(215c) Cluster Instabilities in Gas-Solid Flows Using Direct Numerical Simulation and Two-Fluid Model: Quantifying the Mean Sedimentation Velocity and Particle Velocity Fluctuations

The main purpose of this contribution is to present particle-resolved direct numerical simulations (DNS) and two-fluid modeling (TFM) of sedimentation of spherical particles subjected to cluster instabilities. DNS and TFM of freely evolving sedimenting suspensions in periodic domains were performed at two solid concentrations of 15% and 40% and two density ratios ρ_p/ρ_f = 100:1 and 1000:1. Particle Reynolds number based on the terminal velocity was set to 3 and 30. Both elastic collisions and inelastic collisions were studied. Evolution of cluster instabilities can be qualitatively divided into three stages: initial void and clusters randomly distribute, then dense and dilute phases grow and attract each other, leading to large and clearly divided dilute and dense phases the scales of which encompass the entire domain. In a previous study of a â??one-dimensionalâ? system (L_y â?? 34 d_p, L_x = L_z â?? 8 d_p,), when visible clustering occurred it was typically in the form of a horizontally spanning, vertically traveling structure. In â??two-dimensionalâ? systems (L_x= L_y â?? 34 d_p, L_z â?? 8 d_p,) studied here, clustering instability has additional degrees of freedom and additional patterns are observed. The particle clusters (dense regions) constantly swing in the horizontal direction, leading to more turbulent and transient flows than in the â??1-Dâ? case. Only at the highest mean flow Stokes numbers are horizontal bands observed which signal a significant reduction in the turbulence statistics.

Sedimentation velocities from DNS are compared with existing drag correlations [1-3]. DNS sedimentation velocities are significantly influenced by the clustering instability, and appear to decrease with increasing density ratio as a result of clustering. The influences of the parameters on the instability and the sedimentation velocity are reported. Velocity fluctuations in horizontal and vertical directions are largely driven by the motion of the clusters and depend significantly on the particle-fluid density ratio and the mean flow Stokes number. Vertical fluctuations coming from particle clusters characterized by non-dimensional variance are low when Stokes numbers are medium and high. Under most situations, whether the collisions are elastic or inelastic (normal restitution coefficient = 0.9) did not generate significant difference in the mean sedimentation velocity and variance.

Sedimentation velocities and particle velocity fluctuations are also compared to results from continuum simulations using a recently developed kinetic-theory-based TFM [4]. In general, the KT-TFM results agree both qualitatively and quantitatively with the DNS data. Two significant deviations are observed, however. At intermediate mean flow Stokes numbers, the KT-TFM has a tendency to over-predict the degree of fluctuations, which, in turn, leads to over-prediction of the mean sedimenting velocity. It is believed that this behavior stems from an over-prediction in the degree of segregation and studies are currently underway to either verify or refute this hypothesis. At the lowest mean flow Stokes numbers the KT-TFM becomes essentially uniform, i.e., clustering vanishes. Consequently, the KT-TFM fails to capture the departure from homogeneous velocities (mean and fluctuating) observed in the DNS data. This breakdown has been determined to stem from a violation of one of the underlying assumptions of the KT derivation. In other words, it is not necessarily a failure of the model, but a failure to apply it within its range of validity. One of the most promising results of the comparison is the match in the anisotropy of the fluctuating particle velocity. The fluctuations in the streamwise (y-) dimension are always strongest, followed by the wide transverse (x-) dimension and finally the narrow transverse (z-) dimension, which is near the homogenous value. The KT-TFM is able to accurately capture these trends qualitatively and quantitatively where the theory is valid.

[1] J. F. Richardson, and W. N. Zaki. â??Sedimentation and Fluidization: Part I,â? Trans. Inst., Chem. Eng., 32:35-53 (1954).

[2] R.J. Hill, D.L. Koch, J.C. Ladd. â??The first effects of fluid inertia on flows in ordered and random arrays of spheresâ?, J. Fluid Mech., 448:213â??241 (2001).

[3] Tenneti S, Garg R, Hrenya CM, Fox RO, Subramaniam S. â??Direct numerical simulation of gas-solid suspensions at moderate Reynolds number: Quantifying the coupling between hydrodynamic forces and particle velocity fluctuationsâ?, Powder Tech. 203:57-69 (2010).

[4] Garzó V, Tenneti S, Subramaniam S, Hrenya CM. â??Enskog kinetic theory for monodisperse gasâ??solid flowsâ?, J. Fluid Mech. 712:129â??168 (2012).