Simulated configurational temperature of particles and a model of constitutive relations of rapid-intermediate-dense granular flow based on generalized granular temperature

In gas–solid flat-base spout bed with a jet, the flow of particles must go through an intermediate regime where both kinetic/collisional and frictional contributions play a role. In this paper, the statistical framework is proposed to define the generalized granular temperature which sums up the configurational temperature and translational granular temperature. The configurational temperature, translational and rotational granular temperatures of particles are simulated by means of CFD-DEM (discrete element method) in a 3D flat-base spout bed with a jet. The configurational temperatures of particles are calculated from instantaneous overlaps of particles. The translational and rotational granular temperatures of particles are calculated from instantaneous translational and angular velocities of particles. Roughly, the simulated translational and rotational granular temperatures increase, reach maximum, and then decrease with the increase of solids volume fractions. However, the configurational temperature increases with the increase of solids volume fractions. At high solid volume fraction, the predicted configurational temperatures are larger than the translational and rotational granular temperatures, indicating that the rate of energy dissipation do contributes by contact deformation of elastic particles. The generalized granular temperature is proposed to show the relation between the variance of the fluctuation velocity of deformation and the variance of the translational fluctuation velocity of particles. The constitutive relations of particle pressure, viscosity, granular conductivity of fluctuating energy and energy dissipation in rapid-intermediate-dense granular flows are correlated to the generalized granular temperature. The variations of particle pressure, shear viscosity, energy dissipation and granular conductivity are analyzed on the basis of generalized granular temperature in a flat-base spout bed with a jet. The axial velocities of particles predicted by a gas–solid two-fluid model of rapid-intermediate-dense granular flows agree with experimental results in a spout bed.     

Assessment of the kinetic–frictional model for dense granular flow

This paper aims to quantitatively assess the application of kinetic–frictional model to simulate the motion of dry granular materials in dense condition, in particular, the annular shearing in Couette configuration. The weight of frictional stress was varied to study the contribution of the frictional stress in dense granular flows. The results show that the pure kinetic-theory-based computational fluid dynamics (CFD) model (without frictional stress) over-predicts the dominant solids motion of dense granular flow while adding frictional stress [Schaeffer, D. G. (1987). Instability in the evolution equations describing incompressible granular flow. Journal of Differential Equations, 66(1), 19–50] with the solids pressure of [Lun, C. K. K., Savage, S. B., Jeffrey, D. J., & Chepurniy, N. (1984). Kinetic theories for granular flow: Inelastic particles in Couette flow and slightly inelastic particles in a general flow field. Journal of Fluid Mechanics, 140, 223–256] in the CFD model improves the simulation to better conform available experimental results. The results also suggest that frictional stress transmission plays an important role in dense granular flow and should not be neglected in granular flow simulations. Compatible simulation results to the experimental data are seen by increasing the weight of frictional stress to a factor of 1.25–1.5. These improved simulation results suggest the current constitutive relations (kinetic–frictional model) need to be improved in order to better reflect the real dense granular flow.     

Analysis of instantaneous dynamic states of vibrated granular materials

The behavior of granular materials subjected to continuous vertical vibrations is dependent on a variety of factors, including how energetically the containment vessel is shaken as well as particle properties. Motivation for the investigation reported here is based on phenomenon in which bulk solids attain an increase in density upon relaxation. The results of a detailed, discrete element study designed to examine the dynamic state of a granular material is presented, in which particles are represented as inelastic, frictional spheres. The phase in which the assembly finds itself immediately before vibrations are stopped is quantified by computing depth profiles of the translational energy ratio R in conjunction with profiles of solids fraction ν and granular temperature T. The use of particles that are more frictional tends to hinder or delay thermalization, while particle restitution coefficient plays a role when the flow is collision dominated. The structure before vibrations are applied plays an important role in determining the depth profiles and the phase pattern only at low accelerations. On the other hand, large accelerations can easily dislodge the poured configuration very quickly so that the initial condition is not major factor in the phase pattern.     

