Formation mechanism of shock-induced particle jetting

The shock dissemination of granular rings or shells is characterized by the formation of coherent particle jets that have different dimensions from those associated with the constituent grains. In order to identify the mechanisms governing the formation of particle jets, we carry out the simulations of the shock dispersal of quasi-two-dimensional particle rings based on the discrete-element method. The evolution of the particle velocities and contact forces on the time scales ranging from microseconds to milliseconds reveals a two-stage development of particle jets before they are expelled from the outer surface. Much effort is made to understand the particle agglomeration around the inner surface that initiates the jet formation. The shock interaction with the innermost particle layers generates a heterogeneous network of force chains with clusters of strong contacts regularly spaced around the inner surface. Momentum alongside the stresses is primarily transmitted along the strong force chains. Therefore, the clustering of strong force chains renders the agglomeration of fast-moving particles connected by strong force chains. The fast-moving particle clusters subsequently evolve into the incipient particle jets. The following competition among the incipient jets that undergo unbalanced growth leads to substantial elimination of the minor jets and the significant multiplication of the major jets, the number of jets thus varying with time. Moreover, the number of jets is found to increase with the strength of the shock loading due to an increased number of jets surviving the retarding effect of major jets.

Figure 1

(a) Top view of the particle ring subjected to shock loading with $p_0=20\mathrm{bars}$ at $t=0$ and 0.2 ms after the start of loading (e). The inner surfaces are shaded red. (b) Distribution the particle diameter, which follows the Gaussian distribution. (c) and (d) Profiles of two types of shock loadings. In this study, we employ (d) a sustained shock wave as the shock loading.

Figure 2

(a) Illustration of the mesh of the particle ring. (b) Comparison of the configurations of the shock-loaded particles with and without taking the gas effect into account. Particles clustering around the jet tips represent those under the influence of the penetrating gases.

Figure 3

Trajectories of the inner surface and the shock front in the cases of (a) $p_0=5\mathrm{bars}$ and (b) $p_0=20\mathrm{bars}$. The slopes of both trajectories undergo the transitions almost concurrently.

Figure 4

Evolutions of the configurations of particle rings subjected to shock loading with (a) $p_0=5\mathrm{bars}$ and (b) $p_0=20\mathrm{bars}$. Particles are color coded according to the magnitude of their velocities. The trajectories of the inner surfaces shown in Fig. 3 can be used as the length scales.

Figure 5

The top panel shows the azimuthal distribution of all particle radial velocities around the inner surface of a particle ring subjected to shock loading with $p_0=20\mathrm{bars}$ at $t=0.1\mathrm{ms}$. The second panel from the top to the bottom show the azimuthal distributions of average radial velocities of a particle ring subjected to shock loading with $p_0=5\mathrm{bars}$ at different times after the loading by coarse graining the particle radial velocities of the azimuthal bins $V_r\left(\theta \right)$, here $V_r\left(\theta \right)$ being subtracted from the mean particle radial velocity in the particle ring. Two sets of arrows, namely, the red thick arrows and the blue thin arrows, are used to identify the considerably reduced or even diminished velocity peaks and the enhanced peaks, respectively.

Figure 6

(a) From top to bottom, the azimuthal distributions of the particle radial velocity $V_r\left(\theta \right)$ for the cases of $p_0=5$, 10, 20, and 30 bars at $t=0.75$, 0.4, 0.2, and 0.2 ms, respectively. Also shown are the power spectra of the fast Fourier transformation of the azimuthal distribution of $V_r\left(\theta \right)$ for the cases of (b) $p_0=5\mathrm{bars}$ and (c) $p_0=20\mathrm{bars}$.

Figure 7

Evolution of a segment of the particle ring subjected to shock loading with $p_0=20\mathrm{bars}$. Particles are denoted by arrows that are scaled and shaded according to the magnitude of the particle velocities. Dotted circles indicate the minor jets that are to be eliminated at a later time.

Figure 8

Temporal variations of jet number for particle rings subjected to shock loading with varied $p_0$.

Figure 9

(a) Snapshot of the particle ring subjected to shock loading with $p_0=20\mathrm{bars}$ at $t=0.7\mathrm{ms}$. The dashed red line indicates the location of particle jets. (b) High-speed photo of a flour particle ring subjected to a blast wave with $p_0=2\mathrm{bars}$ at $t=36\mathrm{ms}$ [6] (copyright 3918030359983).

Figure 10

Snapshots of the network of force chains in the bottom layer of the particle ring subjected to shock loading of $p_0=20\mathrm{bars}$ at different times. Red shaded force chains at $t=0.02$ and 0.06 ms indicate the long linear force chains acting as the nuclei of the force chain clustering.

Figure 11

Azimuthal variations of the radial contact force $F_r\left(\theta \right)$ in the particle ring subjected to shock loading with $p_0=20\mathrm{bars}$ at different times.

Figure 12

Azimuthal variations in the radial contact force, $F_r\left(\theta \right)$, and the radial particle velocity, $V_r\left(\theta \right)$, for the cases with $p_0=5$ (a) and 20 bar (b) at $t=0.75$ and 0.38 ms. (c) and (d) Cross correlations between $F_r\left(\theta \right)$ and $V_r\left(\theta \right),\chi (F_r,V_r)$ as a function of the phase shift $\delta \theta$ for cases shown in (a) and (b), respectively. (e) Average velocities of all particles, particles connected by and disconnected from the contact network.

Figure 13

(a) Snapshots of a segment of the particle ring subjected to the shock loading with $p_0=20$ bars at different times in terms of the velocity profile superimposed by the contact network. Particles are denoted by arrows representing the particle velocity vectors, which rescaled and shaded with the magnitude of the velocities. (b) Illustration of the evolution of the jetting pattern as well as the contact network. The red circles, green dashed circles, and blue dotted circles represent the fast-moving particles connected by force chains, slow-moving particles without effective contacts among them, and slow-moving particles connected by transversely oriented force chains, respectively. The times depicted in (b) correspond to the times in (a), namely, the formation of the fast-moving particle clusters (incipient jets), the elimination of the minor jets, and the multiplication of the major jets (from top to bottom).

Figure 14

(a) Illustration of the interplay between adjacent particle jets. Spikes represent the incipient particle jets. The spikes indicated by the red dotted lines denote the minor jets that are encompassed by the compression waves emitted from the neighboring major jets and are to be eliminated soon. (b) Variations of the inner radii of the expanding particle rings subjected to the shocking loadings with varied $p_0$.

Figure 15

Illustration of ${\delta }_n$ and ${\delta }_t$ in the Hertz-Mindlin contact model. The insets show the directions of $\vec{n},\vec{t},{\vec{F}}_n$, and ${\vec{F}}_t$.