Segregation Effects in Granular Collapses
Joshua Caplan
Supervisors: Stuart Dalziel and Nathalie Vriend
Abstract
In this essay we consider the collapse of a cylindrical column comprised of two
layers of differently sized ballotini. We vary the initial mass of the column and
compare the two orientations of the layers with the case of a column of a single
material. The use of differently sized particles leads to so-called granular segregation,
with the larger particles rising towards the surface of the flow. Remarkably this
does not affect the macroscopic flow dynamics and other properties, such as the
final run-out distances, are consistent with previous work.
1 Introduction
Granular materials are ubiquitous in both geophysical and industrial situations and exhibit a wide variety of complex behaviours. Despite much recent research they remain
poorly understood. One particular problem, as illustrated in figure 1, is that of the collapse of a column of granular material under its own weight, and the resulting spreading
flow. Such situations occur in, for example, cliff collapses and industrial accidents, and
can be extremely dangerous and destructive.
ℎ
𝑟
ℎ∞
𝑟∞
Figure 1: Illustration of the column collapse problem.
The problem of granular collapse was first considered by Lube et al. (2004) and Lajeunesse et al. (2004) who independently considered the case of a monodisperse column
1
and varied the initial column height ℎ , the initial column radius 𝑟 , the granular medium
and the base roughness. Both papers found that the key variable was the aspect ratio,
𝑎 ≡ ℎ /𝑟 , of the initial column and that the actual mass (or volume) of the column was
unimportant. Using this they found that there were two main regimes depending on
whether the aspect ratio was greater or less than a critical value 𝑎 , although they disagreed on the exact value. For small 𝑎, only the edge of the column collapsed, leaving
a truncated cone. For larger 𝑎, the collapse reached the centre, leaving a steep central
cone which tapered towards the edges. Lajeunesse et al. refer to this as a ‘Mexican hat’
profile. In both regimes there was an undisturbed region of material in the centre of the
deposit.
Lube et al. (2004) found power laws for the non-dimensional run-out distance
𝑟̃ ≡
⎧1.24𝑎
𝑟∞ − 𝑟
=⎨
/
𝑟
⎩1.6𝑎
𝑎 < 1.7
𝑎 > 1.7
.
In contrast Lajeunesse et al. (2004) derived semi-empirical laws
⎧
𝑟∞
= 𝑟̃+ 1 = ⎨
𝑟
⎩
𝑎 + 4 tan 𝜃 −
𝑎 < 0.74
,
𝑎 > 0.74
where 𝜃 was the angle of repose of the material. These two sets of laws are, however,
similar over the range of aspect ratios considered. Both groups found that the final nondimensional height of the deposit followed power laws
⎧𝑎
ℎ
𝑎<1
ℎ̃ = ∞ = ⎨
(Lube et al., 2004)
/
𝑟
𝑎 > 1.7
⎩0.88𝑎
⎧𝑎
𝑎 < 0.74
=⎨
(Lajeunesse et al., 2004),
⎩0.74 𝑎 > 0.74
where the differences appear to be in the interpretation of very similar data. In addition,
both groups found that, for large aspect ratios, the deposit spread outwards at a constant
velocity proportional to 𝑔𝑟 , with only a short acceleration and deceleration phase.
Lube et al. also found that the time taken for the full collapse was
𝑡∞ = 3.0
ℎ
,
𝑔
which they note is approximately twice the free-fall time for a particle dropped from
height ℎ , i.e. 𝑡∞ ≈ 2(2ℎ /𝑔) / .
Neither group found that the choice of material had any significant effects on the flow,
although later work by Trepanier & Franklin (2010) showed that extremely non-spherical
grains (particle aspect ratio > 4) behave very differently. The choice of base roughness
2
also had little effect on the run-out distance, but Lajeunesse et al. found that it did affect
the final profile shape at large aspect ratios. Both groups also found that there was little
mixing between adjacent horizontal slices of the initial monodisperse column.
