Implementation and Refinement of a Comprehensive Model for
Dense Granular FlowsYile Gu, Ali Ozel, Sebastian Chialvo, and Sankaran Sundaresan
Princeton University
This work is supported by DOE-UCR grant DE-FE0006932.
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Monday, April 29, 2015
Monday, April 27, 2015
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Granular rheological behavior
2Monday, April 27, 2015
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Granular rheological behavior
Ubiquitous in nature and widely encountered in industrial processes,
2Monday, April 27, 2015
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Granular rheological behavior
Ubiquitous in nature and widely encountered in industrial processes,
Complex behavior: multiple regimes of rheology, jamming
2Monday, April 27, 2015
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Granular rheological behavior
Ubiquitous in nature and widely encountered in industrial processes,
Complex behavior: multiple regimes of rheology, jamming
2
Shear flow of frictional particlesin a periodic box
Monday, April 27, 2015
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Granular rheological behavior
Ubiquitous in nature and widely encountered in industrial processes,
Complex behavior: multiple regimes of rheology, jamming
2
Shear flow of frictional particlesin a periodic box
Shear flow of frictional particles
with bounding walls
Shearing plate
Shearing plate
Monday, April 27, 2015
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Computational methodology
• Simulate particle dynamics of homogeneous assemblies under simple shear using discrete element method (DEM).
‣ Linear spring-dashpot withfrictional slider.
‣ 3D periodic domain without gravity
‣ Lees-Edwards boundary conditions
• Extract stress and structuralinformation by averaging.
3LAMMPS code. http://lammps.sandia.gov S. J. Plimpton. J Comp Phys, 117, 1-19 (1995)Monday, April 27, 2015
/25
Dense phase rheology: Questions asked
4Monday, April 27, 2015
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Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
4Monday, April 27, 2015
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Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
Non-cohesive
4Monday, April 27, 2015
/25
Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
Non-cohesive
Cohesive
4Monday, April 27, 2015
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Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
Non-cohesive
Cohesive
Rheological models
4Monday, April 27, 2015
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Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
Non-cohesive
Cohesive
Rheological models
Steady state models that bridge various regimes
4Monday, April 27, 2015
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Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
Non-cohesive
Cohesive
Rheological models
Steady state models that bridge various regimes
Modified kinetic theory (for non-cohesive particles)
4Monday, April 27, 2015
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Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
Non-cohesive
Cohesive
Rheological models
Steady state models that bridge various regimes
Modified kinetic theory (for non-cohesive particles)
Wall Boundary conditions
4Monday, April 27, 2015
/25
Dense phase rheology: Questions asked
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
Non-cohesive
Cohesive
Rheological models
Steady state models that bridge various regimes
Modified kinetic theory (for non-cohesive particles)
Wall Boundary conditions
Implementation of modified kinetic theory in MFIX/openFOAM
4Monday, April 27, 2015
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10 6 10 4 10 2 10010 10
10 8
10 6
10 4
10 2
100
ˆ! ! !d/!
k/("sd)
pd/k
Flow map: Non-cohesive Particles
5
μ=0.5
φc = 0.58710 4 10 2 100 102 104
10 8
10 6
10 4
10 2
100
! !d/!
k/("sd)
p! d
/k
# = 0 .5
# = 0 .52
# = 0 .54
# = 0 .55
# = 0 .56
# = 0 .57
# = 0 .578
# = 0 .584
# = 0 .588
# = 0 .594
# = 0 .6
# = 0 .61
# = 0 .618
p≡
Previous studies
• Computational
‣ C. S. Campbell, J. Fluid Mech. 465, 261 (2002).
‣ T. Hatano, J. Phys. Soc. Japan 77, 123002 (2008).
• Experimental
‣ K. N. Nordstrom et al. Phys. Rev. Lett. 105, 175701 (2010).
Monday, April 27, 2015
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10 6 10 4 10 2 10010 10
10 8
10 6
10 4
10 2
100
ˆ! ! !d/!
k/("sd)
pd/k
Flow map: Non-cohesive Particles
5
μ=0.5
φc = 0.587
Quasi-static
10 4 10 2 100 102 104
10 8
10 6
10 4
10 2
100
! !d/!
k/("sd)
p! d
/k
# = 0 .5
# = 0 .52
# = 0 .54
# = 0 .55
# = 0 .56
# = 0 .57
# = 0 .578
# = 0 .584
# = 0 .588
# = 0 .594
# = 0 .6
# = 0 .61
# = 0 .618
p≡
Previous studies
• Computational
‣ C. S. Campbell, J. Fluid Mech. 465, 261 (2002).
