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8th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries

SINTEF/NTNU, Trondheim Norway

21-23 June 2011

CFD11-124

1

PARCEL-BASED APPROACH FOR THE

SIMULATION OF GAS-PARTICLE FLOWS

Stefan RADL

1

*, Charles RADEKE

2

, Johannes G. KHINAST

2,3

, Sankaran SUNDARESAN

1

1

Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ

2

Research Center Pharmaceutical Engineering GmbH, Graz, Austria

3

Institute for Process and Particle Engineering, Graz University of Technology, Graz, Austria

* E-mail: sradl@princeton.edu

ABSTRACT

We focus on a parcel-based approach, similar to the one

used by O'Rourke and Snider, 2010, which tracks the

motion of a so-called “parcel” of particles. We derive a

scaling law for a linear-spring dashpot interaction model

that enables tracking of clouds of particles through

DEM-based simulation of (scaled) pseudo-particles.

This guarantees convergence to a DEM-based

simulation of the unscaled system, i.e., a system where

all the individual particles are tracked. We use a BGK-

like relaxation term to model collisions between

particles in dilute regions of the flow field. This

combined approach is implemented in an in-house code

that runs on GPUs (Radeke et al., 2010), and is used to

study a granular jet impinging on a plane surface, as

well as a simple shear flow. We find that a BGK-type

relaxation model is necessary when using parcel-based

approaches for capturing some prominent flow features.

Keywords: Discrete element method, simulation, granular

impinging jet, shear flow.

NOMENCLATURE

Latin Symbols

c Damping coefficient of primary particles,

[kg/s].

C Constant in the filter function.

d Diameter, [m].

e

p

Coefficient of restitution.

E Young’s modulus, [N/m²].

f Particle distribution function.

F Force, [N].

g

0

Radial distribution function.

G Filter function.

h Switch-off function.

k Stiffness of primary particles, [N/m].

K Constant.

m Mass, [kg].

N Number of particles in a parcel.

p Pressure, [N/m²].

P Probability distribution function.

R Radius, [m].

r

32

Sauter mean radius of primary particles, [m].

t Time, [s].

T Granular temperature, [m²/s²].

T Torque, [N

.

m].

U Initial velocity, [m/s].

v Velocity, [m/s].

y Vertical distance, [m].

Greek Symbols

α

Scaling ratio.

β

Size ratio of colliding particles.

δ

Overlap, [m].

∆

Dimensionless filter length.

φ

Volume fraction.

γ

Shear rate, [1/s].

η

Damping function.

κ

Exponent.

λ

Experimental parameter for the scattering

angle.

µ Friction coefficient.

Π

Dimensionless parameter.

θ

Scattering angle, [rad].

ρ

Particle density, [kg/m

3

].

σ

σσ

σ Stress, [N/m²].

τ

Shear stress [N/m²].

τ

D

Relaxation time, [s].

ω

ωω

ω Rotation rate, [rad/s].

Sub/superscripts

*, *’ Dimensionless quantities.

BGK Bhatnagar–Gross–Krook.

CP Close packed.

eff Effective.

el Elastic.

eq Equilibrium.

fluct Fluctuation.

half Half.

impact At impact.

jet Jet.

model Modeled term.

n Normal direction.

t Tangential direction.

off Value at switch-off.

p Parcel.

prim Primary particle.

S. Radl, C. Radeke, J.G. Khinast, S. Sundaresan

2

PP Particle-particle.

PW Particle-wall.

roll Rolling.

sample At sample position.

0 Reference value.

INTRODUCTION

Discrete Particle Models (DPM), aim at tracking

individual particles, or parcels thereof, in the flow

domain. By using this approach, the treatment of a

granular assembly made up by particles with different,

size, density, shape, or composition is straightforward.

Particle-based methods offer the possibility to include

complex particle-wall interactions, the handling of rare

events, as well as to extend the method to gas-particle-

droplet systems (Link et al., 2009; O'Rourke et al.,

2009; Zhao et al., 2009). Interest in the latter systems

has been recently motivated by the pharmaceutics

industry (Radeke and Khinast, 2011; Radeke et al.,

2010), and also by classical applications like coking

(typically performed in fluidized beds, see Darabi et al.,

2010; Radmanesh et al., 2008), or catalyst impregnation.

When using a DPM, two major approaches can be

chosen: First, each particle in the system can be tracked.

In this case, relatively simple models based on

instantaneous (“hard-sphere” model), or enduring

collision dynamics (“soft-sphere” model, recently

summarized by Cleary, 2009) can be employed. The

soft-sphere approach is more general, and allows also

for the simulation of dense regions with enduring

contacts, and is usually referred to as “Discrete Element

Method” (DEM).

