Granular material flowing down an inclined chute: a
molecular dynamics simulation
Thorsten Pöschel
To cite this version:
Thorsten Pöschel. Granular material flowing down an inclined chute: a molecular dynamics simulation. Journal de Physique II, EDP Sciences, 1993, 3 (1), pp.27-40. <10.1051/jp2:1993109>.
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Submitted on 1 Jan 1993
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phys.
J,
France
II
(1993)
3
27-40
1993,
JANUARY
27
PAGE
Classification
Physics
Abstracts
05.60
46.lo
02.60
material
simulation
Granular
dynamics
Thorsten
Berlin,
D-1040
(Received
July 1992,
27
Abstract.
in
of
with
1.
of
model
our
accepted
Two-dimensional
flow
different
falling
spheres
is
friction.
shear
1913,
Postfach
Berlin,
zu
Germany
Humboldt-Universitit
surface
down
an
inclined
chute:
molecular
a
P6schel
HLRZ, I(FA Jfilich,
42,
flowing
types
experimental
stream
of
in
final
Molecular
down
an
form
30
Dynamics
inclined
be
to
of
Theoretische
Physik,
Invalidenstrase
September1992)
chute.
simulations
The
subjected to the
Hertzian
particles shows a
characteristic
The
fluidization.
numerically
observed
assumed
The
Jfilich, Germany
Physik, Institut fir
D-5170
FB
are
interaction
contact
force
used
model
to
between
and
height profile,
flow properties
the
normal
free
particles
the
as
well
as
consisting of layers
qualitatively
agree
results.
Introduction.
granular
material
such as sand or powder reveal
macroscopic
behaviour
far from
flowing liquids. Examples of astonishing effects are heap formation
under
vibration
size
segregation
formation
convection
cells
and
density
[10-12],
of
[13-15]
[1-9],
waves
emitted
from
outlets [16, 17] or from flows through
pipes [18] and on surfaces [19-22]
narrow
displacements inside shear cells [23-30]. For its exotic
granular materials have
and
behaviour
decades.
been of interest
to physicists
over
many
carried out concerning the
Recently highly sophisticated experimental investigations were
twc-dimensional
free surface flow of plastic spheres generated in an inclined glass-walled chute
consists of layers of different
[31]. The observed flows at sufficiently low inclination
types of
and
collisional
fluidization.
The flow generally can be divided
into
frictional
regions. At least
of a rough bed the
frictional
region typically consists of a quasi-static zone at the
in the
case
bottom
and above of the so called block-gliding
where
blocks of coherent
moving particles
zone
the
bed.
parallel
to
move
results of our
numerical
investigations concerning
In the
current
want to present the
paper
we
mentioned
Beginning at an initial state of particle positions, velocities and
the problem
above.
angular velocities the dynamics of the system will be
determined
by integrating
Newton's
Flows
the
of
behaviour
of
28
JOURNAL
equation of motion
the particles
upon
have
we
chosen
the
a
of the
evaluation
calculation
of
PHYSIQUE
DE
N°1
II
each particle for all further
time steps.
the forces acting
In our
case
exactly
for
configuration
known
each
of
For this
system.
are
our
reason
order
sixth
predictor-corrector
molecular
dynamics simulation [32] for the
positions of the particles and a fourth order predictor-corrector
method for
the particles angular
numerical
methods to
movement
to be the appropriate
for
problem.
our
The height profiles of the particle's velocities, of the particle density and the segregation of
the particle flow into
different
the
essential
points of interest of our
types of fluidization
were
investigations. As shown below the properties of the particle flow varies sensitively with the
of the bed surface. As we will demonstrate, the
smoothness
numerical
results are in qualitative
with the experimental
accordance
data.
solve
Nuiuei~ical
2.
model.
spherical particles with radius R moving
angle (tY
..30°) which is assumed
10°
in the
direction
conditions
parallel to the bed. Our results did not
to have periodic
with the
number of particles provided that they are enough to yield reliable
vary qualitatively
(N > 250). In simulations with more particles (N
statistics
1000) we observed the same
effects, the profiles of the averaged particle density, the cluster density and the particle velocity
became slightly smoother, while the
calculation
time
substantially. In figure I the setup
rose
The
consists
system
down
chute
a
of
of N
equal
400
=
length L
boundary
50
=
twc-dimensional
by
inclined
R
an
=
=
of
simulations
computer
our
is
drawn.
