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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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Figures

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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