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.

New constitutive relations for filtered two‐fluid models (TFM) of gas‐particle flows are obtained by systematically filtering results generated through highly resolved simulations of a kinetic theory‐based TFM. It was found in our earlier studies that the residual correlations appearing in the filtered TFM equations depended principally on the filter size and filtered particle volume fraction. Closer inspection of a large amount of computational data gathered in this study reveals an additional, systematic dependence of the correction to the drag coefficient on the filtered slip velocity, which serves as a marker for the extent of subfilter‐scale inhomogeneity. Furthermore, the residual correlations for the momentum fluxes in the gas and particle phases arising from the subfilter‐scale fluctuations are found to be modeled nicely using constitutive relations of the form used in large‐eddy simulations of single‐phase turbulent flows. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3265–3275, 2013

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The tendency of granular materials in rapid shear ow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear flow of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

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