periodic boundary
conditions
x
z
ccelerqtiou
to
Fig-I-
The
the
experimental
Computer
particles
Al
mass
accelerated
were
set
was
to
with
respect
The
and
chute
surface
same
material
These
spheres
simulate
a
Fz
a
Fx
and
are
the
parallel
forces
to
it
"
=
acting
gravity (g
=
-9.81
m/s~,
M
g
cos(tY)
ii)
M
g
sin(tY)
(2)
upon
each
particle
in
the
directions
normal
to
the
particles (Rs
0.66 R) of the
elasticity
and
friction
parameters.
same
arranged at fixed positions either regularly and close together in order to
~
bed or randomly out of the interval [2 Rs; 2
simulate
to
the
chute
as
the
moving particles,
were
due to
respectively.
of
smooth
acceleration
the
to
unity):
F>,
inclined
to
setup.
Fz
where
due
gravity
rough bed (Fig. 2).
was
built
up
of
I-e-
spherical
smaller
with
the
"
N°i
Rough
of
case
they
beds
smooth
and
smooth
the
rough
both
spheres
these
bed
consist
are
of
out
arranged
close
CHUTE
INCLINED
AN
bed
smooth
Fig.2.
DOWN
FLOWING
IIATERIAL
GRANULAR
29
bed
spheres with
diameter
Rs
together while in the case
the
of
In
the
rough
bed
R.
0.66
"
arranged irregularly.
are
To
describe
applied
the
the
ansatz
interaction
between
described
in [33] and
~)
F;j
~
~'
l~~
=
particles and
slightly modified
between
the
~)
~~
particles and
the
bed
the
we
in [10]:
~
~ ~
~~
~~ ~~~
~j
°~
(3)
otherwise
0
with
FN
(ri
EN
"
ix;
+ rj
+
xj ()~.~
7N
kj)
iii
Me~
(4)
and
Fs
min(-7s
=
Me~
vrej
where
~rel
Ii>
"
RI
Me~=
R'(hi
+
~
~
~
,
(5)
(FN()
Al
16)
~
Mi+Mi
=
(7)
2
it, hi and M; denote the current position, velocity, angular velocity and mass
includes an elastic
force which corresponds to the
of the ith particle.
The model
restoration
microscopic assumption that the particles can penetrate slightly each other. In order to mimic
The
terms
x;,
three-dimensional
[34].
force"
The
other
this
ternls
particles according
between
for
behaviour
normal
the
and
force
with
the
describe
the
energy
normal
and
shear
to
coefficients.
friction
shear
rises
which is called
power
dissipation
friction.
the system
parameters
of
The
Equation (5) takes
"Hertzian
due
7N
account
into
contact
collision
to
and
7s
the
fact
stand
that
rotational
but slide upon each other in case the
relative
particles will not transfer
energy
velocity at the contact point exceeds a certain value depending on the normal component of
between
simulations
have
the force acting
the particles (Coulomb-relation [35]. For all our
we
the
chosen
the
the
integration
the
time
3.
Results.
3.I
step
way
time
we
step
lattice
obtain
such
JOURNAL
DE
used
we
In
the
above
tile
bed
periodic
boundary
that
center
the
PHYSIQUE
7N
got exactly the
INITiALizATiON.
disturbed
to
parameters
constant
II
T
3. N'
At
same
=
s~~,
0.001s
7s
to
"
300
s~~, kN
provide
"
100
numerical
~~
m
accuracy.
and ~
When
=
0.5.
For
doubling
results.
simulations, the particles are placed initially on a randomly
that they do not touch each other at the beginning. In order
so
conditions
of the bed we have to fit the length of the bed in a
of the
I.
1000
"
JANUARY
first
1993
sphere
of the
bed is at
~
=
0 and
can
reappear
at
z
=
2
L
JOURNAL
30
kinetic
DE
PHYSIQUE
II
N°1
energy
E_TR
ROT
E
25
20
15
io
5
0
50000
0
150000
100000
200000
steps
time
Fig.3.
