Christine M. Hrenya, University of Colorado (Boulder)
Instabilities, such as dynamic particle clusters, occur in both granular and gas-solid rapid flows. In granular flows, such non-uniformities in concentration (bulk density) can be traced to the dissipative nature of collision, whereas in gas-solid flows, both mean drag and viscous damping can also lead to clustering instabilities. Regardless of the source, clustering instabilities impact the performance of industrial operations, such as gas-solid fluidized beds and pneumatic conveyers. Kinetic-theory-based descriptions of rapid particulate flows predict inhomogeneities that are qualitatively similar to previous experiments and discrete-particle simulations. Here, we instead consider the quantitative ability of the kinetic theory applied to a simple granular system.
To date, much qualitative work has been done on instabilities in the homogeneous cooling of granular flows. The homogeneous cooling system (HCS) consists of dissipative particles in a periodic domain. The absence of external forces in the HCS gives rise to a homogeneous system when stable, though homogeneity may be lost due to the formation of velocity vortices (i.e., transversal- or shear-mode instabilitiesb)or particle clusters (i.e., longitudinal- or heat-mode instabilities. More specifically, previous works have shown that the presence of such instabilities is dictated by the restitution coefficient, the solids fraction, and the ratio of the (linear) domain length to particle diameter, L/d. Instabilities are more likely in larger domains, and, in general, velocity vortices manifest more readily than particle clusters. For a given restitution coefficient and solids fraction pairing, a critical dimensionless system scale Lvortex/d demarcates (stable) homogeneous flow from one with velocity-vortex instabilities, while a separate critical value Lcluster/d demarcates a (stable) homogeneous particle distribution from one exhibiting the clustering instability.
Recently, the first quantitative assessment of predictions of the critical system size for velocity-vortex instabilities (Lvortex/d) was made. The results show excellent agreement between molecular dynamics (MD) simulations and a linear stability analysis(Garzó, 2005) of the continuum equations derived from the Enskog kinetic theory (Garzó & Dufty, 1999) over moderate ranges of dissipation (0.6≤restitution coefficient≤0.9) and solids fraction (0.05<solids fraction≤0.4). At this point, it is worth noting the difference between the previous treatment and what is being investigated here. The difference lies in the continuum theory, and in particular how the integral equations defining the transport coefficients were evaluated. In particular, in order to obtain an analytical solution for integrals defining the transport coefficients, Garzó & Dufty (Garzó & Dufty, 1999) approximated the first-order velocity distribution as the product of the Maxwell-Boltzmann distribution and a truncated Sonine polynomial; we will refer henceforth to this procedure as the standard-Sonine approximation. In spite of this simple approximation, such predictions compare well with Direct Simulation Monte Carlo (DSMC) results (Brey & Ruiz-Montero, 2004, Brey, et al., 2005), except for heat flux transport coefficients in highly dissipative systems (e<0.6) at the dilute limit. Motivated by this disagreement, a slight modification to the standard Sonine approximation has been recently proposed for monocomponent (Garzó, et al., 2007) and multicomponent (Garzó, et al., 2009) granular gases. The idea behind the modified-Sonine approximation is to assume that the isotropic part of the first-order distribution function is mainly governed by the HCS distribution rather than by the Maxwellian distribution. This modified-Sonine approach significantly improves the e-dependence of the heat flux transport coefficients and corrects the disagreement observed in dilute systems between kinetic theory and DSMC results. It is worth stressing that DSMC, a numerical solution of the Boltzmann and/or Enskog equations(from which the continuum equations and transport coefficients are derived analytically),is ideal for testing mathematical assumptions used in the derivation process, whereas MD, based on Newton’s equations of motion, is entirely independent of the starting kinetic equation. MD thus serves as an ideal test bed for the kinetic equation itself along with assumptions used to derive the continuum description.
In this work, we have determined Lvortex/d and Lcluster/d via molecular dynamics (MD) simulations to quantitatively assess the ability of the standard (Garzó & Dufty, 1999) and modified-Sonine (Garzó, et al., 2007) approximations of the Enskog equation to predict vortex and cluster instabilities. We build on the earlier comparison(Mitrano, et al., 2011), which was limited to velocity vortices, by considering a previously unexplored range of high dissipation (restitution coefficient = 0.25-0.4) and dilute-to-moderate volume fractions (phi=0.05-0.15),where the new modification (Garzó, et al., 2007) strongly impacts the theoretical transport coefficients. This work also represents the first time that Lcluster/d has been obtained from MD simulations. Our comparisons between the theoretical predictions for the critical length scales and the MD simulations show excellent quantitative agreement between MD and the modified-Sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a (high-dissipation) qualitative mismatch between the standard theory and MD for the instability that forms more readily. Furthermore, the evolution of cluster size was briefly examined via MD, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters.
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