Implementation and Refinement of a Comprehensive Model for Dense Granular Flows Yile Gu, Ali Ozel, Sebastian Chialvo, and Sankaran Sundaresan Princeton University This work is supported by DOE-UCR grant DE-FE0006932. Monday, April 29, 2015 Monday, April 27, 2015
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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
Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?
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
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
/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
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
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)
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
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
/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
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
/25
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
/25
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
/25
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).