The gas-particle flow with multiple dispersed solid phases is associated with a complicated multiphase flow dynamics. In this paper, a unified algorithm is proposed for the gas-particle multiphase flow. The gas-kinetic scheme (GKS) is used to simulate the gas phase and the multiscale unified gas-kinetic wave–particle (UGKWP) method is developed for the multiple dispersed solid particle phase. For each disperse solid particle phase, the decomposition of deterministic wave and statistic particle in UGKWP is based on the local cell's Knudsen number. The method for solid particle phase can become the Eulerian fluid approach at the small cell's Knudsen number and the Lagrangian particle approach at the large cell's Knudsen number. This becomes an optimized algorithm for simulating dispersed particle phases with a large variation of Knudsen numbers due to different physical properties of the individual particle phase, such as the particle diameter, material density, etc. The GKS-UGKWP method for gas-particle flow unifies the Eulerian–Eulerian and Eulerian–Lagrangian methods. The particle and wave decompositions for the solid particle phase and their coupled evolution in UGKWP come from the consideration to balance the physical accuracy and numerical efficiency. Two cases of a gas–solid fluidization system, i.e. one circulating fluidized bed and one turbulent fluidized bed, are simulated. The typical flow structures of the fluidized particles are captured, and the time-averaged variables of the flow field agree well with the experimental measurements. In addition, the shock particle–bed interaction is studied by the proposed method, which validates the algorithm for the polydisperse gas-particle system in the highly compressible case, where the dynamic evolution process of the particle cloud is investigated.

The properties of multiphase flows are challenging to measure, and yet effective properties are fundamental to modelling and predicting flow behaviour. The current study is motivated by rheometric measurements of a gas-fluidized bed using a coaxial rheometer in which the fluidization rate and the rotational speed can be varied independently. The measured torque displays a range of rheological states: quasistatic, dense granular flow behaviour at low fluidization rates and low-to-moderate shear rates; turbulent toroidal-vortex flow at high shear rates and moderate-to-high fluidization rates; and viscous-like behaviour with rate-dependent torque at high shear rates and low fluidization or at low shear rates and high fluidization. To understand the solid-like to fluid-like transitions, additional experiments were performed in the same rheometer using single-phase liquid and liquid–solid suspensions. The fluidized bed experiments are modelled as a Bingham plastic for low fluidization rates, and as a shear-thinning Carreau liquid at high fluidization rates. The suspensions are modelled using the Krieger–Dougherty effective viscosity. The results demonstrate that, by using the effective properties, the inverse Bingham number marks the transition from solid-like to viscous-flow behaviour; a modified gap Reynolds number based on the thickness of the shear layer specifies the transition from solid-like to turbulent vortical flow; and a gap Reynolds number distinguishes viscous behaviour from turbulent vortical flow. The results further demonstrate that these different multiphase flows undergo analogous flow transitions at similar Bingham or Reynolds numbers and the corresponding dimensionless torques show comparable scaling in response to annular shear.

In two-fluid simulations of gas–solid fluidised beds, the gaseous phase and the particulate phase are modelled as continuous media. The stress exerted by the particulate medium on the container walls should be modelled to predict accurately the bed dynamics. This paper addresses the modelling of sliding particle–wall contacts in two-fluid simulations, based on reference simulations coupling computational fluid dynamics with the discrete element method (CFD-DEM), in which the individual movement of the particles is tracked. The analysis of the CFD-DEM highlights the complex near-wall behaviour of the particles, which is not reproduced by two-fluid models. Nevertheless, the particle–wall shear stress can be expressed based on the total granular pressure within the first cell off the wall. The model is validated for the two-fluid simulation of a bubbling gas–solid fluidised bed of olefin particles in the dense-fluidisation regime.

