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Hydrodynamic Modeling of Granular Materials
Abstract
Granular materials are composed by macroscopic solid particles where interactions are characterized by dissipative collisions. This kind of material is present in our daily life, for example, sand, coal, nuts, rocks etc. In this thesis we study different phenomenon presents in rapid granular flows by means of hydrodynamic simulations of the Navier-Stokes granular equations. In Chapter 2, we introduce the Navier-Stokes equations and two different approaches for the transport coefficients for fluidized granular materials. One of the consequences of the inelastic interaction between the particles is that the flow developed is supersonic and sharp-profiles of the hydrodynamic fields arise. To deal with this problem, we use a high-order shock-capturing method. In Chapters 3 and 4 is shown the model used to solve the hydrodynamic equations and its successful parallelization. Due to the inelasticity, granular gases form dense clusters of particles. This phenomenon is studied for a force-free granular gas in Chapter 5. The kinetics of the cluster formation is analyzed and compared with the presently accepted mode-enslaving mechanism. We observed that the mode-enslaving theory cannot explain the process of cluster formation. Nevertheless, a direct correlation between the appearance of the shock waves and formations of clusters is observed. Another effect specific to granular systems is the collapse. Considering gravity, a granular gas will become at rest if there is no energy supply. In Chapter 6, we analyze the evolution of the granular gas throughout different stages before the collapse, where regions of supersonic and subsonic dynamics are observed. In the supersonic regions, the system develops shocks followed by sharp profiles of the temperature and the density fields moving upwards. In the last stage of the sedimentation, the energy decay has been studied and compared with previous studies. In agreement with all of them, we confirmed that the entire system collapse simultaneously. Similarly as is described for liquids, a vertical vibrated granular layer develops characteristic patterns for certain intervals of frequencies and amplitudes of oscillation. This phenomenon called Faraday instability is an interesting example of granular collective behavior. In Chapter 7, we study numerically the formation of Faraday waves using two approaches of the transport coefficients described in Chapter 2 and comparing with event-driven molecular dynamics simulations. We observed that the two approaches work quite well, although there is a discrepancy related with the expression of the heat flux.
... Hydrodynamic models and discrete element methods. The granular media are sometimes modeled by hydrodynamic equations of state; see [41][42][43][44][45][46]. The space-average hydrodynamic equations of motion for the granular medium can be derived by the Chapman -Enskog method [47] and written in the form [51]: ...
... where p is the hydrostatic pressure; is the shear viscosity, is the bulk viscosity; is the thermal conductivity coefficient; is the additional parameter specifying dependence of the heat flux on the number density gradient; the last expression for the specific mechanical energy loss is known as the Goldstein -Shapiro equation [44]. Equations (1.9), (1.10) form the closed form system of hyperbolic equations, solved either by the finite difference algorithms; see [48,49]; or the discrete element methods (DEM); see [45][46][47][48]. Sometimes, hydrodynamic models are used within the meshfree smooth particle hydrodynamics (SPH) method; in this regard see [54][55][56]. ...
A comparative study of the vertical seismic barriers intended for protecting from Rayleigh seismic waves and filled with (1) homogeneous linearly elastic materials and (2) granular metamaterials, is done by the finite-element (FE) modeling. The granular metamaterial obeys the Mohr–Coulomb plasticity model with the associated flow rule, low cohesion value, and small internal friction and dilation angles. The performed numerical analysis reveals a principal ability achieving much higher reduction ratios for magnitudes of accelerations along with much longer shadow zones behind the barrier for barriers filled with metamaterials in comparison with the purely elastic homogeneous barriers.
We examine the hydrodynamics of a granular gas using numerical simulation. We demonstrate the appearance of shearing and clustering instabilities predicted by linear stability analysis, and show that their appearance is directly related to the inelasticity of collisions in the material. We discuss the rate at which these instabilities arise and the manner in which clusters grow and merge.