Second, the particle population can be represented by

so-called “parcels” of particles, i.e., a cloud of particles,

and hence the name parcel-based approach. Another

way of interpreting such an approach is to think of a

(discrete) approximation of the particle distribution

function by test particles. For gas-solid flows, Andrews

and O'Rourke, 1996, proposed the so-called “Multi-

Phase Particle-In-Cell” (MP-PIC) approach, which is a

parcel-approach. MP-PIC has been applied widely to

fluidized beds, sedimentation, hopper flow, as well as

other dense granular flows (O'Rourke et al., 2009;

O'Rourke and Snider, 2010; Snider, 2001; Snider,

2007). In principle, a parcel-based approach does not

require a second-phase to be present; however, MP-PIC

requires an implicit coupling to the fluid phase due to

stability reasons. The MP-PIC approach does not track

collisions between particles directly, but employs a

simple “particle pressure” model to prevent particles

from becoming close-packed. Instead of modeling

particle interaction forces with a particle pressure,

Patankar and Joseph, 2001, explored the use of a simple

soft-sphere model (without friction) in conjunction with

a parcel-based method for a relatively small system. A

similar approach was taken by Sakai et al., 2010; Sakai

and Koshizuka, 2009, as well as Mokhtar et al., 2011.

Only recently Bierwisch et al., 2009 have shown that a

parcel-based approach with contact detection, when

using appropriately scaled interaction parameters, yields

simulation results independent of the number of particles

making up the parcel. Bierwisch et al., 2009 showed that

this scaling must be based on identical (i) particle

density, (ii) coefficient of restitution, (iii) friction

coefficients, and (iv) Young’s modulus for their

Hertzian repulsion force model. The scaling proposed

by Bierwisch et al., 2009 yields stresses in the quasi-

static regime, and parcel velocities in all regimes, that

are scale independent. However, the analysis of

Bierwisch et al., 2009, is based on a single collision, and

studies on parcel behavior in dense to moderately dense

systems (i.e., for particle volume fractions between 0.05

and 0.50) are still lacking.

Our first objective is to provide a scaling for the

parameters of a DEM-based (linear spring-dashpot)

model, such that stresses and velocities in the quasi-

static flow regime are scale independent. Our second

objective is to establish a method that takes the effect of

sub-parcel collisions, not directly tracked when using a

parcel-based approach, into account. We base our

method on the ideas of O'Rourke and Snider, 2010.

However, we aim at a strategy that is consistent with

DEM, i.e., that yields a “pure” DEM-based simulation in

a situation where only one particle is in the parcel. Our

third objective is to demonstrate the effect of the

proposed scaling and the sub-parcel collision model on

the dynamics of two granular flows. Our simulations are

based on an in-house code running on graphic

processing units (GPUs).

MODEL DESCRIPTION

Dense Region

When using a parcel-based approach, one has to prevent

over-packing in dense regions either with an (indirect)

particle pressure model (Andrews and O'Rourke, 1996),

or a model based on the direct detection of particle

collisions (e.g., the strategy followed by Patankar and

Joseph, 2001). In our model we have chosen to account

for collisions by using a linear spring-dashpot model

(see Eqns. 1-2), which is well established for granular

dynamics. The details of our implementation are

provided in Radeke et al., 2010, and here we only

provide the most essential model equations. For our

present work we included a spring-dashpot-slider model

in the tangential direction (see Eqn. 2), as well as a

rolling friction model between particles, and particles

and the wall (see Eqns. 3-5;

roll

v

is a “rolling velocity”

as defined by Luding, 2008).

nnF ⋅⋅+⋅⋅=

nnnnnij

ck

δδ

&

,

(1)

(

)

ttFF ⋅⋅+⋅⋅⋅=

ttttnijij,t

ck

δδµ

&

,min

,

(2)

rollrollnrollroll

vvFF ⋅⋅−=

µ

(3)

[

]

21

ωnωnv

×

−

×

⋅

−

=

effroll

R

(4)

rollroll

R

FnT

×

⋅

=

1

(5)

The central question when using a DEM-based model

for the parcel-based approach is how to connect particle

interaction parameters with parcel interaction

parameters. Specifically, we ask the question, how

Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124

3

spring stiffness and damping coefficient must be scaled,

when a parcel with a diameter

d

p

=

α

.

d

prim

is used in the

simulation (

α

is the ratio of the parcel diameter to the

primary particle diameter).

In the following we describe the details on such a

scaling that yields identical stresses for dense granular

flow using the linear spring-dashpot model (Eqns. 1-5).

