Managing A Computer Simulation of Gravity-Driven Granular Flow

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Managing A Computer Simulation of Gravity-Driven
Granular Flow
Scientific Computing Seminar, May 29, 2007
The University of Western Ontario Department of
Applied Mathematics John Drozd and Dr. Colin
Denniston
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Event-driven Simulation
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Collision Time Algorithm

• Tree data structure for collision times
• Ball with smallest collision time value kept at
  bottom left
• Sectoring
• Time complexity O (log n) vs O (n)
• Buffer zones

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Domain Sectoring Collision Times O(log n) vs
O(n)
Racking balls (use MPI or OpenMP )
Staggered in front and back as well
We only update interacting balls locally in
adjacent sectors, and we only do periodic global
updates for output.
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Ball-Ball Collisions

• Treat as smooth disk collisions
• Calculate Newtonian trajectories
• Calculate contact times
• Adjust velocities

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A Smooth Ball Collision
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Numerical Tricks
Avoid catastrophic cancellation, by rationalizing
the numerator in solving the quadratic formula
for
When comparing floating point numbers, take
their difference and compare to float epsilon
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Coefficient of Restitution

• µ is calculated as a velocity-dependent
  restitution coefficient to reduce inelastic
  collapse and overlap occurrences as justified by
  experiments and defined below
• Here vn is the component of relative velocity
  along the line joining the disk centers, B
  (1??)v0??, ? 0.7, v0?g? and ? varying
  between 0 and 1 is a tunable parameter for the
  simulation.

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Velocity
Savage and Jeffrey J. Fluid Mech. 130, 187,
1983.
q
Collision rules for dry granular media
as modelled by inelastic hard spheres
As collisions become weaker (relative velocity vn
small), they become more elastic.
C. Bizon et. al., PRL 80, 57, 1997.
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300 (free fall region)
Donev et al PRL96 "Do Binary Hard Disks Exhibit
an Ideal Glass Transition?"
?
monodisperse
polydisperse
250 (fluid region)
vy
200 (glass region)
dvy/dt
150
y
Polydispersity means Normal distribution of
particle radii
P??
x
z
y
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The density in the glassy region is a
constant. In the interface between the fluid and
the glass does the density approach the glass
density exponentially?
?0 0.9
?0 0.95
?0 0.99
Interface width seems to increase as ?0 ? 1
?
vy

y

How does ? depend on (1 ? ?0) ?
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Density vs Height in Fluid-Glass Transition
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Length Scale in Transition
Slope 0.42 poly Slope 0.46 mono
"interface width diverges"
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300 (free fall region)
Y Velocity Distribution
Poiseuille flow
250 (fluid region)
Plug flow snapshot
200 (glass region)
Mono kink fracture
Mono-disperse (crystallized) only
150
y
z
x
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300 (free fall region)
Granular Temperature
fluid
250 (fluid region)
235 (At Equilibrium Temperature)
200 (glass region)
equilibrium
150
y
glass
x
z
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Fluctuating and Flow Velocity
?Experiment by N. Menon and D. J.
Durian, Science, 275, 1997.
16 x 16
32 x 32
?v
?Simulation results
In Glassy Region !
J.J. Drozd and C. Denniston
vf
Europhysics Letters, 76 (3), 360, 2006
"questionable" averaging over nonuniform regions
gives 2/3
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? ? 1 in fluid glass transition For ?0
0.9,0.95,0.96,0.97,0.98,0.99
Subtracting of Tg and vc and not averaging over
regions of different ?vx2?
Down centre
Slope ? 1.0
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"Particle Dynamics in Sheared Granular Matter"
Physical Review Letters 85, Number 7, p. 1428
(2000)
Experiment (W. Losert, L. Bocquet, T.C. Lubensky
and J.P. Gollub)
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Velocity Fluctuations vs. Shear Rate
U
Slope 0.406 ? 0.018
Experiment
Slope 0.4
From simulation
Physical Review Letters 85, Number 7, (2000)
Must Subtract Tg !
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Shear Stress
q
y
x
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Viscosity vs Temperature

• can do slightly better

Slope 2 ? 1 1.92 ? 0.084
Transformation from a liquid to a glass
takes place in a continuous manner.
Relaxation times of a liquid and its Shear
Viscosity increase
very rapidly as Temperature is lowered.
"anomalous" viscosity. Is a fluid with
"infinite" viscosity a useful description of the
interior phase?
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Experimental data from the book Sands, Powders,
and Grains An Introduction to the Physics of
Granular Materials By Jacques Duran.
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Related to Forces Impulse Distribution
Most frequent collisions contributing to
smallest impulses
Simulation ?
Impulse defined Magnitude of momentum after
collision minus momentum beforecollision.
Experiment ?
(Longhi, Easwar)
Quasi-1d Theory ?
(Coppersmith, et al)
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Power Laws for Collision Times
Collision time time between collisions
1) spheres in 2d 2) 2d disks 3) 3d spheres
Similar power laws for 2d and 3d simulations!
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Comparison With Experiment
? experiment 1.5 vs. simulation 2.8
Discrepancy as a result of Experimental response
time and sensitivity of detector.
? Experiment
Spheres in 2d 3d Simulation with front and
back reflecting walls separated one diameter
apart ?
Pressure Transducer
? Figure from experimental
paper Large Force Fluctuations in a Flowing
Granular Medium Phys. Rev. Lett. 89, 045501
(2002) E. Longhi, N. Easwar, N. Menon
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Probability Distribution for Impulses vs.
Collision Times (log scale)
? 2.75
? 1.50
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?random packing at early stage ? 2.75
Is there any difference between this glass and a
crystal? Answer Look at Monodisperse grains
crystallization ? at later stage ? 4.3
Disorder has a universal effect on Collision
Time power law.
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Summary of Power Laws
Radius Polydispersity 2d disks Spheres in 2d 3d spheres
0 (monodisperse) 4 4.3 4
15 (polydisperse) 2.75 2.85 2.87
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Conclusions

• A gravity-driven hard sphere simulation was used
  to study the glass transition from a granular
  hard sphere fluid to a jammed glass.
• We get the same 2/3 power law for velocity
  fluctuations vs. flow velocity as found in
  experiment, when each data point is averaged over
  a nonuniform region.
• When we look at data points averaged from a
  uniform region we find a power law of 1 as
  expected.
• We found a diverging length scale at this jamming
  (glass) to unjamming (granular fluid) transition.
• We compared our simulation to experiment on the
  connection between local velocity fluctuations
  and shear rate and found quantitative agreement.
• We resolved a discrepancy with experiment on the
  collision time power law which we found depends
  on the level of disorder (glass) or order
  (crystal).

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Normal Stresses Along Height
Momentum Conservation?k?ik?gi 0
Weight not supported by a pressure gradient.
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Momentum Conservation