‣ T. Hatano, J. Phys. Soc. Japan 77, 123002 (2008).
• Experimental
‣ K. N. Nordstrom et al. Phys. Rev. Lett. 105, 175701 (2010).
Monday, April 27, 2015
/25
10 6 10 4 10 2 10010 10
10 8
10 6
10 4
10 2
100
ˆ! ! !d/!
k/("sd)
pd/k
Flow map: Non-cohesive Particles
5
μ=0.5
φc = 0.587
Quasi-static
Inertial
10 4 10 2 100 102 104
10 8
10 6
10 4
10 2
100
! !d/!
k/("sd)
p! d
/k
# = 0 .5
# = 0 .52
# = 0 .54
# = 0 .55
# = 0 .56
# = 0 .57
# = 0 .578
# = 0 .584
# = 0 .588
# = 0 .594
# = 0 .6
# = 0 .61
# = 0 .618
p≡
Previous studies
• Computational
‣ C. S. Campbell, J. Fluid Mech. 465, 261 (2002).
‣ T. Hatano, J. Phys. Soc. Japan 77, 123002 (2008).
• Experimental
‣ K. N. Nordstrom et al. Phys. Rev. Lett. 105, 175701 (2010).
Monday, April 27, 2015
/25
10 6 10 4 10 2 10010 10
10 8
10 6
10 4
10 2
100
ˆ! ! !d/!
k/("sd)
pd/k
Flow map: Non-cohesive Particles
5
• Critical volume fraction and its flow curve distinguish the three flow regimes.
μ=0.5
φc = 0.587
Quasi-static
Inertial
Intermediate
10 4 10 2 100 102 104
10 8
10 6
10 4
10 2
100
! !d/!
k/("sd)
p! d
/k
# = 0 .5
# = 0 .52
# = 0 .54
# = 0 .55
# = 0 .56
# = 0 .57
# = 0 .578
# = 0 .584
# = 0 .588
# = 0 .594
# = 0 .6
# = 0 .61
# = 0 .618
p≡
φc p = αˆγm
Previous studies
• Computational
‣ C. S. Campbell, J. Fluid Mech. 465, 261 (2002).
‣ T. Hatano, J. Phys. Soc. Japan 77, 123002 (2008).
• Experimental
‣ K. N. Nordstrom et al. Phys. Rev. Lett. 105, 175701 (2010).
Monday, April 27, 2015
/25
10 6 10 4 10 2 10010 10
10 8
10 6
10 4
10 2
100
ˆ! ! !d/!
k/("sd)
pd/k
Flow map: Non-cohesive Particles
5
• Critical volume fraction and its flow curve distinguish the three flow regimes.
μ=0.5
φc = 0.587
Quasi-static
Inertial
Intermediate
10 4 10 2 100 102 104
10 8
10 6
10 4
10 2
100
! !d/!
k/("sd)
p! d
/k
# = 0 .5
# = 0 .52
# = 0 .54
# = 0 .55
# = 0 .56
# = 0 .57
# = 0 .578
# = 0 .584
# = 0 .588
# = 0 .594
# = 0 .6
# = 0 .61
# = 0 .618
p≡
φc p = αˆγm
Previous studies
• Computational
‣ C. S. Campbell, J. Fluid Mech. 465, 261 (2002).
‣ T. Hatano, J. Phys. Soc. Japan 77, 123002 (2008).
• Experimental
‣ K. N. Nordstrom et al. Phys. Rev. Lett. 105, 175701 (2010).
