Multiscale kinetic theory for heterogeneous granular and gas-solid flows

Abstract

It has been recognized that the particle phase stress model derived from classical kinetic theory is valid only when sufficient scale resolution is offered to explicitly resolve the heterogeneous structures during numerical simulations, however, industrial applications prefer to use coarse computational grids where the heterogeneous structures are not explicitly resolved but implicitly modeled. Unfortunately, in this case a kinetic theory for heterogeneous granular and gas–solid flows is not available yet. To this end, an attempt was made to tracking this challenge: The single particle velocity distribution function at the nonequilibrium stationary state with heterogeneous structures was firstly derived by combining the idea of doubly stochastic Poisson processes or superstatistics with the concept of compromise in competition in the EMMS (Energy Minimization Multi-Scale) theory, the standard Chapman-Enskog method was then used to develop the constitutive relations of heterogeneous continuum theory. It was found that (i) seven state variables are needed to quantify the heterogeneous structures as compared to three for homogeneous systems; (ii) the constitutive relations not only include the contribution from microscale particle–particle interactions but also those due to the interactions between mesoscale structures; and (iii) the resultant constitutive relations are much more complex than those of homogeneous systems due to the simultaneous consideration of microscale and mesoscale contributions and the appearance of cross-coupling effects, but they correctly contain the constitutive relations of homogeneous systems as a limiting case. Finally, the theory was coupled with an EMMS drag model to offer a preliminary validation and to provide a unified EMMS-based constitutive relations for heterogeneous gas–solid flows.