Coarse-grained local and objective continuum description of three-dimensional granular flows down an inclined surface

Physics of Fluids - Tập 25 Số 7 - 2013
Thomas Weinhart1, Remco Hartkamp1, Anthony R. Thornton1,2, Stefan Luding1
1University of Twente 1 Multi Scale Mechanics, CTW and MESA+, , P.O. Box 217, 7500AE Enschede, The Netherlands
2University of Twente 2 Mathematics of Computational Science, EWI, , P.O. Box 217, 7500AE Enschede, The Netherlands

Tóm tắt

Dry, frictional, steady-state granular flows down an inclined, rough surface are studied with discrete particle simulations. From this exemplary flow situation, macroscopic fields, consistent with the conservation laws of continuum theory, are obtained from microscopic data by time-averaging and spatial smoothing (coarse-graining). Two distinct coarse-graining length scale ranges are identified, where the fields are almost independent of the smoothing length w. The smaller, sub-particle length scale, w ≪ d, resolves layers in the flow near the base boundary that cause oscillations in the macroscopic fields. The larger, particle length scale, w ≈ d, leads to smooth stress and density fields, but the kinetic stress becomes scale-dependent; however, this scale-dependence can be quantified and removed. The macroscopic fields involve density, velocity, granular temperature, as well as strain-rate, stress, and fabric (structure) tensors. Due to the plane strain flow, each tensor can be expressed in an inherently anisotropic form with only four objective, coordinate frame invariant variables. For example, the stress is decomposed as: (i) the isotropic pressure, (ii) the “anisotropy” of the deviatoric stress, i.e., the ratio of deviatoric stress (norm) and pressure, (iii) the anisotropic stress distribution between the principal directions, and (iv) the orientation of its eigensystem. The strain rate tensor sets the reference system, and each objective stress (and fabric) variable can then be related, via discrete particle simulations, to the inertial number, I. This represents the plane strain special case of a general, local, and objective constitutive model. The resulting model is compared to existing theories and clearly displays small, but significant deviations from more simplified theories in all variables – on both the different length scales.

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Tài liệu tham khảo

1998, Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order, J. Fluid Mech., 361, 41, 10.1017/S0022112098008660

2012, Kinetic theory applied to inclined flows, Granular Matter, 14, 79, 10.1007/s10035-011-0308-x

2009, A non-local rheology for dense granular flows, Phys. Eng. Sci., 367, 5091, 10.1098/rsta.2009.0171

2012, Nonlocal constitutive relation for steady granular flow, Phys. Rev. Lett., 108, 178301, 10.1103/PhysRevLett.108.178301

2010, On the evolution of stress and microstructure during general 3D deviatoric straining of granular media, Geotechnique, 60, 333, 10.1680/geot.2010.60.5.333

2009, Granular solid hydrodynamics, Granular Matter, 11, 139, 10.1007/s10035-009-0137-3

2011, A local constitutive model with anisotropy for various homogeneous 2D biaxial deformation modes, Chem.-Ing.-Tech., 83, 672, 10.1002/cite.201000180

2012, Bridging the rheology of granular flows in three regimes, Phys. Rev. E, 85, 021305, 10.1103/PhysRevE.85.021305

2012, From discrete particles to continuum fields near a boundary, Granular Matter, 14, 289, 10.1007/s10035-012-0317-4

2005, Rheophysics of dense granular materials: Discrete simulation of plane shear flows, Phys. Rev. E, 72, 021309, 10.1103/PhysRevE.72.021309

2004, Granular lubrication: Toward an understanding between kinetic and fluid regime, ASME J. Tribol., 126, 137, 10.1115/1.1633575

GDR MiDi, 2004, On dense granular flows, Eur. Phys. J. E, 14, 341, 10.1140/epje/i2003-10153-0

2006, A constitutive law for dense granular flows, Nature (London), 441, 727, 10.1038/nature04801

2001, Continuum description of avalanches in granular media, Phys. Rev. E, 64, 020301, 10.1103/PhysRevE.64.020301

2002, A frictional Cosserat model for the slow shearing of granular materials, J. Fluid Mech., 457, 377, 10.1017/S0022112002007796

2010, Behavior of pressure and viscosity at high densities for two-dimensional hard and soft granular materials, Prog. Theor. Phys. Suppl., 184, 110, 10.1143/PTPS.184.110

2011, A constitutive model with microstructure evolution for flow of rate-independent granular materials, J. Fluid Mech., 682, 590, 10.1017/jfm.2011.251

2009, Assessing continuum postulates in simulations of granular flow, J. Mech. Phys. Solids, 57, 828, 10.1016/j.jmps.2009.01.009

2005, Statics and Kinematics of Granular Materials

1995, Bifurcation Analysis in Geomechanics

Herrmann, 1998, Non-associated plasticity for soils, concrete and rock, Physics of Dry Granular Media, 163, 10.1007/978-94-017-2653-5

2001, From discrete element simulations to a continuum model, Comput. Methods Appl. Mech. Eng., 191, 21, 10.1016/S0045-7825(01)00242-0

