Modelling of gas–solid turbulent channel flow with non-spherical particles with large Stokes numbers

This paper describes a complete framework to predict the behaviour of interacting non-spherical particles with large Stokes numbers in a turbulent flow. A summary of the rigid body dynamics of particles and particle collisions is presented in the framework of Quaternions. A particle-rough wall interaction model to describe the collisions between non-spherical particles and a rough wall is put forward as well. The framework is coupled with a DNS-LES approach to simulate the behaviour of horizontal turbulent channel flow with 5 differently shaped particles: a sphere, two types of ellipsoids, a disc, and a fibre. The drag and lift forces and the torque on the particles are computed from correlations which are derived using true DNS.The simulation results show that non-spherical particles tend to locally maximise the drag force, by aligning their longest axis perpendicular to the local flow direction. This phenomenon is further explained by performing resolved direct numerical simulations of an ellipsoid in a flow. These simulations show that the high pressure region on the acute sides of a non-spherical particle result in a torque if an axis of the non-spherical particle is not aligned with the flow. This torque is only zero if the axis of the particle is perpendicular to the local direction of the flow. Moreover, the particle is most stable when the longest axis is aligned perpendicular to the flow.The alignment of the longest axis of a non-spherical particle perpendicular to the local flow leads to non-spherical particles having a larger average velocity compared to spherical particles with the same equivalent diameter. It is also shown that disc-shaped particles flow in a more steady trajectory compared to elongated particles, such as elongated ellipsoids and fibres. This is related to the magnitude of the pressure gradient on the acute side of the non-spherical particles. Finally, it is shown that the effect of wall roughness affects non-spherical particles differently than spherical particles. Particularly, a collision of a non-spherical particle with a rough wall induces a significant amount of rotational energy, whereas a corresponding collision with a spherical particle results in mostly a change in translational motion.     

A model for tracking inertial particles in a lattice Boltzmann turbulent flow simulation

A computationally inexpensive model for tracking inertial particles through a turbulent flow is presented and applied to the turbulent flow through a square duct having a friction Reynolds number of Re_τ = 300. Prior to introducing particles into the model, the flow is simulated using a lattice Boltzmann computation, which is allowed to evolve until a steady state turbulent flow is achieved. A snapshot of the flow is then stored, and the trajectories of particles are computed through the flow domain under the influence of this static probability field. Although the flow is not computationally evolving during the particle tracking simulation, the local velocity is obtained stochastically from the local probability function, thus allowing the dynamics of the turbulent flow to be resolved from the point of view of the suspended particles. Particle inertia is modeled by using a relaxation parameter based on the particle Stokes number that allows for a particle velocity history to be incorporated during each time step. Wall deposition rates and deposition patterns are obtained and exhibit a high level of agreement with previously obtained DNS computational results and experimental results for a wide range of particle inertia. These results suggest that accurate particle tracking through complex turbulent flows may be feasible given a suitable probability field, such as one obtained from a lattice Boltzmann simulation. This in turn presents a new paradigm for the rapid acquisition of particle transport statistics without the need for concurrent computations of fluid flow evolution.     

Microstructural transition in an ordered set of magnetic spheres immersed in a carrier liquid

This work explores the physics of an ordered set of interacting spheres immersed in a carrier liquid. We present numerical simulations that compute the translational and rotational motion of N interacting spheres based on classical principles of Stokesian dynamics. The spheres are assumed to be made of a magnetizable material, subjected to magnetic and hydrodynamic long range interactions. We explore structure transition using a Lagragian approach of a continuum volume of fluid containing micrometric magnetic particles. We present local maps of particle volume fraction within the calculation Lattice. In this condition, considering the presence and absence of an applied magnetic field, instantaneous snapshots of the local microstructure are taken. Thus, different possibilities of long range interactions are considered. We also complement these results with meaningful statistics of time series obtained through our simulations, such as the correlation time of velocity fluctuations and their self-correlation functions. The data analyzed in the present work sustain the fact that initially ordered neutrally buoyant suspensions have an anisotropic memory-like behavior in the direction of an applied field. It is also observed that particles tend to form small isotropic clusters in the absence of an external field. However, hydrodynamic interactions tend to disperse the particulate phase, avoiding the formation of clusters. This finding suggests that hydrodynamic interactions may play a relevant role on the magnetization dynamics of ferrofluids.     

