† Corresponding author. E-mail:
A parametric study of the clustering transition of a vibration-driven granular gas system is performed by simulation. The parameters studied include the global volume fraction of the system, the size of the system, the friction coefficient, and the restitution coefficient among particles and among particle–walls. The periodic boundary and fixed boundary of sidewalls are also checked in the simulation. The simulation results provide us the necessary “heating” time for the system to reach steady state, and the friction term needed to be included in the “cooling” time. A gas-cluster phase diagram obtained through Kolmogorov–Smirnov (K–S) test analysis using similar experimental parameters is given. The influence of the parameters to the transition is then investigated in simulations. This simulation investigation helps us gain understanding which otherwise cannot be obtained by experiment alone, and makes suggestions on the determination of parameters to be chosen in experiments.
We regard a system formed by a large number of macroscopic particles with a size larger than micrometers as a granular system. Granular matter is ubiquitous in our daily life. It is the most existing material form on earth except water. Examples such as sand, industrial products, food, grains, etc., are all considered as granular materials. Some natural phenomena such as sandstorm, avalanche, earthquake, debris flow, and other geological disasters are closely related to granular flow. The cognition of these phenomena and their mechanisms has been one of the foci of physicists’ interest.[1]
In the granular system, thermal fluctuation can be neglected when compared to particle kinetic energy. Thus, the granular system is considered as a system far from thermal equilibrium. It is an amorphous, discrete system with intrinsic energy dissipation. According to the volume fraction of the system and the kinetic energy of the particles, we can generally divide granular systems into “granular gas”, “granular liquid”, and “granular solid”. Unlike molecular gases, the particles in granular gas collide with each other in an inelastic way, and sedimentation occurs under gravity. Inelastic collision causes cluster formation in granular gas. The mechanisms of clustering have been of interest to physicists in recent years.[1–3] Understanding the mechanisms of the instability and the spontaneous formation of clusters in granular gas may help physicists establishing non-equilibrium statistical model.
To study the granular gas properties experimentally, energy input is necessary to balance the intrinsic energy dissipation due to the particle–particle inelastic collisions. The most commonly used method is to agitate the system mechanically by oscillation, that is, particles gain momentum through colliding with the oscillating-wall, the so-called “boundary heating” method. The agitated particles will then reach a “thermal” equilibrium through particle–particle collisions. However, sedimentation under gravity makes experimental investigation difficult. Therefore, it is necessary to conduct experimental study of the granular gas behavior either under microgravity[4,5] or by computer simulation.[6,7]
In a pioneering experiment, cluster forming in granular gas was reported by Falcon et al.[8] The system was mechanically driven into a steady-state by shaker to balance the internal energy dissipation. Clustering occurred when the global volume fraction ϕg exceeded a critical value, resulting in a decrease in both pressure and kinetic temperature.[9] The location of the cluster was far from the driving piston and ‘preferred’ to aggregate near the sidewalls.[10–12]
In event-driven simulation by Miller and Luding,[7] the growth of cluster was characterized by the power law decay of kinetic energy of the system. To identify a cluster, a statistical method Kolmogorov–Smirnov (K–S) test was proposed[13] to evaluate the deviation of the spatial distribution of particle from a uniform distribution. In the numerical work of Opsomer et al.,[14] the local Voronoi volume fraction ϕlocal exceeding 0.285 was considered as a criterion for cluster to appear. There seems to be consistency between the K–S test and criterion by ϕlocal.
In this work, we focus on the simulation investigation of the gas-cluster transition in piston driven granular gas system. The “heat” transport from oscillating-walls to the granular gas, and the “cooling” of the kinetic energy due to inelastic collisions among particles to cluster state are investigated. The K–S test and ϕlocal criteria used to establish a phase diagram are checked. We then investigate the influencing parameters of the transition curve in the phase diagram.
The simulation is based on discrete element method (DEM).[15] We use Hertz nonlinear contact model to calculate the force between particles. The interaction between them is considered if and only if two particles collide. Taking the rotation into consideration, the particles’ equations of motion are as follows:
![](cpb_29_5_054502/cpb_29_5_054502_ieqn1.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn2.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn3.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn4.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn5.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn6.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn7.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn8.gif)
The tangential contact force is determined by the tangential component of the relative velocity and the normal contact force
![](cpb_29_5_054502/cpb_29_5_054502_ieqn9.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn10.gif)
The simulation system contains monodisperse spheres with a diameter of 1 mm. Other related parameters are listed in Table
![]() | Table 1.
The parameters used in simulation. . |
![]() | Table 2.
Seven sets of simulation parameters, where μpp, μpw, ε are respectively the coefficient of friction between spheres, the coefficient of friction between spheres and walls, and the coefficient of restitution. No. 1: reference; No. 7: freely cooling. . |
The maximum mean distance between two particles in the cell is given by
One method to set the criterion of cluster from gas is by the two-sample K–S test.[16] In the method, the maximum difference between a cumulative distribution function F(x) of counts along the x-axis and the cumulative distribution function of a uniform function U(x) is defined as
![](cpb_29_5_054502/cpb_29_5_054502_ieqn12.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn13.gif)
![](cpb_29_5_054502/cpb_29_5_054502_ieqn14.gif)
Figure
All the analyses given above are based on the system reaching a steady state in the granular gas phase. We then need to know the time for the system to reach a steady state. Here, in Fig.
In this work, for each set of parameters in Table
In this simulation, we only consider cluster forming due to inelastic collisions among particles. In this section, we investigate the free cooling[20,21] process without energy input and with no sidewall collisions (periodic boundary condition as set No. 7 in Table
The initial temperature of the system is set to be T0 = T(t = 0). The temperature T(t) decreases rapidly over time through particle collisions as shown in Fig.
Considering friction coefficient 0.1 and restitution coefficient 0.9, simulation results are plotted in a δ/r–ϕg diagram, as shown in Fig.
The propagation time τp can be calculated by simplifying the problem as a one-dimensional granular chain with equal intervals Δx
The propagation time τp (Eq. (
Fitting the data by Eq. (
Using K–S test and Voronoi volume fraction, we have analyzed the simulation data and obtained the gas-cluster phase diagram in δ/r–ϕg plot. Simulations with periodic and fixed sidewall boundary conditions are performed. The friction coefficient μpp = 0.1, μpw = 0.1, and the restitution coefficient ε = 0.9 are chosen. In Fig.
The influence of three material parameters, particle–particle friction coefficient μpp, particle–wall friction coefficient μpw, and the restitution coefficient ε, are investigated in the following simulations. In Fig.
In this work, we have performed a simulation study on the parametric influences of the clustering transition in a vibration-driven granular gas system. The parameters studied include the global volume fraction of the system, the size of the system, the friction coefficient, and the restitution coefficient among particles and among particle–walls. Periodic boundary and fixed boundary of sidewalls are also checked in the simulation. It is found that in our model, the restitution coefficient of the particle-particle collision plays the major role in influencing the transition. Friction among particles plays a minor role, and a friction term γ is needed to modify the Haff’s cooling time. This simulation study helps us in the determination of the experimental parameters in the future microgravity experimental preparations.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] |