Single-particle Velocity Distribution Research Articles

Particle–particle drag force in inertial bidisperse gas–particle suspensions

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.

Multiscale kinetic theory for heterogeneous granular and gas-solid flows

It has been recognized that the particle phase stress model derived from classical kinetic theory is valid only when sufficient scale resolution is offered to explicitly resolve the heterogeneous structures during numerical simulations, however, industrial applications prefer to use coarse computational grids where the heterogeneous structures are not explicitly resolved but implicitly modeled. Unfortunately, in this case a kinetic theory for heterogeneous granular and gas–solid flows is not available yet. To this end, an attempt was made to tracking this challenge: The single particle velocity distribution function at the nonequilibrium stationary state with heterogeneous structures was firstly derived by combining the idea of doubly stochastic Poisson processes or superstatistics with the concept of compromise in competition in the EMMS (Energy Minimization Multi-Scale) theory, the standard Chapman-Enskog method was then used to develop the constitutive relations of heterogeneous continuum theory. It was found that (i) seven state variables are needed to quantify the heterogeneous structures as compared to three for homogeneous systems; (ii) the constitutive relations not only include the contribution from microscale particle–particle interactions but also those due to the interactions between mesoscale structures; and (iii) the resultant constitutive relations are much more complex than those of homogeneous systems due to the simultaneous consideration of microscale and mesoscale contributions and the appearance of cross-coupling effects, but they correctly contain the constitutive relations of homogeneous systems as a limiting case. Finally, the theory was coupled with an EMMS drag model to offer a preliminary validation and to provide a unified EMMS-based constitutive relations for heterogeneous gas–solid flows.

LB3D: A parallel implementation of the Lattice-Boltzmann method for simulation of interacting amphiphilic fluids

We introduce the lattice-Boltzmann code LB3D, version 7.1. Building on a parallel program and supporting tools which have enabled research utilising high performance computing resources for nearly two decades, LB3D version 7 provides a subset of the research code functionality as an open source project. Here, we describe the theoretical basis of the algorithm as well as computational aspects of the implementation. The software package is validated against simulations of meso-phases resulting from self-assembly in ternary fluid mixtures comprising immiscible and amphiphilic components such as water–oil–surfactant systems. The impact of the surfactant species on the dynamics of spinodal decomposition are tested and quantitative measurement of the permeability of a body centred cubic (BCC) model porous medium for a simple binary mixture is described. Single-core performance and scaling behaviour of the code are reported for simulations on current supercomputer architectures. Program summaryProgram Title: LB3DProgram Files doi:http://dx.doi.org/10.17632/9g9x2wr8z8.1Licensing provisions: BSD 3-clauseProgramming language: FORTRAN90, Python, CNature of problem: Solution of the hydrodynamics of single phase, binary immiscible and ternary amphiphilic fluids. Simulation of fluid mixtures comprising miscible and immiscible fluid components as well as amphiphilic species on the mesoscopic scale. Observable phenomena include self-organisation of mesoscopic complex fluid phases and fluid transport in porous media.Solution method: Lattice-Boltzmann (lattice-Bhatnagar–Gross–Krook, LBGK) [1, 2, 3] method describing fluid dynamics in terms of the single particle velocity distribution function in a 3-dimensional discrete phase space (D3Q19) [4, 5, 6]. Multiphase interactions are modelled using a phenomenological pseudo-potential approach [7, 8] with amphiphilic interactions utilising an additional dipole field [9, 10]. Solid boundaries are modelled using simple bounce-back boundary conditions and additional pseudo-potential wetting interactions [11].Additional comments including Restrictions and Unusual features: The purpose of the release is the provision of a refactored minimal version of LB3D suitable as a starting point for the integration of additional features building on the parallel computation and IO functionality. [1]S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Oxford University Press, 2001.[2]B. Dünweg, A. Ladd, Lattice Boltzmann simulations of soft matter systems, Adv. Poly. Sci. 221 (2009) 89–166[3]C. K. Aidun, J. R. Clausen, Lattice-Boltzmann Method for Complex Flows, Annual Review of Fluid Mechanics 42 (2010) 439.[4]X. He, L.-S. Luo, A priori derivation of the lattice-Boltzmann equation, Phys. Rev. E 55 (1997) R6333.[5]X. He, L.-S. Luo, Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E 56.[6]Y. H. Qian, D. D’Humiéres, P. Lallemand, Lattice BGK Models for Navier–Stokes Equation, Europhysics Letters 17 (1992) 479.[7]X. Shan, H. Chen, Lattice-Boltzmann model for simulating flows with multiple phases and components, Physical Review E 47 (1993) 1815.[8]X. Shan, G. Doolen, Multicomponent lattice-Boltzmann model with interparticle interaction, Journal of Statistical Physics 81 (1995) 379.[9]H. Chen, B. Boghosian, P.V. Coveney, M. Nekovee, A ternary lattice-Boltzmann model for amphiphilic fluids, Proceedings of the Royal Society of London A 456 (2000) 2043.[10]M. Nekovee, P. V. Coveney, H. Chen, B. M. Boghosian, Lattice-Boltzmann model for interacting amphiphilic fluids, Phys. Rev. E 62 (2000) 8282.[11]N. S. Martys, H. Chen, Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice-Boltzmann method, Phys. Rev. E 53 (1996) 743.