This paper is dedicated to the development of particle charge and velocity second-order moment transport equations for monodisperse particles in gas flow with a tribocharging effect. The full transport equations for the particle charge–velocity covariance and the charge variance are derived in the framework of the kinetic theory of granular flow assuming that the electrostatic interaction does not modify the collision dynamics. The collision integrals are solved without presuming the form of the electric part for the particle probability density function. The full second-order transport equation model is tested in a one-dimensional periodic domain. The results show that this model is able to capture more important physical mechanisms that are neglected by simple algebraic models proposed in the past. An in-depth analysis of the transport equations is also performed. This study reveals that, for sufficiently small covariance characteristic destruction time scales, the transient and third-order moments terms can be safely neglected. In addition, two different reduced-order models are proposed: a more general algebraic model that takes into account the variance effect and a semi-algebraic model that only resolves a transport equation for the charge variance coupled with an algebraic model for the covariance. The former could, however, lead to non-physical predictions in many cases, while the latter can be a suitable alternative only for a sufficiently small interparticle collision time. Finally, a simple chart based on test case simulations is proposed to show under which conditions a semi-algebraic model could be considered as a suitable alternative.

Particle-resolved direct numerical simulations are employed to investigate the particle–particle drag force in the bidisperse gas–particle suspensions where the particles are smooth and the single-particle velocity distribution function is Maxwellian. The particle Reynolds number ranges from 6.7 to 123.8, and in this range the particle inertia is high enough that the lubrication force is not essential. It is found that the relation derived by the kinetic theory of granular flow (KTGF) highly overestimates the particle–particle drag force. This is because the pre-collision velocities of colliding particles are not completely uncorrelated with each other. From the time sequence of collision events, it is observed that the particle pair that has just collided will probably collide again after a short time due to the restriction of the particle motion in dense suspensions. Since the post-collision velocities of the first collision cannot relax entirely in such a short time, the relative velocity before the subsequent collision is statistically smaller than the domain-averaged relative velocity. Consequently, the particle–particle drag force is over-predicted when the domain-averaged relative velocity is used. For this reason, this work assumes that the particle–particle drag force is determined by the relative velocity within a local region near large particles. When the local region is set to be the spherical shells centred on the centres of large particles and with an outer radius of a mean free path of small particles, the KTGF-based relation can reasonably predict the particle–particle drag force.

Recently, the existence of so-called granular bubbles and droplets has been demonstrated experimentally. Granular bubbles and droplets are clusters of particles that respectively rise and sink if submerged in an aerated and vibrated bed of another granular material of different size and/or density. However, currently, there is no model that explains the coherent motion of these clusters and predicts the transition between a rising and sinking motion. Here, we propose an analytical model predicting accurately the neutral buoyancy limit of a granular bubble/droplet. This model allows the compilation of a regime map identifying five distinct regimes of granular bubble/droplet motion.

We study the effects of fluid–particle and particle–particle interactions in a three-dimensional monodispersed reactor with unstable fluidization. Simulations were conducted using the immersed boundary method for particle Reynolds numbers of 20–70 with an Archimedes number of 23 600. Two different flow regimes were identified as a function of the particle Reynolds number. For low particle Reynolds numbers ($20 < Re_p < 40$), the porosity is relatively low and the particle dynamics are dominated by interparticle collisions that produce anisotropic particle velocity fluctuations. The relative importance of hydrodynamic effects increases with increasing particle Reynolds number, leading to a minimized anisotropy in the particle velocity fluctuations at an intermediate particle Reynolds number. For high particle Reynolds numbers ($Re_p > 40$), the particle dynamics are dominated by hydrodynamic effects, leading to decreasing and more anisotropic particle velocity fluctuations. A sharp increase in the anisotropy occurs when the particle Reynolds number increases from 40 to 50, corresponding to a transition from a regime in which collision and hydrodynamic effects are equally important (regime 1) to a hydrodynamic-dominated regime (regime 2). The results imply an optimum particle Reynolds number of roughly 40 for the investigated Archimedes number of 23 600 at which mixing in the reactor is expected to peak, which is consistent with reactor studies showing peak performance at a similar particle Reynolds number and with a similar Archimedes number. Results also show that maximum effective collisions are attained at intermediate particle Reynolds number. Future work is required to relate optimum particle Reynolds number to Archimedes number.