These experiments have also been extended to quasi-two-dimensional collapses along
rectangular channels (Lube et al., 2005; Balmforth & Kerswell, 2005; Lajeunesse et al.,
2005; Lube et al., 2007). As with axisymmetric collapses, two regimes were found depending on the aspect ratio, but the behaviours for large 𝑎 followed a different power
law that, due to the effects of friction at the walls, depended on the width of the channel. A semi-circular geometry was also considered by Lajeunesse et al. (2005), which
behaved the same as the axisymmetric case but allowed more of the internal structure
of the collapses to be observed. Various systems involving sloped beds have also been
considered (Hogg, 2007; Mangeney et al., 2010; Lube et al., 2011; Maeno et al., 2013). Collapses have also been successfully modelled numerically using discrete element methods (DEM) (Staron & Hinch, 2005, 2006; Zenit, 2005), using shallow-water-like systems
of equations (Mangeney-Castelnau, 2005; Kerswell, 2005; Larrieu et al., 2006; Doyle et al.,
2007; Hungr, 2008), and using the non-Newtonian 𝜇(𝐼) rheology (Lagrée et al., 2011).
All of this previous work has focused on monodisperse materials, whereas real systems typically feature significant amounts of polydispersity. This leads to segregation,
with the larger and less dense particles rising to the surface of the flow, and the smaller
and denser particles sinking towards the bottom (Drahun & Bridgwater, 1983). This can
then cause so-called segregation-mobility feedback, which can have dramatic effects on the
flow dynamics (e.g. Pouliquen et al., 1997). For the collapse problem this could lead to
changes in the run-out distance and other flow properties.
Tunyasuvunakool (2011) considered columns of a fixed size composed of large pearl
barley and small glass ballotini collapsing onto a smooth base. He found that columns
with large particles initially on the bottom (normally stratified columns) typically had
larger run-outs than the reverse (inversely stratified columns). In particular the normally
stratified columns had their longest run-out when they comprised around 80% ballotini by volume. Inversely stratified columns only had longer run-outs for columns with
around 10% ballotini, which Tunyasuvunakool attributed to the smaller particles forming rollers for the barley to slide along. Tunyasuvunakool also found that, for certain
combinations of layer orientation and height ratio, a region of barley would separate
from the deposit to give what he termed a ‘detached ring’.
In Caplan (2012), I instead considered the same materials but varied the column height
for a fixed ratio (50%) of the two materials. Overall, I found that inversely stratified
columns behaved almost identically to the monodisperse columns considered by Lajeunesse et al. (2004) and Lube et al. (2004), albeit with a slightly reduced run-out distance
due to the large barley grains sinking into the ballotini layer beneath them. In contrast,
normally stratified columns behaved significantly differently due to the smaller ballotini falling through the barley layer and pushing a ring of barley outwards. At lower
aspect ratios this gave rise to a detached ring, whereas it remained attached to the main
3
deposit for larger columns. In addition, I found that the collapse edge became unstable
leading to asymmetric deposits, most likely due to a fingering instability of the form
discussed in Pouliquen et al. (1997) and Pouliquen & Vallance (1999).
2 Experimental Method
Figure 2: Photograph of the experimental setup. The cylinder used to construct the
columns can be seen in the centre of the image. The laser scanner is the metal
box to the right of the cylinder; the red line on the deposit is the image of the
laser. The high-speed camera can be seen in the background to the left of the
cylinder.
The apparatus used in this experiment can be seen in figure 2. In all the experiments
we used spherical ballotini. For the small particles we used colourless ballotini with
diameters between 300 µm and 425 µm, and for the large particles we used red ballotini
with diameters between 1000 µm and 1300 µm. We considered columns with layers of
equal mass of the two particle types in both the inversely stratified (large ballotini on
top) and normally stratified (small ballotini on top) orientations. In addition, we considered monodisperse columns comprised solely of the smaller ballotini. The columns
were constructed in a plastic cylinder of internal diameter 32 mm and ranged in total
4
initial mass from 200 g to 1200 g. At the start of the experiment the cylinder was lifted
using a pneumatic piston, thereby allowing the column to collapse over a flat base covered with sandpaper. The collapses were filmed with a high-speed camera recording
at 1000 frames per second, and the height profiles were measured using a laser scanner
operating at 100 profiles per second. The high-speed camera footage was then digitally
processed to automatically measure the radius of the final deposit.