Thermal properties of an impurity immersed in a granular gas of rough hard spheres

We study in this work the dynamics of a granular impurity immersed in a low-density granular gas of identical particles. For description of the kinetics of the granular gas and the impurity particles we use the rough hard sphere collisional model. We take into account the effects of non-conservation of energy upon particle collision. We find an (approximate) analytical solution of the pertinent kinetic equations for the single-particle velocity distribution functions that reproduces reasonably well the properties of translational/rotational energy non-equipartition. We assess the accuracy of the theoretical solution by comparing with computer simulations. For this, we use two independent computer data sets, from molecular dynamics (MD) and from Direct Simulation Monte Carlo method (DSMC). Both approach well, with different degrees, the kinetic theory within a reasonable range of parameter values.

Perfect and ε-perfect simulation methods for the one-dimensional Kac equation

Abstract We review the derivation of the Kac master equation model for random collisions of particles, its relationship to the Poisson process, and existing algorithms for simulating values from the marginal distribution of velocity for a single particle at any given time. We describe and implement a new algorithm that efficiently and more fully leverages properties of the Poisson process, show that it performs at least as well as existing methods, and give empirical evidence that it may perform better at capturing the tails of the single particle velocity distribution. Finally, we derive and implement a novel “ε-perfect sampling” algorithm for the limiting marginal distribution as time goes to infinity. In this case the importance is a proof of concept that has the potential to be expanded to more interesting (DSMC) direct simulation Monte Carlo applications.

Modification of kinetic theory of granular flow for frictional spheres, Part I: Two-fluid model derivation and numerical implementation

We derive a kinetic theory of granular flow (KTGF) for frictional spheres in dense systems, including rotation, sliding and sticking collisions. We use the Chapman–Enskog solution procedure of successive approximations, where the single-particle velocity distribution is assumed to be nearly Maxwellian both translationally and rotationally, as assumed by McCoy et al. (1966). An expression for the first-order particle velocity distribution function is derived, which includes the effects of particle rotation and friction. Using a simple moment method, balance equations for mass, momentum and energy are derived with closure equations for viscosities, and thermal conductivities and collisional energy dissipation rates of angular and translational kinetic energy. Because the internal angular momentum changes are coupled to the flow field, the stress tensor contains anti-symmetric components which are associated with a rotational viscosity. In the resulting closure equations, the rheological properties of the particles are explicitly described in terms of the friction coefficient. The model has been incorporated into our in-house two-fluid model (TFM) code for the modeling of dense gas–solid fluidized beds. For verification, a comparison of the present model in the limit of zero friction with the original (frictionless) KTGF model is carried out. Simulation results of both models agree well. In the next part, simulation results obtained with the new model will be compared with experimental data and discrete particle model (DPM) simulations.

A High-Order Accurate Gas-Kinetic Scheme for One- and Two-Dimensional Flow Simulation

AbstractThis paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the “initial” cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.