The dynamics of a one-dimensional granular medium has a finite time singularity if the number of particles in the medium is greater than a certain critical value. The singularity (‘‘inelastic collapse’’) occurs when a group of particles collides infinitely often in a finite time so that the separations and relative velocities vanish. To avoid the finite time singularity, a double limit in which the coefficient of restitution r approaches 1 and the number of particles N becomes large, but is always below the critical number needed to trigger collapse, is considered. Specifically, r↠1 with N∼(1−r)−1. This procedure is called the ‘‘quasielastic’’ limit. Using a combination of direct simulation and kinetic theory, it is shown that a bimodal velocity distribution develops from random initial conditions. The bimodal distribution is the basis for a ‘‘two-stream’’ continuum model in which each stream represents one of the velocity modes. This two-stream model qualitatively explains some of the unusual phenomena seen in the simulations, such as the growth of large-scale instabilities in a medium that is excited with statistically homogeneous initial conditions. These instabilities can be either direct or oscillatory, depending on the domain size, and their finite-amplitude development results in the formation of clusters of particles.
In this paper we investigate a two-dimensional dilute granular flow around an immersed cylinder using discrete element computer simulations. Simulation measurements of the drag force acting on the cylinder, Fd, are expressed in terms of a dimensionless drag coefficient, Cd=Fd/[12rhonu∞U∞2(D+d)], where rho is the upstream particle mass density, nu∞ is the upstream solid fraction, U∞ is the upstream velocity, and (D+d) is the sum of the cylinder diameter, D, and surrounding particle diameter, d. The drag coefficient increases rapidly with decreasing Mach number for subsonic Mach numbers, but remains insensitive to Mach number for supersonic values. The drag coefficient is also a strong function of the flow Knudsen number, with the drag coefficient increasing with increasing Knudsen number and approaching an asymptotic value for very large Knudsen numbers. The drag coefficient decreases with decreasing normal coefficient of restitution and is relatively insensitive to the friction coefficient. Bow shock structures and expansion fans are also observed in the simulations and are compared to similar structures observed in compressible gas flows.
The dynamics of a one-dimensional gas of inelastic point particles is investigated with emphasis on the inelastic collapse phenomenon. In particular, it is shown that, as the coefficient of restitution, r, approaches one, inelastic collapse requires an infinitely increasing number of particles. A simple problem in the kinetic theory of granular media, viz., the cooling of a uniformly excited gas confined between inelastic walls, is analyzed. It is suggested that inelastic collapse and the associated velocity correlations occur in higher dimensions and that they are related to the development of an 'inelastic microstructures' observed in two-dimensional simulations of granular flow.
The density distribution arising at the nonlinear stage of gravitational instability is similar to intermittency phenomena in acoustic turbulence. Initially small-amplitude density fluctuations of Gaussian type transform into thin dense pancakes, filaments, and compact clumps of matter. It is perhaps surprising that the motion of self-gravitating matter in the expanding universe is like that of noninteracting matter moving by inertia. A similar process is the distribution of light reflected or refracted from rippled water. The similarity of gravitational instability to acoustic turbulence is highlighted by the fact that late nonlinear stages of density perturbation growth can be described by the Burgers equation, which is well known in the theory of turbulence. The phenomena discussed in this article are closely related to the problem of the formation of large-scale structure of the universe, which is also discussed.
Molecular dynamics simulations of the inelastic hard sphere model for granular media have been done to study the heat conduction between two parallel plates. The results show that Fourier's law is not valid and a new term proportional to the density gradient must be added to compute the heat flux. The new transport coefficient associated with the density gradient dependence has been measured vanishing in the case of elastic collisions.
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.
We show, in two different experiments on stationary flow past an obstacle, that several features such as Mach cones and shock wave detachment usually observed in supersonic molecular fluids under extreme conditions are also observed for granular fluids. By pursuing this analogy, we measure the speed of sound in these experiments and find it in agreement with predictions from granular kinetic theories. Surprisingly, and in spite of this agreement, measured velocity distributions are far from being Gaussian and display algebraic tails.
Shocks and blasts can be readily obtained in granular flows be they dense or dilute. Here, by examining the propagation of a blast shock in a dilute granular flow, we show that such a front is unstable with respect to transverse variations of the density of grains. This instability has a well-defined wavelength which depends on the density of the medium and has an amplitude which grows as an exponential of the distance traveled. These features can be understood using a simple model for the shock front, including dissipation which is inherent to granular flows. While this instability bears much resemblance to that anticipated in gases, it is distinct and has special features we discuss here.
From a first principles theory for the behavior of smooth granular
systems, we derive the form for the instantaneous dissipative force
acting between two grains. The present model, which is based on the
classical harmonic crystal, reproduces the dependence of the kinetic
energy dissipation on the grain deformation obtained by models that
assume a viscoelastic behavior (without permanent plastic deformations)
during the collision.