=⇒=⇒
k
k
• Role of particle softness:
- Large quasi-static or inertial regime
- Small intermediate regimeMonday, April 27, 2015
/25
10 4 10 2 100 102 10410 10
10 8
10 6
10 4
10 2
100102
! !d/!
k/("sd)
p! d
/k
Pressure scalings for frictional, non-cohesive particles
6
γ∗ = ˆγ/|φ− φc|bp∗ = p/|φ− φc|a
Scaled pressure and shear rate†:
�
Independent of µ
a = 2/3
b = 4/3
Choose exponents:
γ∗p
p∗
μ=0.5
Monday, April 27, 2015
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10 4 10 2 100 102 10410 10
10 8
10 6
10 4
10 2
100102
! !d/!
k/("sd)
p! d
/k
10 4 10 2 100 102 10410 10
10 8
10 6
10 4
10 2
100102
! !d/!
k/("sd)
p! d
/k
Pressure scalings for frictional, non-cohesive particles
6
γ∗ = ˆγ/|φ− φc|bp∗ = p/|φ− φc|a
Scaled pressure and shear rate†:
�
Independent of µ
a = 2/3
b = 4/3
Choose exponents:
γ∗p
p∗
• Three pressure asymptotes: pi|φ− φc|2/3
= αi
�γ
|φ− φc|4/3
�mi
μ=0.5
Monday, April 27, 2015
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10 4 10 2 100 102 10410 10
10 8
10 6
10 4
10 2
100102
! !d/!
k/("sd)
p! d
/k
10 4 10 2 100 102 10410 10
10 8
10 6
10 4
10 2
100102
! !d/!
k/("sd)
p! d
/k
Pressure scalings for frictional, non-cohesive particles
6
γ∗ = ˆγ/|φ− φc|bp∗ = p/|φ− φc|a
Scaled pressure and shear rate†:
• Transitions between regimes blended smoothly
�
Independent of µ
a = 2/3
b = 4/3
Choose exponents:
γ∗p
p∗
• Three pressure asymptotes: pi|φ− φc|2/3
= αi
�γ
|φ− φc|4/3
�mi
μ=0.5
Monday, April 27, 2015
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10 4 10 2 100 102 10410 10
10 8
10 6
10 4
10 2
100102
! !d/!
k/("sd)
p! d
/k
10 4 10 2 100 102 10410 10
10 8
10 6
10 4
10 2
100102
! !d/!
k/("sd)
p! d
/k
Pressure scalings for frictional, non-cohesive particles
6
γ∗ = ˆγ/|φ− φc|bp∗ = p/|φ− φc|a
Scaled pressure and shear rate†:
• Transitions between regimes blended smoothly
�
Independent of µ
a = 2/3
b = 4/3
Choose exponents:
γ∗p
p∗
• Three pressure asymptotes: pi|φ− φc|2/3
= αi
�γ
|φ− φc|4/3
�mi
μ=0.5
S. Chialvo et al., PRE 85, 021305 (2012).
Monday, April 27, 2015
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Pressure in frictional, cohesive particles
7
Bo*=0
Monday, April 27, 2015
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Pressure in frictional, cohesive particles
7
Bo*=0 Bo*=5.0E-06
Monday, April 27, 2015
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Pressure in frictional, cohesive particles
7
Bo*=0 Bo*=5.0E-06
Monday, April 27, 2015
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Pressure in frictional, cohesive particles
7
Bo*=0 Bo*=5.0E-06
Bo*=5.0E-05
Monday, April 27, 2015
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Pressure in frictional, cohesive particles
7
Bo*=0 Bo*=5.0E-06
Quasi-static, inertial and intermediate regimes persist. A new cohesive regime emerges below the jamming conditions for equivalent non-cohesive particles.
Bo*=5.0E-05
Monday, April 27, 2015
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Cohesive particles: Stress ratio
cohesion increases effective stress ratio
8
σ = pI− pηS
Bo*=0 Bo*=5.0E-06
Bo*=5.0E-05
Monday, April 27, 2015
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Dense phase rheology: Summary
Flow regime map:
Rheological models
Steady state models that bridge various regimes
Modified kinetic theory
Wall Boundary conditions
Implementation
9
(completed)
S. Chialvo et al., PRE 85, 021305 (2012).
Y. Gu et al., PRE 90, 032206 (2014).
(completed)
Monday, April 27, 2015
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Kinetic-theory models
10
• Traditionally use kinetic-theory (KT) models for modeling inertial regime
• Most KT models designed for dilute flows offrictionless particles
Monday, April 27, 2015
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Kinetic-theory models
10
• Traditionally use kinetic-theory (KT) models for modeling inertial regime
• Most KT models designed for dilute flows offrictionless particles
• Can KT model be modified to capture dense-regime scalings?