Scaling of Interaction Parameters

We base our analysis on equal energy densities in the

original, and the coarse-grained system. This means that

the density of the particles, as well as the translational

velocity must be invariant. Also, the total rotational

kinetic energy of the original and coarse-grained

particles must be identical. The following analysis is

valid for a linear spring-dashpot model with frictional

slider, and is similar to the analysis for a Hertzian

interaction model by Bierwisch et al., 2009. Since we

base our analysis on an effective mass and radius for the

collision, it is valid for both particle-wall, as well as for

particle-particle collisions with arbitrary size ratios of

the primary particles (i.e., the particles in the original

system). Our analysis does not include differences in

particle densities; however, an extension to these

systems can be easily done in analogy to particle size

differences.

We start with the differential equation for the overlap in

normal direction from Newton’s equation of motion:

nnnnneff

ckm

δδδ

&&&

⋅+⋅=⋅

.

(6)

Here the effective mass is:

( )

3

3

3

13

4

β

βρ

π

+

⋅⋅

=

+

⋅

=

pi

ji

ji

eff

R

mm

mm

m.

(7)

The effective radius is

Inserting these expressions, and using the dimensionless

variables

inn

R

δ

δ

=

*

,

0

*v

nn

δδ

&& =

, and

(

)

0

*vRtt

i

=

,

as well as approximating Young’s modulus with

in

RkE

≈

yields:

*

*

*

0

22

0

1

n

pi

n

pi

nn

n

vR

c

vR

k

K

δ

ρρ

δ

δ

&&& ⋅

⋅⋅

+

⋅⋅

⋅

=⋅ ,

(9)

with

(

)

[

]

33

1

134

ββπ

+⋅=K

. Thus, our scaling is based

on the dimensionless (normal) overlap for the

translational motion of a particle, with the reference

length being the parcel diameter in the parcel approach

and the particle diameter in the original unscaled

problem, i.e., the relative overlap will remain invariant

when scaling the system. From Eqn. 9 the following

dimensionless parameters can be identified:

β

=

Π

1

,

(

)

2

02

vRk

pin

⋅⋅=Π

ρ

,

(

)

0

2

3

vRc

pin

⋅⋅=Π

ρ

(10)

Π

1

requires a constant ratio of the radii of the colliding

particles or parcels. This ratio will remain constant, as

long as each parcel is made up by the same number of

particles N. Π

2

requires that

constRk

in

=

, since we

require also the density and the reference velocity v

0

to

be invariant. This is in agreement with the simulation

results of Chialvo et al., 2010, which found that the

pressure scales with

Rk

n

in a monodisperse, quasi-

static granular flow.

Also, the stresses will be identical during scaling. This is

because the ratio of the elastic normal forces in the

original and scaled system is given by

(

)

(

)

[

]

2

,,

**'

αδαδα

=⋅⋅⋅⋅⋅⋅=

inninn

el

nij

el

nij

RkRkFF

.

(our analysis was based on invariant non-dimensional

overlaps

*

n

δ

). Since the macroscopic contact stress is

given by the sum of the dyadic product of contact forces

and the distances between two particles in a control

volume (see Latzel et al., 2000, for details on the

evaluation of stress tensors), the stresses are invariant

when using this scaling in the dense regime.

Π

3

requires that

constRc

in

=

2

, i.e., c

n

scales with

α

²,

which results in an invariant coefficient of restitution

when scaling the system, as well as a damping force that

scales with

α

². Finally, it is easy to see that the already

dimensionless friction coefficients µ and µ

roll

must be

kept constant when scaling the system.

Dilute Region

Previous Work

To take collisional effects between the particles in a

parcel into account, O'Rourke and Snider, 2010,

proposed a relaxation of the particle distribution

function f to an equilibrium distribution f

eq

in a BGK-

like fashion (in our discrete approximation, f is

represented by parcels having individual velocities v).

Thus, they set:

D

eq

ff

t

f

τ

−

=

∂

∂

(11)

to take into account particle collisions. O'Rourke and

Snider, 2010, suggested the following correction to the

parcel velocity to take collisions of particles within the

parcel into account:

(

)

(

)

( )( )

D

D

t

t

τδ

τδ

2/1

2/

'

+

⋅+

=

vv

v

(12)

Here,

δ

t is the computational time step, and the mass-

averaged particle velocity

v

is given by a summation

over parcels p near the parcel under consideration:

(

)

(

)

∑

∑

⋅⋅⋅=

p

prim

p

prim

mNmN vv

,

(13)

where m

prim

is the mass of primary particles, and N is the

number of particles in each parcel. O'Rourke and Snider,

2010, proposed Eqns. 14 to 16 for calculating the

damping time

τ

D

.

