Managing A Computer Simulation of Gravity-Driven

Granular Flow

The University of Western OntarioDepartment of Applied MathematicsJohn Drozd and Dr. Colin Denniston

Scientific Computing Seminar, May 29, 2007

Event-driven Simulation

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

Domain Sectoring Collision Times: O(log n) vs O(n)

Racking balls (use MPI or OpenMP )

Staggeredin frontand backas well

We only update interacting balls locally in adjacent sectors, and we only do periodic global updates for output.

Ball-Ball Collisions

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

• Adjust velocities

02 2222 ijijijijijijijijijij rtbtvtvrttr

||2/ ijijijji vrbvv

ijijij vrb

1/

0

,

,

2

22

2222

22

2222

ji

jijiji

ijij

iiiijij

iiijii

ijijijijij

jiiijjijji

jijiijji

jiji

jijijiji

ijjjijii

mmm

rbv

rb

vvvrbm

vmvmrCpp

bmvrmCpprCC

ppprCprCCpp

prCprCpp

pppp

pppppppp

rCpprCpp

A Smooth Ball Collision

Numerical Tricks

020 22222 ijijijijijijij rtbtvctbta

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:

floatxxifxxif 2121

acbb

c

acbb

acbba

acbbtbfor ij4

2

4

42

4:022

22

acbb

c

acbb

acbba

acbbtbfor ij4

2

4

42

4:022

22

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.

0

0

,,1

vvvvvB

vn

nnn

ab

v n

0 50 100 150 200 250

0.9

0.92

0.94

0.96

0.98

1

1

2

3

4

5

6

Velocity q

q

qrqr

mmmm

mmv

rr

r

r n

2

1

11

22

212

1

'2

'1 1

.dissipated isEnergy

,,1ˆˆ

,

,11

:lossenergy thedeterminesn restitutio oft coefficiendependent velocity The

012

120

00

0

7.0

00

particleofradiusagavqvvqvv

vv

vvvv

v

v

n

nn

n

n

1r 2rCollision rules for dry granular media asmodelled by inelastic hard spheres

As collisions become weaker(relative velocity vn small),they become more elastic.

C. Bizon et. al., PRL 80, 57, 1997.

Savage and JeffreyJ. Fluid Mech. 130, 187, 1983.

300 (free fall region)

250 (fluid region)

200 (glass region)

150

Polydispersity meansNormal distribution of particle radii

y

x z

vy

dvy/dt

P

Donev et al PRL96"Do Binary Hard DisksExhibit an Ideal GlassTransition?"

monodispersepolydisperse

1/1 cBTkDP

y

0 = 0.9 0 = 0.95 0 = 0.99

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?

Interface width seems to increase as 0 1

y

abcd

v y

vdytd

A

y0 100 200 300 400

ssalg

diulf eerfllaf

0.10.20.30.40.50.60.7

5.

10.

15.

20.

0.

5.

10.

15.

20.

0 100 200 300

0.010.1

110

0

0.5

1

1.5

2

2.5

vy

yAc exp

How does depend on (1 0) ?

Density vs Height in Fluid-Glass Transition

yAc exp

160 180 200 220 240 260 280 300y

4

3

2

1

0

goL01 c

0 0.9990 0.9950 0.990 0.980 0.970 0.960 0.950 0.90 0.8

Length Scale in Transition

yAc exp

44.001

"interface width diverges"

Slope = 0.42 polySlope = 0.46 mono

300 (free fall region)

250 (fluid region)

200 (glass region)

150

Y VelocityDistribution

0 5 1015202530x

0.51

1.52

v y cy150

0 5 1015202530x

0.51

1.52

v y by200

0 5 1015202530x

1

234

v y ay250

Plug flowsnapshot

Monokinkfracture

Poiseuille flow

y

zx

Mono-disperse(crystallized)only

300 (free fall region)

250 (fluid region)

200 (glass region)

150

Granular Temperature

235 (At Equilibrium Temperature)

fluid

equilibrium

glass

0 5 10 15 20 25 30x

0.1

0.2

0.3

0.4

v yv y2v

x2

y200

0 5 10 15 20 25 30x

0.2

0.4

0.6

0.8

1

1.2v yv y2

v

x2 y235

0 5 10 15 20 25 30x

5

10

15

20

25

30

35

v yv y2v

x2

y250

y

x z

2/3flowvv

Experiment by N. Menon and D. J. Durian, Science, 275, 1997.

Simulation results

Fluctuating and Flow Velocity

v

vf

0.5 1 5 10vy

0.1

0.2

0.5

1

2

5

vy

In Glassy Region !