A new finite element formulation for both compressible and nearly incompressible fluid dynamics

A new finite element formulation designed for both compressible and nearly incompressible viscous flows is presented. The formulation combines conservative and non‐conservative dependent variables, namely, the mass–velocity (density * velocity), internal energy and pressure. The central feature of the method is the derivation of a discretized equation for pressure, where pressure contributions arising from the mass, momentum and energy balances are taken implicitly in the time discretization. The method is applied to the analysis of laminar flows governed by the Navier–Stokes equations in both compressible and nearly incompressible regimes. Numerical examples, covering a wide range of Mach number, demonstrate the robustness and versatility of the new method. Copyright © 2000 John Wiley & Sons, Ltd.     

On stability of rapid granular shear flows

We study stability properties of rapid granular flows that are described by the balance laws of mass, momentum and fluctuation energy with phenomenological relationships based upon dimensional arguments [12, 13]. Small disturbances propagating perpendicular to the shear plane are only studied. Calculations show that such flows are stable at small free path lengths of the granules but unstable if free path lengths are large. Received June 17, 1997     

Computational modeling of surface phenomena in soft-wet materials

We consider the problem of extracting tribological information from experimental observations of contact with soft-wet materials. Particular attention is placed on simulating the response of two rotating cylinders of soft specimens placed in frictional contact, with a variable coefficient of friction dependent on the relative sliding velocity. The bulk behavior is modeled by means of a finite deformation viscoelasticity formulation, with constitutive parameters taken to be representative of hydrogels. We focus on the modeling of the surface behavior and employ a mortar-finite element contact formulation. Through a series of numerical studies, we demonstrate the strong sensitivity of the results to the choice of interfacial constitutive parameters. The difficulties of extracting such parameters using only experimental data and approximate analytical expressions are also examined.     

Simulations of granular flow along an inclined plane using the Savage–Hutter model

In this work a velocity-dependent friction is introduced into a depth-averaged Savage–Hutter dynamical model for shallow granular flows. The process of granular material flowing along an inclined plane and then depositing on a horizontal plane is simulated. The surface profiles and evolution of various types of energy are investigated and compared when using the standard Coulomb-type friction versus velocity-dependent friction. Interestingly, there is a small difference between the two different types of friction.     

Shakedown in elastic contact problems with Coulomb friction

Elastic systems with frictional interfaces subjected to periodic loading are sometimes predicted to ‘shake down’ in the sense that frictional slip ceases after the first few loading cycles. The similarities in behaviour between such systems and monolithic bodies with elastic–plastic constitutive behaviour have prompted various authors to speculate that Melan’s theorem might apply to them – i.e., that the existence of a state of residual stress sufficient to prevent further slip is a sufficient condition for the system to shake down.In this paper, we prove this result for ‘complete’ contact problems in the discrete formulation (i) for systems with no coupling between relative tangential displacements at the interface and the corresponding normal contact tractions and (ii) for certain two-dimensional problems in which the friction coefficient at each node is less than a certain critical value. We also present counter-examples for all systems that do not fall into these categories, thus giving a definitive statement of the conditions under which Melan’s theorem can be used to predict whether such a system will shake down.     

Assessment of the kinetic-frictional model for dense granular flow

This paper aims to quantitatively assess the application of kinetic-frictional model to simulate the motion of dry granular materials in dense condition, in particular, the annular shearing in Couette configuration. The weight of frictional stress was varied to study the contribution of the frictional stress in dense granular flows. The results show that the pure kinetic-theory-based computational fluid dynamics （CFD） model （without frictional stress） over-predicts the dominant solids motion of dense granular flow while adding frictional stress [Schaeffer, D. G. （1987）. Instability in the evolution equations describing incompressible granular flow. Journal of Differential Equations, 66（1）, 19-50] with the solids pressure of [Lun, C. NTK., Savage, S. B., Jeffrey, D. J., ＆ Chepurniy, N. （1984）. Kinetic theories for granular flow： Inelastic particles in Couette flow and slightly inelastic particles in a general flow field. Journal of Fluid Mechanics, 140, 223-256] in the CFD model improves the simulation to better conform available experimental results. The results also suggest that frictional stress transmission plays an important role in dense granular flow and should not be neglected in granular flow simulations. Compatible simulation results to the experimental data are seen by increasing the weight of frictional stress to a factor of 1.25-1.5. These improved simulation results suggest the current constitutive relations （kinetic-frictional model） need to be improved in order to better reflect the real dense granular flow.     

