Research on Rapid Granular Flows of Highly Inelastic Particles

Transcription

1 Proc. Natl. Sci. Counc. ROC(A) Vol. 24, No. 5, pp (Invited Review Paper) Research on Rapid Granular Flows of Highly Inelastic Particles CHUEN-SHII CHOU Department of Mechanical Engineering National Pingtung University of Science and Technology Pingtung, Taiwan, R.O.C. (Received June 14, 1999; Accepted September 29, 1999) ABSTRACT We review both kinetic theories and computer simulations for rapid granular flows of highly inelastic particles. Different forms of single particle velocity distribution functions, employed to derive the collisional production of the kinetic theory, are introduced. Three main methods for computer simulation: rigid particle, soft particle, and Monte Carlo models, which have been employed to analyze the flow behavior of granular material, are briefly discussed. In addition, hybrid models are introduced. Comparisons between the theoretical results and the computer simulation results are made. Key Words: rapid granular flows, highly inelastic particles, kinetic theory, computer simulation I. Introduction A granular material is a collection of discrete solid particles immersed in an interstitial fluid; thus, technically, a granular flow is a multiphase process. If the interstitial fluid is a gas, the granular material is said to be dry. When the particles do not adhere to one another, the material is said to be cohesionless. Within the material, these particles may vary in size, shape, and orientation, as well as prolonged or instantaneous contact with their neighbors. Granular material can flow through a hopper, but in contrast to a fluid, it also forms piles on a flat surface. This duality between fluid- and solid-like behavior is the reason why granular media have been of central technological importance since the beginning of civilization: Solids can only be processed in a granular form (Wolf, 1996). Special attention has been drawn to the transition between fluid- and solid-like behavior where surface instabilities like convection cells (Herrmann, 1992; Tahuchi, 1992; Luding et al., 1994a; Jaeger et al., 1994) and heap formation (Clement et al., 1992; Lee, 1994) are observed. Basic physical properties such as: dilatancy (Coniglio and Herrmann, 1996), friction (Poschel and Buchholtz, 1993), cohesion (Hopkins and Louge, 1991), fluidization (Clement and Rajchenbach, 1991; Tsuji et al., 1992; Ichiki and Hayakawa, 1995), particle segregation (Duran et al., 1993; Baumann et al., 1994; Ristow, 1994a, 1994b), dissipative interactions (Luding et al., 1994b), density waves (Lee and Leibig, 1994), fragmentation (Ishii and Matsushita, 1992; Herrmann, 1995), attrition (Ning and Ghadiri, 1996; Ghadiri et al., 1991) etc. make granular materials an interesting research subject. The flow of granular material is an important transport process that occurs in numerous industrial applications and in many geophysical settings. Examples of such granular flows are found in powder technology, grain handling, slurry transport by pipelines, avalanches, and in the transport of sediment and ice in rivers and oceans. Although these examples might appear to be of a very disparate nature, they in fact have many similarities, despite the differences in scale (Hutter and Rajagopal, 1994). In one limiting flow regime, slow deformations of granular assemblies generally do not flow as individual particles but as groups of particles, and the particles within this granular material experience enduring contact with their neighbors. Such flows are often described using methods derived from metal-plasticity theory and fall into the quasi-static (or rate-independent) regime, in which the effects of grain inertia on the transfer of momentum throughout the medium can be neglected. Mroz (1980) and Spencer (1981) reviewed the current state of knowledge in this regard. Bagnold (1954) distinguished three regimes of behavior for the following granular materials. When the pressure forces within an interstitial fluid are balanced by viscous forces, the granular material is in the macro-viscous regime, and the viscous effects of the interstitial fluid on the transfer of momentum are significant. In the 317

2 C.S. Chou grain-inertia (or rate-dependent) regime, particle transport and interactions are the dominant mechanisms by means of which momentum and kinetic energy are transferred. Consequently, the viscous effects of the interstitial fluid may be ignored in describing the flow behavior. The transition regime occurs between the two above-mentioned regimes. In order to determine the appropriate regime for a given flow, Bagnold (1954) introduced a dimensionless ratio of inertia stress to viscous stress, called the Bagnold number. This Bagnold number increases with increasing strain rate and solid fraction and decreases with increasing fluid viscosity. There are two complementary approaches used to model the behavior of granular materials (Richman, 1984). Continuum theories are founded on the assumption that the material properties of interest are continuous functions of position. Balance laws involving these properties are derived, and the physical quantities that are expected to influence the material behavior are identified. General constitutive relations, which depend on these quantities, are then obtained by insuring that neither the principle of material frame indifference nor the entropy inequality are violated. Particulate theories, on the other hand, are based on the mechanics of the interactions between individual particles. In this case, laws governing the entire ensemble emerge only after appropriate statistical averaging. Recent reviews of rapid (or rate-dependent) granular flows were compiled by Savage (1984); continuum theories, microstructural theories, numerical modeling, and rheological test devices and experiments were discussed briefly. Richman (1986) discussed the kinetic theories which govern the flow of granular material and the corresponding boundary conditions. Half (1986) presented a physical and heuristic discussion of the kinetic model for granular material. Jenkins (1987) reviewed recent