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Particle Simulations. Benjamin Glasser. Overview. Physics of a collision Experimental perspective Instantaneous collisions Sustained contacts Particle Simulations Hard particle models Event-driven Soft particle models Time stepping Continuum models. - PowerPoint PPT Presentation
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Particle Simulations
Particle Simulations
Benjamin Glasser
Particle Simulations 2
Overview
• Physics of a collision– Experimental perspective
• Instantaneous collisions• Sustained contacts
• Particle Simulations– Hard particle models
• Event-driven
– Soft particle models• Time stepping
• Continuum models
http://www.ux1.eiu.edu/~cfadd/1150/07Mom/Images/NewtPend.gif
http://www.ux1.eiu.edu/~cfadd/1150/07Mom/Images/boing.gif
Particle Simulations 3
Why?
• To better understand, control, and optimize– Fluidization processes
• Fluid bed reactors• Catalyst manufacture
– Solids handling operations• Powder mixing• Hopper flow
– Geophysical flows• Avalanches• Mudslides
– Geophysical formations• Sand dunes• Martian topography
http://www.rrdc.com/images/ph_peru_rockslide_lrg.jpg
http://photojournal.jpl.nasa.gov/jpegMod/PIA02405_modest.jpg
Particle Simulations 4
History
• Leonardo da Vinci (1452-1519) – first to device a simple and convincing experiment demonstrating dry friction.
• Charles de Coulomb (1736-1806) – Coulomb laws of dry friction between solids – would be extended to granular
materials.• Michael Faraday (1791-1872)
– examined how vibrations affect sand piles.• William Rankine (1820-1872)
– examined friction in granular materials.• H. Jannsen (1880’s)
– model of stresses in silos (granular material in a cylinder)• Lord Rayleigh (1842-1919)
– further work on stresses in containers• Osborne Reynolds (1842-1912)
– dilatency – expansion of material during shear• Ralph Bagnold (1950’s)
– sand dunes, role of particle-particle interactions vs. fluid-particle interactions
Particle Simulations 5
Benefits
• Ability to see “inside” granular flows• Relatively cheap• Permit theoretical investigations• Investigate transitions between fluid-like and solid-like behavior• Safer to run computer simulations• Validate granular experiments• Trace every particle
– Part of a force chain?• Versatility to be used for similar systems• Quick answers
– Industry pleasing• Manipulate parameters
– Coefficient of restitution– Coefficient of friction
Poschel and Scwager. Computational Granular Dynamics. Springer, New York, 2005.
http://www.nature.com/nature/journal/v435/n7045/images/4351041a-f1.2.jpg
Particle Simulations 6
A Simple Scenario
Two particles approach one another with known initial velocity in a frontal (normal) impact v1
m1 m2
v2
Before:
W. Goldsmith, Impact: The Theory of Physical Behavior of Colliding Solids, 1960
m1
u1
m2
u2
After:
22112211 umumvmvm
2 unknowns (u1 and u2)
Conservation of momentum!