( )

eps

g

pCPp

CPp

p

+−

=

φφ

φ

φ

,

,

0

(14)

(

)

21

p

e

+

=

η

(15)

( )

β

β

+

⋅

=

+

⋅

=1

i

ji

ji

eff

R

RR

RR

R.

(8)

S. Radl, C. Radeke, J.G. Khinast, S. Sundaresan

4

( )

( )

( )

( )

( )

( )

∑

∑

−⋅+⋅

−⋅+⋅

⋅−⋅⋅⋅⋅

⋅

⋅

=

p

prim

p

prim

p

p

D

rdN

rdN

g

r

22

32

24

32

0

3

32

2

2

1

3

281

vv

vv

ηηφ

φ

πτ

(16)

Here eps is a small number (taken to be 10

-5

), r

32

is the

local Sauter-mean radius, and d

prim

is the (primary)

particle diameter of the particles making up the parcel.

Eqn. 16 is based on the kinetic theory of granular flow

with corrections for particle inelasticity, as well as local

particle volume fraction. Also, one can take frictional

contacts between particles into account by using an

effective coefficient of restitution as proposed by

Jenkins and Berzi, 2010. However, it should be kept in

mind that the velocity relaxation proposed by O'Rourke

and Snider, 2010, applies for an approach where

collisions between parcels are not detected; when such

collisions are detected, as in our present study, one must

modify Eqn. 16 to avoid overdamping.

Modified Relaxation Model for Parcel Velocities

The equation for the collision frequency proposed by

O'Rourke and Snider, 2010, models the effect of

collisions between particles in different parcels.

However, since we are already accounting for collisions

between parcels in our implementation (via the spring-

dashpot model), we need to model only collisions that

occur with a frequency 1/

τ

D,model

, i.e.,

pD,DmodelD,

111

τ

τ

τ

−

=

.

(17)

Assuming that the latter collision frequency between

parcels follows the expression in Eqn. 16, and by

requiring that the fluctuation velocities of particles and

parcels are identical, one obtains:

(

)

α

τ

τ

1111

DmodelD,

−

=

.

(18)

Hence, the frequency of collisions 1/

τ

D,model

that

requires modeling in addition to the “resolved”

collisions of the parcels, is somewhat lower in our

model than the one proposed by O'Rourke and Snider,

2010. Note, that Eqn. 18 is consistent with a “full”

simulation of the original system, where all particles are

tracked, i.e.,

α

= 1 and 1/

τ

D,model

equals zero.

Furthermore, we have to consider a correction in the

dense limit, where parcel-parcel collisions become more

frequent, and Eqn. 16 is no longer valid. This is because

in the limit of close packing, contact forces due to

enduring contacts (calculated directly with the spring-

dashpot model) will already lead to a velocity

relaxation. Thus, we have to reduce the modeled

collision frequency at a certain volume fraction

φ

p,off

to

avoid the divergence of 1/

τ

D,model

at

φ

p

=

φ

p,CP

. We do

this by multiplying Eqn. 18 with a factor h(

φ

p

) that

approaches zero when

φ

p

equals

φ

p,CP

:

( )

−

−

=

off

offpCPp

pCPp

p

h

κ

φφ

φφ

φ

,,

,

,1min

.

(19)

Here

κ

off

is an exponent that controls how fast the

collision frequency model is switched off. We choose

κ

off

to be 8 for a rapid switch off, and

φ

p,off

= 0.60. This

rather abrupt switch off is motivated by the publications

of Silbert et al., 2007, Reddy and Kumaran, 2007, as

well as Chialvo et al., 2011. These authors show that

there is an abrupt switch between an inertial regime

(where the model given by Eqns. 14 and 18 is

appropriate) and a quasi-static regime, for which a more

complex rheological model has to be used (in our case

the DEM-based model shown in Eqns. 1-5).

In summary, our modified relaxation model is consistent

with a DEM-based simulation in the dense regime, and

with a “full” DEM-based simulation in case the original

system is modeled, i.e., a situation where

α

equals 1. In

the dilute limit, and when the parcel size is much larger

than the particle size, our relaxation model is identical to

the model based on kinetic theory shown in O'Rourke

and Snider, 2010.

Local Particle Volume Fraction and Velocity

For the relaxation of parcel velocities, one needs to

know the local mean particle volume fraction

p

φ

, as

well as the mean velocity of the parcels v. Previous

work (e.g., O'Rourke and Snider, 2010) used a Cartesian

grid for this purpose. This requires an interpolation of

particle volume and velocity from and to this grid. We,

however, reconstruct

p

φ

and v from a spatial filtering

on a spherical domain surrounding each parcel. Our

filter function has the form:

( )

(

)

(

)

( )

∆⋅+≥

∆⋅+<+

=

BGKji

BGKji

RRR

RRRRRC

RG 0

2

02

, (20)

with parameters C

2

, and R

0

chosen such that the integral

of G(R) in a spherical domain is unity. This filter

guarantees that the mean quantities are evaluated at the

center of the parcel. For our simulations we choose ∆

BGK

= 3, i.e., filtering was performed in a sphere with a

diameter three times the parcel diameter.