3 Results
3.1 Collapse dynamics
The dynamics of all three column types were very similar over the range of aspect ratios
(1.3 to 8.0) considered, as shown in figure 3. As the column collapses it spreads axisymmetrically outwards at the base, with the upward regions falling straight down. When
the upper layer reaches the base of the collapse it spreads outwards over the lower layer.
At this stage no segregation can be observed in any initial stratification. As the collapse
progresses, the upper layer moves outwards faster than the lower layer, reaching the deposit edge as the collapse ends. During the very end of the collapse, some large ballotini
can be seen rising to the surface of the normally stratified columns due to segregation.
In figure 4 we can see the height profiles taken during the collapse of a typical inversely
stratified column; the other column types are qualitatively and quantitatively similar at
the same aspect ratios. This shows how the deposit only spreads at the base, rather than
along its whole height. We can also see how material moves from the centre of collapse
to the deposit edge as the collapse occurs and the deposit thins.
An example of the surface radial velocities during a collapse can be seen in figure 5.
As with the height profiles this does not vary significantly between collapses of different types with the same aspect ratio. As the collapse begins, the deposit accelerates
outwards before reaching a roughly constant spreading velocity. Material that was initially higher in the column moves outwards faster than the front, as was seen in figure 3.
When the column of material has completely collapsed (at 𝑡 ≈ 400 ms in this example),
the flow begins to decelerate before stopping. This leads to a region of stationary material, which can be seen expanding from the centre of the deposit towards the edge.
Figure 6 shows the deposit radius against time for the full range of collapses considered. In this the constant spreading speed can be clearly seen along with shorter
acceleration and deceleration phases. In particular, the spreading velocity is independent of the initial height of the column and the composition of the column. For the initial column radius used in these experiments, the spreading velocity is approximately
0.5 m s− .
Figure 7 shows the run-out against time for the columns of largest aspect ratio after
5
(a) Inverse (𝑎 = 7.9)
(b) Monodisperse (𝑎 = 8.0)
(c) Normal (𝑎 = 7.9)
Figure 3: Images for column collapses of different types. Images are taken 200 ms,
300 ms, 400 ms and 600 ms after the start of the collapse. These correspond
to non-dimensional times of approximately 0.33, 0.95, 1.6 and 2.8.
6
60
0.25 s
0.35 s
0.37 s
0.39 s
0.41 s
0.45 s
0.55 s
50
h/mm
40
30
20
10
0
−40
−20
0
20
40
60
80
100
x/mm
Figure 4: Deposit profiles taken at various times during the collapse of an inversely stratified column of aspect ratio 𝑎 = 7.9 as measured using the laser scanner.
t/ms
500
0.8
400
0.6
300
0.4
200
0.2
100
0
50
100
150
200
Radial velocity/ms−1
1
600
0
250
r/mm
Figure 5: Plot of the radial velocity at the surface of the flow throughout the collapse for
the same collapse as in figure 4.
non-dimensionalising by
𝑟̂ =
𝑟̃
𝑎
/
=
𝑟∞ − 𝑟
,
𝑟
𝑡̃=
𝑡−𝑡
,
ℎ /𝑔
where 𝑡 is the time at the start of the constant speed phase. As can be seen, all the data
collapses onto a single curve thereby showing that these scalings are appropriate. In
particular, the time scaling is that for a freely falling object (free fall time = (2ℎ/𝑔) / ),
demonstrating that the flow is controlled by the influx of new material as the column
collapses. For smaller aspect ratios the acceleration and deceleration phases dominate
(as can be seen in figure 6) and the scalings break down.