Evolution
sharp
The
bed.
regime
and
the
fitted
initial
were
freely
down
kinetic
the
SAIOOTH
their
finds
kinetic
its
movement
shows
the
fastest
the
evolution
energy
steady
state
before
the
particle.
of the
by
the
rotational
particle
for the
energy
particle
the
of
flow
called
movement
Suppose the length of the
was
rises.
as
we
of the
case
by
regimes
3.2.
in
and
After
~vill
beginning
very
inclined
different
the
on
smooth
a
low
energy
chute
chosen
was
be Lo it
to
3.3.
of
run
rotational
energy is
of the
amount
the
In all
our
energy
while
sii~iulations
coincides
figure
bed
smooth
20°.
=
the
shown
are
figure
below.
In
displayed.
The
small
line
a
below.
a
The
due
gravity,
saturation
snapshot
of the
drawn
of
kink
numerical
to
into
comes
4
energy
significant
accelerated
system
in 3A,
kinetic
the
shows
curve
as
particles
time
shows
3
The
movement
the
waiting enough
demonstrate
simulation.
of the
angle tY
an
the particle
kink of the
and
direction
the
at
For
chute
reported
simulations
neighbours.
its
energy
two
and
regimes
diierent
conditions.
The particles'
L E [Lo Lo + Rs) to hold periodic boundary
After starting the
simulation
the particles gain kinetic
energy since
zero.
collisions, I-e- without
dissipation. This is the
for a
without
any
reason
BED.
system. The
which separates
the
(transversal)
two
regime.
energy
with
to L
velocities
they fall
peak of
3.2
high
kinetic
separates
colliding with
without
was
of the
kink
and
grains'
within each particle
velocity of the particle, its length was scaled due to
rotational
the
energy is negligible. In figure 3 the
with
the
tii~ie-axis.
velocity arrow
inside the particles the flux divides into the lower
drawn
block gliding zone, where nearly all the particles move coherently with an equal velocity forming
single block of particles. The particles which belong to one block are drawn bold faced. More
a
precisely to identify which particles form a block we applied the following rule: each two
R) for a time of at least 5000 time steps
particles moving in the neighbourhood ((ri
rj <
distance
of (10.
which
corresponds to a covered
30) R approximately belong to the same
As
indexed
the
FigA.
Snapshot
cluster
them
drawn
are
above
block.
while
anymore
cluster
two
when
grains
in the
was
All
belong
chosen
was
90.000
faced.
bold
block
the
If I
after
neighbouring particles
the
MATERIAL
GRANULAR
N°i
timesteps
grains at
small
to
11
the
particles moving
chosen
lower part
behaviour
the
within
the
DOWN
low
(h
bottom
energy
<
100)
CHUTE
INCLINED
AN
regime.
form
Particles
one
31
which
large block.
belong to a
Only few of
clusters.
to
very
leads
FLOWING
too
wrong
with
large.
move
<
2.I)
then
small
a
the
With
coherently
as
that
velocity
relative
value
a
particles
the
=
2.2
block in the
we
were
got
case
bouncing
elastical
small
very
even
interpretation
are
detected
correct
of low
to
results.
energy
of
two
neighboured
not
belong
As
distinct
to
a
visible
from
figure 8. The upper particles move irregularly.
energy
as
Figures 5, 6 and 7 show the particle-density, velocity in horizontal
direction
due to the bed
cluster-density depending on the vertical
and
distance
from the bed h corresponding to the
snapshot given in figure 4.
direction
The velocity
distribution
in horizontal
separates into two regions, the lower particles
including the block and the grains close to the block move with nearly the same velocity. The
particles on the surface of the flow move with larger velocity.
There is a sharp
transition
between the
velocities of particles which belong to the block and the grains moving at the
the portion of particles which belong to one of the
surface.
The term
cluster density
means
density as a function of the distance from the bed h does not vary from
clusters.
The
cluster
the bed to the top of the block and lowers then fastly as expected from the snapshot (Fig. 4)
clusters
because
there
almost
smaller
The
velocities
in figure 6 have
except the block.
are
no
negative sign because the particles move from the right to the left I-e- in negative direction.