RESULTS

Granular Jet

Cheng et al., 2007, have experimentally investigated the

normal impact of 0.050 mm to 2.1 mm glass and brass

particles (monodispers and spherical) on a circular disc.

They showed that the characteristic scattering angle

θ

half

of the rebounding particles is influenced by the ratio of

particle and jet diameter d

p

/d

jet

:

(

)

jetphalf

dd

⋅

⋅

=

2atan

λ

θ

(21)

with

λ

= 1.05 ± 0.05. Thus, for a small particle-to-jet

ratio the granular jet behaves like a liquid without a

scattering of particle velocities. When using our parcel-

approach, the goal is that only the size of the primary

particles should affect the scattering angle, and not the

parcel diameter itself.

We performed simulations using (i) an unscaled system

(i.e., the interaction parameters in Eqns. 1-5 were not

Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124

5

changed with parcel size); (ii) a scaled system (scaling

based on the dimensionless parameters displayed in Eqn.

10), but without relaxation model; as well as (iii) a

scaled system with the relaxation model shown in Eqn.

18. Also, a simulation of the original system, for which

the motion of the primary particles was calculated

directly, was performed. The simulation parameters are

listed in Table 1. The damping coefficient c

n

was

calculated from the coefficient of restitution (see

Luding, 2008), and we set c

t

= c

n

.

Figure 1: Particle velocities near the impact region of a

granular jet. The jet (which has a circular cross section with

diameter d

jet

) impacts the planar surface under an angle of

θ

impact

= π/2.

The scattering patterns of these three systems are shown

in Figure 1a-c; a magnified version of a slightly smaller

region near the impact point for the original system is

shown in Figure 1d. Here we show front views of the

particle jet, which approaches the planar surface from

the top. For the results displayed in Figure 1 we have

chosen the scaling factor

α

as 6.3, i.e., we have grouped

250 single particles into one parcel.

As can be easily seen from Figure 1a, the unscaled

system represents a case with a much larger (effective)

particle diameter, and the characteristic scattering angle

is much larger than that observed in the original system

(see Figure 1d). This behavior is expected, since we do

not perform any scaling, and the scattering angle is in

good agreement with the experimental results of Cheng

et al., 2007 (see the comparison of simulation results for

the unscaled system indicated by dots with the

correlation indicated by the bold line in Figure 3).

Table 1: Parameters for the granular jet simulations.

Parameter Value

d

jet

7.3 [mm]

y

jet

25 [mm]

d

jet

/ d

p

3.5; 14.6; 29.2; 73

d

p

0.10 - 2.19 [mm]

U

p,0

1 - 8 [m/s]

φ

p

0.60

ρ

p

2500 (glass)

e

p

0.75, 0.95

µ

PP

0.10

µ

PW

0.15

µ

roll

0.01

k

n

306 [N/m]

k

t

88 [N/m]

R

sample

63.5 [mm], 20 [mm]

Appropriate scaling of the system significantly lowers

the scattering angle (see Figure 1b), even in the case no

relaxation is employed. However, some scattering of

parcels still occurs. Employing the BGK-like relaxation

(Figure 1c) finally yields a parcel behavior similar to the

one observed in the original system with 250-times more

particles.

Figure 2: The scattering angle distribution - comparison of

the original system (red line), with results using the parcel

approach (symbols).

To quantify the effect of the BGK-like relaxation, we

compare the distribution of particle scattering angles in

Figure 2. Specifically, we plot the number density

distribution P(

θ)

of the scattering angle

θ

(i.e., the angle

measured between the impact surface and the particle

position as illustrated in Figure 1a). To calculate P(

θ)

,

we only consider particles located at a radial distance of

R

sample

= 63.5 ± 2.5 mm from the origin, as per Cheng et

(a) unscaled system

d

prim

= 100 µm, d

p

= 630 µm

(c) scaled system

with relaxation

d

prim

= 100 µm, d

p

= 630 µm

(d) original system

d

prim

= d

p

= 100 µm

0,p

U

r

θ

θθ

θ

d

jet

(b) scaled system

without relaxation

d

prim

= 100 µm, d

p

= 630 µm

R

sample

origin

S. Radl, C. Radeke, J.G. Khinast, S. Sundaresan

6

al., 2007; for the original system, though, we set R

sample

= 20 ± 2.5 mm, since we were unable to simulate the full

system. As can be seen from Figure 2, the unscaled and

scaled systems give a wide distribution of scattering

angles, with scattering half angles

θ

half

(defined as

( )