Influence of particle shape on the microstructure evolution and the mechanical properties of granular materials

In order to study the influence of particle shape on the microstructure evolution and the mechanical properties of granular materials, a two-dimensional DEM analysis of samples with three particle shapes, including circular particles, triangular particles, and elongated particles, is proposed here to simulate the direct shear tests of coarse-grained soils. For the numerical test results, analyses are conducted in terms of particle rotations, fabric evolution, and average path length evolution. A modified Rowe's stress–dilatancy equation is also proposed and successfully fitted onto simulation data.     

Development of Gas-Particle Euler-Euler LES Approach: A Priori Analysis of Particle Sub-Grid Models in Homogeneous Isotropic Turbulence

A new large eddy simulation (LES) approach for particle-laden turbulent flows in the framework of the Eulerian formalism for inertial particle statistical modelling is developed. Local instantaneous Eulerian equations for the particle cloud are first written using the mesoscopic Eulerian formalism (MEF) proposed by Février et al. (J Fluid Mech 533:1–46, 2005), which accounts for the contribution of an uncorrelated velocity component for inertial particles with relaxation time larger than the Kolmogorov time scale. Second, particle LES equations are obtained by volume filtering the mesoscopic Eulerian ones. In such an approach, the particulate flow at larger scales than the filter width is recovered while sub-grid effects need to be modelled. Particle eddy-viscosity, scale similarity and mixed sub-grid stress (SGS) models derived from fluid compressible turbulence SGS models are presented. Evaluation of such models is performed using three sets of particle Lagrangian results computed from discrete particle simulation (DPS) coupled with fluid direct numerical simulation (DNS) of homogeneous isotropic decaying turbulence. The two phase flow regime corresponds to the dilute one where two-way coupling and inter-particle collisions are not considered. The different particle Stokes number (based on Kolmogorov time scale) are initially equal to 1, 2.2 and 5.1. The mesoscopic field properties are analysed in detail by considering the particle velocity probability function (PDF), correlated velocity power spectra and random uncorrelated velocity moments. The mesoscopic fields measured from DPS+DNS are then filtered to obtain large scale fields. A priori evaluation of particle sub-grid stress models gives comparable agreement than for fluid compressible turbulence models. It has been found that the standard Smagorinsky eddy-viscosity model exhibits the smaller correlation coefficients, the scale similarity model shows very good correlation coefficient but strongly underestimates the sub-grid dissipation and the mixed model is on the whole superior to pure eddy-viscosity model.     

Modeling of the random packing of a loose layer of polydisperse spherical particles

A method for calculating the loose packing structure of polydisperse spherical particles with a predetermined size distribution function is proposed. The coordinates of the particle centers in the loose layer are determined as the result of random fall of single spheres on a substrate under the action of gravity, assuming the inelastic collision of the spheres and considering the force of their adhesive interaction, and also assuming that the motion of one sphere on the surface of the other is pure slip. Numerical simulation is used to obtain the pattern of arrangement of polydisperse spherical particles in the loose powder layer, whose porosity depends on the particle size distribution function. The results are compared with experimental data.     

Separation efficiency of liquid–solid undergoing vibration based on breakage of liquid bridge

Vibrating separation is a significant method for liquid–solid separation. A typical example is the vibrating screen to dewater wet granular matter. The properties of granular matter and the vibrating parameters significantly affect the separation efficiency. This study investigates the effect of vibration parameters in separation based on the breakage of large-scale liquid bridge numerically by using a calibrated simulation model. Through analysing the simulation results, the liquid bridge shape and the volume between two sphere particles for various particle sizes and particle distances were studied in the static condition under the effect of gravity. The results show a general reducing trend of liquid bridge volume when the radius ratio of two particles increases, particularly when the ratio increases to 5. Additionally, a set of vibrating motion was applied to the liquid bridge in the simulation model. A group of experiments were also performed to validate the simulation model with vibration. Then, the effect of vibrating peak acceleration, distance between spheres and radius on the separation efficiency which was reflected by the residual water were investigated. It is found that separation efficiency increased obviously with the peak acceleration and the increase slowed down after the peak acceleration over 1 m/s².     