Visualization of collisional substructure in granular shock waves

We study shock wave formation and propagation in an experimental vertically driven quasi-two-dimensional granular gas. We measure the moments of the single particle velocity distribution as a function of space and time. The space-time fields of the velocity moments show acoustic waves with a serrated substructure on the scale of a particle diameter. We show that this substructure is the result of collisional transport in which sequential collisions each transport momentum and energy by one particle diameter.

Modified gas-kinetic scheme for shock structures in argon

The gas-kinetic scheme (GKS) is a finite-volume method in which the fluxes are constructed from the single particle velocity distribution function f. The distribution function f is obtained from the linearised Boltzmann equation and is retained only to the Navier-Stokes order in terms of the Chapman-Enskog expansion. Higher order non-equilibrium effects are included in a variable local particle collision time depending on local gradients of the hydrodynamic variables up to second order. The fluxes so constructed possess the non-equilibrium information beyond the linear constitutive relations and Fourier law. The modified gas-kinetic scheme with a local particle collision time λ* depending on the gradients of hydrodynamic variables is used to compute shock structures in argon for Mach numbers between 1.2 and 25.0. We validate our results with existing numerical and experimental ones. Our results show that the modified gas-kinetic scheme is effective and efficient for shock structure calculations.

Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport

A hydrodynamic description for an s -component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

Incomplete Equilibrium in Long-Range Interacting Systems

We use Hamiltonian dynamics to discuss the statistical mechanics of long-lasting quasistationary states particularly relevant for long-range interacting systems. Despite the presence of an anomalous single-particle velocity distribution, we find that the central limit theorem implies the Boltzmann expression in Gibbs' Gamma space. We identify the nonequilibrium submanifold of Gamma space characterizing the anomalous behavior and show that by restricting the Boltzmann-Gibbs approach to this submanifold we obtain the statistical mechanics of the quasistationary states.

Molecular dynamics simulations of vibrated granular gases.

We present molecular dynamics simulations of monodisperse or bidisperse inelastic granular gases driven by vibrating walls, in two dimensions (without gravity). Because of the energy injection at the boundaries, a situation often met experimentally, density and temperature fields display heterogeneous profiles in the direction perpendicular to the walls. A general equation of state for an arbitrary mixture of fluidized inelastic hard spheres is derived and successfully tested against numerical data. Single-particle velocity distribution functions with non-Gaussian features are also obtained, and the influence of various parameters (inelasticity coefficients, density, etc.) are analyzed. The validity of a recently proposed random restitution coefficient model is assessed through the study of projected collisions onto the direction perpendicular to that of energy injection. For the binary mixture, the nonequipartition of translational kinetic energy is studied and compared both to experimental data and to the case of homogeneous energy injection ("stochastic thermostat"). The rescaled velocity distribution functions are found to be very similar for both species.

INELASTIC HARD SPHERES WITH RANDOM RESTITUTION COEFFICIENT: A NEW MODEL FOR HEATED GRANULAR FLUIDS

We consider a vertically shaken granular system interacting elastically with the vibrating boundary, so that the energy injected vertically is transferred to the horizontal degrees of freedom through inter-particle collisions only. This leads to collisions which, once projected onto the horizontal plane, become essentially stochastic and may have an effective restitution coefficient larger than unity. We therefore introduce the model of inelastic hard spheres with random restitution coefficient α (larger or smaller than unity) to describe granular systems heated by vibrations. In the non-equilibrium steady state, we focus in particular on the single particle velocity distribution f(v) in the horizontal plane, and on its deviation from a Maxwellian. We use Molecular Dynamics simulations and Direct Simulation Monte Carlo (DSMC) to show that, depending on the distribution of α, different shapes of f(v) can be obtained, with very different high energy tails. Moreover, the fourth cumulant of the velocity distribution (which quantifies the deviations from Gaussian statistics) is obtained analytically from the Boltzmann equation and successfully tested against the simulations.