∫

=

half

dP

θ

θθ

0

5.0

) of 0.28 and 0.14, respectively. This

result is in distinct contrast to the very narrow

distribution of the original system (shown by the red line

in Figure 2). However, when we employ the relaxation

model, all simulations using a larger parcel diameter

(see the symbols for different values of d

p

/d

jet

in Figure

2) agree reasonably well with behavior of the original

system. The improved agreement of the BGK cases

indicates the importance of the relaxation model in

replicating the behavior of the much smaller particles

within a parcel.

Figure 3: The scattering half angle

θ

half

as a function of

dimensionless parcel size (symbols represent simulation

results, lines are theoretical predictions).

In Figure 3 we show a summary of our results for the

granular jet for various scalings, expressed as d

p

/d

jet

.

Clearly, when the system is unscaled (filled circles and

crosses), the parcels behave like large particles. In this

systems the scattering half angle is close to the

correlation established by Cheng et al., 2007, and given

in Eqn. 21. For the unscaled system we find that the

coefficient of restitution has no significant effect at large

dimensionless parcel diameters. Only for the smallest

system studied (i.e., d

p

/d

jet

= 0.04) there is a large

relative difference for our results involving e

p

= 0.75

and e

p

= 0.95. The cause for this behavior is unclear, as

previous computational studies (e.g., that of Huang et

al., 2010) could not afford to simulate that large systems

(the simulations for d

p

/d

jet

= 0.04 required us to track

0.97 million particles) and the experiments of Cheng et

al., 2007, only included particles that had a coefficient

of restitution equal to 0.75.

The scaled system using the BGK-like relaxation model

(blue filled squares in Figure 3) shows a more realistic

behavior, as we obtained the small scattering angles

typical for a system made up by particles of 100 µ m (the

scattering angle for 100 µm particles as expected from

Eqn. 21 is shown as green dashed line Figure 3). Our

simulation results for the original system are indicated

by the leftmost black circle at d

p

/d

jet

= 0.0137. These

simulations involved already 2.7 million particles in a

domain half the size as the one used for the other cases.

Our results when using the relaxation model (blue filled

squares in Figure 3) are somewhat below the expected

results from Eqn. 21, as well as our simulations of the

original system. This behavior is also visible in Figure 2,

where the simulation results using the BGK relaxation

do not match perfectly with the original system (red line

in Figure 2). It seems that the chosen expression for the

relaxation time slightly over-predicts the collision

frequency, even though we already subtract the collision

frequency between parcels. More work, and possibly a

more suitable model, is required to explain this

difference, and to make the scattering angles in both

systems collapse. Another explanation could be that the

switch off particle volume fraction of

φ

p,off

= 0.60 in our

simulations is somewhat too high. Indeed, simulations

with lower values for

φ

p,off

showed that such a parameter

adjustment could be used to improve the collapse.

However, there is no rationale for choosing a different

value for

φ

p,off

, and hence we suggest to first improve the

collision frequency model, rather than to perform a

parameter fit.

Simple Shear

We performed simple shear flow simulations using a

cubic box of particles. Lees-Edwards boundary

conditions (LEBCs, Lees and Edwards, 1972) were

employed on two sides of the box, whereas on the other

four sides periodic boundaries were set. The LEBCs

impose a steady shear motion on the particles. Stresses,

fabric tensors, velocities, as well as particle rotation

rates were recorded during the simulation, and then

averaged over the box. Stresses were probed by

calculating the contact and streaming stress tensor for

each particle, and then summing up the contributions

from each particle. A similar procedure was employed

for the fabric tensor (results not shown). Both

procedures (i.e., for the stress and the fabric) follow the

work of Latzel et al., 2000. The granular temperature in

the system was estimated by calculating the velocity

fluctuation tensor

2

,fluctp

v

, and then setting

3

2

,

=

fluctp

trT v

.

(22)

The velocity fluctuations were calculated from the

instantaneous particle velocity and the local average

velocity. The latter was calculated under the assumption

of a linear velocity profile.

Similar to the granular jet, we performed shear flow

simulations in (i) an unscaled system (i.e., the

interaction parameters for the contact model were not

changed with parcel size); (ii) a scaled system (scaling

based on Eqn. 10), but without relaxation model; as well

as (iii) a scaled system with relaxation model. For our

simulations we used the same particle interaction

parameters as shown in Table 1, except for µ

roll

, which

was set to zero. Simulations were performed with

φ

P

=

Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124

7

0.55, e

p

= 0.75, and the shear rate was chosen such that

(

)

4

10

−

=⋅⋅=∗

primpprim

dkd

ργγ

was constant.