Drag forces of interacting spheres in power-law fluids

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6 to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.     

A three‐dimensional Cartesian cut cell method for incompressible viscous flow with irregular domains

A three‐dimensional Cartesion cut cell method is presented for the simulations of incompressible viscous flows with irregular domains. A new model (referred to as ‘6+N’ model) is proposed to describe arbitrarily shaped cut cells and treat all the cells as polyhedrons with 6+N faces. The finite volume discretization of the Navier–Stokes equation is then implemented by using the ‘6+N’ model to separate the surface flux integrals into two parts, that is, the fluxes through the basic face of the hexahedron and those through the cutting surfaces. The previously proposed Kitta Cube algorithm and volume computer‐aided design platform (J. Comput. Aided. Des. 2005; 37(4): 1509–1520. Doi:10.1016/j.cad.2005.03.006) are adopted to generate cut cells and provide shape data and physical attributes for the numerical analysis. A modified SIMPLE‐based smoothing pressure correction scheme is applied to suppress checkerboard pressure oscillations caused by the collocated arrangement of velocities and pressure. The calculation accuracy of the numerical method expressed by L₁ and L _∞ norm errors is first demonstrated by the simulation of a pipe flow. Then its feasibility, efficiency, and potential in engineering applications are verified by applying it to solve natural convections between concentric spheres and between eccentric spheres. The heat transfer patterns in eccentric spheres are also obtained by using the numerical method. Copyright © 2011 John Wiley & Sons, Ltd.     

Particle flow simulations with homogenised lattice Boltzmann methods

An alternative approach to simulating arbitrarily shaped particles submersed in viscous fluid in two dimensions is proposed, obtained by adapting the velocity parameter of the equilibrium distribution function of a standard lattice Boltzmann method (LBM). Comparisons of exemplifying simulations to results in the literature validate the approach as well as the convergence analysis. Pressure fluctuations occurring in Ladd’s approach are greatly reduced. In comparison with the immersed boundary method, this approach does not require cost intensive interpolations. The parallel efficiency of LBM is retained. An intrinsic momentum transfer is observed during particle–particle collisions. To demonstrate the capabilities of the approach, sedimentation of particles of several shapes is simulated despite omitting an explicit particle collision model.     

Simulation of the rheological properties of suspensions of oblate spheroidal particles in a Newtonian fluid

A simulation algorithm was developed to predict the rheological properties of oblate spheroidal suspensions. The motion of each particle is described by Jeffery’s solution, which is then modified by the interactions between the particles. The interactions are considered to be short range and are described by results from lubrication theory and by approximating locally the spheroid surface by an equivalent spherical surface. The simulation is first tested on a sphere suspension, results are compared with known experimental and numerical data, and good agreement is found. Results are then presented for suspensions of oblate spheroids of two mean aspect ratios of 0.3 and 0.2. Results for the relative viscosity η _r, normal stress differences N ₁ and N ₂ are reported and compared with the few available results on oblate particle suspensions in a hydrodynamic regime. Evolution of the orientation of the particles is also observed, and a clear alignment with the flow is found to occur after a transient period. A change of sign of N ₁ from negative to positive as the particle concentration is increased is observed. This phenomenon is more significant as the particle aspect ratio increases. It is believed to arise from a change in the suspension microstructure as the particle alignment increases.     