快速颗粒流碰撞应力的动理模型

颗粒碰撞是快速颗粒流的动量传递,能量传递及耗散的主要机制。因此,碰撞应力是快速颗粒流研究的主要内容。本文基于大小均匀、光滑圆球颗粒的非弹性碰撞模型,类比气体分子动理论中的处理方法得出快速颗粒流积分形式的碰撞应力,能量传递通量及能量耗散率等关系。在颗粒速度分布函数的一级近似式符合Maxwell分布的假设下,对碰撞应力积分得出相应于Boltzmann方程二级近似解的碰撞应力表达式。在简单快速颗粒流动条件下,本文的理论结果与实验资料符合较好。     

An Eulerian model for the motion of granular material with a large Stokes number in fluid flow

This study introduced a novel Euler–Euler approach for modeling granular multiphase flow. The motion of particles with a large Stokes number was investigated assuming that granular material has unilateral compressibility. Solid pressure in the momentum equations for granular multiphase flow was determined so that the unilateral incompressibility condition was satisfied. Using the continuity condition of the granular phase, the equation was rewritten in the optimal form to calculate the solid pressure. A discrete formulation of smoothed particle hydrodynamics was applied for the convective terms so that the discrete matrix was positive semidefinite for the convergence and the discretization for an unstructured mesh was allowed. Frictional stress was then determined from solid pressure and, by using the solid pressure and frictional stress, momentum equations for the granular phase were solved. The method was incorporated into ANSYS FLUENT by a UDF (user defined function). Model validation was performed through a comparison with two previous results, and efficacy of the proposed model was confirmed.     

Quadratic programming with parametric vectors in three-dimensional contact problems

The present study is concerned with 3-D contact problems by parametric quadratic programming(PQP). The stiffness matrix of a 3-D contact element is introduced by means of penalty function expression of the contact pressure and frictional force, and two constitutive relations of 3-D contact problems can be obtained, similar to elasto—plastic problems. The penalty factors can be eliminated by using a definite numerical technique, so that a parametric linear complementary problem is obtained. As an application of the theory presented, some numerical examples are illustrated.     

Accounting for finite-size effects in simulations of disperse particle-laden flows

A numerical formulation for Eulerian–Lagrangian simulations of particle-laden flows in complex geometries is developed. The formulation accounts for the finite-size of the dispersed phase. Similar to the commonly used point-particle formulation, the dispersed particles are treated as point-sources, and the forces acting on the particles are modeled through drag and lift correlations. In addition to the inter-phase momentum exchange, the presence of particles affects the fluid phase continuity and momentum equations through the displaced fluid volume. Three flow configurations are considered in order to study the effect of finite particle size on the overall flowfield: (a) gravitational settling, (b) fluidization by a gaseous jet, and (c) fluidization by lift in a channel. The finite-size formulation is compared to point-particle representations, which do not account for the effect of finite-size. It is shown that the fluid displaced by the particles plays an important role in predicting the correct behavior of particle motion. The results suggest that the standard point-particle approach should be modified to account for finite particle size, in simulations of particle-laden flows.     

Gas-particle two-phase turbulent flow in a vertical duct

Two-phase gas-phase turbulent flows at various loadings between the two vertical parallel plates are analyzed. A thermodynamically consistent turbulent two-phase flow model that accounts for the phase fluctuation energy transport and interaction is used. The governing equation of the gas-phase is upgraded to a two-equation low Reynolds number turbulence closure model that can be integrated directly to the wall. A no-slip boundary condition for the gas-phase and slip-boundary condition for the particulate phase are used. The computational model is first applied to dilute gas-particle turbulent flow between two parallel vertical walls. The predicted mean velocity and turbulence intensity profiles are compared with the experimental data of Tsuji et al. (1984) for vertical pipe flows, and good agreement is observed. Examples of additional flow properties such as the phasic fluctuation energy, phasic fluctuation energy production and dissipation, as well as interaction momentum and energy supply terms are also presented and discussed.

Applications to the relatively dense gas-particle turbulent flows in a vertical channel are also studied. The model predictions are compared with the experimental data of Miller & Gidaspow and reasonable agreement is observed. It is shown that flow behavior is strongly affected by the phasic fluctuation energy, and the momentum and energy transfer between the particulate and the fluid constituents.     

