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

Abstract

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.

An adaptive timestepping algorithm for particle time integration in coupled CFD-DEM simulations.

“…Fig. 2(d) shows the comparison between the computed and analytic solutions for the temperature (T = 1 3 v 2 ) decay as predicted by the Haff's law [24,25], indicating reasonable agreement. Slight deviations are observed due to the idealized assumptions (such as constant collision cross-section) used in the analytic derivation of Haff's law.…”