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Kinetic-theory predictions of clustering instabilities in granular flows: beyond the small-Knudsen-number regime

  • Peter P. Mitrano (a1), John R. Zenk (a1), Sofiane Benyahia (a2), Janine E. Galvin (a2), Steven R. Dahl (a1) and Christine M. Hrenya (a1)...

In this work we quantitatively assess, via instabilities, a Navier–Stokes-order (small-Knudsen-number) continuum model based on the kinetic theory analogy and applied to inelastic spheres in a homogeneous cooling system. Dissipative collisions are known to give rise to instabilities, namely velocity vortices and particle clusters, for sufficiently large domains. We compare predictions for the critical length scales required for particle clustering obtained from transient simulations using the continuum model with molecular dynamics (MD) simulations. The agreement between continuum simulations and MD simulations is excellent, particularly given the presence of well-developed velocity vortices at the onset of clustering. More specifically, spatial mapping of the local velocity-field Knudsen numbers ( $K{n}_{u} $ ) at the time of cluster detection reveals $K{n}_{u} \gg 1$ due to the presence of large velocity gradients associated with vortices. Although kinetic-theory-based continuum models are based on a small- $Kn$ (i.e. small-gradient) assumption, our findings suggest that, similar to molecular gases, Navier–Stokes-order (small- $Kn$ ) theories are surprisingly accurate outside their expected range of validity.