7
250
Inverse
Monodisperse
Normal
r/mm
200
150
100
50
0
0
100
200
300
400
500
600
700
t/ms
Figure 6: Measurements of deposit radius 𝑟 against time 𝑡 for a range of collapses. Collapses of the same type range in initial mass from 200 g to 1200 g, with the
larger columns travelling a further distance.
2.5
2
r̂
1.5
1
Inverse
Monodisperse
Normal
0.5
0
0
0.5
1
1.5
2
t̃
2.5
3
3.5
4
Figure 7: Graph showing the modified non-dimensional run-out distance 𝑟 ̂ = 𝑟𝑎̃ − /
against the non-dimensional time 𝑡 ̃ for collapses with large aspect ratio (𝑎 > 5).
3.2 Deposit details
Figure 8 shows the interiors of an inversely and a normally stratified collapse. Both
types of collapse feature a central cone where the initial orientation is preserved. As expected, there is negligible mixing anywhere in the inversely stratified column as the stable stratification prevents any segregation and, as shown in the work on monodisperse
collapses, there is no other mechanism for mixing. In contrast, the normally stratified
collapse shows some signs of segregation but this process is incomplete. For segregation to occur there needs to be shearing across the interface between the layers. The
experimental set-up means that this can only occur for a short period of time and the
amount of shearing is too weak to allow complete segregation.
Figure 9 shows the central height of the deposit, as measured using the laser scan-
8
(a) Inversely stratified column.
(b) Normally stratified column.
Figure 8: Cross-sections of the final deposit for typical examples of the two types of column. Photograph was taken by rapidly inserting a piece of clear plastic and
removing the material from one side of it.
32
Inverse
Monodisperse
Normal
h∞ /mm
30
28
26
24
22
20
1
2
3
4
a
5
6
7
8
Figure 9: Graph of the final deposit height ℎ∞ against the initial aspect ratio 𝑎.
ner, against the initial aspect ratio of the column. Monodisperse columns appear to
be slightly taller than either type of stratified column. In addition, inversely stratified
columns appear to be slightly shorter at larger aspect ratios. The cause of this is not
fully understood and is likely, in part, to be a measurement issue. The deposit height
appears to decrease slightly at larger aspect ratios but the trend is unclear.
Figure 10 shows a log-log plot of the final non-dimensional radius for the collapses
9
r̃
8
7
6
5
4
Inverse
Monodisperse
Normal
3
2
1
1
2
3
4
5
6
7
8
9 10
a
Figure 10: Log-log plot of non-dimensional run-out 𝑟 ̃ against initial aspect ratio 𝑎. The
solid line is 𝑟 ̃ = 2.1𝑎 / and the dashed line is 𝑟 ̃ = 1.25𝑎.
considered. In addition, the best fitting half-power laws
⎧1.25𝑎
𝑟̃ = ⎨
/
⎩2.1𝑎
𝑎<2
𝑎>4
,
have been plotted which are consistent with those previously found for monodisperse
columns. While all three types of column have very similar final run-outs, the monodisperse columns travel slightly further. In addition, the inversely stratified columns travel
a slightly shorter distance than the normally stratified ones. The exact cause of this is
unclear and may just be due to differences in how the radius measurement program
detected the edge of the deposit.
3.3 Effect of ratio of particle types
In addition we also considered a range of columns with varying proportions of small
particles Π . These were all of total mass 2000 g and of initial radius 47 mm. Only normally stratified columns were considered: inversely stratified columns should not be
affected by column make-up due to their stable initial stratification.
Although figure 11 appears to show that columns with a large proportion of small
ballotini go further, this is misleading. The main difference is due to the type of particles present at the edge of the collapse when it begins to decelerate. As this happens,
particles at the edge are able to separate from the bulk and spread out as discrete objects.
If these are small particles then they cannot get very far due to the surface roughness,
whereas large ballotini are able to bounce across the surface. This means that the large
particles are not included in the final deposit, causing it to appear smaller. Figure 12
shows the final deposits for these collapse, showing the different edge behaviour.