On the approximate zeroth and first‐order consistency in the presence of 2‐D irregular boundaries in SPH obtained by the virtual boundary particle methods

In this paper, a new method to impose 2‐D solid wall boundary conditions in smoothed particle hydrodynamics is presented. The wall is discretised by means of a set of virtual particles and is simulated by a local point symmetry approach. The extension of a previously published modified virtual boundary particle (MVBP) method guarantees that arbitrarily complex domains can be readily discretised guaranteeing approximate zeroth and first‐order consistency. To achieve this, three important new modifications are introduced: (i) the complete support is ensured not only for particles within one smoothing length distance, h, from the boundary but also for particles located at a distance greater than h but still within the support of the kernel; (ii) for a non‐uniform fluid particle distribution, the fictitious particles are generated with a uniform stencil (unlike the previous algorithms) that can maintain a uniform shear stress on a particle‐moving parallel to the wall in a steady flow; and (iii) the particle properties (density, mass and velocity) are defined using a local point of symmetry to satisfy the hydrostatic conditions and the Cauchy boundary condition for pressure. The extended MVBP model is demonstrated for cases including hydrostatic conditions for still water in a tank with a wedge and for curved boundaries, where significant improved behaviour is obtained in comparison with the conventional boundary techniques. Finally, the capability of the numerical scheme to simulate a dam break simulation is also shown. Copyright © 2015 John Wiley & Sons, Ltd.     

Turbulent thermal diffusion in a multi-fan turbulence generator with imposed mean temperature gradient

We studied experimentally the effect of turbulent thermal diffusion in a multi-fan turbulence generator which produces a nearly homogeneous and isotropic flow with a small mean velocity. Using particle image velocimetry and image processing techniques, we showed that in a turbulent flow with an imposed mean vertical temperature gradient (stably stratified flow) particles accumulate in the regions with the mean temperature minimum. These experiments detected the effect of turbulent thermal diffusion in a multi-fan turbulence generator for relatively high Reynolds numbers. The experimental results are in compliance with the results of the previous experimental studies of turbulent thermal diffusion in oscillating grid turbulence (Buchholz et al. 2004; Eidelman et al. 2004). We demonstrated that the turbulent thermal diffusion is an universal phenomenon. It occurs independently of the method of turbulence generation, and the qualitative behavior of particle spatial distribution in these very different turbulent flows is similar. Competition between turbulent fluxes caused by turbulent thermal diffusion and turbulent diffusion determines the formation of particle inhomogeneities.     

CFD-DEM simulation of the hole cleaning process in a deviated well drilling: The effects of particle shape

We investigate the effect of particle shape on the transportation mechanism in well-drilling using a three-dimensional model that couples computational fluid dynamics (CFD) with the discrete element method (DEM). This numerical method allows us to incorporate the fluid–particle interactions (drag force, contact force, Saffman lift force, Magnus lift force, buoyancy force) using momentum exchange and the non-Newtonian behavior of the fluid. The interactions of particle−particle, particle−wall, and particle−drill pipe are taken into account with the Hertz–Mindlin model. We compare the transport of spheres with non-spherical particles (non-smooth sphere, disc, and cubic) constructed via the multi-sphere method for a range of fluid inlet velocities and drill pipe inclination angles. The simulations are carried out for laboratory-scale drilling configurations. Our results demonstrate good agreement with published experimental data. We evaluate the fluid–particle flow patterns, the particle velocities, and the particle concentration profiles. The results reveal that particle sphericity plays a major role in the fluid–solid interaction. The traditional assumption of an ideal spherical particle may cause inaccurate results.     

Galilean invariance in the lattice-Boltzmann method and its effect on the calculation of rheological properties in suspensions

The lattice-Boltzmann method (LBM) provides an efficient simulation technique for the study of particle suspensions. These simulations provide an important tool in elucidating the effect of suspended particles on the rheology of suspensions. The most common solid–fluid boundary condition used in the LBM is the bounce-back operation, and as such, the errors introduced by this operation to the dynamics of the particles and the calculation of relevant rheological quantities must be quantified. This paper derives the Galilean invariant term in the standard bounce-back operation and shows the effect of this error on the calculation of particle dynamics and stresslet. In particular, an error is found in the calculation of normal stresses that may be significant in magnitude compared with typical values found in suspensions of rigid spherical particles. A correction is proposed, and simulation results are shown that verify the original assessment and show the reduction of error when using the proposed correction.     