We study the effects of Archimedes number $Ar$ and volume fraction $\phi$ in three-dimensional, high concentration and monodispersed particle suspensions. Simulations were conducted using the immersed boundary method with direct forcing for triply periodic cases and with $Ar = 21 - 23\,600$ and $\phi = 0.22 - 0.43$. We find that cluster formation is strongly dependent on the Archimedes number but weakly dependent on the volume fraction for concentrated suspensions. Particles in low $Ar$ cases are characterized by less frequent but long-lived clusters, resulting in higher hindered settling, while high $Ar$ cases consist of more frequent but short-lived clusters, leading to reduced hindered settling. By quantifying the effects of collisions on the hydrodynamic fluctuations, we show that the lifespan of clusters for the low $Ar$ cases is longer because particles are subject to appreciable wake interactions without collisions. On the other hand, clusters for high $Ar$ cases are broken before being subject to appreciable wake interactions due to frequent collisions, leading to a shorter cluster lifespan. The results imply that there exists an $Ar$ for particles in fluidized bed reactors that can reduce short circuiting due to clustering and enhance performance by maximizing flow–particle interactions. This result is consistent with existing reactor studies demonstrating that optimal particle diameters and $Ar$ values correspond to cases with short-lived clusters, although more thorough experimental studies are needed.

Gas-fluidized beds of flexible fibres, which have been rarely studied before, are investigated in this work using the coupled approach of the discrete element method and computational fluid dynamics. In the present numerical method, gas–fibre interaction is modelled by calculating the interaction force for each constituent element in the fibre, and the composition of the interaction forces on the constituent elements generates a resultant hydrodynamic force and a resultant hydrodynamic torque on the fibre. Pressure drops and fibre orientation results from the present simulations with various fibre aspect ratios are in good agreement with previous experimental and simulation results. Some novel results are obtained for the effects of fibre flexibility. Larger hydrodynamic forces on fibres (before the bed is fluidized) and smaller minimum fluidization velocities (MFVs) are observed for more flexible fibre beds due to the smaller porosities, while smaller hydrodynamic forces are obtained for the more flexible fibres when the beds are fluidized with significant fibre motion. By scaling the superficial gas velocity using the MFVs, the data of pressure drop can collapse onto the Ergun correlation for stiff fibres of various aspect ratios; however, the pressure drop curves deviate from the Ergun correlation for very flexible fibres, due to the significant fibre bed expansion before the MFV is reached. The fibre aspect ratio and flexibility both have an impact on the solids mixing rate, and it is found that the solids mixing rates are essentially determined by the ratio of the superficial gas velocity to MFV.

Due to triboelectric charging, the solid phase in gas–particle flows can become electrically charged, inducing an electrical interaction among all the particles in the system. Because this force decays rapidly, many of the current models neglect the contribution of this electrostatic interaction. In this work, an Eulerian particle model for gas–particle flow is proposed in order to take into consideration the electrostatic interaction among the particles. The kinetic theory of granular flows is used to derive the transport equation for the mean particle electric charge. The collision integrals are closed without presuming the form of the electric part for the particle probability density function. A linear model for the mean electric charge conditioned by the instantaneous particle velocity is proposed to account for the charge–velocity correlation. First, a transport equation is written for the charge–velocity correlation. Then, a gradient dispersion model is derived from this equation by using some simplifying hypotheses. The model is tested in a three-dimensional periodic box. The results show that the dispersion phenomenon has two contributions: a kinetic contribution due to the electric charge transport by the random motion of particles and a collisional contribution due to the electric charge transfer during particle–particle collisions. Another phenomenon that contributes to the mean electric charge transport is a triboelectrical current density due to the tribocharging effect by particle–particle collisions in the presence of a global electric field. The corresponding electric charge flux is written as equal to the product of the electric field by a triboconductivity coefficient.