2011, The critical-state yield stress (termination locus) of adhesive powders from a single numerical experiment, Granular Matter, 13, 109, 10.1007/s10035-010-0241-4

2011, Jamming by shear, Nature (London), 480, 355, 10.1038/nature10667

1980, Observation of the coupling of concentration fluctuations to steady-state shear flow, Phys. Rev. Lett., 44, 1005, 10.1103/PhysRevLett.44.1005

1980, Shear-flow-induced distortion of the pair-correlation function, Phys. Rev. A, 22, 2844, 10.1103/PhysRevA.22.2844

1983, Shear-flow induced distortion of the radial distribution function, test of a model kinetic equation in the nonlinear flow regime, Physica A, 118, 444, 10.1016/0378-4371(83)90228-5

1983, Similarities and differences in the nonlinear flow behavior of simple and of molecular liquids, Physica A, 118, 79, 10.1016/0378-4371(83)90178-4

1983, Pressure tensor and viscosity coefficients of a soft sphere liquid under shear, Int. J. Thermophys., 4, 97, 10.1007/BF00500134

1989, The motion of a finite mass of material down a rough incline, J. Fluid Mech., 199, 177, 10.1017/S0022112089000340

2001, Regularization of the Burnett equations for rapid granular flows via relaxation, Physica D, 150, 207, 10.1016/S0167-2789(00)00224-4

2003, Rheology of bidisperse granular mixtures via event-driven simulations, J. Fluid Mech., 476, 69, 10.1017/S002211200200263X

Garcia-Rojo, 2005, Non-Newtonian granular fluids: Simulation and theory, Powders and Grains 2005, 1141, 10.1201/NOE0415383486

2011, Suspensions in a tilted trough: Second normal stress difference, J. Fluid Mech., 686, 26, 10.1017/jfm.2011.315

2011, Dense suspensions in rotating-rod flows: Normal stresses and particle migration, J. Fluid Mech., 686, 5, 10.1017/jfm.2011.272

2012, A study of the anisotropy in a fluid confined in a nanochannel, J. Chem. Phys., 137, 044711, 10.1063/1.4737927

2012, Closure relations for shallow granular flows from particle simulations, Granular Matter, 14, 531, 10.1007/s10035-012-0355-y

2008, Cohesive, frictional powders: Contact models for tension, Granular Matter, 10, 235, 10.1007/s10035-008-0099-x

2001, Granular flow down an inclined plane: Bagnold scaling and rheology, Phys. Rev. E, 64, 051302, 10.1103/PhysRevE.64.051302

1977, A numerical approach to testing the fission hypothesis, Astron. J., 82, 1013, 10.1086/112164

2010, Stress, stress asymmetry and couple stress: From discrete particles to continuous fields, Granular Matter, 12, 239, 10.1007/s10035-010-0181-z

2006, Continuum mechanics for small systems and fine resolutions, Handbook of Theoretical and Computational Nanotechnology, 330

2010, Constitutive relations for the isotropic deformation of frictionless packings of polydisperse spheres, C. R. Mec., 338, 570, 10.1016/j.crme.2010.10.004

2013, Hydrostatic and shear behavior of frictionless granular assemblies under different deformation conditions, KONA Powder Part. J., 30, 84, 10.14356/kona.2013011

2010, Quasi-static simulations of compact polydisperse particle systems, Particuology, 8, 119, 10.1016/j.partic.2009.07.007

2013, Density dependence of the stress relaxation function of a simple fluid, Phys. Rev. E, 87, 032155, 10.1103/PhysRevE.87.032155

2006, Scale separation in granular packings: Stress plateaus and fluctuations, Phys. Rev. Lett., 96, 168001, 10.1103/PhysRevLett.96.168001

2000, Macroscopic material properties from quasi-static, microscopic simulations of a two-dimensional shear-cell, Granular Matter, 2, 123, 10.1007/s100350000048

2001, Scale dependence, correlations, and fluctuations of stresses in rapid granular flows, Phys. Fluids, 13, 407, 10.1063/1.1338543

2010, A continuum approach applied to a strongly confined Lennard-Jones fluid, Proceedings of the Fifth International Conference on Multiscale Materials Modeling (MMM2010), Freiburg, Germany

1954, Experiments on a gravity-free dispersion of large spheres in a Newtonian fluid under shear, Proc. R. Soc. London, Ser. A, 225, 49, 10.1098/rspa.1954.0186

2011, A local constitutive model with anisotropy for ratcheting under 2D axial-symmetric isobaric deformation, Granular Matter, 13, 225, 10.1007/s10035-011-0266-3

2001, Flow of variably fluidized granular masses across three-dimensional terrain. 1. Coulomb mixture theory, J. Geophys. Res., 106, 537, 10.1029/2000JB900329

2012, Friction dependence of shallow granular flows from discrete particle simulations, Eur. Phys. J. E, 35, 127, 10.1140/epje/i2012-12127-5

2013, A constitutive framework for the non-Newtonian pressure tensor of a simple fluid under planar flows, J. Chem. Phys., 138, 244508, 10.1063/1.4810746

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Physics of Fluids

Tập 25 Số 7

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