theories for idealized granular material experiencing rapid deformation. Campbell (1990) discussed various modeling techniques used to describe the motion of the bulk material. Jaeger and Nagel (1992) discussed theories related to fluctuations, noise, and instabilities. Hutter and Rajagopal (1994) reviewed the behavior of material made up of a large assemblage of solid particles under rapid and quasi-static deformations. Ristow (1994b) reviewed recent molecular dynamics simulations of granular materials. Wolf (1996) reviewed the modeling and computer simulation of granular media. Massoudi and Boyle (1991) compiled a review of theories for flows of granular material with applications to fluidized beds and the transport of solids. Nedderman et al. (1982) reviewed theoretical and experimental work with an emphasis on chemical engineering applications. This paper is concerned with rapid granular flows of highly inelastic particles within the grain inertia regime. The transfer of momentum and kinetic energy within these flows results from both collisions between particles and the transport of particles between collisions. Since these flows are highly dissipative, more than a small amount of energy is dissipated in a collision. Here, we discuss the behavior of granular flows of highly inelastic particles observed by means of computer simulations carried out by Hopkins and Shen (1992) for three-dimensional systems of smooth spheres, and by Campbell (1989) for threedimensional systems of frictional spheres. Also discussed are kinetic theories developed by Jenkins and Richman (1988) for two-dimensional systems of smooth disks in dilute and dense limits, by Richman (1989) for threedimensional systems of smooth spheres in a dilute limit, by Richman and Marciniec (1989) and Chou and Richman (1998) for three-dimensional systems of smooth spheres, and by Chou (1995) for three-dimensional systems of smooth spheres in a dense limit. The rest of this paper is divided into three parts. The first part describes the typical formulation of the kinetic theory of granular flows. The second part discusses the kinetic theories developed to describe the behavior of granular flows of highly inelastic particles. The third part briefly introduces three general classes of computer models that have been applied to granular systems and hybrid models. II. Formulation of Kinetic Theory In developing recent theories for the rapid flow of granular materials, workers have exploited the analogy between the fluctuating nature of rapid granular motion and the random molecular motion within a dense gas. The grains are assumed to interact vigorously with their neighbors through energy dissipating binary collisions while the effects of enduring contact and static friction between grains are ignored. By considering a statistical description of the velocities of particles, it is possible to define mean fields, such as density, velocity and granular temperature, and to derive balance equations corresponding to each one. Constitutive quantities can then be identified through Fig. 1. A schematic drawing of a particle s kinetic transport mode and collisional transport mode. 318

3 Research on Granular Flow of Inelastic Particle their appearance in these balance equations, and constitutive relations can be obtained through appropriate statistical averaging. The appearance in these constitutive theories of a rate of dissipation due to inelastic collisions is the most striking departure from the kinetic theory of dense gases. In the rapid granular flow regime, the principal transport mechanisms are kinetic transport by means of particle fluctuation and collisional transport due to particle interaction. A schematic drawing of kinetic transport between collisions and of collisional transport between two particles is shown in Fig Mean Value of a Particle Property Following the kinetic theory for dense gases proposed by Chapman and Cowling (1970), a statistical description of the velocities of particles is adopted. A single particle distribution function f(c,r,t) is defined such that f(c,r,t)dcdr gives the number of particles with velocity c within the range dc whose centers are located at r within dr at time t. The number of particles per unit area (or per unit volume for a three-dimensional system) n(r,t), which is the integral of f(c,r,t) taken over all velocities, is determined using nrt (,) = f(,,) crtdc. If each particle is of mass m and diameter σ, then the density ρ of the flow is mn. The solid fraction is nπσ 3 /6 for an assembly of spheres and nπσ 2 /4 for a disk. The mean value of a particle property ψ(c) weighted average is ψ 1 n and the mean velocity is defined as u c. The fluctuation velocity C is the velocity of a particle relative to the mean, C c u. 2. Balance Equations Referring to a direct analysis of the flow through a differential control volume fixed in space conducted by Reif (1965), the general governing equation for the rate of change of the mean amount <nψ> of particle property ψ(c) within the volume element is nψ = nz t ψ() c f(,,) c r t dc, ψ c ncψ +ℵ( ψ). Here, z is the body force per unit mass, and ℵ(ψ) is the rate of change of ψ per unit volume due to collisions. (1) (2) (3) Equation (3) expresses the fact that at any fixed spatial location, the rate of change of <nψ> is determined by three facts: (1) the external force (e.g., body force acting on each particle to change c and, therefore, ψ(c)), (2) the net flux of particles, bearing the property ψ(c), into the volume element and (3) the collisional production of ψ, which can be decomposed into the sum ℵ ( ψ) = χ( ψ) θ( ψ). Equation (4) was demonstrated by Jenkins and Savage (1983). In this decomposition, χ(ψ) and θ(ψ) are the collisional source term and the collisional flux term, respectively. Both χ(ψ) and θ(ψ) are expressible as integrals over all possible collisions. They are, essentially, statistical averages of the change per collision in the appropriate particle properties weighted by the frequency of each collision. The frequency of each collision is related to the pair distribution function at impact, and the pair distribution function