Continuum modelling of dense gas–solid flows strongly depends on the constitutive relations used, including the interphase drag force, particle phase stress and the boundary condition for particle–wall interactions. The lack of scale separation is usually claimed to cause the breakdown of the Navier–Stokes (NS) order continuum theory. In this study, computational fluid dynamics–discrete element method (CFD-DEM) simulations of bubbling, turbulent and fast fluidization of smooth, inelastic spheres were conducted to systematically analyse the valid range of NS theory. An entropy-based criterion $I_{s}$ and Knudsen numbers defined using different characteristic length scales ($Kn_{frac}$, $Kn_{gran}$ and $Kn_{vel}$) were quantified. It was found that (i) except at the centre of bubbles where the solid concentration is quite low, NS theory for discrete particles was valid for bubbling fluidization irrespective of the breakdown criterion. Even at the boundary of the bubbles, values of $I_{s}$ and $Kn$ were small; and (ii) the conclusion depended on the criterion used in turbulent and fast fluidization. If $Kn_{frac}$ was used, NS theory would be generally valid. If $I_{s}$ was chosen, NS theory would be still valid but with lower confidence. However, if $Kn_{vel}$ or $Kn_{gran}$ was selected, NS theory broke down. Because $I_{s}$ includes the non-equilibrium effects caused by the gradient of hydrodynamic fields and particle inelasticity, we may conclude that NS theory was valid for all tested cases. This means that the continuum description of discrete particles is not the main source of the breakdown of NS theory.

In nature and in many industrial applications, the boundary of a channel flow is made of solid particles which form a porous wall, so that there is a mutual influence between the free flow and the subsurface flow developing inside the pores. While the influence of the porous wall on the free flow has been well studied, less well characterized is the subsurface flow, due to the practical difficulties in gathering information in the small spaces given by the pores. It is also not clear whether the subsurface flow can host turbulent events able to contribute significantly to the build-up of forces on the particles, potentially leading to their dislodgement. Through large eddy simulations, we investigate the interface between a free flow and a bed composed of spherical particles in a cubic arrangement. The communication between surface and subsurface flow is in this case enhanced, with relatively strong turbulent events happening also inside the pores. After comparing the simulation results with a previous experimental work from a similar setting, the forces experienced by the boundary particles are analysed. While it remains true that the lift forces are largely dependent on the structure of the free flow, turbulence inside the pores can also give a significant contribution. Pressure inside the pores is weakly correlated to the pressure in the free flow, and strong peaks above and below a particle can happen independently. Ignoring the porous layer below the particle from the computations leads then in this case to an underestimation of the lift forces.

The effects of triboelectricity in a small-scale fluidized bed of polyethylene particles were investigated by imaging the particle layer in the vicinity of the column wall and by measuring the pressure drop across the bed. The average charge on the particles was altered by changing the relative humidity of the gas. A triboelectric charging model coupled with a computational fluid dynamics–discrete element method (CFD-DEM) model was utilized to simulate gas–particle flow in the bed. The electrostatic forces were evaluated based on a particle–particle particle–mesh method, accounting for the surface charge on the insulating walls. It was found that simulations with fixed and uniform charge distribution among the particles capture remarkably well both the agglomeration of the particles on the wall and the associated decrease in the pressure drop across the bed. With a dynamic tribocharging model, the charging rate had to be accelerated to render the computations affordable. Such simulations with an artificial acceleration significantly over-predict charge segregation and the wall becomes rapidly sheeted with a single layer of strongly charged particles.

Particles subjected to flow are known to acquire electrostatic charges through repeated contacts with each other and with other surfaces. These charges alter gas–particle flow behaviour at different scales. In this work, we present a continuum framework for analysing the interplay between tribocharging and the flow of a monodisperse assembly of particles characterized by a single effective work function. Specifically, we have derived the continuum, kinetic theory transport equations for gas–particle flow and local-averaged charge on particles directly from the Boltzmann equation. We also derive the auxiliary conditions to capture tribocharging at bounding conducting walls. The resulting two-fluid model with tribocharging and boundary conditions has then been validated against results from discrete element simulations that have been specially designed to probe specific terms in the models.