Europhysics Letters, 76 (3), 360, 2006

J.J. Drozd and C. Denniston

16 x 1632 x 32

"questionable" averaging over nonuniform regions gives 2/3

cygx vvTvv 2 1 in fluid glass transitionFor 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

1.5 1 0.5 0 0.5log10vy vc

0.25

0

0.25

0.5

0.75

1

1.25

1.5

gol01v x2

Tg Slope = 1.0

50 100 150 200 250 300 350 400y

3

2

1

0

1

gol01v x2

Tg

0 0.9950 0.990 0.980 0.970 0.960 0.950 0.90 0.8

Down centre

Experiment:(W. Losert, L. Bocquet,T.C. Lubensky andJ.P. Gollub)

Physical Review Letters 85, Number 7, p. 1428 (2000)

"Particle Dynamics inSheared Granular Matter"

Velocity Fluctuations

vs. Shear Rate

Physical Review Letters 85, Number 7, (2000)From simulation

Experiment

Must Subtract Tg !

3.5 3 2.5 2 1.5 1log10xvyU log10U2.4

2.2

2

1.8

1.6

v x2 T

g12

h 240, yg 185, 0 0.9

Slope = 0.406 0.018

Slope = 0.4

5 10 15 20 25 30x

1.21.31.41.51.61.7

v yh 240

U

yx

yx v

Shear Stress

yqxqqrrf

yqxqqrrt

c

xy

ˆˆˆˆˆ121

ˆˆˆˆˆ1211

21

21collisions

xyx0.15554674 slope0.00165986slope

qrrfc )(121 slopes of ratio 21

5 10 15 20 25 30x

0.02

0.01

0

0.01

0.02

q.x

q.y

y100

5 10 15 20 25 30x

3

2

1

0

1

2

3

yx

y100

q

x

y

Viscosity vs Temperature…can do slightly better…

…"anomalous" viscosity.Is a fluid with "infinite" viscosity a useful description of the interior phase?

1.5 1 0.5 0 0.5 1 1.5log10vx

2 Tg3.5

3

2.5

2

1.5

1

0.5

gol

01

gol01 xy xv y

x16

0 0.990 0.980 0.970 0.960 0.950 0.9

Slope ~ 2 11.92 0.084

yxyx

yx

yx

v

v

/

Transformation from aliquid to a glass takesplace in a continuousmanner. Relaxationtimes of a liquid and itsShear Viscosity increase

very rapidly asTemperature islowered.

Experimental data from the book:“Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials”By Jacques Duran.

Simulation

Experiment

Quasi-1d Theory (Coppersmith, et al)

(Longhi, Easwar)

Impulse defined: Magnitude ofmomentumafter collisionminus momentum beforecollision.

Related to Forces:Impulse Distribution

Most frequent collisions contributing to smallest impulses

Power Laws for Collision Times

regionglassy in 2.87 to2.75 0.06 2.81

%15

grainssepolydisperP

Similar power laws for 2d and 3d simulations!

Collision time= time between collisions

0.01 0.02 0.05 0.1 0.2 0.5

1. 107

0.00001

0.001

0.1

10

P

1) spheres in 2d2) 2d disks3) 3d spheres

Comparison With Experiment

Figure from experimental paper:“Large Force Fluctuations in a Flowing Granular Medium”Phys. Rev. Lett. 89, 045501 (2002)E. Longhi, N. Easwar, N. Menon

: 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 withfront and backreflecting wallsseparated onediameter apart

Pressure Transducer

P

15 12.5 10 7.5 5 2.5 0ln

10

8

6

4

2

0

2

nlI

= 2.75

= 1.50

Probability Distribution forImpulses vs. Collision Times (log scale)

random packingat early stage = 2.75

crystallization at later stage = 4.3

Is there any difference between this glass and a crystal? Answer: Look at Monodisperse grains

Disorder has a universal effect on Collision Time power law.

Radius Polydispersity

2d disks Spheres in 2d 3d spheres

0 %(monodisperse)

4 4.3 4

15 %(polydisperse)

2.75 2.85 2.87

Summary of Power Laws

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).

Normal Stresses Along Height

Weight not supported by a pressure gradient.

0xyyy

gyyyyxx

Momentum Conservationkik+gi = 0

0 100 200 300 400y

12

10

8

6

4

2

0

yy

Momentum Conservation

stressshear by supportedWeight 0yxyx

gyyyyxx

5 10 15 20 25 30x

3

2

1

0

1

2

3

yx

y100

5 10 15 20 25x

0.050.1

0.150.2

0.25

xxy y

yy,

g y100