Shock wave propagation arising from steady one-dimensional motion of a piston in a granular gas composed of inelastically colliding particles is treated theoretically. A self-similar long-time solution is obtained in the strong shock wave approximation for all values of the upstream gas volumetric concentration v 0 . Closed form expressions for the long-time shock wave speed and the granular pressure on the piston are obtained. These quantities are shown to be independent of the particle collisional properties, provided their impacts are accompanied by kinetic energy losses. The shock wave speed of such non-conservative gases is shown to be less than that for molecular gases by a factor of about 2.
The effect of particle kinetic energy dissipation is to form a stagnant layer (solid block), on the surface of the moving piston, with density equal to the maximal packing density, v M . The thickness of this densely packed layer increases indefinitely with time. The layer is separated from the shock front by a fluidized region of agitated (chaotically moving) particles. The (long-time, constant) thickness of this layer, as well as the kinetic energy (granular temperature) distribution within it are calculated for various values of particle restitution and surface roughness coefficients. The asymptotic cases of dilute ( v 0 [Lt ] 1) and dense ( v 0 ∼ v M ) granular gases are treated analytically, using the corresponding expressions for the equilibrium radial distribution functions and the pertinent equations of state. The thickness of the fluidized region is shown to be independent of the piston velocity.
The calculated results are discussed in relation to the problem of vibrofluidized granular layers, wherein shock and expansion waves were registered. The average granular kinetic energy in the fluidized region behind the shock front calculated here compared favourably with that measured and calculated (Goldshtein et al. 1995) for vibrofluidized layers of spherical granules.
The hydrodynamic equations for a gas of hard spheres with dissipative dynamics are derived from the Boltzmann equation. The heat and momentum fluxes are calculated to Navier-Stokes order and the transport coefficients are determined as explicit functions of the coefficient of restitution. The dispersion relations for the corresponding linearized equations are obtained and the stability of this linear description is discussed. This requires consideration of the linear Burnett contributions to the energy balance equation from the energy sink term. Finally, it is shown how these results can be imbedded in simpler kinetic model equations with the potential for analysis of more complex states. [S1063-651X(98)15709-4].
Steady simple shear flows of smooth inelastic spheres are studied by means of a model kinetic equation and also of a direct Monte Carlo simulation method. Both approaches are based on the Enskog equation and provide for each other a test of consistency. The dependence of the granular temperature and of the shear and normal stresses on both the solid fraction and the coefficient of restitution is analysed. Quite a good agreement is found between theory and simulations in all cases. Also, simplified expressions based on the analytical solution of the model for small dissipation are shown to describe fairly well the simulation results even for not small inelasticity. A critical comparison with previous theories is carried out.
Recent theories for rapid deformations of granular materials have attempted to exploit the similarities between the grains of deforming granular mass and the molecules of a disequilibrated gas. Methods from the kinetic theory may then be used to determine, for example, the form of the balance laws for the means of density, velocity, and energy and to calculate specific forms for the mean fluxes of momentum and energy and, in these dissipative systems, the mean rate at which energy is lost in collisions.
The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.
An experimental study of the behavior of one bead bouncing repeatedly off a static flat horizontal surface is presented. We
observe that the number of bounces made by the bead is finite. When the duration between two successive bounces becomes of
the order of the impact duration, the bead no longer bounces but oscillates on the elastically deformed surface before coming
to rest. This transition is explained with a modified Hertz interaction law in which gravity is taken into account during
the interaction. For each bounce, measurement of both the duration of collision and the restitution coefficient have been
done. The effective restitution coefficient is essentially constant and close to 1 during almost all bounces before decreasing
to zero when the impact velocity vanishes. This is due to an interplay between gravity and viscoelastic dissipation.