depends on the two particles velocities and positions. With ψ = m in Eq. (3), the balance equations for mass becomes ρ+ ρ u = 0, where the overdot denotes the derivative during travel with the mean flow. Similarly, setting ψ in Eq. (3) equal to mc and eliminating the intermediate results using Eq. (5), the balance equation for momentum becomes ρu = P + ρz, where the negative sign indicates that the compressive stress acts on the granular material; therefore, P ª is called the pressure tensor by most researchers. The pressure tensor P ª measures the flux of momentum within the flow and is the sum for a collisional contribution θ(mc) and a particle kinetic transport contribution <ρcc>. A collisional source term does not appear here because linear momentum is conserved in binary collisions. The balance equation for the energy associated with velocity fluctuations, found by setting ψ = mc 2 2, replacing <C 2 >/3 with the granular temperature T and eliminating the intermediate results using Eqs. (5) and (6), is provided by 3 2 ρ T = Q tr ( P u ) γ. Here, Q_ is the flux of kinetic energy associated with the velocity fluctuations, and γ = χ(mc 2 2) is the rate per unit volume at which energy becomes dissipated due to inelastic collision. In general, the energy flux Q_ is composed of contributions coming from particle collisions as (4) (5) (6) (7) 319

4 C.S. Chou well as from kinetic transport of particles between collisions. Consequently, the total energy flux Q_ is the sum of the collisional part of the energy flux vector θ(mc 2 2) and the kinetic transport part of the energy flux vector <ρcc 2 /2>. 3. Integral Expression of Collisional Production The constitutive quantities θ and χ are, respectively, the collisional flux and collisional source. Both are statistical averages of the change per collision in the appropriate particle property weighted by the frequency of each collision. The frequency of any binary collision is governed by the pair distribution function f (2), defined such that f (2) (c 1, r 1, c 2, r 2 )dc 1 dc 2 dr 1 dr 2 gives the number of pairs of particles whose velocities are in the range dc 1 and dc 2 centered about c 1 and c 2, and whose center positions are within the volume element dr 1 and dr 2 centered about r 1 and r 2. As a special case of the general integral expression first derived by Jenkins and Savage (1983) for the collisional flux and later written in closed form by Jenkins and Richman (1988), θ is determined using mσ 12 / ( 2 θ = δf ) 1 [ c1, r+ ( w ) σk, c2, r 2 12 / ( w + ) σ k] dwk( g k) dkdc1dc2, 2 where α = 2 is for two-dimensional systems (or α = 3 for three-dimensional systems), the change δ in any particle s property is experienced by the first particle, g _ c 1 c 2, and the integration is taken over g _ k 0. As a special case of the general integral expression derived by Jenkins and Savage (1983) for the collisional source in systems of inelastic particles, χ is determined using α α (8) mσ 2 χ = f ( ) ( c1, r σ k, c2, r)( g k) dkdc1dc2, (9) 2 where α = 1 is for two-dimensional systems (or α = 2 for three-dimensional systems), the total change in any particle s property, for example, the second moment of the velocity is experienced by both particles, and the integration is taken over g _ k 0. III. Kinetic Theory for Highly Inelastic Particles 1. Motivation for Treating the Second Moment as the Mean Field The most systematic approach used in formulating theories for dry rapid granular flows, as reviewed, for example, by Savage (1984), Richman (1986), Jenkins (1987) and Campbell (1990), has been to extend the methods employed in developing the kinetic theory of dense and disequilibrated gases. Typically, the effect of dissipative collisions on the statistical description of the velocities of particles has been treated as a small perturbation from the near-maxwellian description of molecular velocities in a disequilibrated dense gas. When the particles are smooth and the amount of energy dissipated in a collision is small, standard arguments for the kinetic theory, slightly modified, may be employed to derive balance laws for the means of the mass density, velocity, and energy of the velocity fluctuations; to determine the velocity distribution function; and to calculate the stress, the flux of fluctuation energy, and its collisional rate of dissipation. This has been done for spheres by Lun et al. (1984), Jenkins and Richman (1985a) and Richman and Chou (1989), and for plane flows of circular disks by Jenkins and Savage (1983), Jenkins and Richman (1985b) and Richman and Chou (1988). However, the kinetic theories noted above are only applied in circumstances where the collisions between grains are nearly energy conserving. This serious restriction derives from the common assumption that the granular temperature, an isotropic measure of the velocity fluctuation (or kinetic energy of fluctuation velocity per unit mass) is sufficient to completely characterize the fluctuation throughout the flow. In the simple kinetic theories, for example, the deviatoric part of the second moment of velocity fluctuation is entirely ignored compared to its isotropic part. As a consequence of this inexact treatment of the full second moment of fluctuation velocity, each of these theories predicts that under all circumstances, the normal stresses in homogeneous shear flows will be equal. When collisions between particles involve more significant dissipation, the numerical simulations of the detailed particle dynamics in a steady, homogeneous shear flow carried out by Walton and Braun (1986a) and Campbell and Gong (1986) for frictional disks, by Walton and Braun (1986b) and Hopkins and Shen (1992) for smooth spheres, and by Campbell (1989) and Lun and Bent (1994) for frictional spheres have demonstrated that, at least in relatively dilute systems, the deviatoric part of the second moment of the velocity fluctuation is of the same order as its isotropic part. In particular, the simulations predict that the difference between the normal stress in the direction of mean velocity and the normal stress in the direction of the velocity gradient increases as the flows become more dilute. This normal stress difference indicates that, in general, the mean squares of the three fluctuation velocity components are unequal, and that an isotropic measure of 320

