Influence of particle dynamics on the instability for pattern formation in shallow pulsed beds

A granular layer can form standing-wave patterns, such as squares, stripes, and hexagons, when it is fluidized with a pulsed gas flow. These patterns resemble the well-known patterns formed in vertically vibrated granular layers, but are governed by different dimensionless numbers. Recent research [de Martín et al., Phys. Rev. Fluids 3, 034303 (2018)] reveals that the onset to pattern formation in shallow pulsed beds can be understood in terms of the dimensionless number ${\mathrm{\Gamma }}_p=u_a/u_t\overline{\varphi }$, where $u_a$ is the amplitude of the gas velocity, $u_t$ is the terminal velocity of the particles, and $\overline{\varphi }$ is the average solids volume fraction. In contrast, pattern formation in vertically vibrated granular layers in vacuo is governed by the dimensionless number ${\mathrm{\Gamma }}_v=4{\pi }^2{f}^{2}d/g$, where $f$ and $d$ are the frequency and displacement of the vibrated plate, respectively, and $g$ is the gravitational acceleration. In addition, the threshold for pattern formation in pulsed beds exhibits a strong dependence with the frequency of the excitation that is not observed in the threshold for pattern formation in vibrated systems. This work explores the origin of these differences by simulating the dynamics of a one-dimensional pulsed array of particles. Simulations reproduce well the experimental stability curves, and reveal that the criterion for instability in shallow pulsed and vibrated systems is actually the same; the layer flight time must be equal to $1/f$. In pulsed beds, this criterion is determined by the traveling time of the kinematic wave that forms in each flow pulse. These results provide a theoretical basis to the recent experimental observations and highlights commonalities between the mechanisms behind pattern formation in thin vibrated granular layers and shallow pulsed fluidized beds.

Figure 1

Experimental stability curve for pattern formation in shallow pulsed beds, adapted from [4]: $◯$, glass 237; $\blacksquare$, glass 130; and $★$, polystyrene. The parameters are ${\mathrm{\Gamma }}_p=u_{a,c}/u_t\overline{\varphi }$ and $f_n=\frac{1}{\pi }\sqrt{\frac{g}{h}}$.

Figure 2

Scheme to estimate the local solids volume fraction ${\varphi }_i$ from the interparticle distance. Black particles represent the array of particles simulated with PAM. The red particle represents the top particle. Gray particles represent ghost particles included in the calculation of ${\varphi }_i$.

Figure 3

Trajectories for the top particle between $t=3$ and $t=6$ s in an array of glass 237. A small increase in the amplitude of the oscillating gas velocity induces an abrupt change in the particle dynamics, from (a) disordered, $u_a/u_{\mathrm{MF}}=0.34$, to (b) ordered, $u_a/u_{\mathrm{MF}}=0.37$. The parameters are $N=17$ and $f=15$ Hz.

Figure 4

Simulated stability curve for pattern formation in shallow pulsed beds, with the same conditions as for Fig. 1: $◯$, glass 237; $\blacksquare$, glass 130; $★$, polystyrene; and $▴$, D1000. The inset shows the evolution of $\tau$ with ${\mathrm{\Gamma }}_p$ for glass 237, with $N=21$ and $f=14$ Hz.