As pointed out above the sharp kink of the energy in figure 3 corresponds to different regimes.
Figures 8- ii are analogous to figures 4-7 for a snapshot short time after the transition into the
high velocity regime at time step 110.000.
the
high
shown
below
in
JOURNAL
32
PHYSIQUE
DE
II
N°1
h
~~~
OOO~
OO%o&oo
o
o
o
oo
~~o%°o$ /
o~$~od~
o
o
o
o
O'*$o
150
o~$~~~
o
oo~p~
l~~~
Q@
o@$~
~°°
4~o$yo~
°
@o
WW(
og g#p
~oo o
oW§~w
°4o~~j~$ooo
5°
'$~oo
o~
~o
~
°
O
~
~
~
~
particle
0
Fig-S-
density depending
partical density
Particle
Within
block
the
on
the
does
the
vertical
not
density
distance
8
~
~
o
~
°
o
o
~~oo
o
(normalizedl
from
the
bed
1
(low
h
regime).
energy
vary.
h
250
200
o
iso
ioo
50
o
o
Fig.6.
move
other.
Averaged
velocity (low
horizontal
energy
equal velocity, I-e- there is really only
The velocity changes desultory at the
upper
with
particles
are
1.3
to
i-s
times
as
fast
as
the
block.
regime).
one
(normalizedl
velocity
horizontal
0
block
boundary
grains which belong
The
but
1
not
ofthe
different
block
(h
m
to
blocks
gliding
100),
the
block
the
free
on
each
moving
N°i
GRANULAR
&IATERIAL
DOWN
FLOWING
AN
CIIUTE
INCLINED
33
h
250
200
oo~
150
~
~
~j~~l~°'9Q~~*
»~"~o
o
o
oo
o
ooo~~
~
loo
~
~
~
~
~~mmo
o
o
50
o
o
0
Fig.7.
almost
are
no
density
cluster
density (low energy
particles which belong
regime).
Cluster
to
a
cluster
Above
the
except
of
block
the
(normalizedl
block
at
the
1
density
cluster
the
vanishes.
There
bed.
o
~~o
Q
~oo
sm°° ~~
~~&
~o
o
Fig.8.
within
lies
still
a
Snapsliot after
relatively short
close
to
the
bed.
l10.000
time
It
into
also
timesteps in the high energy
independently moving
many
vanishes
after
some
time.
regime.
smaller
The
block
clusters.
(Fig. 4) separated
The
rest
of the
block
JOURNAL
34
PIIYSIQUE
DE
N°1
II
h
200
o
o
oo~~~~
m~~~
oum~~ ~'~~°°
°Qt%~d~
15°
w~$°&
~~o~
~o~oj
~p
v
~na
ioo
~
~
j (Roo&
q~O~[°~OiW
ooc
~~~j°9*
~~~
°S~~j~
&
50
W
~
~
o
a~~8%qt°° /
°
iiiim
o
o
o
o
o
°
o
Fig.9.
Particle
density depending on the vertical
Although the properties of the flow are very different
differs not significantly.
of the particle density
(normalized)
density
particle
0
distance
from
from
the
~
o
flow in
the
bed
1
h
the low
(high
energy
energy
regime
regime).
the
profile
value while the grains
accelerated
due to gravity
move
into
block
and
longer
divide
the
the
free
moving
particles
not
any
independently
moving
but the block
into
smaller
clusters.
This
transition
separates
occurs
interval
which corresponds to a short time from the beginning of the
within a short
energy
distance
simulation.
As shown in figure 9 the density as a function
of the
from the bed h
(Fig.
function
low
6).
Nevertheless
varies only slightly from the
the profile of
at
energy
same
the
horizontal
velocity (Fig, 10) differs significantly from the corresponding profile in the low
regime (Fig. 6). The velocity of the grains rises continually from the bed to the top
energy
The profile of the
horizontal
velocity shows the transition into another
of the partical flow.
the small block of grains near
regime of movement.
