Dry granular flows: gas, liquid or solid? Figure 1: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 1 Characterizing size and size distribution Grains are not uniform (size, shape, ) Statistical analysis of particle sample: Mean diameter: 50% Standard deviation: 16-84% Most earth materials deviate: Lack of small and large particles sorting Create skewness and kurtosis 2 1
Packing: 2D theoretical packing fraction Square packing: Hexagonal packing: = /4 = 0.7854 = 2/(3) 1/2 = 0.9069 3 Packing: 3D theoretical packing fraction Body-centered packing: = 0.6802 Hexagonal & facecentered packing: = 0.7405 Figures from: http://www.ndt-ed.org 4 2
Topography dunes 2D view Dune topography profile: Leeward face: at angle of repose Windward face: S-shape (from sand flux q) Topography dune: detail 3
Angle of repose (1) Methods to measure angle of repose: material on verge of sliding Funnel (point-source) Static angle of repose Tilting box Rotating cylinder: Dynamic angle of repose 7 Angle of repose (2) Physical interpretation: Static angle of repose due to cohesive forces related to friction coefficient: s = arctan( s ) Dynamic angle of repose due to dilatation and # of contacts difference ( s - d ) is dilation angle Characteristic values: Angular grains (e.g. sand, gravel): s 40 Rounded grains (e.g. ballotini): s 25 From: Santamarina & Cho, Proc. Skempton Conf., 2004 8 4
Angle of repose (3) Effect of reduced gravity (e.g. on Mars: a = 0.1 g): Static angle increases: s, 0.1g = s, 1g + 5 Dynamic angle decreases: d, 0.1g = d, 1g - 10 Dilation angle & mobility of flow increase! Low slopes on Mars can create large dry granular flows! From: Kleinhans, et al., JGR, 2011 Dundas, et al., GRL, 2010 9 Mixing & segregation Mixing of a granular material: Homogeneous (re)distribution of different particles reducing entropy creating uniform material Segregation of a granular material: Separation of grains (size, density, shape) due to a variety of physical processes: shear gravity vibration 10 5
Segregation in a wedge Two parallel plates forming a silo: Point-source & triangular pile white (0.5 mm) sugar crystals dark (0.34 mm) spherical iron powder Difference static & dynamic friction angle: discrete avalanches forming a roll-wave kinetic sieving upslope propagating shock wave at wall frozen inverse grading pattern pine-tree pattern, alternating sides Stratification pattern: sandwich: coarse-fine-coarse coarse rich flow front strongly inversely graded behind From: Gray & Hutter, Cont. Mech. & Therm., 1997, Gray & Ancey, JFM, 2009 11 Segregation in an avalanche (1) Flows in nature carve their own path: Coarse material in the levees and the flow front Fine material in the centre and the back of the flow From: Gray and Kokkelaar, GRC, 2010 12 6
Segregation in an avalanche (2) 2D chute with side walls: Rough-bottomed with smooth walls 3 m long, 2 cm wide chute avoids 3D effects in segregation pattern, has sidewall friction bidisperse mixture: 1 mm & 2 mm, same density Experiment (from gray scale) Theory (from continuum model) From: Wiederseiner et al., Phys. of Fluids, 2011 13 Segregation in an avalanche (3) 3D chute with rough bottom: Experiment (25/08/2009) at the USGS debris-flow flume: large-scale debris-flow experiments: 10 m 3 sand, gravel & water size segregation: laterally strongly graded, vertical weakly graded Coarse material: flow front basal slip & shear down & sidewards From: Johnson et al., JGR, 2012 14 7
Segregation in an avalanche (4) Fingering instability in a bimodal mixture: Segregation-mobility feedback mechanism Creates fingers and self-channelizes to form lateral levees Particles: large irregular (black) and small spherical (white): Velocity shear & size segregation: large grains to flow front Lateral instability: uniform front breaks up Flow degenerates into distinct fingers Numerical studies: grid-dependency Linear stability analysis: perturbations grow Experimental studies: triggering Thanks to Perry Harwood: reproducibility! 15 Physical processes of segregation (1) Transfer of particles between layers: Kinetic sieving: gravity-induced, size-dependent, void-filling smaller particles fall easier into holes Squeeze expulsion: imbalance on contact forces on individual particle more space frees up when larger particle moves to adjacent layer not size-preferential, no preferential direction of transfer From: Savage and Lun, JFM, 1988 16 8
Physical processes of segregation (2) Modeling segregation with a phenomenological model: Segregation-remixing equation: Hyperbolic equation: D = 0 sharp concentration jumps Parabolic equation: D > 0 smooth transitions Volume fraction small particles: Segregation rate S r : speed of segregation Diffusive remixing D r : speed of remixing From: Gray & Kokkelaar, GRC conference, 2010 17 Physical processes of segregation (3) Non-dimensionalization expressions for S r & D r Segregation rate (with percolation velocity q): Diffusive remixing (with diffusion D): Dependence on: particle size ratio, shear rate, slope angle? No slope gradients: time, S r, D r Steady-state, u = u(z), S r, no diffusion From: Gray, IUTAM conference proceedings, 2010 18 9