The formation of inhomogeneities within fluidized beds, both in terms of the particle configurations and flow structures, have a pronounced effect on the interaction force between the fluid and particles. While recent numerical studies have begun to probe the effects of inhomogeneities on the drag force at the particle scale, the applicability of prior microscale constitutive drag relations is still limited to random, homogeneous distributions of particles. Since an accurate model for the drag force is needed to predict the fluidization behaviour, the current study utilizes the lessons of prior inhomogeneity studies in order to derive a robust drag relation that is both able to account for the effect of inhomogeneities and applicable as a constitutive closure to larger-scale fluidization simulations. Using fully resolved lattice Boltzmann simulations of systems composed of fluid and monodisperse spherical particles in the low-Reynolds-number (Re) regime, the fluid–particle drag force, normalized by the ideal Stokes drag force, is found to significantly decrease, over a range of length scales, as the extent of inhomogeneities increases. The extent of inhomogeneities is found to most effectively be quantified through one of two subgrid-scale quantities: the scalar variance of the particle volume fraction or the drift flux, which is the correlation between the particle volume fraction and slip velocity. Scale-similar models are developed to estimate these two subgrid measures over a wide range of system properties. Two new drag constitutive models are proposed that are not only functions of the particle volume fraction and the Stokes number ($St$), but also dependent on one of these subgrid measures for the extent of inhomogeneities. Based on the observed, appreciable effect of inhomogeneities on drag, these new low-Re drag models represent a significant advancement over prior constitutive relations.

Fluidisation is the process by which the weight of a bed of particles is supported by a gas flow passing through it from below. When fluidised materials flow down an incline, the dynamics of the motion differs from their non-fluidised counterparts because the granular agitation is no longer required to support the weight of the flowing layer. Instead, the weight is borne by the imposed gas flow and this leads to a greatly increased flow mobility. In this paper, a framework is developed to model this two-phase motion by incorporating a kinetic theory description for the particulate stresses generated by the flow. In addition to calculating numerical solutions for fully developed flows, it is shown that for sufficiently thick flows there is often a local balance between the production and dissipation of the granular temperature. This phenomenon permits an asymptotic reduction of the full governing equations and the identification of a simple state in which the volume fraction of the flow is uniform. The results of the model are compared with new experimental measurements of the internal velocity profiles of steady granular flows down slopes. The distance covered with time by unsteady granular flows down slopes and along horizontal surfaces and their shapes are also measured and compared with theoretical predictions developed for flows that are thin relative to their streamwise extent. For the horizontal flows, it was found that resistance from the sidewalls was required in addition to basal resistance to capture accurately the unsteady evolution of the front position and the depth of the current and for situations in which sidewall drag dominates, similarity solutions are found for the experimentally measured motion.

We propose a two-phase two-thin-layer model for fluidized debris flows that takes into account dilatancy effects, based on the closure relation proposed by Roux & Radjai (Physics of Dry Granular Media, 1998, Springer, pp. 229–236). This relation implies that the occurrence of dilation or contraction of the granular material depends on whether the solid volume fraction is respectively higher or lower than a critical value. When dilation occurs, the fluid is sucked into the granular material, the pore pressure decreases and the friction force on the granular phase increases. On the contrary, in the case of contraction, the fluid is expelled from the mixture, the pore pressure increases and the friction force diminishes. To account for this transfer of fluid into and out of the mixture, a two-layer model is proposed with a fluid layer on top of the two-phase mixture layer. Mass and momentum conservation are satisfied for the two phases, and mass and momentum are transferred between the two layers. A thin-layer approximation is used to derive average equations, with accurate asymptotic expansions. Special attention is paid to the drag friction terms that are responsible for the transfer of momentum between the two phases and for the appearance of an excess pore pressure with respect to the hydrostatic pressure. For an appropriate form of dilatancy law we obtain a depth-averaged model with a dissipative energy balance in accordance with the corresponding three-dimensional initial system.