Between
the chute and the freely moving
particles in the collisional zone there is a region with an approximately
cluster density
constant
(Fig.ll). This behaviour is also observed in the experiment [31]. Neighbouring clusters move
If the
rises
energy
flow does
over
critical
a
particle
the
with
3.3
relative
velocities
ROUGH
same
set
of
between
this
In
BED.
parameters
as
each
section
in the
other.
we
previous
want
to
section
discuss
but
the
with
the
results
of the
rough
bed.
simulations
As
shown
in
with
the
figure 12,
behaviour
for the case of the
energy-tii~ie-diagram for this case does not show the same
as
qualitatively over a
rough bed but has roughly a parabolic shape. This shape is maintained
long time corresponding to a wide range of kinetic energy of the particles until the kinetic
will be discussed
into the region of
saturation.
The problem of energy
saturation
energy
coi~ies
the
in
the
more
detail
in
Corresponding
flow, figures
section
to
the
14-16
3A.
previous section, figure
sho~v the
particle-density,
13
the
snapshot of a typical situation of
a
averaged velocity in horizontal
direction
shows
N°I
GRANULAR
DOWN
FLOiVING
IIATERIAL
CHUTE
INCLINED
AN
35
h
250
o
200
o
~
iso
ioo
50
o
o
fact
that
Averaged
the particle
and
the
cluster-density
flow
as
snapshot figure
These layers
The
values.
(Fig. IS)
velocity (high
changed into an other
horizontal
13
shows
the
regime).
It is
the
1
noticeable
most
index
to
the
regime.
distance
situation
it
as
experimentally
found
were
of the
function
a
energy
(normalizedl
velocity
horizontal
0
Fig,10.
from
be
can
[31].
the
bed
h.
observed
The
flow
over
a
divides
wide
into
of
range
three
energy
different
particles close to the chute whose velocity rises continously with the
h, the particles in the block gliding zone which have approximately
the
velocity and a few fast moving particles in the collisional
at the top of the flow.
same
zone
into three regions of different
Also figure 16 shows evidently that the flow divides
types of
The slow particles close to the bed and the fast particles in the
fluidization.
collisional
zone
This
call block gliding
separated by a zone, where clusters
It was
zone
are
occur.
we
zone.
first investigated experimentally in [31].
zones
distance
from
the
the
slow
chute
nui~ierical
The
experiments shown above were performed in the accelSTATE.
reach a steady state
regime. Since our model includes friction the system will of course
simulations
above, this
after waiting a long enough time. For the parameter
sets we used in the
results
steady state is incident to very high particle velocities.
To obtain
numerical
accurate
in the steady state regime therefore
has
choose
high
resolution
I.e.
time
to
one
a
very
a
very
small tii~ie step hi.
Hence the computation
time rises strongly with the kinetic
energy of the
system.
STEADY
3.4
erated
demonstrate
To
angle
the
of
inclination
energy
of the
that
tY
our
=
system
model
and
10°
as
a
is able
the
function
to
reach
time
step
of
time.
At
the
=
steady
0.001
s.
state
Figure
regime
17
we
shows
have
the
chosen
evolution
the
of
JOURNAL
36
PIIYSIQUE
DE
N°1
II
h
200
°°O°°°°°m°
iso
+M+°Q~h
°
~o
oar
o
~_
~~
*Wom~~ogqoo°
°~~
~OS° ~~o@V°
ioo
o
°
°
4°~f~~
50
°
°
~
o
o
~y
o#
~~
~ooo
ooooo>oww_o
~
om~_~
wo
o
o
density
cluster
0
Fig.ll.
(normalized)
density (high energy regime). Between the rest
the free moving particles on the top of the flow lies a region of
and
Cluster
of the
1
gliding on
cluster-density.
block
constant
the
chute
Discussion.
4.
the surface of an
inclined
chute
investigated through a
on
was
dynamics
simulation.
of
smooth
bed
simulated
by
For the
case
a
regularly arranged small spheres on the surface of the chute we found two qualitatively different
corresponding to low and high kinetic energies. The velocity as
regimes of particle
movement
well as the density profiles agree with experimental results [31]. For the case of the rough bed
behaviour.
bed the density and velocity
did not find such a
As in the
of the
smooth
case
we
motion of groups
profiles agree with the measured data. In both cases we observed a coherent
observed
in the experiment.