Require another equation
Particle Simulations 7
A Simple Scenario
If kinetic energy were conserved (elastic spheres):
222
211
222
211 2
1
2
1
2
1
2
1umumvmvm
221
21
21
211
2v
mm
mv
mm
mmu
221
121
21
12 v
mm
mmv
mm
mu
and
Then:
However, energy is not conserved
inelastic collisions
Particle Simulations 8
Inelastic Collisions
• Initial velocity is v, rebound velocity is –v• Unique to granular materials
• Why?– Permanent deformation
• Microcracks• Deformed surface
– Acoustic Waves• Dissipated through heat http://
www.mathworks.com/access/helpdesk/help/techdoc/math/ballode.gif
Particle Simulations 9
Coefficient of Normal Restitution
• ε is the coefficient of normal restitution•
• Ratio of pre-collisional to post-collisional velocities
• Change in Kinetic Energy
• Function of approaching velocity
• Common values:
1
21
21
vv
uu
221
212
1vvmEkin
Glass spheres 100 cm/s 0.95 - 9
Steel spheres 100 cm/s 0.91 - 0.95
Brass spheres 100 cm/s 0.9
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Particle Simulations 10
Matrix Equations
• Reference: System center of gravity– Particle - Particle
– Particle – Wall (infinite mass, rigid body)
2
1
2
1
2
1
2
12
1
2
1
v
v
u
u
1
0
1
0
1
01
v
v
u
u
u0: velocity of wall
u1: particle velocity
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Particle Simulations 11
Normal Collision - No Friction
Only translational motion x and y are components of linear
momentum• ux = -vx
•
• uy = vy
• 1 = 0
Xvum xx
x
y
Particle Simulations 12
Spin and Friction
Spherical, Spinning Ball - Vertical Wall Precollisional velocities
• vx
• vy
• 0
Compute:• ux
• uy
• 1
x
y
x
y
Particle Simulations 13
Normal Collision - Friction
Case I: Gliding velocity remains positive
and non-zero Xvum xx
Xvum yy
Xama 012
5
2
xx vu
: coefficient of friction between the ball and the wall
Can distinguish between two cases based on
x
y
x
y
v
av
01
2
7
True when:
Coulomb
Rotation
4 Equations, 4 Unknowns (X, ux, uy, 1)
a: radius
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Particle Simulations 14
Normal Collision - Friction
Case II: Gliding velocity drops to 0 during the
collision
aYma 012
5
2
Xvum xx
Yvum yy
xx vu 01 au y
x
y
x
y
v
av
01
2
7
True when:
Rotation
Pure rotation
Particle Simulations 15
Contribution from spin
Non-Frontal Collision with Friction
• Two particles– Diameter d1 and d2
– Mass m1 and m2
– Unit vector normal to contact•
– Relative velocity at point of contact•
21
21
rr
rrn
ndd
vvvc
2
21
121 22
Relative linear velocity
The magnitude of the relative velocity |vc| increases when the individual velocities point in opposite directions and when the rotations are in the same direction
12
v2
v1
n
1
2
Particle Simulations 16
Tangential Velocity - Rotational Motion
Normal component of vc
• Tangential component of vc
• Tangential unit vector
•
Angle of impact• From normal vector n to
relative velocity vc
•
nnvv cnc
ncc
tc vvv
tc
tc
v
vt
Normal collision: =
Glancing collision: = /2
vc
n
vc,, n
12
v2
v1
vc
n
2
Particle Simulations 17
Momentum Change
• Linear change of momentum from a collision
• Tangential contribution to angular momentum (torque)– Normal component of P has no contribution
for i=1,2
– Ii: moment of inertia of particle i
– The change in angular momentum is the same for both particles
222111 vumvumP
iii
i
d
IPn '2
Particle Simulations 18
Outcome of the CollisionMaking use of the equations, one can compute the outcome of the collision
Linear Velocity
PnI
d
1
11
'1 2
PnI
d
2
22
'2 2
Angular Velocity
111 m
Pvu
222 m
Pvu
nc
nc vu
ndd
vvvc
2
21
121 22
Where:
nc
n vmP 112
21
2112 mm
mmm
cos112 tvmP ct
21
21
rr
rrn
nt PPP The translation velocities
before and after the
collision are still related by
this
Particle Simulations 19
Rolling
• The previous equations describe all collisions on the basis of Coulomb’s Law () and and nothing else
• Ignored an important physical mechanism: rolling– Heuristic model to agree with experiments
– Fire two spheres together with initial spin and examine the outcomes
– Coefficient of tangential restitution, • Equal to the smallest of two values:
• 0 – Rolling; 1 – Sliding/Gliding
1 2
Particle Simulations 20
Critical Angle – 0
Applies for this form of the momentum equation
tc
nc vmvmP 1
7
21 1212
0
0 1
1
2
7tan
Applies for this form of the momentum equation
For < 0:Sliding/Gliding regime
1,10
cot12
711
S.F. Forrester et al., Phys. Fluids, 6, p.1108 (1994)
Small values of correspond to dry friction
(previous result)
For > 0:Rolling regime
Particle Simulations 21
Sustained ContactsWhat about particles that are not completely rigid?