5 Research on Granular Flow of Inelastic Particle the velocity fluctuation is, therefore, not sufficient to characterize the energetics within highly dissipative flows. On the other hand, Goldreich and Sela (1996) discovered that the normal stress differences are due to Burnett effects, which can be directly obtained from the systematic Chapman-Enskog expansion of the Boltzmann equation for granular fluids. This theory is limited to relatively dilute two-dimensional fluids and nearly elastic collisions between smooth disks. Because of the measurement difficulty, only a few experimental measurements of granular temperature have been made. Ann et al. (1991) and Hsiau and Hunt (1993) employed fiber-optic probe technology to measure velocity fluctuations and calculated the granular temperature in a streamwise direction. Drake (1991) employed highspeed photography and image technology to measure the two-dimensional fluctuations in gravity-driven chute flows. Natarajan et al. (1996) employed image technology to measure the two-dimensional fluctuations in gravitydriven granular flows in a vertical channel. Hsiau and Jang (1998) employed image processing technology and particle tracking methods to analyze the velocity fluctuations of Couette granular flows. All of the above experimental studies demonstrated that the fluctuations were anisotropic. In order to account for this finding, the kinetic theory may be extended by treating the full second moment as a field variable; therefore, the velocity statistical distribution function is expected to depend upon the full second moment. Such extensions of the theory for dilute and dense systems of spheres have been made by, respectively, Goldreich and Tremaine (1978), Araki and Tremaine (1986) and Araki (1988) in studies on the dynamics of planetary rings. 2. Homogeneous Shear Flows of Disks (Jenkins- Richman Model) In order to analyze the steady, plane and homogeneous shear flows of identical, smooth, highly inelastic circular disks, Jenkins and Richman (1988) assumed a single particle distribution to be anisotropic Maxwellian, and the equilibrium radial distribution function g 0 evaluated at the point of contact: ( 2 f ) ( c, r σk, c, r) = g f( c, r σk) f( c, r), (11) where σ is the diameter of the particle and k is the unit vector directed from the center of the first particle to that of the second particle. With Eq. (11), the assumptions of molecular chaos, and with Eq. (10), the single particle velocity distribution function, the collisional fluxes and collisional sources can then be calculated as functions of the mean fields and their spatial derivatives. Jenkins and Richman (1988) substituted the assumptions of molecular chaos, Eq. (11), single particle velocity distribution function, Eq. (10), and the total change in the second moment of the velocity experienced by both particles into the collisional source, Eq. (9). Consequently, the collisional source of the second moment for a plane, homogeneous shear flows of identical, smooth, highly inelastic circular disks was obtained. Using the same method, the collisional flux of the momentum for the same granular flows was obtained by substituting the assumptions of molecular chaos, Eq. (11), the single particle velocity distribution function, Eq. (10), and the change δ in velocity experienced by the first particle into the collisional flux, Eq. (8). In the limits of the dilute and dense steady granular plane, homogeneous shear flows, Jenkins and Richman (1988) combined the constitutive theory, which was obtained using the above procedure, with the balance equation for the full second moment. Consequently, they obtained the approximated analytical solutions that resulted in stresses whose qualitative behavior and magnitudes were in good agreement with the results of numerical simulations carried out by Walton and Braun (1986a). 3. Dilute Flows of Spheres (Richman Model) Richman (1989) extended the work of Jenkins and Richman (1988) and assumed the single particle distribution function to be anisotropic Maxwellian, which is the three-dimensional analog of Eq. (10), n f( c, r, t) = exp 1 1 / C K C 12, 2π 2 (10) n f( c, r, t) = exp 1 1 C K C, 3 8π 2 (12) where c and r are the particle s velocity and position, respectively. The second moment of the fluctuation velocity is defined as K ª <CC>, and its determinant is. In this model, the assumption of molecular chaos is employed to relate the complete pair distribution function at collision to the single particle velocity distribution function. Then, f (2) for a colliding pair can be written as the product of the f(c,r,t) of each disk, evaluated at its center, in order to obtain a general constitutive relationship for the collisional source of the second moment in dilute flows of identical, smooth and highly inelastic spheres. In Eq. (9), the integral expression derived by Jenkins and Savage (1983) for the collisional source of any particle property, Richman (1989) replaced the pair distribution function with the product of the corresponding single particle distribution functions to obtain 321