Monday, April 27, 2015
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Kinetic-theory models
10†Garzó, V., Dufty, J.W. Phys. Rev. E 59, 5895 (1999).
• Seek modifications to KT model of Garzó-Dufty (1999)†
• Traditionally use kinetic-theory (KT) models for modeling inertial regime
• Most KT models designed for dilute flows offrictionless particles
• Can KT model be modified to capture dense-regime scalings?
Monday, April 27, 2015
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Kinetic theory equations
Garzó-Dufty kinetic theory for simple shear flow
11
Pressure
Steady-state energy balance
Energy dissipation rate
Shear stress
p = ρsH(φ, g0(φ))T
τ = ρsdγJ(φ)√T
Γ =ρsdK(φ, e)T 3/2
Γ− τ γ = 0
Monday, April 27, 2015
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Kinetic theory equations
Garzó-Dufty kinetic theory for simple shear flow
11
Pressure
Steady-state energy balance
Energy dissipation rate
Shear stress
p = ρsH(φ, g0(φ))T
τ = ρsdγJ(φ)√T
Γ =ρsdK(φ, e)T 3/2
Γ− τ γ = 0
Important quantities:
• Radial distribution function at contact
‣ Measure of packing
‣ Diverges at random close packing
• Restitution coefficient
‣ Measure of dissipation
‣ Has strong effect on temperature
g0 = g0(φ)
e
Monday, April 27, 2015
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Kinetic theory equations
Garzó-Dufty kinetic theory for simple shear flow
11
Pressure
Steady-state energy balance
Energy dissipation rate
Shear stress
p = ρsH(φ, g0(φ))T
τ = ρsdγJ(φ)√T
Γ =ρsdK(φ, e)T 3/2
Γ− τ γ = 0
Γ =ρsdK(φ, eeff(e, µ))T
3/2δΓ
τ = τs + ρsdγJ(φ)√T δτ
Γ− (τ − τs)γ = 0
Modifications (in red)
p = ρsH(φ, g0(φ,φc(µ)))T
Monday, April 27, 2015
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Dense phase rheology: Summary
Flow regime map:
Rheological models
Steady state models that bridge various regimes
Modified kinetic theory
Wall Boundary conditions
Implementation
12
(completed)
S. Chialvo et al., PRE 85, 021305 (2012).
Y. Gu et al., PRE 90, 032206 (2014).
(completed)
S. Chialvo & S. Sundaresan, Phy. of Fluids, 25, 070603 (2013).
(completed)
Monday, April 27, 2015
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Boundary vs. core regions
• comprises the bulk of the flow
• exhibits uniform flow properties
• obeys local, inertial-number rheological models*†
Core region
20 10 0 10 200.4
0.2
0
0.2
0.4
y/d
v/v
w
20 10 0 10 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
y
!
20 10 0 10 200
2
4
6
8
10
12
14
yTVolume fraction Temperature
Scaled velocity
*S. Chialvo et al. PRE 85, 021305 (2012). †F. da Cruz et al. PRE 72, 021309 (2005).
• lies within ~10d of each wall
• exhibits large variations in field variables
• due to nonlocal conduction of pseudothermal energy
Boundary layer
Monday, April 27, 2015
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Boundary vs. core regions
• comprises the bulk of the flow
• exhibits uniform flow properties
• obeys local, inertial-number rheological models*†
Core region
20 10 0 10 200.4
0.2
0
0.2
0.4
y/d
v/v
w
20 10 0 10 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
y
!
20 10 0 10 200
2
4
6
8
10
12
14
yTVolume fraction Temperature
Scaled velocity
*S. Chialvo et al. PRE 85, 021305 (2012). †F. da Cruz et al. PRE 72, 021309 (2005).
• lies within ~10d of each wall
• exhibits large variations in field variables
• due to nonlocal conduction of pseudothermal energy
Boundary layer
Questions:
• How to define the slip velocity to get simple scaling to work? • What if we want to avoid the need to resolve the small boundary layer?