Spheres can deform Can have interpenetration of the spheres, resulting in long contact times
Consider 2 identical spheres– Mass, M– Radius, R relative velocity v
Fan and Zhu, Principles of Gas-Solid Flow, 1998
Hertz’s elastic energy
2
5
2
1 kEe
where
RE
k2115
24
E – Young’s modulus – Poison’s ratio
R R
Stored by each sphere during contact
Ratio of transverse strain to longitudinal strain
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Particle Simulations 22
Sustained Contact Quantities
• Upon impact the kinetic energy is converted into a reduced kinetic energy and stored elastic energy– Energy balance:
2
2
52
dt
dmkmv
• Velocity drops to zero when the two spheres have overlapped by a distance 0
5
45
2
0 vk
m
• The entire duration of the collision (up to 0 and recoil)
0
0 2
52
2
mk
v
d 5
1
2
2
94.2
vk
m
0dt
d
Important: only depends weakly on v – exponent is 1/5. Thus the duration of impact is only a weak function of the initial relative velocity
Particle Simulations 23
Example Calculation
Consider aluminum beads ( = 2.70 g/cc) 1.5 mm in diameter moving towards one another at 5 cm/s.
E = 6 x 1011 dynes/cm2
= 0.3What is the duration of contact?
D=1.5 mm
Answer:
k = 7 x 1010 dynes/cm3/2RE
k2115
24
3
3
4Rm m = 4.77 x 10-3 g
5
1
2
2
94.2
vk
m = 1.15 x 10-5 s
Particle Simulations 24
Issues
• Exceed elasticity limit– Plastic, not elastic deformations– Model can be adjusted to handle this
• Energy dissipation– Sound waves– Heat
• Spin complications– Can handle similar to rigid spheres
Particle Simulations 25
Particle Simulations
• Follow trajectories of individual particles– Incorporate statics and dynamics
• Methods– Particle dynamics
• Hard Particles• Soft Particles
– Cellular automata• Motion evolves according to simple rules based on lattice
sites– Monte-Carlo
• Analogous to molecules but change probabilities to match particles
• Assumption of molecular chaos
Baxter and Behringer (1990) Cellular Automata of Granular Flows. Phys. Review A., 42, 1017-1020
Rosato et al. (1987) Monte Carlo simulation of particulate matter segregation. Powder Technology, 49, 59-69
Source: Jacques Duran, Sands, Powders, and Grains, Springer, 1999.
Source: Thorsten Poschel, Thomas Schwager, Computational Granular Dynamics, Springer, 2005.
Particle Simulations 26
Boundary Conditions
• Wall constructed from individual particles• Containers do not follow Newton’s
equation of motion– Predetermined path as a function of time
• Vibrated bed• Moving plate• Rotating vessel
• Periodic boundaries– Can mimic infinitely-wide regions
Particle Simulations 27
Initial Conditions
• Depending on the algorithm (predictor-corrector), you may need to define higher order derivatives
• Most long-term behavior is independent of initial conditions
• Often, random (non-overlapping) positions and velocities are assigned
Particle Simulations 28
Hard Particles
• Event Driven (ED)• Strictly binary collisions• No integration of the equations of motion
– More efficient
• Without gravity – straight line paths• With gravity – parabolas• Time to collision is identical
With gravityWithout gravity
Collision at (xc, yc + 1/2gt2)Collision at (xc, yc)
v10 v2
0
Particle Simulations 29
di
Particle Motion
• Consider a particle– Position vector xi
– Velocity vector vi
– Initial position (t=0) • xi = xi
0
• vi = vi0
– Undeterred position at time t200
2
1attvxx iii
x
yxi
0 = (x, y)vi
0 = (vx, vy)
Particle Simulations 30
Collision Prediction
Two particles collide if:
i
jdj/2
di/2
22ji
ji
ddxx
Insert the equation of motion:(same force on each particle, so the accelerations are
equal as well and cancel)
22 0000 ji
jiji
ddtvvxx
2
22000000200
22 2
ji
jijijiji
ddtvvtvvxxxx
Expand : Real root > 0 collision
Particle Simulations 31
Scheduling• Can schedule the events for N particles in a box
– 4 walls of a box 4N events– N particles N(N-1) events
• An example stack from t=0
• March to t = 0.1, execute the collision, then recalculate the stack– The whole stack, or– Just events with particles 5 or 7 (much faster)
time wall event particle event type
0.1 no yes particle 5 and 7
0.25 yes no particle 3 and wall 1
0.33 no yes particle 2 and 5
…
Particle Simulations 32
Gridding
Predicting collisions between all particles wastes time
• Black arrows will rarely collide
Divide region into cells• Search within cells for collisions• Include cell crossings as events• Track the cell location of each
particle
Particle Simulations 33
Inelastic collapse
Left and right particle collide with the middle particles alternately until the motion of all three particles is zero
Only for a constant coefficient of restitution– Not valid for real materials– Experiments suggest is a function of v
Poschel and Scwager. Computational Granular Dynamics. Springer, New York, 2005.
