Interplay of air and sand: Faraday heaping unravelled

We report on numerical simulations of a vibrated granular bed including the effect of the ambient air, generating the famous Faraday heaps known from experiment. A detailed analysis of the forces shows that the heaps are formed and stabilized by the airflow through the bed while the gap between bed and vibrating bottom is growing, confirming the pressure gradient mechanism found experimentally by Thomas and Squires [Phys. Rev. Lett. 81, 574 (1998)], with the addition that the airflow is partly generated by isobars running parallel to the surface of the granular bed. Importantly, the simulations also explain the heaping instability of the initially flat surface and the experimentally observed coarsening of a number of small heaps into a larger one.

Figure 1

Numerically simulated Faraday heaping in a vibrated granular bed after 0, 30, 55, and 240 driving cycles. After a few cycles, some slight surface ripples start to grow into small heaps, which coarsen into larger heaps until a steady state with a single Faraday heap is reached.

Figure 2

Circulation in the Faraday heap: The arrows indicate the particle displacement (multiplied by 2 for clarity) during one driving cycle, averaged over 25 cycles. The part of the heap shown here corresponds to the black box in Fig. 1.

Figure 3

The stable Faraday heap (see Fig. 2) at four successive phases during one driving cycle. The arrows indicate the air drag force on the particles averaged per CFD cell (one out of every four arrows is shown); at $234°$, when the drag force is very strong, the arrows are scaled by a factor of 1/3 compared to those at the other phase angles. For comparison, the gravitational force on a particle is shown in the $90°$ figure (and also at $234°$); the ratio of gravity and air drag forces, which is comparable to the Bagnold number [14], is of order one in this simulation. The dynamics of the rectangular ensemble of dark particles is further analyzed in Fig. 4. A video of one driving cycle can be found here [13].

Figure 4

(a) The averaged horizontal components of the contact and drag forces (per unit mass) on the rectangular ensemble of dark particles in Fig. 3, as a function of time during a complete driving cycle. Positive forces point towards the center of heap. (b) The virtual displacements due to the contact and drag forces individually, and their sum, i.e., the actual horizontal displacement of the ensemble. The net displacement during one cycle ($0.43\mathrm{mm}$ towards the center of the heap) is already reached at $260°$. After that, the heap is fully compactified and the particles in the ensemble are locked until a new cycle starts.

Figure 5

(Color online) Evolution of the heap pattern in a box three times as wide as that in Fig. 1. A video of this evolution can be found here [13].

Figure 6

(a) Averaged horizontal particle displacement in a heap, due to the inward air drag (black) and outward avalanching (gray), as a function of the growing slope angle, during the first stage when the heaps are formed. The displacements balance each other at a slope of $18.5°$. (b) Particle displacement during one cycle in the heap indicated by the frame in Fig. 5, which is about to merge with the neighboring heap to the right. The displacements are averaged over five cycles around 240 cycles and multiplied by two for clarity.

Figure 7

(Color online) (a) Fourier transform of the first 45 cycles of Fig. 5 as a function of the wave number $k$ (made dimensionless by the width of the box $W$). The color indicates the relative value of the Fourier transform per cycle, the height is the absolute value. The initial number of seven heaps is already present in the initial condition and grows stronger as the heaps grow during the first 30 cycles. (b) Fourier transform of the first 45 cycles of a simulation with different initial conditions, resulting in a landscape of heaps without any specific wavelength.

Figure 8

(a) Heap angle as a function of the air drag factor. (b) Heap angle as a function of restitution and friction coefficient. The measurements are obtained from simulations in which one parameter is changed slightly after each vibration cycle. The initial condition for each simulation, corresponding to the steady state heap of Fig. 1, is indicated with a marker.

Figure 9

(Color online) (a) Straight lines: Heap angle as a function of vibration amplitude $a$ and dimensionless acceleration $\Gamma$. The color bar indicates the heap angle in degrees. The measurements are obtained from simulations at five different frequencies as indicated. Contour lines: Impact velocity obtained from the model. The increment between the contour lines is $0.1\mathrm{m}∕\mathrm{s}$. (b) Straight lines: Inward particle motion obtained from the same simulations. The color bar indicates the inward displacement per particle per vibration cycle in mm. The inward displacement increases in the direction of the arrow. Contour lines: Inward motion of the particles obtained from the model. The maximum value of the inward motion in the model is scaled to match the maximum value in the simulation.