Of particular
of grains, so called
clusters, which has also been
of the flow on a chute with regular (smooth)
that the
behaviour
interest
to us the fact
seenls
flow
The
granular
of
twc-dimensional
significantly
differs
surface
By simulating
model
a
does
include
not
show
static
qualitatively correctly for
the
the
flow
on
si~ialler
a
steady
a
results
from
with
system
reach
to
numerical
The
which
able
is
material
molecular
state
that
our
friction
system
at
chute
a
irregular (rough)
with
inclination
angle
constant
kinetic
a
dimensional
two
is able
gave
we
surface.
that
our
numerical
energy.
ansatz
describe
to
evidence
the
for
the
force
acting
experimentally
on
the
measured
grains
scenario
considered.
Acknowledgements.
author
The
the
sions.
I
wants
Jiilicli
KFA
thank
Dynamics.
to
express
his
gratitude
to
and to him as well as to Stefan
Gerald
Ristow for his support
Hans
Ilerrmann
Sokolowski
for
concerning
the
invitation
for the
many
fruitful
technical
to
and
problenls
the
HLRZ
patient
of
of
discus-
Molecular
N°I
GRANULAR
kinetic
IIATERIAL
FLOiVING
DOWN
AN
CHUTE
INCLINED
37
energy
E_TR
E_ROT
25
20
15
io
5
0
0
50000
100000
200000
150000
steps
time
Fig,12.
The
over
Evolution
energy
a
wide
shows
no
of the
kinetic
(transversal)
kink
in the
flow
as
the
on
and
smooth
rotational
bed
energy
(Fig. 4)
but
for
the
its
shape
case
of the
is close
rough bed.
parabolic
to
range.
~oo
~o
~oo
~o
o
o
Fig.13.
blocks
Snapshot
are
drawn
bold.
after
80.000
timesteps.
As in
figure
4
the
particles
which
belong
to
one
of
the
JOURNAL
38
PHYSIQUE
DE
II
N°1
h
200
°°°°°
°
°f&oowu
o
o~o~pop
o
*~i°~6ij
150
°~'~
~a~o
°°~
ioo
a~~
o~
°W
50
~°°°~io$*°
~~oo°g
%4°
oftj$"
~O~O
o
~o
o
o
o
o
density
Particle
different
than
behaviour
depending on
corresponding
the
the
vertical
density
distance
figures (Figs. 5,9)
(normalized)
from
for
oooo
o
o
o
particle
0
Fig.14.
o
o
the
the
case
1
bed
It
h.
of the
shows
no
very
bed.
smooth
h
250
200
o
iso
ioo
50
o~~
o
o
o
o
oco
o
0
Fig,
to
Averaged
lb-
the
chute,
continually
the
with
velocity
horizontal
block
the
horizontal
gliding
distance
velocity.
zone
from
and
the
The
the
bed
flow
consists
collisional
h.
zone
of
near
diierent
the
(normalized)
regions:
top of the
1
the
flow.
slow
The
particles close
velocity rises
FLOWING
MATERIAL
GRANULAR
N°I
DOWN
AN
INCLINED
CHUTE
39
h
250
200
~~~
~
~
~
~
~
150
~
~
~~
~
~~~~~~~°°°°°°j#~
~
~~l~f8qw
*°"~i~o~o
o
100
/$
~
~fll~gq~~
o
O~~°~
5°
~~~
~o
oo
O
°
oo~
~
o
o
woo
o
~
°
~~oo~l4"
~
~
o
0
Fig.16.
called
cluster
density.
Cluster
gliding
block
is
zone
a
of
significiant higher
(normalizedl
cluster
1
density.
This
region
will
be
zone.
kinetic
1
There
density
energy
E_ROT
8
1.6
1
4
1
2
1
o
s
0.6
0.4
0.2
0
0
100000
50000
200000
150000
300000
250000
350000
400000
time
Fig.17.
was
inclined
Saturation
by
an
of the
angle
of
kinetic
a
=
energy
lo° only.
after
long
time
for
a
flow of
particles
down
steps
a
chute
which
PIIYSIQUE
DE
JOURNAL
40
II
N°1
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