Particle Simulations 34
2D - Couette Flow
U
-U
Particle Simulations 35
3D - Couette Flow
Particle Simulations 36
Soft Particles
• Force-based• Time stepping• Small overlap allowed• Useful for
– Statics– Dense quasi-static flows
• To follow particles:
F is the normal force on particles proportional to amount of overlap
m
Ftvv ii '
''iii tvxx
Particle Simulations 37
Advanced Algorithms
• Verlet Algorithm– Position from acceleration
42)()()(2 tOttattrtrttr
?
program termination
data output
initialization
predictor
force computation
corrector
Poschel (2005) Computational Granular Dynamics. Springer, New York.
• Predictor – Corrector (left)– Predict future acceleration using
previous position time derivatives– Acceleration from force– Adjust the predicted value
• Acceleration from force
maF
– Back out velocity
t
ttrttrtv
2
)(
Particle Simulations 38
Contact Models
• Spring and Dashpot
0 jijiij rrRR
nn
nij kF
– may be related to kn and n
• Mutual compression of particles i and j
otherwise 0
0 if ijt
ijn
ijij
FFF
Poschel (2005) Computational Granular Dynamics. Springer, New York.
Particle Simulations 39
Force Calculation
• In addition to Fn
– Gravity– Wall forces– Interstitial fluid– Spin– Cohesion
Total of all is the resultant force on the particle, F
Fw
Fg
F2
F3
Particle Simulations 40
More Contact Models
Hertz2/3n
nij kF
2/12/3nn
nij kF
)(unloading 0 ,
(loading) 0 ,
02
1
k
kF n
ij
Kuwabara and Kono
Walton and Braun (right)
Can also include friction in the mechanism
Poschel (2005) Computational Granular Dynamics. Springer, New York.
Particle Simulations 41
Discrete ModelsShape Issues:
Spheres
Collections of spheres
Arbitrary surfaces
Simplest
Most Complex
Needles
Flakes
Particle Simulations 42
Results: Constant inter-particle force Model
• Optimized Condition: 20,000 particles, 2mm diameter, in an axially smaller rotating drum of 9 cm radius and 1 cm length are considered. The sidewalls are made frictionless to avoid end wall effects.
• Inter-particle or particle-wall cohesive force is varied such that K=45-75
Fast-Flo Lactose
Size: 100 micron
RPM = 7 K = 45 ; RPM = 20
sp = 0.8 ; dp = 0.1 ;
sw = 0.5 dw = 0.5 ;
• Avalanches start appearing at K = 30 and become bigger at K =45.
• Distinct angles of repose are visible at top and bottom of “cascade” layer.
Credit: F. Muzzio
Particle Simulations 43
Similarities:Increase in size of avalanches.Mixture of splashing and timid bulldozing.
Avicel-101; Size : 50 m; RPM = 7
Results: Constant inter-particle force Model
Dynamic friction within the particles and the cohesion are increased to simulate the flow of more cohesive material. Wall friction is increased.
