Technical Brief

The Homogeneous Cooling State as a Verification Test for Kinetic Theory-Based Continuum Models of Gas–Solid Flows

[+] Author and Article Information
William D. Fullmer

Department of Chemical and Biological Engineering,
University of Colorado,
596 UCB,
Boulder, CO 80309
e-mail: william.fullmer@colorado.edu

Christine M. Hrenya

Department of Chemical and Biological Engineering,
University of Colorado,
596 UCB,
Boulder, CO 80309
e-mail: hrenya@colorado.edu

1Corresponding author.

Manuscript received August 17, 2016; final manuscript received December 23, 2017; published online January 25, 2018. Assoc. Editor: Amit Shukla.

J. Verif. Valid. Uncert 2(4), 044501 (Jan 25, 2018) (5 pages) Paper No: VVUQ-16-1024; doi: 10.1115/1.4038916 History: Received August 17, 2016; Revised December 23, 2017

Granular and multiphase (gas–solids) kinetic theory-based models have emerged a leading modeling strategy for the simulation of particle flows. Similar to the Navier–Stokes equations of single-phase flow, although substantially more complex, kinetic theory-based continuum models are typically solved with computational fluid dynamic (CFD) codes. Under the assumptions of the so-called homogeneous cooling state (HCS), the governing equations simplify to an analytical solution describing the “cooling” of fluctuating particle velocity, or granular temperature. The HCS is used here to verify the implementation of a recent multiphase kinetic theory-based model in the open source mfix code. Results from the partial verification test show that the available implicit (backward) Euler time integration scheme converges to the analytical solution with the expected first-order rate. A second-order accurate backward differentiation formula (BDF) is also implemented and observed to converge at a rate consistent with its formal accuracy.