PACS. 46.10.+z Mechanics of discrete systems - 83.70.Fn Granular solids
An intriguing phenomenon displayed by granular flows and predicted by kinetic-theory-based models is the instability known as particle "clustering," which refers to the tendency of dissipative grains to form transient, loose regions of relatively high concentration. In this work, we assess a modified-Sonine approximation recently proposed [Garzó, Santos, and Montanero, Physica A 376, 94 (2007)] for a granular gas via an examination of system stability. In particular, we determine the critical length scale associated with the onset of two types of instabilities--vortices and clusters--via stability analyses of the Navier-Stokes-order hydrodynamic equations by using the expressions of the transport coefficients obtained from both the standard and the modified-Sonine approximations. We examine the impact of both Sonine approximations over a range of solids fraction φ<0.2 for small restitution coefficients e = 0.25-0.4, where the standard and modified theories exhibit discrepancies. The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations, of which a small percentage were not considered due to inelastic collapse. Results show excellent quantitative agreement between MD and the modified-Sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a high-dissipation qualitative mismatch between the standard theory and MD for the instability that forms more readily. Furthermore, the evolution of cluster size is briefly examined via MD, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters.
The goal of this note is to provide most of the technical details involved in
the application of the Chapman-Enskog method to solve the revised Enskog
equation to Navier-Stokes order. Explicit expressions for the transport
coefficients and the cooling rate are obtained in terms of the coefficient of
restitution and the solid volume fraction by using a new Sonine approach. This
new approach consists of replacing, where appropriate in the Chapman-Enskog
procedure, the local equilibrium distribution (used in the standard first
Sonine approximation) by the homogeneous cooling state distribution. The
calculations are performed in an arbitrary number of dimensions.
A numerical study is presented to analyze the thermal mechanisms of unsteady,
supersonic granular flow, by means of hydrodynamic simulations of the
Navier-Stokes granular equations. For this purpose a paradigmatic problem in
granular dynamics such as the Faraday instability is selected. Two different
approaches for the Navier-Stokes transport coefficients for granular materials
are considered, namely the traditional Jenkins-Richman theory for moderately
dense quasi-elastic grains, and the improved Garz\'o-Dufty-Lutsko theory for
arbitrary inelasticity, which we also present here. Both solutions are compared
with event-driven simulations of the same system under the same conditions, by
analyzing the density, the temperature and the velocity field. Important
differences are found between the two approaches leading to interesting
implications. In particular, the heat transfer mechanism coupled to the density
gradient which is a distinctive feature of inelastic granular gases, is
responsible for a major discrepancy in the temperature field and hence in the
diffusion mechanisms.
Kinetic Theory of Granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics — a theory which has mainly evolved over the last decade. The book is aimed at readers from the advanced undergraduate level upwards and leads on to the present state of research. Throughout, special emphasis is put on a microscopically consistent description of pairwise particle collisions which leads to an impact-velocity-dependent coefficient of restitution. The description of the many-particle system, based on the Boltzmann equation, starts with the derivation of the velocity distribution function, followed by the investigation of self-diffusion and Brownian motion. Using hydrodynamical methods, transport processes and self-organized structure formation are studied. An appendix gives a brief introduction to event-driven molecular dynamics. A second appendix describes a novel mathematical technique for derivation of kinetic properties, which allows for the application of computer algebra. The text is self-contained, requiring no mathematical or physical knowledge beyond that of standard physics undergraduate level. The material is adequate for a one-semester course and contains chapter summaries as well as exercises with detailed solutions. The molecular dynamics and computer-algebra programs can be downloaded from a companion web page.
A granular gas in gravity heated from below develops a certain stationary density profile. When the heating is switched off, the granular gas collapses. We investigate the process of sedimentation using computational hydrodynamics, based on the Jenkins-Richman theory, and find that the process is significantly more complex than generally acknowledged. In particular, during its evolution, the system passes several stages which reveal distinct spatial regions of inertial (supersonic) and diffusive (subsonic) dynamics. During the supersonic stages, characterized by Mach>1, the system develops supersonic shocks which are followed by a steep front of the hydrodynamic fields of temperature and density, traveling upward.
We investigate a phase separation instability that occurs in a system of nearly elastically colliding hard spheres driven by a thermal wall. If the aspect ratio of the confining box exceeds a threshold value, granular hydrostatics predict phase separation: the formation of a high-density region coexisting with a low-density region along the wall that is opposite to the thermal wall. Event-driven molecular dynamics simulations confirm this prediction. The theoretical bifurcation curve agrees with the simulations quantitatively well below and well above the threshold. However, in a wide region of aspect ratios around the threshold, the system is dominated by fluctuations, and the hydrostatic theory breaks down. Two possible scenarios of the origin of the giant fluctuations are discussed.