Reg. Lactose; Size : 60 m; RPM = 7
K = 75 ; RPM = 20
sp = 0.8 ; dp = 0.6 ;
sw = 0.8 ; dw = 0.8
K = 60 ; RPM = 20
sp = 0.8 ; dp = 0.6 ;
sw = 0.8 ; dw = 0.8
Similarities:Mixture of chugging and bulldozing. Periodic AvalanchesBigger Avalanches
Credit: F. Muzzio
Particle Simulations 44
Uniform Binary SystemComparison of model and experiment
• Optimized Condition: 10,000 red and 10,000 green particles of same size (radius: 1mm) are loaded side by side along the axis of the drum. The drum of radius 9 cm and length of 1 cm is considered. The sidewalls are made frictionless to avoid wall effects.
• Inter-particle or particle-wall cohesive force is varied such that K=0 – 120Glass beads (40 m)
RPM=10
K=0
RPM = 20
•No avalanches. •Mixes well in 3-4 revolutions
•Avalanches appearing•Slower mixing
K=60
RPM = 20
Colored Avicel (50 m)
RPM=10
Credit: F. Muzzio
Particle Simulations 45
Non-uniform Binary System
Experiment: Blue (30 mm) and Red glass beads (50 mm) of equal mass are axially loaded side by side in a drum of radius = 7.5 cm and length 30 cm.
Simulation: 8000 blue particles (1mm) and 2370 red particles(1.5mm) of same density are loaded side by side along the axis of the drum. Red and blue particles of are of the same total mass.
Inter-particle Force ModelFRR = KRR WB (red-red pair)FRB = KRB WB (red-blue pair)FBB = KBB WB (blue-blue pair)WB is the weight of a blue particle.
Non-cohesive binary mixture : Axial Size Segregation is evident in both the simulation and experiments.
RPM =12 RPM =20
(K.M.Hill et.al Phys. Rev. E.,49,1994).Credit: F. Muzzio
Particle Simulations 46
Example –A Particle/Wall Contact
2D Disk - Flat, Vertical Wall• R = 0.5 mm
• kn = 10 N mm-3/2
• t = 0.25 s• m = 0.148 kg
• vp0 = (vx
0, vy0) = (1 mm s-1, 0 mm s-1)
• xp0 = (xx
0, xy0) = (1.2 mm, 1 mm)
• xw = 0 (line from origin to +∞)
xy
xy
Particle Simulations 47
Results
2/3nn
ij kF
0 jijiij rrRR
maF
Force
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
Overlap
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Overlap Rate
-1.2-1
-0.8-0.6-0.4-0.2
00.20.40.6
0 0.5 1 1.5 2
Time (s)
Collision!!
tavvii 0
tvxx ii i 0
Position
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Velocity
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Acceleration
00.5
11.5
22.5
33.5
44.5
0 0.5 1 1.5 2
Time (s)
Wall
Particle Simulations 48
Hard vs. Soft
• When to use hard (event) instead of soft (force)– Average collision duration <<
Time between collisions• Granular gases cosmic dust
clouds
– Unknown interaction force• Non-linear materials• Complicated particle shapes• Can experimentally determine pre- and post-collisional
velocities
Poschel (2005) Computational Granular Dynamics. Springer, New York.
Particle Simulations 49
Drawbacks
• Computationally expensive– t << to calculate forces– 20,000 particles
• Real time of seconds to minutes
• Dynamic issues– Strain hardening– Contact erosion over time
Particle Simulations 50
Continuum
• Discrete particles replaced (averaged out) with continuous medium
• Quantities such as velocity and density are assumed to be smooth functions of position and time
• Volume element (dv) contains multiple particles• Time (dt) should be large compared to time
required for a particle to cross dv
Truesdall, C. and Muncaster, R.G. (1980) – Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatonic Gas. Academic Press, pp. xvi.
