Meso-scale structures that take the form of clusters and streamers are commonly observed in dilute gas–particle flows, such as those encountered in risers. Continuum equations for gas–particle flows, coupled with constitutive equations for particle-phase stress deduced from kinetic theory of granular materials, can capture the formation of such meso-scale structures. These structures arise as a result of an inertial instability associated with the relative motion between the gas and particle phases, and an instability due to damping of the fluctuating motion of particles by the interstitial fluid and inelastic collisions between particles. It is demonstrated that the meso-scale structures are too small, and hence too expensive, to be resolved completely in simulation of gas–particle flows in large process vessels. At the same time, failure to resolve completely the meso-scale structures in a simulation leads to grossly inaccurate estimates of inter-phase drag, production/dissipation of pseudo-thermal energy associated with particle fluctuations, the effective particle-phase pressure and the effective viscosities. It is established that coarse-grid simulation of gas–particle flows must include sub-grid models, to account for the effects of the unresolved meso-scale structures. An approach to developing a plausible sub-grid model is proposed.

Continuum model equations for unsteady gas-particle flows in devices such as fluidized beds and circulating fluidized bed risers contain unstable modes whose length scale is of the order of 10 particle diameters. Yet, because of limited computational resources, these flows are routinely simulated by solving the discretized version of continuum models over coarse spatial grids. These simulations resolve the large-scale flow structures but not the finer scale structures. In most industrial applications involving large devices, it is impractical to resolve all the fine-scale structures, and therefore the effects of the unresolved structures must be addressed through suitable subgrid models. Using gas-particle flows in a wide and very tall vertical channel as an example, we have demonstrated in this study that the results obtained in coarse-grid integration of the microscopic equations for gas-particle flows change appreciably if subgrid corrections to account for the effects of unresolved structures are included. The addition of a simple time-averaged subgrid closure for the effective drag coefficient and particle phase viscosity and pressure led to a qualitative change in the simulation results. Our simulations also revealed a lack of separation of time scales between the resolved and unresolved structures. This led us to formulate a simple stochastic subgrid closure for the drag coefficient and investigate its consequence. The addition of a stochastic correction made quantitative, but not qualitative, changes to the simulation results.

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