Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-14T11:35:42.821Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  04 December 2013

Peter P. Mitrano ,
John R. Zenk ,
Sofiane Benyahia ,
Janine E. Galvin ,
Steven R. Dahl  and
Christine M. Hrenya
Peter P. Mitrano
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309, USA
John R. Zenk
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309, USA
Sofiane Benyahia
Affiliation:
National Energy Technology Laboratory, Morgantown, WV 26507, USA
Janine E. Galvin
Affiliation:
National Energy Technology Laboratory, Morgantown, WV 26507, USA
Steven R. Dahl
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309, USA
Christine M. Hrenya*
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309, USA
*
†Email address for correspondence: hrenya@colorado.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Granular media: Granular media Rarefied Gas Flow: Kinetic theory Instability: Nonlinear instability
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 738 , 10 January 2014 , R2
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agarwal, R. K., Yun, K. Y. & Balakrishnan, R. 2001 Beyond Navier–Stokes: Burnett equations for flows in the continuum-transition regime. Phys. Fluids 13, 3061–3085.CrossRefGoogle Scholar
Allen, M. P. & Tildesley, D. J. 1989 Computer Simulation of Liquids. Clarendon.Google Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Brey, J. J., Ruiz-Montero, M. J. & Moreno, F. 2001 Hydrodynamics of an open vibrated granular system. Phys. Rev. E 63, 061305.Google Scholar
Brilliantov, N., Saluena, C., Schwager, T. & Pöschel, T. P. 2004 Transient structures in a granular gas. Phys. Rev. Lett. 93, 134301.CrossRefGoogle Scholar
Brito, R. & Ernst, M. 1998 Extension of Haff’s cooling law in granular flows. Europhys. Lett. 43, 497.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. 1970 The Mathematical Theory of Non-Uniform Gases. Cambridge University Press.Google Scholar
Conway, S. L. & Glasser, B. J. 2004 Density waves and coherent structures in granular Couette flows. Phys. Fluids 16, 509.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Garg, R., Galvin, J. E., Li, T. & Pannala, S. 2010 Documentation of open-source MFIX-DEM software for gas–solids flows. From https://mfix.netl.doe.gov/documentation/dem.doc.Google Scholar
Garzó, V. 2005 Instabilities in a free granular fluid described by the Enskog equation. Phys. Rev. E 72, 021106.Google Scholar
Garzó, V. & Dufty, J. W. 1999 Dense fluid transport for inelastic hard spheres. Phys. Rev. E 59, 5895.CrossRefGoogle ScholarPubMed
Gidaspow, D. 1994 Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic.Google Scholar
Gidaspow, D. & Jiradilok, V. 2009 Computational Techniques: The Multiphase CFD Approach to Fluidization and Green Energy Technologies. Nova Science.Google Scholar
Glasser, B. J., Sundaresan, S. & Kevrekidis, I. G. 1998 From bubbles to clusters in fluidized beds. Phys. Rev. Lett. 81, 1849.CrossRefGoogle Scholar
Goldhirsch, I. 2003 Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Goldhirsch, I. & Zanetti, G. 1993 Clustering instability in dissipative gases. Phys. Rev. Lett. 70, 1619.CrossRefGoogle ScholarPubMed
Goldreich, P. & Tremaine, S. 1982 The dynamics of planetary rings. Annu. Rev. Astron. Astrophys. 20, 249.Google Scholar
Herman, A. 2011 Molecular-dynamics simulation of clustering processes in sea-ice floes. Phys. Rev. E 84, 056104.Google Scholar
Hopkins, M. A. & Louge, M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47.Google Scholar
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. 102–127. IGI Global.Google Scholar
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.Google Scholar
Jackson, R. 2000 The Dynamics of Fluidized Particles. Cambridge University Press.Google Scholar
Kunii, D. & and Levenspiel, O. 1991 Fluidization Engineering, vol. 2, Butterworth-Heinemann.Google Scholar
Li, J. & Kuipers, J. A. M. 2003 Gas-particle interactions in dense gas-fluidized beds. Chem. Engng Sci. 58, 711.CrossRefGoogle Scholar
Lissauer, J. J. 1993 Planet formation. Annu. Rev. Astron. Astrophys. 31, 129.Google Scholar
Luding, S. & Herrmann, H. J. 1999 Cluster-growth in freely cooling granular media. Chaos 9, 673.CrossRefGoogle ScholarPubMed
Martin, T. W., Huntley, J. M. & Wildman, R. D. 2005 Hydrodynamic model for a vibrofluidized granular bed. J. Fluid Mech. 535, 325.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere.Google Scholar
Pöschel, T. P. & Schwager, T. 2005 Computational Granular Dynamics: Models and Algorithms. Springer.Google Scholar
Rericha, E., Bizon, C., Shattuck, M. & Swinney, H. 2001 Shocks in supersonic sand. Phys. Rev. Lett. 88, 14302.CrossRefGoogle ScholarPubMed
Rice, R. B. & Hrenya, C. M. 2009 Characterization of clusters in rapid granular flows. Phys. Rev. E 79 (2), 021304.CrossRefGoogle ScholarPubMed
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41.Google Scholar
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.CrossRefGoogle Scholar
Soto, R., Mareschal, M. & Mansour, M. M. 2000 Nonlinear analysis of the shearing instability in granular gases. Phys. Rev. E 62, 3836.Google Scholar
Sundaresan, S. 2000 Modelling the hydrodynamics of multiphase flow reactors: current status and challenges. AIChE J. 46, 1102–1105.CrossRefGoogle Scholar
Sundaresan, S. 2003 Instabilities in fluidized beds. Annu. Rev. Fluid Mech. 35, 63.Google Scholar
Syamlal, M. 1998 MFIX documentation: Numerical technique, p. 31346. US Department of Energy.Google Scholar
Wassgren, C. R., Cordova, J. A., Zenit, R. & Karion, A. 2003 Dilute granular flow around an immersed cylinder. Phys. Fluids 15, 3318.CrossRefGoogle Scholar
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.Google Scholar
Wylie, J. J. & Koch, D. L. 2000 Particle clustering due to hydrodynamic interactions. Phys. Fluids 12, 964.CrossRefGoogle Scholar
Xu, H., Louge, M. & Reeves, 2003 Solutions of the kinetic theory for bounded collisional granular flows. Contin. Mech. Thermodyn. 15, 321.Google Scholar
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.Google Scholar