Heated granular fluids: The random restitution coefficient approach

We introduce the model of inelastic hard spheres with random restitution coefficient $\alpha$, in order to account for the fact that, in a vertically shaken granular system interacting elastically with the vibrating boundary, the energy injected vertically is transferred to the horizontal degrees of freedom through collisions only, which leads to heating through collisions, i.e. to inelastic horizontal collisions with an effective restitution coefficient that can be larger than 1. This allows the system to reach a non-equilibrium steady state, where we focus in particular on the single particle velocity distribution $f(v)$ in the horizontal plane, and on its deviation from a Maxwellian. Molecular Dynamics simulations and Direct Simulation Monte Carlo (DSMC) show that, depending on the distribution of $\alpha$, different shapes of $f(v)$ can be obtained, with very different high energy tails. Moreover, the fourth cumulant of the velocity distribution quantifying the deviations from Gaussian statistics is obtained analytically from the Boltzmann equation and successfully tested against the simulations.

Velocity distributions in homogeneous granular fluids: the free and the heated case

Non-Gaussian properties (cumulants, high energy tails) of the single particle velocity distribution for homogeneous granular fluids of inelastic hard spheres or disks are studied, based on the Enskog-Boltzmann equation for the unforced and heated case. The latter is in a steady state. The non-Gaussian corrections have small effects on the cooling rate, and on the stationary temperature in the heated case, at all inelasticities. The velocity distribution in the heated steady state exhibits a high energy tail ˜exp(-A c3/2), where c is the velocity scaled by the thermal velocity and A˜ 1/ with e the inelasticity. The results are compared with molecular dynamics simulations, as well as direct Monte Carlo simulations of the Boltzmann equation.

The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials

Following the granular flow kinetic theory of Lun, Savage, Jeffrey and Chepurniy, a moment method is used to obtain the approximate form for the single particle velocity distribution function for the case of smooth, slightly inelastic, uniform spherical particles in which the coefficient of restitutione depends upon the particle impact velocity. Constitutive equations for stress are derived and the theory is applied to the case of a simple shear flow. Theoretical predictions of stresses are compared with experimental results. The effect of the impact velocity dependente is to cause the stresses to vary with the shear rate raised to a power less than two; this is consistent with the experimental observations. On the basis of the present theory and comparisons with experimental data it is concluded that theoretical models which include both surface friction and an impact velocity dependente will lead to improved agreement between the theoretical predictions and the measurements.

Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks

Grad’s method of moments is employed to derive balance laws and constitutive relations for plane flows of a dense gas consisting of identical, rough, inelastic, circular disks. Two temperatures are involved; these are proportional to the kinetic energies associated with fluctuations in translational velocity and spin, respectively. When the single particle velocity distribution function is assumed to be close to a two-temperature Maxwellian, two distinct theories are obtained. One applies when the particles are almost smooth and the collisions between them are nearly elastic; the other applies to nearly elastic particles that, in a collision, almost reverse the relative velocity of their points of contact. I both cases energy is nearly conserved in collisions.

The stress tensor in a granular flow at high shear rates

The stress tensor in a granular shear flow is calculated by supposing that binary collisions between the particles comprising the granular mass are responsible for most of the momentum transport. We assume that the particles are smooth, hard, elastic spheres and express the stress as an integral containing probability distribution functions for the velocities of the particles and for their spatial arrangement. By assuming that the single-particle velocity distribution function is Maxwellian and that the spatial pair distribution function is given by a formula due to Carnahan & Starling, we reduce this integral to one depending upon a single non-dimensional parameter R: the ratio of the characteristic mean shear velocity to the root mean square of the precollisional particle-velocity perturbation. The integral is evaluated asymptotically for R [Gt ] 1 and R [Lt ] 1 and numerically for intermediate values. Good agreement is found between the stresses measured in experiments on dry granular materials and the theoretical predictions when R is given the value 1·7. This case is probably the one for which the present analysis is most appropriate. For moderate and large values of R, the theory predicts both shear and normal stresses that are proportional to the square of the particle diameter and the square of the shear rate, and depend strongly on the solids volume fraction. A provisional comparison is made between the stresses predicted in the limit R → ∞ and the experimental results of Bagnold for shear flow of neutrally buoyant wax spheres suspended in water. The predicted stresses are of the correct order of magnitude and yield the proper variation of stress with concentration. When R [Lt ] 1, the shear stress is linear in the shear rate, and the analysis can be applied to shear flow in a fluidized bed, although such an application is not developed further here.