6 C.S. Chou σ X = m f( c, r) f( c, r)( g k) dkdc dc. 2 (13) 2.0 By substituting the total change in the second moment of the velocity experienced by both particles into the collisional source, Eq. (13), a general constitutive relationship for the collisional source of the second moment in dilute flows of identical, smooth, highly inelastic spheres was obtained, and this was then combined with the constitutive relationship with the balance equation for the full second moment to determine both exact numerical and approximated closed-form solutions for the second moment and pressure tensor in the case of homogeneous shear flows. Most striking are the resulting normal pressure differences, which are predicted by this theory but not by kinetic theories for nearly elastic particles. Figure 2 demonstrates the variations with the dissipation parameter, ε, such that 1 ε is equal to the coefficient of restitution e, of the normal pressure difference P xx P yy and P yy P zz normalized by the average pressure (P xx + P yy + P zz )/3. The normal pressure differences, both P xx P yy and P yy P zz, increase with increasing particle inelasticity, and the normal pressure difference P xx P yy vanishes in the dense limit. 4. Dense Flows of Spheres (Chou Model) Chou (1995) extended the work of Richman (1989) by substituting the assumptions of molecular chaos, Eq. (11), the single particle velocity distribution function, Eq. (12), and the total change in the second moment of the Fig. 2. The variations with the dissipation parameter ε of the normal pressure differences P xx P yy and P yy P zz normalized by the average pressure P = (P xx + P yy + P zz )/3. ε Fig. 3. The variations with the dissipation parameter ε of the dimensionless collisional normal pressure ratio Γ xx Γ zz. velocity experienced by both particles into the collisional source, Eq. (9). Consequently, the collisional source of the second moment for a homogeneous shear flow of identical, smooth, highly inelastic spheres was obtained. Using the same method, the collisional flux of momentum for the same granular flows was obtained by substituting the assumptions of molecular chaos, Eq. (11), the single particle velocity distribution function, Eq. (12), and the change δ in velocity experienced by the first particle into the collisional flux, Eq. (8). In the limit of dense flows, the approximated analytical solutions for the balance e- quation for the second moment of the velocity and the corresponding stress (or pressure) tensor, whose qualitative behavior and magnitudes were in good agreement with the results of computer simulations conducted by Hopkins and Shen (1992), were obtained by combining the constitutive theory with the balance equations for the full second moment. Figure 3 shows the variations with the dissipation parameter ε of the collisional normal pressure ratio Γ xx Γ zz. The pressure ratio Γ xx Γ zz increases dramatically from 1 to as ε increases from 0 to 1. Such a trend was observed by Campbell (1989) for three-dimensional systems of rough spheres, and by Hopkins and Shen (1992) for three-dimensional systems of smooth spheres. In general, both theories by Jenkins and Richman (1988), Richman (1989) and Chou (1995), as well as computer simulations by Walton and Braun (1986a), Campbell and Gong (1986), Walton and Braun (1986b), Campbell (1989) and Hopkins and Shen (1992) demonstrated that the normal pressure differences increase with increasing particle inelasticity. In addition, the difference between ε 322

7 Research on Granular Flow of Inelastic Particle the normal pressure in the direction of the mean velocity and the normal pressure in the direction of the velocity gradient increases as the flows become more dilute. Also, the difference between the normal pressure in the direction of the velocity gradient (or the normal pressure in the direction of the mean velocity) and the normal pressure in the direction perpendicular to the plane of flow increases as the flows become more dense. 5. Homogeneous Shear Flow of Spheres (Richman- Marciniec Model) The theoretical works cited thus far have one feature in common. Each does not attempt to obtain solutions for the balance equations of the second moment, valid for a full range of solid fraction. Richman and Marciniec (1989) solved this problem, which is a steady, homogeneous, granular shear flow of identical, smooth spheres, valid for a full range of solid fraction, by combining the balance equations with the constitutive theory based upon the single particle velocity distribution function determined using Fig. 4. The variations with the solid fraction v of the dimensionless total normal pressure ratio P xx P yy for e = 0.3, 0.6 and 0.9. Also shown are data from the Monte Carlo simulation of Hopkins and Shen (1992) for e = 0.3 (croosses), e = 0.6 (squares), and e = 0.9 (diamonds). f = Kˆ 1 ij qijk f 2 c i c j 3 ci cj c 0, k (14) in which f 0 is the Maxwellian distribution, f 2 = n C 2 T / exp 2T, ( π ) 0 32 (15) ˆK ij is the deviatoric components of K ª, q ijk are functions of position and time equal to the mean values <C i C j C k >/2, and the repeated indices are summed from one to three. Figure 4 shows the variations with solid fraction v of the normal pressure ratio P xx P yy. In comparison with computer simulations by Hopkins and Shen (1992), the magnitudes of the normal pressure ratio P xx P yy obtained by means of theoretical prediction deviate from the simulation prediction as the particles become more inelastic. Figure 5 shows the variations with solid fraction v of the normal pressure ratio P yy P zz. When compared with the results of computer simulations conducted by Hopkins and Shen (1992), neither the qualitative behavior nor the quantitative magnitudes of the normal pressure ratio P yy P zz show good agreement. Besides comparison with the results of computer simulations, based upon a Monte Carlo method, conducted by Hopkins and Shen (1992), we also compared the theoretic results obtained by Richman and Marciniec (1989) with the results of computer simulations conducted by Campbell (1989). In the left hand side panels in Figs. 6 Fig. 5. The variations with the solid fraction v of the dimensionless total normal pressure ratio P yy P zz for e = 0.3, 0.6 and 0.9. Also shown are data from the Monte Carlo simulation of Hopkins and Shen (1992) for e = 0.3 (croosses), e = 0.6 (squares), and e = 0.9 (diamonds). and 7, we have plotted the variations with v for P xx P yy and P yy P zz obtained by Richman and Marciniec (1989) for smooth spheres. In the right hand side panels in each figure, we show the corresponding simulation results obtained by Campbell (1989) for rough spheres. In order to account, in a crude manner, for the additional dissipation due to particle roughness, we compared the results predicted by Richman and Marciniec (1989) for e = 0.6, 0.4, 323

