Implementation and Reﬁnement of a Comprehensive Model … Library/Events/2015/crosscutting... · Implementation and Reﬁnement of a Comprehensive Model for Dense Granular Flows

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


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Granular rheological behavior

2Monday, April 27, 2015

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Ubiquitous in nature and widely encountered in industrial processes,


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Complex behavior: multiple regimes of rheology, jamming


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2

Shear flow of frictional particlesin a periodic box


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2

Shear flow of frictional particlesin a periodic box

Shear flow of frictional particles

with bounding walls

Shearing plate

Shearing plate


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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

http://lammps.sandia.gov

http://lammps.sandia.gov

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Dense phase rheology: Questions asked


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Flow regime map: What regimes of flow are observed in shear flow of soft, frictional particles?


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Non-cohesive


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Non-cohesive

Cohesive


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Non-cohesive

Cohesive

Rheological models


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Non-cohesive

Cohesive

Rheological models

Steady state models that bridge various regimes


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Non-cohesive

Cohesive

Rheological models


Modified kinetic theory (for non-cohesive particles)


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Non-cohesive

Cohesive

Rheological models



Wall Boundary conditions


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Non-cohesive

Cohesive

Rheological models




Implementation of modified kinetic theory in MFIX/openFOAM


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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).


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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


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



• Experimental



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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


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



• Experimental



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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


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



• Experimental



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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


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



• Experimental


=⇒=⇒

k

k

• Role of particle softness:

- Large quasi-static or inertial regime

- Small intermediate regimeMonday, 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

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


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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


6



�

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


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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


6



• Transitions between regimes blended smoothly

�

Independent of µ

a = 2/3

b = 4/3

Choose exponents:

γ∗p

p∗


= αi

�γ

|φ− φc|4/3

�mi

μ=0.5


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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


6



• Transitions between regimes blended smoothly

�

Independent of µ

a = 2/3

b = 4/3

Choose exponents:

γ∗p

p∗


= αi

�γ

|φ− φc|4/3

�mi

μ=0.5

S. Chialvo et al., PRE 85, 021305 (2012).


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Pressure in frictional, cohesive particles

7

Bo*=0


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7

Bo*=0 Bo*=5.0E-06


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7

Bo*=0 Bo*=5.0E-06


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7

Bo*=0 Bo*=5.0E-06

Bo*=5.0E-05


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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


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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


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Dense phase rheology: Summary

Flow regime map:

Rheological models


Modified kinetic theory


Implementation

9

(completed)


Y. Gu et al., PRE 90, 032206 (2014).

(completed)


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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


/25


10



• Can KT model be modified to capture dense-regime scalings?


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10†Garzó, V., Dufty, J.W. Phys. Rev. E 59, 5895 (1999).

• Seek modifications to KT model of Garzó-Dufty (1999)†



• Can KT model be modified to capture dense-regime scalings?


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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


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11

Pressure



Shear stress




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


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11

Pressure



Shear stress




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


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Flow regime map:

Rheological models




Implementation

12

(completed)


Y. Gu et al., PRE 90, 032206 (2014).

(completed)

S. Chialvo & S. Sundaresan, Phy. of Fluids, 25, 070603 (2013).

(completed)


/2513

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


/2513

Boundary vs. core regions




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


• 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?


/2514

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:


0 0.1 0.2 0.3 0.40.1

0

0.1

0.2

0.3

Icore

!core!

!s




‣ 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


/2515

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�


/2516


• 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



Velocity of wall


v(·)

vsurfslip = vsurf − vw

Rotational velocity

vsurf = vtr ± ωd/2


/2516


• 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



Velocity of wall


v(·)

vsurfslip = vsurf − vw

Question:

• Is one (or more) of these slip velocities amenable to a scaling collapse?

Rotational velocity

vsurf = vtr ± ωd/2


/2517

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).


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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


/25


Flow regime map:

Rheological models




Implementation of modified kinetic theory in MFIX/openFOAM

19

(completed)


Y. Gu et al., PRE 90, 032206 (2014).

(completed)

S. Chialvo & S. Sundaresan, Phy. of Fluids, 25, 070603 (2013).

(completed)

(manuscript under preparation)


/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


/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(

!",&'!"-+&

.&

!*"12+%("%01&,1#.(2#%72/%.&5(%$(

!"#& .&

!*"12+%(3%.0/&'5(90( (%!!& "#$!%&:*0(;/02#0/&'5(8<( -"$#/.%& "#$!'(&:*0(3%.0/&'5(9<( -"((*& "#$!%&

&

0/12345&625/7824&97:9/7;/<"&=45/7>2;?/&?243/<&24<:&/@94:7/A&B/7/&27/&8>C43A/A&8>&927/>5B/</<"&&


/25

Granular Discharge

Unphysical results with free-slip BC. Meaningful trends with no-slip BC.


/25

Granular Discharge



/25

Granular Discharge



/25

Effect of grid resolution

23

•  !"#$%&'()*+,-((–  ./"(/0*+102'$&(*"30&%402(

•  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*+<:"(


/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!


/25

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.


Documents

Implementation and Reﬁnement of a Comprehensive Model … Library/Events/2015/crosscutting... · Implementation and Reﬁnement of a Comprehensive Model for Dense Granular Flows