Rheology: inclined plane (1) Avalanches on an inclined plane: Steady uniform flows: constant V & h Non-steady flow if or h acceleration of flow No flow if h = h stop by decreasing h or h stop () curve: resistance is higher closer to surface From: Pouliquen et al., Physics of Fluids, 1999 19 Rheology: inclined plane (2) Observations on velocity: Continuous transition between inclined plane & surface flow? Thick layers (h >> h stop ): Bagnold velocity profile: accurate in core layer not accurate close to base or free surface (where I is not constant) Thin layers (h ~ h stop ): linear velocity profile: Empirical flow rule for depth-averaged velocity <V>: From: Jop et al., 2005, Pouliquen et al., 2006 20 10
Local rheology (1) Rheology from dimensional arguments & simulations: Shear stress proportional to pressure: Volume fraction: Inertial number defines flow regime: Microscopic (inertial) time scale: Macroscopic (deformation) time scale: Transition regimes for increasing I: quasi-static dense inertial collisional regime Kinetic regime: Friction not relevant Quasi-static regime: Grain-inertia not relevant From: da Cruz et al., PRE, 2005 & From: Jop et al., JFM, 2005 21 Local rheology (2) Friction & dilatancy laws from empirical evidence: Correct for 2D configurations: plane shear & inclined plane Friction law: Volume fraction: I is rate-dependent in intermediate regime flow law Dissipation dominates sliding: I From: Pouliquen et al., J. of Stat. Mech., 2006 & da Cruz et al., PRE, 2005 22 11
Local rheology (3) Bagnold s experiments and scaling: For all shear rates and regimes, for perfectly hard grains Normal stress: Shear stress: Ratio: Rewriting friction & volume fraction in terms of f 1 & f 2 : Friction: Volume fraction: f 1 and f 2 diverge quickly near maximum packing fraction friction () and dilatancy () laws are decoupled From: Lois et al., PRE, 2005, Forterre & Pouliquen, Annu. Rev. Fluid Mech., 2008 23 Constitutive law for granular liquids? (1) Constitutive relations? 2D 2D Valid for other geometries? Simulations & experiments: a) Plane-shear b) Rotating drum c) Inclined planes d) Annular shear cell Yes, collapse! 3D 3D Relevant parameter: I a) Theoretical fit (red) & kinetic theory (blue) model model All dense granular flows: local friction and dilatation laws From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 24 12
Constitutive law for granular liquids? (2) Rheology for all geometries (not only plane shear): Visco-plastic (Bingham) model (Jop et al., 2006) Flow threshold viscosity instead of yield stress Shear rate dependence viscous behavior Analogy to Bingham fluids Nonlinear elasto-plastic model (Kamrin, 2010), includes: Granular elasticity (Jiang & Liu, 2003) for stagnant zones Rate-sensitive fluid-like flow (Jop et al., 2006) for flowing regions 25 Visco-plastic model (1) 3D geometries -- shear from different directions: Non-Newtonian incompressible fluid: assume volume fraction is constant in dense regions co-linearity between shear stress and shear rate Form of a visco-plastic law: isotropic pressure P shear stress:, with viscosity: second invariant of shear rate tensor Flow threshold (Drucker-Prager criterion): second invariant of stress tensor goes to zero viscosity diverges (difficult in some simulations!) Predicts correctly: critical angle & constant volume fraction Bagnold velocity profile From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 26 13
Visco-plastic model (2) Limitations on using a visco-plastic approach: Lack of link with microscopic grain properties: shape of friction law is measured, not derived Shear bands (quasi-static regions) are incorrectly described: modifying plasticity models in shear-rate independent regime explicitly describing nonlocal effects (e.g. jamming) Flow threshold: Coulomb criterion, does not capture hysteresis and finite size effects Transition to kinetic regime: gaseous regime is not captured in visco-plastic approach kinetic theory does not capture correct behavior in dense regime Theoretical fit (red) & kinetic theory (blue) From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 27 Shallow water equations (1) Alternative constitutive relation for thin flows: Interfacial law between bottom and granular layer dynamics of flowing layer without knowing details internal structure Depth-averaged or Saint-Venant equations: assuming incompressible flow variations are on a scale larger than flow thickness Mass conservation: Momentum conservation: with basal friction coefficient b, velocity coefficient and stress ratio K Gravity parallel Tangential Pressure force to plane stress From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 28 14
Shallow water equations (2) Limitations on using shallow water equations: Coulomb-type basal friction may not be sufficient rough inclines steady uniform flow for different inclination angles solid friction is not constant, complicated basal friction laws necessary Second-order effects are not captured: longitudinal and lateral momentum diffusion are not included necessary to control instabilities and lateral stresses Additional equation necessary for erodible layers: exchange of mass and momentum between solid-liquid interface From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 29 Shallow water equations (3) Debris avalanche: Montserrat, December 1997 Failure of south flank of Soufriere Hills volcano numerical simulations: gravitational flow of a homogeneous continuum Coulomb-type basal friction with a dynamic friction coefficient From: Heinrich et al., GRL, 2001 30 15