Systems with different parcel sizes where investigated

by holding the box size (0.015 m) and the primary

particle diameter (d

prim

= 100 µ m) constant, and

grouping between 4 and 8192 particles into one parcel.

Also, a limited number of simulations were performed

with

φ

P

= 0.62 to show the system behavior in the quasi-

static flow regime. Each of these simulations involved

approximately 4,000 parcels in a box of variable size to

investigate the effect of the parcels size.

Granular Temperature

Already previous work (Benyahia and Galvin, 2010)

showed that the granular temperature in a wall-bounded

shear flow dramatically increases with parcel size. This

finding is expected, since a simple calculation based on

the kinetic theory of granular flow would predict that

(see, e.g., Sangani et al., 1996):

( )

(

)

[

]

( )

p

p

p

e

g

d

T

−⋅

⋅⋅+⋅+

=

⋅115

851121

2

0

2

φπ

γ

(23)

Here g

0

is the radial distribution function at contact, for

which we use the expression in Eqn. 15. Thus, by just

scaling the particle diameter, and performing a naive

DEM-based simulation leads to a significant

overestimation of the granular temperature in the

system.

Figure 4: Granular temperature scaled by the velocity

fluctuation in the original system (d

prim

.

γ

)² (symbols represent

simulation results, lines are theoretical predictions,

φ

P

= 0.55

unless otherwise stated).

The results for the granular temperature (made

dimensionless with the shear rate and the primary

particle diameter d

prim

) are shown in Figure 4. The red

bold line represents the expected granular temperature

of the original system for

φ

P

= 0.55 estimated from Eqn.

23, which should remain constant when using the

relaxation model. Also, we have included a line

representing the theoretical prediction using Eqn. 23 for

the increase in granular temperature in the simulations

without relaxation model (i.e., the scaled and unscaled

system should behave like a system consisting of large

particles, and the granular temperature should increase

proportional to (d

p

/d

prim

)²). Clearly, the results for the

scaled system and

φ

P

= 0.55 agree reasonably well with

the theory of Sangani et al., 1996, which slightly

overpredicts the granular temperature. The unscaled

system, as well as the scaled system with

φ

P

= 0.62

shows a higher granular temperature, but the same

principal trend as the theoretical prediction of Sangani et

al., 1996. This is because the unscaled system represents

a case with a higher coefficient of restitution - this

increase can be easily understood by inspection of Eqn.

23. The scaled system with

φ

P

= 0.62 is in the quasi-

static regime, for which the theory of Sangani et al.,

1996, is no longer applicable.

The simulations using the BGK-like relaxation in Figure

4 show a lower granular temperature, i.e., the system

behavior is closer to the theoretical prediction of the

original system (bold red line). Still the granular

temperature is significantly overpredicted using the

relaxation model given by Eqns. 14 and 18. Thus, the

relaxation to the local mean velocity is too week to

dampen the velocity fluctuations of the parcels. Also,

the scaled granular temperature increases with parcel

size, indicating that the dependency of the current

relaxation model on the parcel size is too weak.

Stresses

Figure 5: Pressure (top) and shear stress (bottom) for different

parcel diameters and filter sizes in a scaled and unscaled

system, as well as a in a system with BGK-like relaxation

(symbols represent simulation results, lines are theoretical

predictions,

φ

P

= 0.55 unless otherwise stated).

S. Radl, C. Radeke, J.G. Khinast, S. Sundaresan

8

The results for the sum of contact and streaming stresses

are shown in Figure 5. Here we have defined the

pressure as p = (

σ

xx

+

σ

yy

+

σ

zz

) / 3, and the shear stress

τ

is the stress component pointing in the shearing

direction and acting on the surface normal to the

gradient direction (the other shear stress components are

much smaller).

As can be seen from Figure 5, with increasing parcel

size both pressure and shear stress dramatically increase

for the systems with

φ

P

= 0.55. This is explained with

the fact that these systems correspond to a system with a

higher dimensionless shear rate

γ

*’, in case we define

this dimensionless shear rate with

(

)

psp

dkd ⋅⋅=∗′

ργγ

, i.e., we use the parcel

diameter instead of the primary particle diameter. Thus,

γ

*’ increases with (d

p

/d

prim

)