Monday, April 27, 2015
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Core rheology
Core region
20 10 0 10 200.4
0.2
0
0.2
0.4
y/d
v/v
w
Scaled velocity
core region
Inertial number:
Shear stress ratio:
*S. Chialvo et al. PRE 85, 021305 (2012). †F. da Cruz et al. PRE 72, 021309 (2005).
0 0.1 0.2 0.3 0.40.1
0
0.1
0.2
0.3
Icore
!core!
!s
• comprises the bulk of the flow
• exhibits uniform flow properties
• obeys local, inertial-number rheological models*†
‣ interparticle friction coefficient affects yield stress ratio
‣ wall friction coefficient has no effect on rheological model
µ
µw
ηs
ηcore = ηs(µ) + αIcore
Icore ≡γcored�pcore/ρs
ηcore ≡τcorepcore
Icore ≈ f(φ) φ < φc(µ)for
4 2 0 2 4 6 8 10 121
0
1
2
3
4
5
6
7
0.02 0 0.02 0.04 0.060
0.2
0.4
0.6
0.8
1
!c ! !core
vapp
sli
p/v
wall
µ = 1 .0
µ = 0 .5
µ = 0 .4
µ = 0 .3
µ = 0 .2
0.02 0 0.02 0.04 0.060
0.2
0.4
0.6
0.8
1
!c ! !core
vapp
sli
p/v
wall
µ = 1 .0
µ = 0 .5
µ = 0 .4
µ = 0 .3
µ = 0 .2
µw=1.0
µw=0.7
µw=0.5
µw=0.4
Monday, April 27, 2015
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Definitions of slip velocity
• Slip velocity:
20 10 0 10 200.4
0.2
0
0.2
0.4
y/d
v/v
w
a) ‘Standard’ slip velocity:based on translational velocity of particles at wall
• Options for velocity :
b) ‘Apparent’ slip velocity:based on extrapolated velocity from core region to wall
vapp ≡ γcoreH/2
vappslip = vapp − vw
Some solids velocity at the wall
Velocity of wall
v(·)slip = v(·) − vw
v(·)
vtrslip = vtr − vw
Real velocity
Apparent velocity
= vtr − v�
Monday, April 27, 2015
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Definitions of slip velocity
• Slip velocity:
20 10 0 10 200.4
0.2
0
0.2
0.4
y/d
v/v
w
Real velocity
Apparent velocity
c) ‘Surface’ slip velocity:based on relative velocity of particle surface at wall
• Options for velocity :
Some solids velocity at the wall
Velocity of wall
v(·)slip = v(·) − vw
v(·)
vsurfslip = vsurf − vw
Rotational velocity
vsurf = vtr ± ωd/2
Monday, April 27, 2015
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Definitions of slip velocity
• Slip velocity:
20 10 0 10 200.4
0.2
0
0.2
0.4
y/d
v/v
w
Real velocity
Apparent velocity
c) ‘Surface’ slip velocity:based on relative velocity of particle surface at wall
• Options for velocity :
Some solids velocity at the wall
Velocity of wall
v(·)slip = v(·) − vw
v(·)
vsurfslip = vsurf − vw
Question:
• Is one (or more) of these slip velocities amenable to a scaling collapse?
Rotational velocity
vsurf = vtr ± ωd/2
Monday, April 27, 2015
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Velocity scales
• Dimensionless slip velocity:
• Options for : Some characteristic velocity in the core
Some slip velocityI(·)slip =
v(·)slip
vcharvchar
vchar = γd
vchar =�p/ρs
�τ/ρsor
vchar = ν/ρsd = τ/ρsγd
a) shear-rate-based†:
b) stress-based*:
c) viscosity-based:
†Artoni et al. PRL 108, 238002 (2012). *Artoni et al. PRE 79, 031304 (2009).
Monday, April 27, 2015
/25
0 0.1 0.2 0.3 0.4 0.50
5
10
15
(!core! !s)/(µw ! (!s ! µe))
vsurf
sli
p/("
core/#
sd)
18
DEM results: dimensionless slip velocity
• Possible model form:
• Full collapse achieved by scaling of :
‣
‣ Critical wall friction coefficient separates partial- and full-slip regimes†
µ∗w
†Z. Shojaaee et al. PRE 86, 011302 (2012).