Particle Simulations 51
Continuum
Stay in this region where the average quantities are equal to the bulk
10-10 10-6 10-5 10-4 10-3 10-2 10-110-710-810-9
= mass/volume
l (in meters)
Particle Simulations 52
Continuum Simulation for Granular Flows
Consider a mass balance over a stationary volume element of size ΔxΔyΔz. Rate of mass accumulation = rate of mass in – rate of mass out
]|)(|)[(
]|)(|)[(
]|)(|)[(
zzzszzs
yyysyys
xxxsxxss
vuvuyx
vuvuzx
vuvuzyt
vzyx
Δx Δy
ΔzGranular Flow
Dividing by ΔxΔyΔz and taking limits, we obtain
z
vu
y
vu
x
vu
t
v zsysxss
or )( vut
vs
We can extend this approach to calculate momentum and energy balance. Particle Dynamic Simulations – Limited by computational resources Continuum Simulations with physically realistic closures (Use a set of equations for design, control and optimization) More efficient design, scaling & control through Continuum modeling
Average over a small region in a granular flow.
Particle Simulations 53
Hydrodynamic Model
Boundary Conditions (Johnson and Jackson, 1987)
Force Balance
Energy Balance
0)(
uvt
v
gvvut
uv sss
][
Momentum Balance
Pseudo-Thermal Energy Balance
uquTv
t
vTcss
s
:)2
3(
)23
( ''
3
1vvm
Mass Balance
0))/(1(6
||3'
|| 3/1maxmax
2/1
vvv
uvT
u
nusls
sl
ssl
0])/(1[
)3()]1(
4
1[
3/1maxmax
2/12
sslws u
vvv
TveTqn
u solids velocityv solids fractionT granular temperature
Particle Simulations 54
Computational Approaches
Steady State Simulations
time
1 Steady state solution is augmented by a small perturbation.The small perturbation has a periodic form.
s = - i determines the rate of growth or decay of the perturbation waves.
)exp()exp()(1̂1 xiksty x
Linear Stability Analysis
Transient Integration on Linear Instabilities
Direct Integration
Bifurcation Analysis
Bubble formation in a gas-particle system.
( Anderson et al. 1995)
Particle Simulations 55
Couette and Channel GeometryFrame 001 29 Oct 2002 | | | | | | | | |Frame 001 29 Oct 2002 | | | | | | | | |
u0
u0
Couette Flow
Channel F
low
Particle Simulations 56
Steady State Solutions for Couette Flows
(Nott et al. 1999)
The structure of fully developed solutions (a) Particle volume fraction, (b) pseudo-thermal temperature, (c) axial velocity.
Broken lines: wall is a source of energy.Solid lines: wall is a sink of energy.
T*
Particle Simulations 57
Linear Stability for Couette Flows
(Alam and Nott, 1998)
Eigenfunctions Associated with Solids Fractions
Instability dominated by symmetric patterns
Instability dominated by anti-symmetric patterns
Particle Simulations 58
Transient Integration in Couette Flows
(Wang and Tong 1998)
Particle Simulations 59
Couette Flow with Binary Particles (Steady State)
A: large/heavy particles
B: small/light particles
(R=dA/dB M=mA/mB)
y
x
u0
-u0
Non-uniform Solids DistributionSpecies SegregationNon-linear Velocity Distribution
(Equal Density: R=2, ep=0.9 Mean solids fraction=0.1)
Particle Simulations 60
Steady State Solutions for Channel Flows
(Wang et al. 1997)
u* dimensionless velocityT* dimensionless granular temperaturev solids fraction
Solid line: wall is a sink of energy.Broken line: wall is a source of energy.
Particle Simulations 61
Linear Stability for Channel Flows
Instability dominated by symmetric patterns
Instability dominated by anti-symmetric patterns
(Wang et al. 1997)
Eigenfunctions Associated with Solids Fractions
Particle Simulations 62
Transient Integration for symmetric patterns
Transient Integration for anti-symmetric patterns
(Wang and Tong 2001)
Solids Fraction Distribution in the Channel
Transient Integration in Channel Flows
Particle Simulations 63
Channel Flow with Binary Particles (Steady State)
A: large/heavy particles
B: small/light particles
(R=dA/dB M=mA/mB)
y
x
(Equal Density: R=2, ep=0.9 Mean solids fraction=0.1)