8 C.S. Chou Fig. 6. Variation of P xx P yy with v predicted by Richman and Marciniec (1989) for smooth spheres (left panel); predicted by Campbell (1989) for rough spheres (right panel). Fig. 7. Variation of P yy P zz with v predicted by Richman and Marciniec (1989) for smooth spheres (left panel); predicted by Campbell (1989) for rough spheres (right panel). 0.2 and 0 with Campbell s results (Campbell, 1989) for e = 1.0, 0.8, 0.6 and 0.4. From Fig. 7, the qualitative behavior of the normal pressure ratio P yy P zz is not in good agreement with that obtained through computer simulations. After comparing the theoretic results obtained by Richman and Marciniec (1989) with the results of computer simulations conducted by Campbell (1989) and Hopkins and Shen (1992), we arrive at the conclusion that, strictly speaking, the simple theory developed by Richman and Marciniec (1989) can not precisely predict the flow behavior of granular material in highly dissipative systems. 6. Homogeneous Shear Flow of Spheres (Chou- Richman Model) Chou and Richman (1998) extended the work of Chou (1995) to solve the problem of a steady, homogeneous, granular shear flow of identical, smooth spheres, valid for the full range of the solid fraction by substituting the assumptions of molecular chaos, Eq. (11), the single particle velocity distribution function, Eq. (12), and the total change in the second moment of the velocity experienced by both particles into the collisional source, Eq. (9). Consequently, the collisional source of the second moment for a homogeneous shear flow of identical, smooth, highly inelastic spheres was obtained. Using the same method, the collisional flux of momentum for the same granular flow was obtained by substituting the assumptions of molecular chaos, Eq. (11), the single particle velocity distribution function, Eq. (12), and the change δ in velocity experienced by the first particle into the collisional flux, Eq. (8). The pressure tensor was obtained by summing the collisional flux of the momentum θ(mc) and the kinetic transport contribution <ρcc>. Finally, the constitutive theory was combined with the balance equation for the full second moment so as to determine each component of the second moment and the pressure tensor. Figure 8 shows the variations with the solid fraction v of the normal pressure ratio P xx P yy. Figure 9 shows the variations with the solid fraction v of the normal pressure ratio P yy P zz. If we compare the theoretical results obtained by Chou and Richman (1998) with the results of numerical simulations carried out by Hopkins and Shen (1992), the qualitative behavior and quantitative magnitudes of the theoretical results show good agreement with the results of the numerical simulations. As mentioned above, the results obtained through computer simulations of rapid granular flows were compared with those obtained based on the kinetic theories. The computer simulations will be discussed briefly in the next section. IV. Computer Simulation Computer simulations of rapidly flowing granular material, as reviewed, for example, by Campbell (1986), are currently being used as research tools to investigate the macromechanics and micromechanics of granular flows. Based on well understood models of particle interaction, a mechanical system is set up on a computer on which experiments are performed by statistically averaging the system properties. In a computer simulation, the instantaneous positions and velocities of the particles are known, and these collectively describe the complete state of the system. From this information, literally anything can be determined about the simulated system. Three main methods: rigid particle, soft particle, and Monte Carlo models, as well as hybrid models have been em- 324

9 Research on Granular Flow of Inelastic Particle streaming mode is dominant under dilute (or disperse) packing, and that the collisional mode is dominant under dense packing. The friction coefficient, the ratio of shear to normal forces, has been shown to decrease under dense packing for both the streaming and collisional modes. Normal pressure differences, reflecting anisotropy in the granular temperature, have been observed, and a distinct layered microstructure has been found to develop in highdensity granular flows. Also, the effect of the boundary on granular flows has been discussed by Campbell (1993a) for flat boundaries, and by Campbell (1993b) for roughened boundaries. 2. Soft Particle Model Fig. 8. Variation of P xx P yy with v for e = 0, 0.3, 0.6 and 0.9. Also shown are data from the Monte Carlo simulation of Hopkins and Shen (1992) for e = 0.3 (croosses), e = 0.6 (squares) and e = 0.9 (diamonds). ployed to analyze the flow behavior of granular material. Here, only computer simulations related to rapid granular flows of highly inelastic particles will be briefly discussed. 1. Rigid Particle Model Rigid particle collision models are synonymous with the assumption that all particle interactions are instantaneous collisions (similar assumptions are made in all theoretical calculations), and that only two-particle or binary collisions can be accounted for. Also, the simulation tries to model the long duration contacts as a infinite number of instantaneous collisions. The procedure is as follows: when the simulation is started, the time at which the first collision occurs is computed from the particle trajectories. The positions and velocities of all the particles are updated to that time. The collision results are computed, the time when the next collision will occur is found, and the process is repeated. Computer simulation programs were developed for detailed study of the complete stress tensor by Campbell and Gong (1986) for two-dimensional disks, and by Campbell (1989) for three-dimensional spheres. Two modes of microscopic momentum transport produce the final macroscopic stress tensor: the streaming (or kinetic) mode, by means of which particles carry the momentum as they move through the bulk material, and the collisional modes, by means of which the momentum is transported by interparticle collisions. The results show that the For the soft particle contact model with overlap, a contact between particles, which may be a long duration contact, is modeled by assuming that the colliding particles