The recent avalanche of research activity in the field of granular matter has yielded much progress. The use of state-of-the-art (and other) computational and experimental methods has led to the discovery of new states and patterns and enabled detailed tests of theories and models. The application of statistical mechanical methods and phenomenology has contributed to the understanding of the microscopic a nd macroscopic properties of granular systems. Some previously open problems seem to be solved. Fluidized granular systems (rapid granular flows), recently referred to as granular gases, are often modeled by hydrodynamic equations of motion, some of which are based on systematic expansions applied to the pertinent Boltzmann equation. The undeniable success of granular hydrodynamics is somewhat surprising in view of the lack of scale separation in these systems and the neglect of certain correlations in most derivations of the hydrodynamic equations. Microstructures have been recognized as key features of granular gases; explanations for their existence have been proposed, and many of their properties elucidated. Granular-gas multistability can often be traced back to microstructure dynamics. In spite of these and other impressive advances, this field still poses serious challenges.
A freely cooling granular gas in a gravitational field undergoes a collapse to a multicontact state in a finite time. Previous theoretical [D. Volfson et al., Phys. Rev. E 73, 061305 (2006)] and experimental work [R. Son et al., Phys. Rev. E 78, 041302 (2008)] have obtained contradictory results about the rate of energy loss before the gravitational collapse. Here we use a molecular dynamics simulation in an attempt to recreate the experimental and theoretical results to resolve the discrepancy. We are able to nearly match the experimental results, and find that to reproduce the power law predicted in the theory we need a nearly elastic system with a constant coefficient of restitution greater than 0.993. For the more realistic velocity-dependent coefficient of restitution, there does not appear to be a power-law decay and the transition from granular gas to granular solid is smooth, making it difficult to define a time of collapse.
The clustering of granular assemblies is studied under the influences of (i) a temperature-dependent coefficient of restitution and in (ii) a central gravitational force field. Our stability analyses of the constitutive equations as well as numerical experiments show that in case (i) clusters are still formed even though collisions become more elastic as the temperature decreases. In case (ii) the clusters appear as a transient phenomenon during the establishment of a quasiequilibrium. These transient patterns rotate driven by the shear and then they “melt” away with elapsed time.
Previous work has indicated that inelastic grains undergoing homogeneous cooling may be unstable, giving rise to the formation of velocity vortices, which may also lead to particle clustering. In this effort, molecular dynamics (MD) simulations are performed over a wide parameter space to determine the critical system size demarcating the stable and unstable regions. Specifically, a system of monodisperse, frictionless, inelastic hard spheres is simulated for restitution coefficients e >= 0.6 and solids fractions J
One of the possible phases of a sheared system of inelastically colliding rigid smooth disks is one in which relatively dense strips aligned at 45° to the streamwise direction are interspersed among similarly aligned dilute strips. The dense strips may have secondary microstructures in the form of elongated clusters. The latter are formed by an instability, following which they are convected, stretched, and rotated by the shear field. This process causes cluster–cluster collisions, a result of which is the partial destruction of the colliding clusters, followed by the emergence of new clusters. In addition, it is demonstrated that clustering dynamics can be responsible for hysteresis and multistability in granular systems. The studies presented in this paper involve molecular dynamics simulations complemented by theoretical analysis.
Shock wave evolution arising during a steady one-dimensional motion of a piston in a granular gas, composed of inelastically colliding particles is treated by a computational fluid dynamics (CFD) method. It is shown that the flow reaches an asymptotic stationary (final) stage after large evolution time. At this stage particle kinetic energy dissipation leads to formation of two regions within the upstream flow: a fluidized region adjacent to the shock front and a ``solid'' region adjacent to the piston. In the latter the density is close to the maximum packing density and the kinetic energy of chaotic granular motion is almost zero. The shock wave velocity, mass, and the granular kinetic energy in the fluidized region are found to be constant values, whereas the mass of the solid region grows linearly with time. All properties calculated for the final evolution stage are in excellent agreement with the predictions of the asymptotic solution of the problem obtained earlier. We extend the existing hydrodynamic model of granular gas consisting of hard spheres to provide a plausible description of the static pressure and the speed of sound in the solid block. This is achieved by introducing a cut density rhocut such that dissipation is allowed only if the hydrodynamic density rho is smaller than rhocut. The computational model reveals that all the above quantities tend to their limiting values in a nonmonotonic manner. In particular, the fluidized mass reaches its maximum when the density at the piston reaches its maximal value. The kinetic energy reaches its maximum earlier than does the fluidized mass. The maximal values of the fluidized mass and the kinetic energy are shown to exceed the comparable long-time limiting values with the difference amounting to about 1.5-fold factor. The calculated results are discussed in relation to several granular flows, characterized by formation of fluidized and compacted regions.
