters. The role of air is investigated in this phenomenon by changing the ambient air pressure down to 1/5000th atm. Clustering is observed down to the lowest pressure and the presence of air leads to larger clusters but does not initiate the cluster formation. The analysis shows that the cluster size is set by fluctuations on the order of the size of the particles at the nozzle.

Figure 1

100 micron glass spheres draining from the $4\mathrm{mm}$ wide circular nozzle at atmospheric pressure. This porous funnel is the same as used in the experiment. Pictures are taken at different depths $z$ below the nozzle. (a) At the nozzle, (b) $z=25\mathrm{cm}$, (c) $z=55\mathrm{cm}$, (d) $z=75\mathrm{cm}$, (e) $z=130\mathrm{cm}$.

Figure 2

Left-hand side: The setup. The granular jet emerges from the reservoir at the top of the $14\mathrm{cm}$ wide acrylic tube. The tube is connected to a vacuum pump and the pressure is measured with a pressure gauge mounted on the lid. The optical platform can be adjusted to any height between the nozzle and $2.1\mathrm{m}$ below. The light from a $5\mathrm{mW}$ $(\lambda =650\mathrm{nm})$ laser diode is focused by a lens, then passed through a glass rod to spread it out and finally refocussed with a cylindrical lens onto a photodiode. An identical laser sheet (not shown) is mounted $5.1\mathrm{mm}$ below the first sheet. The signals of the two photodiodes are passed to two current amps that also invert the signals. The two signals are cross correlated with a spectrum analyzer and one of them is sent to an A/D converter card. Right-hand side: Schematics of the reservoir and the funnel. The reservoir consists of a $9\mathrm{cm}$ wide acrylic tube which feeds a $16\mathrm{mm}$ wide porous tube. At the bottom of the porous tube a metal disc is attached with a $4\mathrm{mm}$ circular aperture. The inside is covered with metal mesh and is grounded. The reservoir sits on a ring that has several holes in it to allow for pressure equalization between the reservoir and the lower part of the tube. The depth $z$ is measured from the nozzle.

Figure 3

Velocity of the jet versus depth at different pressures. In all panels, $v_0=0.36\mathrm{m}∕\mathrm{s}$. The solid lines are linear fits to the data. The slope represents the acceleration $a$. (a) $p=0.027\mathrm{kPa}$, $a=9.8\mathrm{m}∕\mathrm{s}$; (b) $p=0.67\mathrm{kPa}$, $a=9.8\mathrm{m}∕\mathrm{s}$; (c) $p=49\mathrm{kPa}$, $a=9.8\mathrm{m}∕\mathrm{s}$; (d) $p=101\mathrm{kPa}$, $a=9.7\mathrm{m}∕\mathrm{s}$.

Figure 4

Blockage at different depths, $z$, at $p=0.027\mathrm{kPa}$ (left-hand panel) and $p=101\mathrm{kPa}$ (right-hand panel). The length of the arrow equals the dip position of the corresponding autocorrelation.

Figure 5

(a) Blockage fluctuations as a function of depth at two different pressures: $(◻)$ $p=0.027\mathrm{kPa}$; $(∎)$ $p=101\mathrm{kPa}$. (b) Mean of the blockage as a function of depth at the respective pressures. The error bar delineates the standard deviation $\mathrm{\Delta }B$.

Figure 6

Histogram of clusters and gaps at $p=0.027\mathrm{kPa}$. In both panels the solid line corresponds to $z=150\mathrm{cm}$ and the dashed line to $z=50\mathrm{cm}$. The threshold to discriminate between clusters and gaps is the average of the signal. (a) Cluster histogram; (b) gap histogram.

Figure 7

Autocorrelation at different heights. Left-hand panels show data at $p=0.027\mathrm{kPa}$, the right-hand panels at $p=101\mathrm{kPa}$. The depth at which the signal was recorded is shown in each panel. The vertical dotted lines indicate the dip positions used to calculate ${\lambda }_0$. At $p=0.027\mathrm{kPa}$ the dip position does not change with depth.

Figure 8

Dip length vs depth at different pressures. Each dip position has been converted into a length scale by multiplying it with the local velocity. The solid lines are fits from Eq. (3). The error bars are the size of the symbols. (a) $p=0.027\mathrm{kPa}$; (b) $p=0.67\mathrm{kPa}$; (c) $p=49\mathrm{kPa}$; (d) $p=101\mathrm{kPa}$.

Figure 9

Dip position as a function of pressure at $z=20\mathrm{cm}$.