are connected by a linear spring and a linear dashpot connected in parallel in both the normal and tangential direction to the contact point. The spring provides a restoring force that tries to push the particles apart, and the dashpot provides energy dissipation that makes the collisions inelastic. In addition, a frictional slider with an associated friction coefficient, µ, is added in series with the tangential-direction spring and dashport so that there is no tangential slippage between the particles as long as the tangential force is smaller than µ times the normal force. If this value is exceeded, the particles will slip with a force equal to µ times the normal force. A simulation based upon the soft particle method may be thought of as the simultaneous numerical integration of an equation of motion for each particle in the system. Each motion equation relates the change in a particle s position to its velocity and the forces to which it is subjected. The total force on a particle consists of a constant gravitational acceleration and the forces exerted over the points of contact with its immediate neighbors and the bounding walls of the simulation. Walton and Braun (1986b) for inelastic spheres in uniform shear, and Walton and Braun (1986a) for shearing assemblies of inelastic, frictional disks, investigated whether particles with highly dissipative interactions produce an anisotropic pressure and velocity distribution in the assembly. The calculated ratio of the shear stress to the normal stress varies significantly with the density, and the inclusion of surface friction (and thus particle rotation) decreases the shear stress at low density but increases the shear stress under steady shearing at high densities. Zhang and Campbell (1992) employed the soft particle model to study the effective phase change from fluid behavior to solid behavior. This simulation exhibited the full range of granular flow behavior, from a stagnant solid-like material, through a quasi-static transition zone, 325

10 C.S. Chou to a rapid granular flow. Chou (1994, 1999) employed the kinetic theory and corresponding boundary conditions to study the behavior of gravity-driven granular flows of smooth spheres, which consisted of a bottom region of an amorphous solid-like granular material interacting with a top region of rapidly agitated grains. It was found that the boundary could either supply energy to the flow or absorb energy from the flow. In addition, since both the fluid-like region and boundary could supply energy to the solid-like region at the same time, the flows only had with a flow depth that was large enough to ensure enough energy dissipation through particle collisions in the solid-like region. 3. Monte Carlo Model Technically, the Monte Carlo method derived by Hopkins (1985) and Hopkins and Shen (1992) should fall into the category of rigid particle models since the particle interactions are modeled as instantaneous collisions. It may be a bit extreme to refer to the constituents of the system as particles as they possess neither a position nor an absolute velocity. Instead they only exist as a series of array entries containing a random component of velocity. The velocity is determined through a large number of virtual collisions between randomly chosen particles. The particle velocities that are generated are assumed to be the results of random sampling of the velocity distribution function. As the simulation proceeds, the particles yield additional sampling, and the principle product of the simulation is, effectively, the velocity distribution function that corresponds to the modeled flow. In a statistical sense, the current state of the velocity array in the Monte Carlo simulation is analogous to the current velocities of a like number of particles in a real system. Once the velocity distribution function is known, all of the transport processes can, in principle, be determined from integral relationships derived from the theory of non-equilibrium gases. In addition, the velocity distribution function generated by the Monte Carlo simulation is a numerical solution to the simplified Boltzmann equation. Consequently, the Monte Carlo simulation defines the limits for the accuracy of analytical granular flow theories based on the kinetic theory and the assumption of molecular chaos. Hopkins and Shen (1992) demonstrated that the normal pressure differences increase with increasing particle inelasticity. The difference between the normal pressure in the direction of the mean velocity and the normal pressure in the direction of the velocity gradient increases as the flows become more dilute. In addition, the difference between the normal pressure in the direction of the velocity gradient (or the normal pressure in the direction of the mean velocity) and the normal pressure in the direction perpendicular to the plane of flow increases as the flows become more dense. 4. Hybrid Models In general, the rigid (or hard) particle collision model is best for very dilute systems where the average time step, which is the reciprocal of the collision frequency, is very large. The soft particle contact model with overlap is most suitable for systems with enduring contacts. Hopkins (1987) developed a hard particle model with overlap which combines the hard particle collision model and the overlap strategy, and possibly is the most efficient technique for simulating rapid granular flows. Hopkins and Louge (1991) employed the hard particle model with overlap and a two-dimensional Fourier analysis of the concentration field to describe the formation of a distinct inelastic microstructure in rapid granular flows of disks undergoing simple shear. When the coefficient of restitution is low, the relative velocity in the direction normal to the point of contact is reduced after a collision. In this case, disks tend to remain together after collision. Thus, they tend to cluster at low concentration or create open voids at high concentration. At low concentration, the clusters are regions of lower stress surrounded by more dilute regions of energetic disks that work to break the clusters apart. At