3/2

in the unscaled system,

and with (d

p

/d

prim

) in the scaled system since our scaling

is based on k/d

p

= const. As our simulations were

performed with µ

PP

= 0.1, and

γ

*’< 10

-2

, the systems

with

φ

p

= 0.55 were in the inertial regime, and the

systems with

φ

p

= 0.62 were in the quasi-static regime

(for an exact regime definition refer to Chialvo et al.,

2011). Previous studies, and simple theory tells us that

pressure and shear stress scale with the dimensionless

shear rate squared in the inertial regime (at constant

volume fraction and friction coefficient). In the quasi-

static regime the stresses are expected to remain

constant, and only a moderate increase for

γ

*’> 10

-3

(which corresponds to d

p

/d

prim

> 10 in our scaled system)

is anticipated due to a regime transition into an

intermediate flow regime (see the findings of Chialvo et

al., 2011). We have included lines showing the expected

trends for the stresses in the quasi-static and inertial flow

regime in Figure 5 (see the red solid lines).

As can be seen, the simulation results for the scaled

system agree well with the expected scalings in the

inertial and quasi-static regime. For the latter regime our

simulation results for the stresses (black diamonds in

Figure 5) slightly increase with parcel size above d

p

/d

prim

= 10 due to the transition to an intermediate flow

regime. This increase has also been observed by Chialvo

et al., 2011, for flows at different dimensionless shear

rates.

Also the results for the unscaled system with

φ

p

= 0.55

follow the theoretical predictions for flow in the inertial

flow regime; however, they show somewhat higher

stresses. This is explained by the fact that for the

unscaled system also the coefficient of restitution

increases with parcel size, which results in higher

granular temperature and stresses.

In case the BGK-like relaxation is applied, the sum of

contact and streaming stress stays nearly unaffected (see

the filled blue boxes in Figure 5), and only for d

p

/d

prim

>

15 a rather small drop in the stresses is observed. Thus,

even in case we perform a BGK-like relaxation, the

stresses do not significantly change compared to a

scaled system.

CONCLUSION

Even though parcel-based approaches have been in use

for more than fifteen years since the pioneering work of

Andrews and O'Rourke, 1996, there is still a strong need

to analyze the basic features of this simulation approach.

Recent literature highlights this need (see the work of

Benyahia and Galvin, 2010; Benyahia and Sundaresan,

2011).

In this work we performed a relaxation of parcel

velocities to a local mean velocity to account for

collisions of particles making up the parcels. We claim

that such a relaxation is necessary in any DEM-based

simulation that uses bigger (pseudo) particles, and has

the ambition to model the original system consisting of a

much higher number of primary particles. Even in cases

where a proper scaling of particle interaction parameters

is performed (i.e., a scaling like the one proposed by

Bierwisch et al., 2009, or the one presented in this work

for the linear spring-dashpot model), a relaxation seems

necessary. Thus, our simulations involving the

relaxation model gave a nearly parcel-sized independent

behavior for a granular jet impinging on a planar

surface, while simulations without relaxation did not.

The challenge for parcel-based methods is to correctly

compute the collision rate, the granular temperature and

the inter-parcel stress in a dilute to moderately dense

region. Our simulations for a simple shear flow in the

inertial regime agree with the kinetic theory of granular

flow that predicts a massive increase in granular

temperature and stress with parcel size when no

relaxation is used. For the stress this is explained with

the higher dimensionless shear rate when parcels are

used, i.e., the shear rate increases with (d

p

/d

prim

)

3/2

in an

unscaled system, and with d

p

/d

prim

in a system with

scaled interaction parameters to yield identical behavior

in quasi-static granular flow. Thus, there is no easy way

of scaling parcel interaction parameters to yield identical

flow behavior in the inertial flow regime. For the quasi-

static flow regime, however, scaling is less problematic

and we observe a nearly constant stress in our shear flow

simulations. Only the transition to an intermediate flow

regime, as defined by Chialvo et al., 2011, will lead to

an increase in the stresses in a correctly scaled system.

For simple shear flow in the inertial regime the use of

the relaxation model based on O'Rourke and Snider,

2010, used in a slightly modified form in our work,

leads to a granular temperature for the parcel-based

approach closer to the original system, but this is not

true for the stress. It is still unclear how one can obtain

an identical stress for flows in the inertial regime when

using a parcel-based approach. Clearly, more work is

required, especially in connection with the relaxation

model, which is subject of ongoing work.

ACKNOWLEDGEMENT

We thank Sebastian Chialvo for reviewing the

manuscript. SR acknowledges the financial support of

the Austrian Science Foundation and Princeton

University through the Erwin-Schrödinger fellowship J-

3072.

Parcel-Based Approach for the Simulation of Gas-Particle Flows / CFD11-124

9

APPENDIX

List of animation files:

(A1) “1000k_boxLEBC_PrincetonLogoMix.avi“

(simple shear of 10

6

particles in a box with Lees-

Edwards boundary conditions)

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