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Agarwal R. K., Yun K. Y. & Balakrishnan R. 2001 Beyond Navier–Stokes: Burnett equations for flows in the continuum-transition regime. Phys. Fluids 13, 30613085.
Allen M. P. & Tildesley D. J. 1989 Computer Simulation of Liquids. Clarendon.
Almazán L., Carrillo J. A., Salueña V., Garzó & Pöschel T. 2013 A numerical study of the Navier–Stokes transport coefficients for two-dimensional granular hydrodynamics. New J. Phys. 15, 043044.
Bartels-Rausch T., Bergeron V., Cartwright J. H. E., Escribano R., Finney J. L., Grothe H., Gutiérrez P. J., Haapala J., Kuhs W. F., Pettersson J. B. C., Price S. D., Sainz-Díaz C. I., Stokes D. J., Strazzulla G., Thomson E. S., Trinks H. & Uras-Aytemiz N. 2012 Ice structures, patterns, and processes: A view across the icefields. Rev. Mod. Phys. 84, 885.
Brey J. J., Ruiz-Montero M. J. & Cubero D. 1999 Origin of density clustering in a freely evolving granular gas. Phys. Rev. E 60, 3150.
Brey J. J., Ruiz-Montero M. J. & Moreno F. 2001 Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, 061305.
Brilliantov N., Saluena C., Schwager T. & Pöschel T. P. 2004 Transient structures in a granular gas. Phys. Rev. Lett. 93, 134301.
Brito R. & Ernst M. 1998 Extension of Haff’s cooling law in granular flows. Europhys. Lett. 43, 497.
Chapman S. & Cowling T. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.
Conway S. L. & Glasser B. J. 2004 Density waves and coherent structures in granular Couette flows. Phys. Fluids 16, 509.
Galvin J. E., Hrenya C. M. & Wildman R. D. 2007 On the role of the Knudsen layer in rapid granular flows. J. Fluid Mech. 585, 73.
Garg R., Galvin J. E., Li T. & Pannala S. 2010 Documentation of open-source MFIX-DEM software for gas–solids flows. From
Garzó V. 2005 Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, 021106.
Garzó V. & Dufty J. W. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895.
Gidaspow D. 1994 Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic.
Gidaspow D. & Jiradilok V. 2009 Computational Techniques: The Multiphase CFD Approach to Fluidization and Green Energy Technologies. Nova Science.
Glasser B. J., Sundaresan S. & Kevrekidis I. G. 1998 From bubbles to clusters in fluidized beds. Phys. Rev. Lett. 81, 1849.
Goldhirsch I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267.
Goldhirsch I., Tan M. L. & Zanetti G. 1993 A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. J. Sci. Comput. 8, 1.
Goldhirsch I. & Zanetti G. 1993 Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 1619.
Goldreich P. & Tremaine S. 1982 The dynamics of planetary rings. Annu. Rev. Astron. Astrophys. 20, 249.
Herman A. 2011 Molecular-dynamics simulation of clustering processes in sea-ice floes. Phys. Rev. E 84, 056104.
Hopkins M. A. & Louge M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47.
Hrenya C. M. 2011 Kinetic theory for granular materials: polydispersity. In Computational Gas-Solids Flows and Reacting Systems: Theory, Methods and Practice (ed. Pannala S., Syamlal M. & O’Brien T.), pp. 102127. IGI Global.
Hrenya C. M., Galvin J. E. & Wildman R. D. 2008 Evidence of higher-order effects in thermally driven rapid granular flows. J. Fluid Mech. 598, 429.
Jackson R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press.
Kunii D. & and Levenspiel O. 1991 Fluidization Engineering, vol. 2, Butterworth-Heinemann.
Li J. & Kuipers J. A. M. 2003 Gas-particle interactions in dense gas-fluidized beds. Chem. Engng Sci. 58, 711.
Lissauer J. J. 1993 Planet formation. Annu. Rev. Astron. Astrophys. 31, 129.
Luding S. & Herrmann H. J. 1999 Cluster-growth in freely cooling granular media. Chaos 9, 673.
Martin T. W., Huntley J. M. & Wildman R. D. 2005 Hydrodynamic model for a vibrofluidized granular bed. J. Fluid Mech. 535, 325.
Mitrano P. P., Dahl S. R., Cromer D. J., Pacella M. S. & Hrenya C. M. 2011 Instabilities in the homogeneous cooling of a granular gas: a quantitative assessment of kinetic-theory predictions. Phys. Fluids 23, 093303.
Mitrano P. P., Dahl S. R., Hilger A. M., Ewasko C. J. & Hrenya C. M. 2013 Dual role of friction in granular flows: attenuation versus enhancement of instabilities. J. Fluid Mech. 729, 484.
Mitrano P. P., Garzó V., Hilger A. M., Ewasko C. J. & Hrenya C. M. 2012 Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases. Phys. Rev. E 85, 041303.
Patankar S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere.
Pöschel T. P. & Schwager T. 2005 Computational Granular Dynamics: Models and Algorithms. Springer.
Rericha E., Bizon C., Shattuck M. & Swinney H. 2001 Shocks in supersonic sand. Phys. Rev. Lett. 88, 14302.
Rice R. B. & Hrenya C. M. 2009 Characterization of clusters in rapid granular flows. Phys. Rev. E 79 (2), 021304.
Sela N. & Goldhirsch I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41.
Schmidt J., Salo H., Spahn F. & Petzschmann O. 2001 Viscous overstability in Saturn’s B-ring: II. Hydrodynamic theory and comparison to simulations. Icarus 153, 316.
Soto R., Mareschal M. & Mansour M. M. 2000 Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, 3836.
Sundaresan S. 2000 Modelling the hydrodynamics of multiphase flow reactors: current status and challenges. AIChE J. 46, 11021105.
Sundaresan S. 2003 Instabilities in fluidized beds. Annu. Rev. Fluid Mech. 35, 63.
Syamlal M. 1998 MFIX documentation: Numerical technique, p. 31346. US Department of Energy.
Wassgren C. R., Cordova J. A., Zenit R. & Karion A. 2003 Dilute granular flow around an immersed cylinder. Phys. Fluids 15, 3318.
Wildman T., Martin T., Huntley J., Jenkins J., Viswanathan H., Fen X. & Parker D. 2008 Experimental investigation and kinetic-theory-based model of a rapid granular shear flow. J. Fluid Mech. 602, 63.
Wylie J. J. & Koch D. L. 2000 Particle clustering due to hydrodynamic interactions. Phys. Fluids 12, 964.
Xu H., Louge M. & Reeves 2003 Solutions of the kinetic theory for bounded collisional granular flows. Contin. Mech. Thermodyn. 15, 321.
Yin X., Zenk J. R., Mitrano & Hrenya C. M. 2013 Impact of collisional versus viscous dissipation on flow instabilities in gas–solid systems. J. Fluid Mech. 727, R2.
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Journal of Fluid Mechanics
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