Simulating orthokinetic heterocoagulation and cluster growth in destabilizing suspensions

Using direct numerical simulation, we investigate the coagulation behavior of non-Brownian colloidal particles as exemplified by Al₂O₃ particles. This yields the so-called capture efficiency, for which we give an analytical expression, as well as other time-dependent variables such as the cluster growth rate. Instead of neglecting or strongly approximating the hydrodynamic interactions between particles, we include hydrodynamic and non-hydrodynamic interactions in a Stokesian dynamics approach and a comprehensive modeling of the interparticle forces. The resulting parallelized simulation framework enables us to investigate the dynamics of polydisperse particle systems composed of several hundred particles at the same high level of modeling we used for a close investigation of the coagulation behavior of two unequal particles in shear flow. Appropriate cluster detection yields all the information about large destabilizing systems, which is needed for models used in flow-sheet simulations. After non-dimensionalization, the results can be generalized and applied to other systems tending to secondary coagulation.     

Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

Gas–solid momentum transfer is a fundamental problem that is characterized by the dependence of normalized average fluid–particle force F on solid volume fraction ? and the Reynolds number based on the mean slip velocity Re_m. In this work we report particle-resolved direct numerical simulation (DNS) results of interphase momentum transfer in flow past fixed random assemblies of monodisperse spheres with finite fluid inertia using a continuum Navier–Stokes solver. This solver is based on a new formulation we refer to as the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). The principal advantage of this formulation is that the fluid stress at the particle surface is calculated directly from the flow solution (velocity and pressure fields), which when integrated over the surfaces of all particles yields the average fluid–particle force. We demonstrate that PUReIBM is a consistent numerical method to study gas–solid flow because it results in a force density on particle surfaces that is reconcilable with the averaged two-fluid theory. The numerical convergence and accuracy of PUReIBM are established through a comprehensive suite of validation tests. The normalized average fluid–particle force F is obtained as a function of solid volume fraction ? (0.1 ? ? ? 0.5) and mean flow Reynolds number Re_m (0.01 ? Re_m ? 300) for random assemblies of monodisperse spheres. These results extend previously reported results of  and  to a wider range of ?, Re_m, and are more accurate than those reported by Beetstra et al. (2007). Differences between the drag values obtained from PUReIBM and the drag correlation of Beetstra et al. (2007) are as high as 30% for Re_m in the range 100–300. We take advantage of PUReIBM’s ability to directly calculate the relative contributions of pressure and viscous stress to the total fluid–particle force, which is useful in developing drag correlations. Using a scaling argument, Hill et al. (2001b) proposed that the viscous contribution is independent of Re_m but the pressure contribution is linear in Re_m (for Re_m > 50). However, from PUReIBM simulations we find that the viscous contribution is not independent of the mean flow Reynolds number, although the pressure contribution does indeed vary linearly with Re_m in accord with the analysis of Hill et al. (2001b). An improved correlation for F in terms of ? and Re_m is proposed that corrects the existing correlations in Re_m range 100–300. Since this drag correlation has been inferred from simulations of fixed particle assemblies, it does not include the effect of mobility of the particles. However, the fixed-bed simulation approach is a good approximation for high Stokes number particles, which are encountered in most gas–solid flows. This improved drag correlation can be used in CFD simulations of fluidized beds that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior.     

The Dilute Rheology of Swimming Suspensions: A Simple Kinetic Model

A simple kinetic model is presented for the shear rheology of a dilute suspension of particles swimming at low Reynolds number. If interparticle hydrodynamic interactions are neglected, the configuration of the suspension is characterized by the particle orientation distribution, which satisfies a Fokker-Planck equation including the effects of the external shear flow, rotary diffusion, and particle tumbling. The orientation distribution then determines the leading-order term in the particle extra stress in the suspension, which can be evaluated based on the classic theory of Hinch and Leal (J Fluid Mech 52(4):683–712, 1972), and involves an additional contribution arising from the permanent force dipole exerted by the particles as they propel themselves through the fluid. Numerical solutions of the steady-state Fokker-Planck equation were obtained using a spectral method, and results are reported for the shear viscosity and normal stress difference coefficients in terms of flow strength, rotary diffusivity, and correlation time for tumbling. It is found that the rheology is characterized by much stronger normal stress differences than for passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.