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Dry granular flows: gas, liquid or solid?
Figure 1: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008
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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
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Packing: 2D theoretical packing fraction
Square packing:
= /4 = 0.7854
Hexagonal packing:
= 2/(3)1/2 = 0.9069
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Packing: 3D theoretical packing fraction
Body-centered packing:
= 0.6802
Hexagonal & face-centered packing:
= 0.7405
Figures from: http://www.ndt-ed.org
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Topography dunes
Dune topography profile:
Leeward face: at angle of repose
Windward face: S-shape (from sand flux q)
2D view
Topography dune: detail
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Angle of repose (1)
Methods to measure angle of repose:
“material on verge of sliding”
Funnel (point-source)
Tilting box
Rotating cylinder: Dynamic angle of repose
Static angle of repose
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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
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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
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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
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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
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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
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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
From: Wiederseiner et al., Phys. of Fluids, 2011
Experiment (from gray scale)
Theory(from continuum model)
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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 m3 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
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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!
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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
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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 Sr:
speed of segregation
Diffusive remixing Dr:
speed of remixing
From: Gray & Kokkelaar, GRC conference, 2010
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Physical processes of segregation (3)
Non-dimensionalization expressions for Sr & Dr
Segregation rate (with percolation velocity q):
Diffusive remixing (with diffusion D):
Dependence on: particle size ratio, shear rate, slope angle?
From: Gray, IUTAM conference proceedings, 2010
No slope gradients: time, Sr, Dr Steady-state, u = u(z), Sr, no diffusion
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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 = hstop by decreasing h or
hstop() – curve: resistance is higher closer to surface
From: Pouliquen et al., Physics of Fluids, 1999
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Rheology: inclined plane (2)
Observations on velocity:
Continuous transition between inclined plane & surface flow?
Thick layers (h >> hstop): Bagnold velocity profile:
accurate in core layer
not accurate close to base or free surface (where I is not constant)
Thin layers (h ~ hstop): linear velocity profile:
Empirical flow rule for
depth-averaged velocity <V>:
From: Jop et al., 2005, Pouliquen et al., 2006
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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
From: da Cruz et al., PRE, 2005 & From: Jop et al., JFM, 2005
Quasi-static regime: Grain-inertia not relevant
Kinetic regime: Friction not relevant
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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
From: Pouliquen et al., J. of Stat. Mech., 2006 & da Cruz et al., PRE, 2005
Dissipation dominates sliding: I
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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 f1 & f2:
Friction:
Volume fraction:
f1 and f2 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
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Constitutive relations?
Valid for other geometries?
Simulations & experiments:a) Plane-shear
b) Rotating drum
c) Inclined planes
d) Annular shear cell
Yes, collapse!
Relevant parameter: Ia) Theoretical fit (red) &
kinetic theory (blue)
All dense granular flows: local friction and dilatation laws
From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008
Constitutive law for granular liquids? (1) 2D2D
3D 3D
model
model
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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
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
Analogy to Bingham fluids
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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
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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
From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008
Theoretical fit (red) & kinetic theory (blue)
Visco-plastic model (2)
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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
From: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008
Gravity parallel to plane
Tangential stress
Pressure force
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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
Shallow water equations (2)
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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
Shallow water equations (3)
