Closure relations for shallow granular flows from particle simulations

The discrete particle method (DPM) is used to model granular flows down an inclined chute with varying basal roughness, thickness and inclination. We observe three major regimes: arresting flows, steady uniform flows and accelerating flows. For flows over a smooth base, other (quasi-steady) regimes are observed: for small inclinations the flow can be highly energetic and strongly layered in depth; whereas, for large inclinations it can be non-uniform and oscillating. For steady uniform flows, depth profiles of density, velocity and stress are obtained using an improved coarse-graining method, which provides accurate statistics even at the base of the flow. A shallow-layer model for granular flows is completed with macro-scale closure relations obtained from micro-scale DPM simulations of steady flows. We obtain functional relations for effective basal friction, velocity shape factor, mean density, and the normal stress anisotropy as functions of layer thickness, flow velocity and basal roughness.

[1]  S. B. Savage,et al.  Two-dimensional spreading of a granular avalanche down an inclined plane Part I. theory , 1993 .

[2]  Richard M. Iverson,et al.  Flow of variably fluidized granular masses across three‐dimensional terrain: 2. Numerical predictions and experimental tests , 2001 .

[3]  George Keith Batchelor,et al.  An Introduction to Fluid Dynamics. , 1969 .

[4]  Jaap J. W. van der Vegt,et al.  hpGEM---A software framework for discontinuous Galerkin finite element methods , 2007, TOMS.

[5]  M. Louge,et al.  Model for dense granular flows down bumpy inclines. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  J. Gray,et al.  Deflecting dams and the formation of oblique shocks in snow avalanches at Flateyri, Iceland , 2007 .

[7]  Marcus I. Bursik,et al.  Input uncertainty propagation methods and hazard mapping of geophysical mass flows , 2006 .

[8]  R. L. Braun,et al.  Viscosity, granular‐temperature, and stress calculations for shearing assemblies of inelastic, frictional disks , 1986 .

[9]  Onno Bokhove,et al.  From discrete particles to continuum fields near a boundary , 2011, 1108.5032.

[10]  J. Jenkins.  Dense inclined flows of inelastic spheres , 2007 .

[11]  Wolfram Volk,et al.  From discrete element simulations to a continuum model , 2000 .

[12]  S. Luding.  Introduction to discrete element methods , 2008 .

[13]  Steven J. Plimpton,et al.  Boundary effects and self-organization in dense granular flows , 2002 .

[14]  S. Luding.  Micro-macro models for anisotropic granular media , 2004 .

[15]  J. Jenkins,et al.  Hydraulic theory for a debris flow supported on a collisional shear layer. , 1999, Chaos.

[16]  Olivier Pouliquen,et al.  SCALING LAWS IN GRANULAR FLOWS DOWN ROUGH INCLINED PLANES , 1999 .

[17]  E Weinan,et al.  Heterogeneous multiscale methods: A review , 2007 .

[18]  Satya N. Atluri,et al.  Atomic-level Stress Calculation and Continuum-Molecular System Equivalence , 2004 .

[19]  C. Brooks.  Computer simulation of liquids , 1989 .

[20]  N R Morgenstern,et al.  Experiments on the flow behaviour of granular materials at high velocity in an open channel , 1984 .

[21]  C. Goujon,et al.  Monodisperse dry granular flows on inclined planes: Role of roughness , 2003, The European physical journal. E, Soft matter.

[22]  G. Grest,et al.  Granular flow down an inclined plane: Bagnold scaling and rheology. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[24]  Y O ¨ E L F O R T E,et al.  Long surface wave instability in dense granular flows , 2003 .

[25]  Flow rule of dense granular flows down a rough incline. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  A. Kulikovskii,et al.  Two-dimensional problem of the motion of a snow avalanche along a slope with smoothly changing properties: PMM vol. 37, n≗5, 1973, pp. 837–848 , 1973 .

[27]  A. U.S.  Hydraulic Flow through a Contraction : Multiple Steady States , 2007 .

[28]  R. Bagnold.  Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[29]  Kolumban Hutter,et al.  Channelized free-surface flow of cohesionless granular avalanches in a chute with shallow lateral curvature , 1999, Journal of Fluid Mechanics.

[30]  R. Iverson,et al.  U. S. Geological Survey , 1967, Radiocarbon.

[31]  Onno Bokhove,et al.  Hydraulic flow through a channel contraction: Multiple steady states , 2008 .

[32]  James W. Landry,et al.  Granular flow down a rough inclined plane: Transition between thin and thick piles , 2003 .

[33]  Rapid granular flows on a rough incline: phase diagram, gas transition, and effects of air drag. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  J. Kirkwood,et al.  The Statistical Mechanical Theory of Transport Processes. IV. The Equations of Hydrodynamics , 1950 .

[35]  A. W. Vreman,et al.  Supercritical shallow granular flow through a contraction: experiment, theory and simulation , 2007, Journal of Fluid Mechanics.

[36]  J. Henderson,et al.  Statistical mechanics of inhomogeneous fluids , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[37]  Stefan Luding,et al.  From molecular dynamics and particle simulations towards constitutive relations for continuum theory , 2009, CSE 2009.

[38]  R. Hayward.  Stress , 2005, The Lancet.

[39]  G. Midi,et al.  On dense granular flows , 2003, The European physical journal. E, Soft matter.

[40]  Olivier Pouliquen,et al.  Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane , 2001, Journal of Fluid Mechanics.

[41]  M. Sheridan,et al.  Evaluation of the Titan2D two-phase flow model using an actual event: Case study of the 2005 Vazcún Valley Lahar , 2008 .

[42]  J. Jenkins,et al.  Density inversion in rapid granular flows: the supported regime , 2007, The European physical journal. E, Soft matter.

[43]  J. Jenkins.  Dense shearing flows of inelastic disks , 2006 .

[44]  Michel Y. Louge,et al.  On dense granular flows down flat frictional inclines , 2001 .

[45]  I. Goldhirsch,et al.  Stress, stress asymmetry and couple stress: from discrete particles to continuous fields , 2010 .

[46]  A. Hogg,et al.  Oblique shocks in rapid granular flows , 2005 .

[47]  B. Dalloz-Dubrujeaud,et al.  Bidisperse granular avalanches on inclined planes: A rich variety of behaviors , 2007, The European physical journal. E, Soft matter.

[48]  M. Babić,et al.  AVERAGE BALANCE EQUATIONS FOR GRANULAR MATERIALS , 1997 .

[49]  S. Savage,et al.  The motion of a finite mass of granular material down a rough incline , 1989, Journal of Fluid Mechanics.

[50]  J. Jenkins,et al.  Dense inclined flows of inelastic spheres: tests of an extension of kinetic theory , 2010 .

[51]  Kolumban Hutter,et al.  Gravity-driven free surface flow of granular avalanches over complex basal topography , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[52]  Sebastian Noelle,et al.  Shock waves, dead zones and particle-free regions in rapid granular free-surface flows , 2003, Journal of Fluid Mechanics.

[53]  J. Gray,et al.  Weak, strong and detached oblique shocks in gravity-driven granular free-surface flows , 2007, Journal of Fluid Mechanics.

[54]  Evans,et al.  Pressure tensor for inhomogeneous fluids. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[55]  Editors , 1986, Brain Research Bulletin.

[56]  Stéphane Douady,et al.  Two types of avalanche behaviour in granular media , 1999, Nature.