Fluctuations in the gas-phase velocity can contribute significantly to the total gas-phase kinetic energy even in laminar gas–solid flows as shown by Mehrabadi et al. (J. Fluid Mech., vol. 770, 2015, pp. 210–246), and these pseudo-turbulent fluctuations can also enhance heat transfer in gas–solid flow. In this work, the pseudo-turbulent heat flux arising from temperature–velocity covariance, and average fluid-phase conduction during convective heat transfer in a gas–solid flow are quantified and modelled over a wide range of mean slip Reynolds number and solid volume fraction using particle-resolved direct numerical simulations (PR-DNS) of steady flow through a random assembly of fixed isothermal monodisperse spherical particles. A thermal self-similarity condition on the local excess temperature developed by Tenneti et al. (Intl J. Heat Mass Transfer, vol. 58, 2013, pp. 471–479) is used to guarantee thermally fully developed flow. The average gas–solid heat transfer rate for this flow has been reported elsewhere by Sun et al. (Intl J. Heat Mass Transfer, vol. 86, 2015, pp. 898–913). Although the mean velocity field is homogeneous, the mean temperature field in this thermally fully developed flow is inhomogeneous in the streamwise coordinate. An exponential decay model for the average bulk fluid temperature is proposed. The pseudo-turbulent heat flux that is usually neglected in two-fluid models of the average fluid temperature equation is computed using PR-DNS data. It is found that the transport term in the average fluid temperature equation corresponding to the pseudo-turbulent heat flux is significant when compared to the average gas–solid heat transfer over a significant range of solid volume fraction and mean slip Reynolds number that was simulated. For this flow set-up a gradient-diffusion model for the pseudo-turbulent heat flux is found to perform well. The Péclet number dependence of the effective thermal diffusivity implied by this model is explained using a scaling analysis. Axial conduction in the fluid phase, which is often neglected in existing one-dimensional models, is also quantified. As expected, it is found to be important only for low Péclet number flows. Using the exponential decay model for the average bulk fluid temperature, a model for average axial conduction is developed that verifies standard assumptions in the literature. These models can be used in two-fluid simulations of heat transfer in fixed beds. A budget analysis of the mean fluid temperature equation provides insight into the variation of the relative magnitude of the various terms over the parameter space.

In a fluidized bed, the drag force acts to oppose the downward force of gravity on a particle, and thus provides the main mechanism for fluidization. Drag models that are employed in large-scale simulations of fluidized beds are typically based on either fixed-particle beds or the sedimentation of particles in liquids. In low-Reynolds-number ($Re$) systems, these two types of fluidized beds represent the limits of high Stokes number ($St$) and low $St$, respectively. In this work, the fluid–particle drag behaviour of these two regimes is bridged by investigating the effect of $St$ on the drag force in low-$Re$ systems. This study is conducted using fully resolved lattice Boltzmann simulations of a system composed of fluid and monodisperse spherical particles. In these simulations, the particles are free to translate and rotate based on the effects of the surrounding fluid. Through this work, three distinct regimes in the characteristics of the fluid–particle drag force are observed: low, intermediate and high $St$. It is found that, in the low-$Re$ regime, a decrease in $St$ results in a reduction in the fluid–particle drag. Based on the simulation results, a new drag relation is proposed, which is, unlike previous models, dependent on $St$.

A novel analytical equation for the assessment of the accuracy of filters used for the interpolation and differentiation of scattered experimental data is presented. The equation takes into account the statistical nature of the filter output resulting from both the arbitrary positions of the data points and the randomness and noise present in the experimental data. Numerical estimation of the accuracy of the filter, using a Monte Carlo procedure, shows good agreement with the deduced analytical equation. This numerical procedure was also used to determine the accuracy of variance filters aimed at calculating the mean-square fluctuation of experimental data. The combination of the numerical results and analytical equations reveals the exact sources of inaccuracy arising in scattered point filters, namely: (i) the spectral inaccuracy of the weighting function; (ii) the noise or stochastic signal amplification; and (iii) the error arising from the random collocation of points within the filter window. The results also demonstrate that the use of the local mean in the calculation of the quadratic fluctuation leads to smaller estimation errors than the central mean. Finally, all these filters are used and critically evaluated in the framework of the stochastic position, diameter, and velocity of bubbles in a gas-fluidized bed. It is shown that the empirical coefficient of bubble coalescence in the two-dimensional bed tested, $\bar {\lambda } $, is in the range 2.0–2.4 when incorporating only the visible flow of bubbles. Here, the vertical distance over which a bubble survives without coalescing is $\bar {\lambda } {L}_{c} $, where ${L}_{c} $ is the characteristic separation between neighbouring bubbles in the horizontal direction prior to coalescence. It was also seen that the relative mean-square-root fluctuation of both bubble diameter and velocity is more than 50 % at the centre of the bed and remains nearly constant along the height of the bed.