10
300
Π =0
r/mm
s
250
Πs= 0.25
200
Πs= 0.5
Πs= 0.75
150
Πs= 1
100
50
0
0
100
200
300
400
t/ms
500
600
700
800
Figure 11: Measurements of the deposit radius 𝑟 against time 𝑡 for a range of collapses
of differing proportions of small particles Π .
This behaviour at the edge also causes problems when measuring the deposit radius.
When the edge becomes disperse it is hard to determine the edge in a consistent manner.
This means that the radius measurement becomes inaccurate at late times, with the level
of inaccuracy depending on the choice of particle type ratio.
4 Conclusions
The key finding of my work is that, contrary to expectations, segregation is surprisingly
unimportant in the macroscopic dynamics of the systems considered: the bidisperse
columns behaved almost identically to the monodisperse ones previously considered.
While segregation does occur, it is incomplete and has no effect on the flow dynamics. The few differences present between the collapses considered can be adequately
explained by the different behaviour of the two sizes of grain when they act as discrete
particles, as opposed to changes in the behaviour of the bulk flow.
It is also important to reconcile these findings with those of Caplan (2012) and Tunyasuvunakool (2011) who found significant differences between their columns. The significant difference in experimental methods is that we both used barley grains for the
larger grained material. Barley has a lower density than the ballotini and means that
columns of the same height no longer have the same potential energy. We can compensate for this by rescaling the initial height to that of a column comprised purely of the
material in the lower layer with the same initial potential energy. As seen in figure 13,
this causes the differences in run-out to mostly disappear.
The increased run-out from the ‘outside’ measurement can be explained in a similar
11
(a) Π = 1
(b) Π = 0.75
(c) Π = 0.5
(d) Π = 0.25
(e) Π = 0
Figure 12: Final deposits for normally stratified columns of mass 2000 g, initial radius
47 mm and a range of different proportions of small particles Π .
12
7
6
5
r̃
4
3
Inverse
Monodisperse
Normal (inside)
Normal (outside)
2
1
0
0
1
2
3
4
5
a
(a) Graph showing the non-dimensional run-out 𝑟 ̃ against the non-rescaled aspect ratio 𝑎.
7
6
5
r̃
4
3
Inverse
Monodisperse
Normal (inside)
Normal (outside)
2
1
0
0
1
2
3
ã
4
5
6
(b) Graph showing the non-dimensional run-out 𝑟 ̃ against the rescaled aspect ratio 𝑎.̃
Figure 13: Graphs showing effect of rescaling the aspect ratios of the collapses considered by Caplan (2012) in order to take into account the differing initial potential energies. For the distinction between the two normal measurements see
the original paper.
manner to the discussion in section 3.3. In the normally stratified collapses a region of
barley detached from the edge of the deposit and, due to the smooth base, was able
to slide a significant distance. In contrast, the other collapses had ballotini at the edge
which could slide a much shorter distance. As the outer measurement includes this
sliding distance it is not surprising that it fails to collapse onto the same curve.
The ‘detached rings’ observed are likely due late stage separation of barley from the
edge of the collapse. Due to the smooth base the barley slides as opposed to the bouncing observed in the set of experiments discussed earlier. This is a less random process
and so a distinct ring is formed as opposed to the diffuse edge.
This sliding is likely why Tunyasuvunakool found that, for normal columns, there
13
was a peak in run-out at roughly 80% ballotini. For any higher proportions there is less
barley at the front edge during the deceleration phase and so less can detach (compare
with the results in section 3.3). For lower proportions, the reduction in column energy
is likely to depress the run-out, eventually reducing it below that of a column of pure
ballotini. Tunyasuvunakool also found a peak in run-out for inverse columns at around
5% ballotini. This is, again, likely due to the edge being comprised only of the larger
barley but is possibly aided by the ballotini acting as rollers, helping the barley to slide
further than would otherwise be expected.
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