• This form still requires solving for rotational velocity and boundary layer
µ∗w ≈ 0.33
vsurf
slip/(ν c
ore/ρ
sd)
4 2 0 2 4 6 8 10 121
0
1
2
3
4
5
6
7
0.02 0 0.02 0.04 0.060
0.2
0.4
0.6
0.8
1
!c ! !core
vapp
sli
p/v
wall
µ = 1 .0
µ = 0 .5
µ = 0 .4
µ = 0 .3
µ = 0 .2
0.02 0 0.02 0.04 0.060
0.2
0.4
0.6
0.8
1
!c ! !corev
app
sli
p/v
wall
µ = 1 .0
µ = 0 .5
µ = 0 .4
µ = 0 .3
µ = 0 .2
µw=1.0
µw=0.7
µw=0.5
µw=0.4
vsurfslip = vsurf − vw{vsurf = vtr ± ωd/2
ηcore − ηs
ηwall = µw + µ∗w
y =1.5x2/3
(1− x)5
ηcore − ηsµw + µ∗
w − ηs
Monday, April 27, 2015
/25
Dense phase rheology: Summary
Flow regime map:
Rheological models
Steady state models that bridge various regimes
Modified kinetic theory
Wall Boundary conditions
Implementation of modified kinetic theory in MFIX/openFOAM
19
(completed)
S. Chialvo et al., PRE 85, 021305 (2012).
Y. Gu et al., PRE 90, 032206 (2014).
(completed)
S. Chialvo & S. Sundaresan, Phy. of Fluids, 25, 070603 (2013).
(completed)
(manuscript under preparation)
Monday, April 27, 2015
/25
MKT Model implemented in openFOAM
Implemented modified kinetic theory in MFIX
Ran into convergence issues
Implemented MKT in openFOAM
After a few months of efforts, resolved convergence issues
Model solves for both gas and particles
Algebraic form of MKT
Will show sample findings in the next few slides
Will return to MFIX implementation soon
20Monday, April 27, 2015
/25
Granular Discharge
21S. Schneiderbauer, A. Aigner, S. Pirker, Chemical Engineering Science, 80, 279-‐292, (2012)
!"#$%"&'( )*+,%( -./&0(
!*"12+%(3/*4%&%"5(3$( !"#$%&'()&*+&
!!&
!*"12+%(6"/21#.(2#%72/%.&5(8(
!",&'!"-+&
.&
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Monday, April 27, 2015
/25
Granular Discharge
Unphysical results with free-slip BC. Meaningful trends with no-slip BC.
22Monday, April 27, 2015
/25
Granular Discharge
Unphysical results with free-slip BC. Meaningful trends with no-slip BC.
22Monday, April 27, 2015
/25
Granular Discharge
Unphysical results with free-slip BC. Meaningful trends with no-slip BC.
22Monday, April 27, 2015
/25
Effect of grid resolution
23
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• 5678(99(+2('/"(,+*":402(0#('/"(;+,'/(0#('/"(*":'$2)%&$*(0*+<:"(• 768(99(+2('/"(,+*":402(0#('/"(&"2)'/(0#('/"(0*+<:"6((
– ./"(="*4:$&(*"30&%402(+3(768(996(• >0$*3"*()*+,-(>0$*3"2",(?@($(#$:'0*(0#(7(+2("$:/(,+*":402(• >0$*3"*()*+,(;+'/(<2"(="*4:$&(*"30&%402(0#('/"(0*+<:"-(
– A&&()*+,3($*"(:0$*3"2",("B:"C'(#0*('/"(="*4:$&(*"30&%402(0#('/"(0*+<:"(
Monday, April 27, 2015
/25
Comparison with experimental data
For 0.875 mm particles, excellent agreement with friction coefficient of 0.3
Discharge rate varies significantly with friction coefficient
For the 2 and 4 mm particles, good agreement if the friction coefficient is chosen to be 0.1.
Granular discharge experiments may be a simple way of tuning friction coefficient!
24Monday, April 27, 2015
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Summary and future work
• Developed rheological model spanning three regimes of dense granular flow
• Proposed modified kinetic theory to capture rheological behavior for dense and dilute systems
• Developed effective boundary conditions for dense flows
• Implementation in openFOAM completed; implementation in MFIX is ahead of us.
25Monday, April 27, 2015