1. The beautiful series of forms assumed by sand, filings, or other grains, when lying upon vibrating plates, discovered and developed by Chladni, are so striking as to be recalled to the minds of those who have seen them by the slightest reference. They indicate the quiescent parts of the plates, and visibly figure out what are called the nodal lines. 2. Afterwards M. Chladni observed that shavings from the hairs of the exciting violin bow did not proceed to the nodal lines, but were gathered together on those parts of the plate the most violently agitated, i. e. at the centres of oscillation. Thus when a square plate of glass held horizontally was nipped above and below at the centre, and made to vibrate by the application of a violin bow to the middle of one edge, so as to produce the lowest possible sound, sand sprinkled on the plate assumed the form of a diagonal cross; but the light shavings were gathered together at those parts towards the middle of the four portions where the vibrations were most powerful and the excursions of the plate greatest.
Large scale, three-dimensional computer simulations were performed to investigate flow dynamics of monosized, viscoelastic, spherical solid particles past a stationary wedge located in the middle of an inclined duct. At low flow rates of solid particles, a continuous flow was observed similar to that excited by steadily and rapidly adding particles to the top of a heap. However, at high flow rates, a totally different situation arises, where a flow with a different nature was established in the duct. In this case, the granular flow within the upper part of the duct accelerates adjacent to the pointed tip of the wedge, and develops into vast masses of solid particles thrust and folded over each other. This is similar to the supercritical nappes in an open-channel flow of a liquid. In addition, some experimental evidences have been presented that suggest the existence of supercritical nappes in flowing grains over a stationary wedge within an inclined duct at high flow rates.
Granular materials are ubiquitous in the world around us. They have properties that are different from those commonly associated with either solids, liquids, or gases. In this review the authors select some of the special properties of granular materials and describe recent research developments.[S0034-6861(96)00204-8]
The dissipative nature of the particle interactions is responsible for an inherent lack of scale separation in granular systems which is not related to the typical grain/container size ratios. It is demonstrated that rapid granular flows are typically supersonic, shear rates in these systems are nearly always ``large,'' the mean free times are comparable with the macroscopic time scale(s), and the mean free paths can be of macroscopic dimensions, the latter indicating nonlocality. Additional physical and computational implications are discussed.
Computer simulations of two-dimensional rapid granular flows of uniform smooth inelastic disks under simple shear reveal a dynamic microstructure characterized by the local, spatially anisotropic agglomeration of disks. A spectral analysis of the concentration field suggests that the formation of this inelastic microstructure is correlated with the magnitude of the total stresses in the flow. The simulations confirm the theoretical results of Jenkins and Richman (J. Fluid Mech. {bold 192}, 313 (1988)) for the kinetic stresses in the dilute limit and for the collisional stresses in the dense limit, when the size of the periodic domain used in the simulations is a small multiple of the disk diameter. However, the kinetic and, to a lesser extent, collisional stresses both increase significantly with the size of the periodic domain, thus departing from the predictions of the theory that assumes spatial homogeneity and isotropy.
Direct Monte Carlo simulation is used to investigate the stability of a dilute freely evolving granular gas of hard disks. The boundary between stability and instability in the plane (alpha,L), where alpha is the restitution coefficient and L the size of the system, has been delineated. Instability is associated with the buildup of spatial correlations, which describes the formation of velocity vortices in the system. The simulation results are compared with theoretical predictions presented recently, and a good agreement is found.
The goal of this article is to provide a somewhat critical introduction to the concept of granular temperature and some of its applications. A brief history of the concept is followed by a presentation of several of its major properties and implications thereof. A number of misconceptions concerning this concept is presented and discussed. Certain open questions are described and some recent developments are briefly outlined.
The Enskog theory for a dense fluid of rigid disks is developed. The collisional contribution, which dominates in liquids, is derived and added to the kinetic term, which describes a dilute gas. Expressions for shear and bulk viscosity and for thermal conductivity are obtained. The initial correlations are evaluated via the autocorrelation function approach, and the exponentially decaying functions which result are related to the Enskog theory.