Numerical simulations of granular free-surface flows using smoothed particle hydrodynamics

This paper presents a two-dimensional SPH model designed to simulate free-surface flows of dense granular materials. Smoothed particle hydrodynamics (SPH) is a mesh-free numerical method based on a Lagrangian discretization of the continuum mass and momentum conservation equations. The rheology of dense granular materials is modelled using a new local constitutive law recently proposed by Jop et al. (Nature, 2006). Of the viscoplastic class, this law is characterized by an apparent viscosity depending both on the local strain rate and local pressure. Validation test cases of the model in steady and unsteady configurations are presented. For steady cases (vertical chute flow and uniform free-surface layers on inclines), excellent agreement with analytical predictions is obtained. In the unsteady case, the simulations satisfactorily capture the dynamics of gravity-driven surges observed in experiments, including behaviours that are very specific to granular materials. Among the various parameters involved in the computations, the influence of SPH particle configuration within the flow and of the threshold viscosity used in the regularization of the constitutive yield criterion are particularly discussed.     

Differentially weighted direct simulation Monte Carlo method for particle collision in gas-solid flows

In gas-solid flows,particle-particle interaction(typical,particle collision) is highly significant,despite the small particles fractional volume.Widely distributed polydisperse particle population is a typical characteristic during dynamic evolution of particles(e.g.,agglomeration and fragmentation) in spite of their initial monodisperse particle distribution.The conventional direct simulation Monte Carlo(DSMC)method for particle collision tracks equally weighted simulation particles,which results in high statistical noise for particle fields if there are insufficient simulation particles in less-populated regions.In this study,a new differentially weighted DSMC(DW-DSMC) method for collisions of particles with different number weight is proposed within the framework of the general Eulerian-Lagrangian models for hydrodynamics.Three schemes(mass,momentum and energy conservation) were developed to restore the numbers of simulation particle while keeping total mass,momentum or energy of the whole system unchanged respectively.A limiting case of high-inertia particle flow was numerically simulated to validate the DW-DSMC method in terms of computational precision and efficiency.The momentum conservation scheme which leads to little fluctuation around the mass and energy of the whole system performed best.Improved resolution in particle fields and dynamic behavior could be attained simultaneously using DW-DSMC,compared with the equally weighted DSMC.Meanwhile,computational cost can be largely reduced in contrast with direct numerical simulation.     

The rational mechanics and thermodynamics of polymeric fluids based upon the concept of a variable relaxed state

The conceptual framework of polymer continuum mechanics based upon Eckart's idea of a variable relaxed state is developed. No constitutive models are explicitly used. The theory admits four constitutive functions only, the scalar specific internal energy, the vectorial heat flux, and two tensorial fluxes representing non-elastic stress and flow (slippage). The non-linearity of the constitutive relations includes self-induced anisotropy (Leonov) with Reiner-Rivlin's equation representing a special example for this. — The effectiveness of this non-linear theory is demonstrated by treating elongational flows of polymer melts.     

A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

Differentially weighted direct simulation Monte Carlo method for particle collision in gas–solid flows

In gas–solid flows, particle–particle interaction (typical, particle collision) is highly significant, despite the small particles fractional volume. Widely distributed polydisperse particle population is a typical characteristic during dynamic evolution of particles (e.g., agglomeration and fragmentation) in spite of their initial monodisperse particle distribution. The conventional direct simulation Monte Carlo (DSMC) method for particle collision tracks equally weighted simulation particles, which results in high statistical noise for particle fields if there are insufficient simulation particles in less-populated regions. In this study, a new differentially weighted DSMC (DW-DSMC) method for collisions of particles with different number weight is proposed within the framework of the general Eulerian–Lagrangian models for hydrodynamics. Three schemes (mass, momentum and energy conservation) were developed to restore the numbers of simulation particle while keeping total mass, momentum or energy of the whole system unchanged respectively. A limiting case of high-inertia particle flow was numerically simulated to validate the DW-DSMC method in terms of computational precision and efficiency. The momentum conservation scheme which leads to little fluctuation around the mass and energy of the whole system performed best. Improved resolution in particle fields and dynamic behavior could be attained simultaneously using DW-DSMC, compared with the equally weighted DSMC. Meanwhile, computational cost can be largely reduced in contrast with direct numerical simulation.