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The Enskog kinetic theory is used as a starting point to model a suspension of solid particles in a viscous gas. Unlike previous efforts for similar suspensions, the gas-phase contribution to the instantaneous particle acceleration appearing in the Enskog equation is modelled using a Langevin equation, which can be applied to a wide parameter space (e.g. high Reynolds number). Attention here is limited to low Reynolds number flow, however, in order to assess the influence of the gas phase on the constitutive relations, which was assumed to be negligible in a previous analytical treatment. The Chapman–Enskog method is used to derive the constitutive relations needed for the conservation of mass, momentum and granular energy. The results indicate that the Langevin model for instantaneous gas–solid force matches the form of the previous analytical treatment, indicating the promise of this method for regions of the parameter space outside of those attainable by analytical methods (e.g. higher Reynolds number). The results also indicate that the effect of the gas phase on the constitutive relations for the solid-phase shear viscosity and Dufour coefficient is non-negligible, particularly in relatively dilute systems. Moreover, unlike their granular (no gas phase) counterparts, the shear viscosity in gas–solid systems is found to be zero in the dilute limit and the Dufour coefficient is found to be non-zero in the elastic limit.

Gas-phase velocity fluctuations due to mean slip velocity between the gas and solid phases are quantified using particle-resolved direct numerical simulation. These fluctuations are termed pseudo-turbulent because they arise from the interaction of particles with the mean slip even in ‘laminar’ gas–solid flows. The contribution of turbulent and pseudo-turbulent fluctuations to the level of gas-phase velocity fluctuations is quantified in initially ‘laminar’ and turbulent flow past fixed random particle assemblies of monodisperse spheres. The pseudo-turbulent kinetic energy $k^{(f)}$ in steady flow is then characterized as a function of solid volume fraction ${\it\phi}$ and the Reynolds number based on the mean slip velocity $\mathit{Re}_{m}$. Anisotropy in the Reynolds stress is quantified by decomposing it into isotropic and deviatoric parts, and its dependence on ${\it\phi}$ and $Re_{m}$ is explained. An algebraic stress model is proposed that captures the dependence of the Reynolds stress on ${\it\phi}$ and $Re_{m}$. Gas-phase velocity fluctuations in freely evolving suspensions undergoing elastic and inelastic particle collisions are also quantified. The flow corresponds to homogeneous gas–solid systems, with high solid-to-gas density ratio and particle diameter greater than dissipative length scales. It is found that for the parameter values considered here, the level of pseudo-turbulence differs by only 15 % from the values for equivalent fixed beds. The principle of conservation of interphase turbulent kinetic energy transfer is validated by quantifying the interphase transfer terms in the evolution equations of kinetic energy for the gas-phase and solid-phase fluctuating velocity. It is found that the collisional dissipation is negligible compared with the viscous dissipation for the cases considered in this study where the freely evolving suspensions attain a steady state starting from an initial condition where the particles are at rest.

Gas-solid flows in nature and industrial applications are characterized by multiscale and nonlinear interactions that manifest as rich flow physics and pose unique modeling challenges. In this article, we review particle-resolved direct numerical simulation (PR-DNS) of the microscale governing equations for understanding gas-solid flow physics and obtaining quantitative information for model development. A clear connection between a microscale realization and meso/macroscale representation is necessary for PR-DNS to be used effectively for model development at the meso- and macroscale. Furthermore, the design of PR-DNS must address the computational challenges of parameterizing models in a high-dimensional parameter space and obtaining accurate statistics of flow properties from a finite number of realizations at acceptable grid resolution. This review also summarizes selected recent insights into the physics of momentum, kinetic energy, and heat transfer in gas-solid flows obtained from PR-DNS. Promising future applications of PR-DNS include the study of the effect of number fluctuations on hydrodynamics, instabilities in gas-solid flow, and wall-bounded flows.

The acceleration of an inertial particle in a gas-solid flow arises from the particle's interaction with the gas and from interparticle interactions such as collisions. Analytical treatments to derive a particle acceleration model are difficult outside the Stokes flow regime, but for moderate Reynolds numbers (based on the mean slip velocity between gas and particles) particle-resolved direct numerical simulation (PR-DNS) is a viable tool for model development. In this study, PR-DNS of freely-evolving gas-solid suspensions are performed using the particle-resolved uncontaminated-fluid reconcilable immersed-boundary method (PUReIBM) that has been extensively validated in previous studies. Analysis of the particle velocity variance (granular temperature) equation in statistically homogeneous gas-solid flow shows that a straightforward extension of a class of mean particle acceleration models (drag laws) to their corresponding instantaneous versions, by replacing the mean particle velocity with the instantaneous particle velocity, predicts a granular temperature that decays to zero, which is at variance with the steady particle granular temperature that is obtained from PR-DNS. Fluctuations in particle velocity and particle acceleration (and their correlation) are important because the particle acceleration-velocity covariance governs the evolution of the particle velocity variance (characterized by the particle granular temperature), which plays an important role in the prediction of the core annular structure in riser flows. The acceleration-velocity covariance arising from hydrodynamic forces can be decomposed into source and dissipation terms that appear in the granular temperature evolution equation, and these have already been quantified in the Stokes flow regime using a combination of kinetic theory closure and multipole expansion simulations. From PR-DNS data we show that the fluctuations in the particle acceleration that are aligned with fluctuations in the particle velocity give rise to a source term in the granular temperature evolution equation. This approach is used to quantify the hydrodynamic source and dissipation terms of granular temperature from PR-DNS results for freely-evolving gas-solid suspensions that are performed over a wide range of solid volume fraction (0.1 φ 0.4), Reynolds number based on the slip velocity between the solid and the fluid phase (10 Re m 100) and solid-to-fluid density ratio (100 ρ p /ρ f 2000). The straightforward extension of drag law models does not give rise to any source in the granular temperature due to hydrodynamic effects. This motivates the development of † Email address for correspondence: shankar@iastate.edu ‡ Present address: CD-adapco, Lebanon, NH 03766, USA. 696S. Tenneti, M. Mehrabadi and S. Subramaniam better Lagrangian particle acceleration models that can be used in Lagrangian-Eulerian formulations of gas-solid flow. It is found that a Langevin equation for the increment in the particle velocity reproduces PR-DNS results for the stationary particle velo...