Both large-scale three-dimensional molecular-dynamics-type simulation and integration of hydrodynamic equations relevant to dense granular flows were performed to investigate the formation of shock waves in granular flows past a stationary wedge in a rectangular duct. No signature of supersonic flows was found at grain speeds as small as 0.1 m/s. Upon increasing the speed of the grains, the flow became supersonic where complex solid structures were observed to form in both upstream and downstream flow regions. In addition, the compression shock formed on the upper edges of the channel adjacent to the wedge and the expansion fan formed adjacent to the rear face of the wedge. Both the compression shock and expansion fan appear to be analogous to those in supersonic gases with some fundamental differences arising due to the inelastic collisions between particles in granular media. © 2004 American Institute of Physics.
Collisional motion of a granular material composed of rough inelastic spheres is analysed on the basis of the kinetic Boltzmann–Enskog equation. The Chapman–Enskog method for gas kinetic theory is modified to derive the Euler-like hydrodynamic equations for a system of moving spheres, possessing constant roughness and inelasticity. The solution is obtained by employing a general isotropic expression for the singlet distribution function, dependent upon the spatial gradients of averaged hydrodynamic properties. This solution form is shown to be appropriate for description of rapid shearless motions of granular materials, in particular vibrofluidized regimes induced by external vibrations.
The existence of the hydrodynamic state of evolution of a granular medium, where the Euler-like equations are valid, is delineated in terms of the particle roughness, β, and restitution, e , coefficients. For perfectly elastic spheres this state is shown to exist for all values of particle roughness, i.e. − 1≤β≤1. However, for inelastically colliding granules the hydrodynamic state exists only when the particle restitution coefficient exceeds a certain value e m (β)< 1.
In contrast with the previous results obtained by approximate moment methods, the partition of the random-motion kinetic energy of inelastic rough particles between rotational and translational modes is shown to be strongly affected by the particle restitution coefficient. The effect of increasing inelasticity of particle collisions is to redistribute the kinetic energy of their random motion in favour of the rotational mode. This is shown to significantly affect the energy partition law, with respect to the one prevailing in a gas composed of perfectly elastic spheres of arbitrary roughness. In particular, the translational specific heat of a gas composed of inelastically colliding ( e = 0.6) granules differs from its value for elastic particles by as much as 55 %.
It is shown that the hydrodynamic Euler-like equation, describing the transport and evolution of the kinetic energy of particle random motion, contains energy sink terms of two types (both, however, stemming from the non-conservative nature of particle collisions) : (i) the term describing energy losses in incompressibly flowing gas; (ii) the terms accounting for kinetic energy loss (or gain) associated with the work of pressure forces, leading to gas compression (or expansion). The approximate moment methods are shown to yield the Euler-like energy equation with an incorrect energy sink term of type (ii), associated with the ‘dense gas effect’. Another sink term of the same type, but associated with the energy relaxation process occurring within compressed granular gases, was overlooked in all previous studies.
The speed of sound waves propagating in a granular gas is analysed in the limits of low and high granular gas densities. It is shown that the particle collisional properties strongly affect the speed of sound in dense granular media. This dependence is manifested via the kinetic energy sink terms arising from gas compression. Omission of the latter terms in the evaluation of the speed of sound results in an error, which in the dense granular gas limit is shown to amount to a several-fold factor.
The Chapman-Enskog expansion is generalized in order to derive constitutive relations for flows of inelastically colliding spheres in three dimensions - to Burnett order. To this end, the pertinent (nonlinear) Boltzmann equation is perturbatively solved by performing a (double) expansion in the Knudsen number and the degree of inelasticity. One of the results is that the normal stress differences and the 'temperature anisotropy', characterizing granular fluids, are Burnett effects. The constitutive relations derived in this work differ, both qualitatively and quantitatively, from those obtained in previous studies. In particular, the Navier-Stokes (order) terms have a different dependence on the degree of inelasticity and the number density than in previously derived constitutive relations; for instance, the expression for the heat flux contains a term which is proportional to epsilon del logn, where epsilon is a measure of the degree of inelasticity and n denotes the number density. This contribution to the heat flux is of zeroth order in the density; a similar term, i.e. one that is proportional to epsilon del n, has been previously obtained by using the Enskog correction but this term is O(n) and it vanishes in the Boltzmann limit. These discrepancies are resolved by an analysis of the Chapman-Enskog and Grad expansions, pertaining to granular flows, which reveals that the quasi-microscopic rate of decay of the temperature, which has not been taken into account heretofore, provides an important scale that affects the constitutive relations. Some (minor) quantitative differences between our results and previous ones exist as well. These are due to the fact that we take into account an isotropic correction to the leading Maxwellian distribution, which has not been considered before, and also because we consider the full dependence of the corrections to the Maxwellian distribution on the (fluctuating) speed.