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Haff, P. K. , 1983, “ Grain Flow as a Fluid-Mechanical Problem,” J. Fluid Mech., 134, pp. 401–430. [CrossRef]
Goldhirsch, I. , and Zanetti, G. , 1993, “ Clustering Instability in Dissipative Gases,” Phys. Rev. Lett., 70(11), pp. 1619–1622. [CrossRef] [PubMed]
Wylie, J. J. , and Koch, D. L. , 2000, “ Particle Clustering Due to Hydrodynamic Interactions,” Phys. Fluids, 12(5), pp. 964–970. [CrossRef]
Garzó, V. , 2005, “ Instabilities in a Free Granular Fluid Described by the Enskog Equation,” Phys. Rev. E, 72(2), p. 021106.
Garzó, V. , Fullmer, W. D. , Hrenya, C. M. , and Yin, X. L. , 2016, “ Transport Coefficients of Solid Particles Immersed in a Viscous Gas,” Phys. Rev. E, 93(1), p. 012905.
Fullmer, W. D. , and Hrenya, C. M. , 2017, “ The Clustering Instability in Rapid Granular and Gas-Solid Flows,” Annu. Rev. Fluid Mech., 49, pp. 485–510.
Brito, R. , and Ernst, M. H. , 1998, “ Extension of Haff's Cooling Law in Granular Flows,” Europhys. Lett., 43(5), pp. 497–502. [CrossRef]
Pathak, S. N. , Jabeen, Z. , Das, D. , and Rajesh, R. , 2014, “ Energy Decay in Three-Dimensional Freely Cooling Granular Gas,” Phys. Rev. Lett., 112(3), p. 038001.
Yin, X. , Zenk, J. R. , Mitrano, P. P. , and Hrenya, C. M. , 2013, “ Impact of Collisional Versus Viscous Dissipation on Flow Instabilities in Gas-Solid Systems,” J. Fluid Mech., 727, p. R2. [CrossRef]
Brey, J. J. , Ruiz-Montero, M. J. , and Cubero, D. , 1999, “ On the Validity of Linear Hydrodynamics for Low-Density Granular Flows Described by the Boltzmann Equation,” Europhys. Lett., 48(4), pp. 359–364. [CrossRef]
Montanero, J. M. , and Garzó, V. , 2002, “ Monte Carlo Simulation of the Homogeneous Cooling State for a Granular Mixture,” Granular Matter, 4(1), pp. 17–24. [CrossRef]
Chamorro, M. G. , Reyes, F. V. , and Garzo, V. , 2013, “ Homogeneous Steady States in a Granular Fluid Driven by a Stochastic Bath With Friction,” J. Stat. Mech., 2013, p. P07013.
Mitrano, P. P. , Dahl, S. R. , Cromer, D. J. , Pacella, M. S. , and Hrenya, C. M. , 2011, “ Instabilities in the Homogeneous Cooling of a Granular Gas: A Quantitative Assessment of Kinetic-Theory Predictions,” Phys. Fluids, 23(9), p. 093303.
Mitrano, P. P. , Zenk, J. R. , Benyahia, S. , Galvin, J. E. , Dahl, S. R. , and Hrenya, C. M. , 2014, “ Kinetic-Theory Predictions of Clustering Instabilities in Granular Flows: Beyond the Small-Knudsen-Number Regime,” J. Fluid Mech., 738, pp. R2–R12. [CrossRef]
Brilliantov, N. , Saluena, C. , Schwager, T. , and Poschel, T. , 2004, “ Transient Structures in a Granular Gas,” Phys. Rev. Lett., 93(13), p. 134301.
Mitrano, P. P. , Dahl, S. R. , Hilger, A. M. , Ewasko, C. J. , and Hrenya, C. M. , 2013, “ Dual Role of Friction on Dissipation-Driven Instabilities in Granular Flows,” J. Fluid Mech., 729, pp. 484–495. [CrossRef]
Rubio-Largo, S. M. , Alonso-Marroquin, F. , Weinhart, T. , Luding, S. , and Hidalgo, R. C. , 2016, “ Homogeneous Cooling State of Frictionless Rod Particles,” Physica A, 443, pp. 477–485. [CrossRef]
Liu, P. , Brown, T. , Fullmer, W. D. , Hauser, T. , Hrenya, C. M. , Grout, R. , and Sitaraman, H. , 2016, “ A Comprehensive Benchmark Suite for Simulation of Particle Laden Flows Using the Discrete Element Method With Performance Profiles From the Multiphase Flow With Interface Exchanges (MFiX) Code,” National Renewable Energy Laboratory, Golden, CO, Technical Report No. NREL/TP-2C00-65637. https://www.nrel.gov/docs/fy16osti/65637.pdf
Garzó, V. , Tenneti, S. , Subramaniam, S. , and Hrenya, C. M. , 2012, “ Enskog Kinetic Theory for Monodisperse Gas-Solid Flows,” J. Fluid Mech., 712, pp. 129–168. [CrossRef]
Ma, D. , and Ahmadi, G. , 1988, “ A Kinetic-Model for Rapid Granular Flows of Nearly Elastic Particles Including Interstitial Fluid Effects,” Powder Technol., 56(3), pp. 191–207. [CrossRef]
Koch, D. L. , and Sangani, A. S. , 1999, “ Particle Pressure and Marginal Stability Limits for a Homogeneous Monodiperse Gas-Fluidized Bed: Kinetic Theory and Numerical Simulations,” J. Fluid Mech., 400(1), pp. 229–263. [CrossRef]
Wylie, J. J. , Koch, D. L. , and Ladd, A. J. , 2003, “ Rheology of Suspensions With High Particle Inertia and Moderate Fluid Inertia,” J. Fluid Mech., 480, pp. 95–118. [CrossRef]
Roache, P. J. , 2002, “ Code Verification by the Method of Manufactured Solutions,” ASME J. Fluids Eng., 124(1), pp. 4–10. [CrossRef]
Choudhary, A. , Roy, C. J. , Dietiker, J. F. , Shahnam, M. , Garg, R. , and Musser, J. , 2016, “ Code Verification for Multiphase Flows Using the Method of Manufactured Solutions,” Int. J. Multiphase Flow, 80, pp. 150–163. [CrossRef]
Drikakis, D. , and Rider, W. , 2005, High Resolution Methods for Incompressible and Low-Speed Flows, 1st ed., Springer-Verlag, Berlin.
Burg, C. O. E. , and Murali, V. K. , 2006, “ The Residual Formulation of the Method of Manufactured Solutions for Computationally Efficient Solution Verification,” Int. J. Comput. Fluid Dyn., 20(7), pp. 521–532. [CrossRef]


Grahic Jump Location
Fig. 1

Comparison of the decay of granular temperature between the analytical solution, Eq. (11), and numerical solutions using the backward Euler method with decreasing time-step size

Grahic Jump Location
Fig. 2

Convergence rates of the L1 and L2 norms for the Euler and second-order BDF time stepping schemes compared to the ideal p = 1 and 2 orders of accuracy




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