high concentration, dilute regions are surrounded by denser aggregates, which tend to fill the voids by exerting collisional pressure upon them. At intermediate concentrations where both kinetic and collisional stresses are low, strong inelastic microstructures develop. Lun and Bent (1994) developed a numerical program to simulate an assembly of inelastic frictional spheres by combining the deterministic approach of molecular dynamics, in which the particle positions and velocities are known at all times, and a sticking-sliding collision model, which is used to emulate binary collisions of real particles. The simulation results show that the stresses are anisotropic and decrease with the decreasing coefficient of restitution and increasing friction coefficient. V. Conclusion We have reviewed recent kinetic theories and computer simulations for rapidly deforming granular flows of highly inelastic particles. Although these theories or numerical simulations may be derived in apparent generality, we should note their limitations. From Figs. 4 7, we find that the theoretical work done by Richman and Marciniec (1989) could not precisely predict the flow behavior of granular material in highly dissipative systems. On the other hand, comparison of the theoretical results obtained by Chou and Richman (1998) as shown in 326

11 Research on Granular Flow of Inelastic Particle Figs. 8 and 9 with the results of numerical simulations carried out by Hopkins and Shen (1992) shows that the qualitative behavior and quantitative magnitudes of the theoretical results are in good agreement with those of the numerical simulations. Consequently, the sophisticated theories based upon the work of Chou and Richman (1998) that apply to a full range of solid fraction and corresponding boundary conditions for rapid inhomogeneous granular shear (or chute) flows of identical, smooth, highly dissipative spheres may be an interesting and important direction for future work. Although the theoretical works cited thus far were able to predict the normal pressure differences, there are no good theoretical models for describing the development of the microstructures: a distinct layered microstructure developing in high-density granular flows as demonstrated by Campbell and Gong (1986) and Campbell (1989); an inelastic microstructure for the formation of particle clusters as demonstrated by Hopkins and Louge (1991). Since either the distinct layered microstructure or the inelastic microstructure affect the collisional stresses, it will be worthwhile to derive a theoretical model for describing the development of microstructures in granular flows. So far, the majority of the theoretical, numerical, and even experimental studies assumed that the granular material was composed of particles of uniform size. However, size reduction in particles results from the repeated interactions between particles during the comminution or transport process, and a wide distribution in particle size is expected. Kinetic theories of homogeneous granular shear flows of identical particles that experience gradual grain size reduction were developed by Richman and Akkoc (1987) for disks, and by Richman and Chou (1989) for spheres. They assumed that, in these flows, repeated collisions between particles result in tiny fractures on their peripheries over time, which effectively reduces their average diameter, and that these particles remain identical to one another as the grains become smaller. In order to describe the flow behavior of inhomogeneous granular flows that experience grain size reduction, an elaborate theoretical model (or computer simulation model) should be developed to account for the particle size distribution. The development of continuum theoretical models for highly dissipative granular flows, which include the effects of particle size and shape, interstitial fluid, and material properties, such as the velocity-dependent coefficient of restitution, will be important. These effects are very likely to be important in many practical flows, but their inclusion in any theoretical framework is a complex as well as difficult task. There is still a great need for further experiments which can provide information about the constitutive equation. New instrumentation and experimental techniques for the measurement of global properties, such as stress and strain rate, as well as microscopic properties, such as granular temperature, would be extremely useful. It is possible that numerical modeling techniques will help to provide the kinds of information that is so difficult to obtain in physical experiments. Acknowledgment The author would like to thank the National Science Council, R.O.C., for its support of this work through grants NSC E and NSC E References Fig. 9. Variation of P yy /P zz with v for e = 0, 0.3, 0.6 and 0.9. Also shown are data from the Monte Carlo simulation of Hopkins and Shen (1992) for e = 0.3 (croosses), e = 0.6 (squares) and e = 0.9 (diamonds). Ahn, H., C. E. Brennen, and R. H. Sabersky (1991) Measurements of velocity, velocity fluctuations, density and stresses in chute flows of granular materials. J. Appl. Mech., 58, Araki, S. (1988) The dynamics of particle disks II: effects of spin degrees of freedom. Icarus., 76, Araki, S. and S. Tremaine (1986) The dynamics of dense particle disks. Icarus., 65, Bagnold, R. A. (1954) Experiment on a gravity-free dispersion of largesolid spheres in a newtonian fluid under shear. Proc. R. Soc. London, A255, Baumann, G., I. M. Janosi, and D. E. Wolf (1994) Particle trajectories and segregation in a two-dimensional rotating drum. Europhys. Lett., 27(3), Campbell, C. S. (1986) Computer simulation of rapid granular flows. 10th US Natl. Congr. Appl. Mech., Austin, TX, U.S.A. Campbell, C. S. (1989) The stress tensor for simple shear flows of a granular material. J. Fluid Mech., 203, Campbell, C. S. (1990) Rapid granular flows. Annu. Rev. Fluid Mech., 22, Campbell, C. S. (1993a) Boundary interactions for two-dimensional granular flows. Part 1. Flat boundaries, asymmetric stresses and couple stresses. J. Fluid Mech., 247,

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