We performed molecular dynamics simulations to investigate the clustering
instability of a freely cooling dilute gas of inelastically colliding disks in
a quasi-one-dimensional setting. We observe that, as the gas cools, the shear
stress becomes negligibly small, and the gas flows by inertia only. Finite-time
singularities, intrinsic in such a flow, are arrested only when close-packed
clusters are formed. We observe that the late-time dynamics of this system are
describable by the Burgers equation with vanishing viscosity, and predict the
long-time coarsening behavior.
On the basis of elastic theory the restitution coefficient e was derived as a function of the elastic constants, radii, masses and colliding velocities of two spheres by taking the visco-elastic property into consideration. For a coefficient close to one, the value of (1-e) was found to be proportional to (velocity)1/5. This result was compared with a treatment based on the plastic property of solids.
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/Ö{e}\sqrt{\epsilon} with ) the inelasticity. The results are compared with molecular dynamics simulations, as well as direct Monte Carlo simulations of the Boltzmann equation.
Results of a numerical study of the dynamics of a collection of disks colliding inelastically in a periodic two-dimensional enclosure are presented. The properties of this system, which is perhaps the simplest model for rapidly flowing granular materials, are markedly different from those known for atomic or moleclar gases, in which collisions are of elastic nature. The most prominent feature characterizing granular systems, even in the idealized situation in which no external forcing exists and the initial condition is statistically homogeneous, is their inherent instability to inhomogeneous fluctuations. Granular gases are thus generically nonuniform, a fact that suggests extreme caution in pursuing direct analogies with molecular gases. We find that once an inhomogeneous state sets in, the velocity distribution functions differ from the classical Maxwell-Boltzmann distribution. Other characteristics of the system are different from their counterparts in molecular systems as well. For a given value of the coefficient of restitution,e, a granular system forms clusters of typical separationL
0l/(1-e
2)1/2, wherel is the mean free path in the corresponding homogeneous system. Most of the fluctuating kinetic energy then resides in the relatively dilute regions that surround the clusters. Systems whose linear dimensions are less thenL
0 do not give rise to clusters; still they are inhomogeneous, the scale of the corresponding inhomogeneity being the longest wavelength allowed by the system's size. The present article is devoted to a detailed numerical study of the above-mentioned clustering phenomenon in two dimensions and in the absence of external forcing. A theoretical framework explaining this phenomenon is outlined. Some general implications as well as practical ramifications are discussed.
Large scale, three dimensional computer simulations of monosized, viscoelastic, spherical glass particles flowing in an inclined
duct were performed using a phenomenological model based on the modified Kelvin–Maxwell model. The particle flow rate in the
model duct was regulated using a stationary wedge located in the middle of the duct. At low flow rates of glass particles,
a continuous flow was observed similar to that excited by steadily and rapidly adding glass particles to the top of a heap.
However, at high flow rates, a totally different situation arises where a flow with a different nature was established in
the duct. The situation was found to be analogous to the case of a supersonic gas flow in a duct, where a curved-bow shock
was observed to have formed on the upper edges of the duct adjacent to the wedge. In addition, in supersonic granular flows
the viscous and conductance effects spread the shock changes over a finite shock layer.
We investigate numerically the interaction of a stream of granular particles with a resting obstacle in two dimensions. For the case of high stream velocity we find that the force acting on the obstacle is proportional to the square of the stream velocity, the density and the obstacle size. This behaviour is equivalent to that of non-interacting hard spheres. For low stream velocity a gap between the obstacle and the incoming stream particles appears which is filled with granular gas of high temperature and low density. As soon as the gap appears the force does not depend on the square of velocity of the stream but the dependency obeys another law.