Blast Shocks in Quasi-Two-Dimensional Supersonic Granular Flows

In a thin, dilute, and fast flowing granular layer, the impact of a small sphere generates a fast growing hole devoid of matter. The growth of this hole is studied in detail, and its dynamics is found to mimic that of blast shocks in gases. This dynamics can be decomposed into two stages: a fast initial stage (the blast) and a slower growth regime whose growth velocity is given by the speed of sound in the medium used. A simple model using ingredients already invoked for the case of blast shocks in gases but including the inelastic nature of collisions between grains accounts accurately for our results. The system studied here allows for a detailed study of the full dynamics of a blast as it relaxes from a strong to a weak shock and later to an acoustic disturbance.

Figure 1

Images of the expanding hole right after the impact of a steel sphere. (a) An image of the flow produced from a funnel of 2.4 mm in diameter. (b) impact of a 16 mm diameter steel sphere. (c) impact of a 2 mm diameter steel sphere. The hole is surrounded by a dense rim seen here as bright since the images use the backscattered light from the layer. This hole expands fast with a circular rim with a roughly constant width at first but which broadens at later times.

Figure 2

Spatiotemporal plot of the angle averaged intensity profile of the hole versus time. This image is color coded in such a way that dense (bright) regions such as the rim appear yellow while less dense regions appear dark.

Figure 3

The radius of the rim $R$ and its width $W$ versus time. Measurements use the backscattered or the transmitted light. The solid line in this graph is a fit to the model proposed in the text while the dashed line indicates propagation at the speed of sound. ($\mathrm{\text{diameter of steel sphere}}=8\mathrm{mm}$, $\mathrm{\text{Volume fraction}}=0.2$). For the fit, $E_0/E_{\mathrm{sphere}}=10\%$ and $N=1.67{\mathrm{mm}}^{-1}$. Inset: angle averaged profiles of the rim for different times after impact. The rim profile has a well-defined maximum and has a width which increases with elapsed time.

Figure 4

(a) $R(t)$ along with a fit to the model. Diamonds: steel sphere of diameter 4 mm, volume fraction of 0.2, and the fit parameters are $E_0/E_{\mathrm{sphere}}=9.4\%$ and $N=1.6{\mathrm{mm}}^{-1}$. Open circles: steel sphere of diameter 8 mm, volume fraction of 0.2, and the fit parameters are $E_0/E_{\mathrm{sphere}}=8\%$ and $N=1.45{\mathrm{mm}}^{-1}$. Closed circles: glass sphere of diameter 16 mm, volume fraction of 0.1, and the fit parameters are $E_0/E_{\mathrm{sphere}}=20\%$ and $N=2{\mathrm{mm}}^{-1}$. (b) Collapse of data from different flow volume fractions (from 6 to 22%), different sphere diameters (from 2 to 16 mm), and materials (steel and glass spheres are used). The dashed line is the Taylor-Sedov prediction while the solid line is a fit to our model. The data deviate from the universal regime when the expansion speed approaches the sound velocity. The fit parameters $N$ and $E_0/E_{\mathrm{sphere}}$ are between 1.4 and $2.5{\mathrm{mm}}^{-1}$ and between 1 and 20%, respectively, for all the runs examined. Inset: $E/E_0$ versus time. Two sphere diameters (8 and 3 mm) are used. The exponential dependence gives a value of $N$ of $1.67{\mathrm{mm}}^{-1}$ and a value of $E_0/E_{\mathrm{sphere}}$ of 5 and 10%.