Third Term Report
Joshua Caplan
Supervisors: Stuart Dalziel and Nathalie Vriend
Problem Statement
Granular materials are ubiquitous in both geophysical and industrial situations
and exhibit a wide variety of complex behaviours. One particular problem 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. A
significant body of research on granular collapse exists (see below), but this
has focused on columns of a single type of material (so called monodisperse
materials), whereas real problems feature many types of material (polydisperse
materials). These materials then separate, with large particles rising and dense
ones sinking (Drahun & Bridgwater, 1983). The effects of this ‘granular segregation’ on the flow dynamics are still poorly understood. My PhD is currently
focused on trying to understand these more complex collapses through lab scale
experiments.
Previous Work
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. Both papers found that the key variable was the aspect ratio a
of the initial column and that the actual mass (or volume) of the column was
unimportant. Using this they found power laws for the non-dimensional runout
Figure 1: Illustration of column collapse problem.
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distances
r∞ − ri
r̃ =
∝
ri
{
a
a1/2
a < a0
,
a > a0
where both a0 and the constants of proportionality varied between the two
papers. Neither paper found that the choice of material had any significant
effects on the flow. The choice of base for the collapse had little effect on the
runout distance, but Lajeunesse et al. (2004) found that it did affect the final
profile shape at large aspect ratios.
These experiments have been extended to other geometries (Lube et al.,
2005; Balmforth & Kerswell, 2005; Lajeunesse et al., 2005; Lube et al., 2007),
to sloped beds (Hogg, 2007; Mangeney et al., 2010; Lube et al., 2011), flows
where the surrounding fluid is important (Rondon et al., 2011), and to more
extreme particle shapes (Trepanier & Franklin, 2010). Collapses have also
been studied numerically using discrete element methods (DEM) (Staron &
Hinch, 2005, 2006; Zenit, 2005), with shallow-water-like equations (MangeneyCastelnau, 2005; Kerswell, 2005; Larrieu et al., 2006; Doyle et al., 2007; Hungr,
2008), and by modelling as a non-Newtonian fluid (Lagrée et al., 2011).
The case of a polydisperse column, however, has not been studied in as
much detail with work focusing on the case of a column comprised of two layers
of materials of different sizes. In these columns it is expected that the larger
particles will rise to the surface during the collapse and that this will affect
the dynamics of the collapse. Tunyasuvunakool (2011) considered columns of
a fixed size composed of large pearl barley and small glass Ballotini. He found
that both the initial orientation of the layers and the height ratio of the layers
had a significant effect on the runout and dynamics of the collapse. He also
found that certain combination of parameters lead to the formation of so-called
‘detached rings’ where a region of barley separated from the main deposit. In
contrast, and as described in Caplan (2012), I considered the same materials
but varied the aspect ratio of the column for a fixed layer height ratio. I found
that columns with the larger material on top (inversely stratified) behaved similarly to monodisperse columns whereas columns with the smaller material on
top (normally stratified) had a larger runout and different dynamics. As well
as detached rings, I also found non-axisymmetric deposits for larger normally
stratified columns.
Current Work
My work this year has been to refine the experimental set-up used by Tunyasuvunakool (2011) and Caplan (2012) and to extend the range and quality of
measurements taken. The first change was to replace the barley with a second size of Ballotini to remove any possible density and size based effects. As
hypothesised by Caplan (2012), this prevented any asymmetries from forming.
The Ballotini, however, was able to roll away from the deposit at late times in
an extreme version of the previous observed detached rings. To prevent this I
changed from a smooth base to a rough one, which completely stopped any ring
detachment.
In addition I changed from specifying the height of the column and the height
ratio of the layers, to specifying the mass and mass ratio. This was easier to
2
250
Radius\mm
200
150
100
50
0
0
200
400
600
Time/ms
800
1000
Figure 2: Deposit radius against time for a selection of collapses of different
masses and same initial radius. Masses range from 200 g to 1200 g.
control accurately as the heights could vary for a given amount of material due
to inadvertent compaction of the column. As the heights could still be recorded
after filling this did not result in any loss of data.
To increase the accuracy of the runout distances, I have developed a semiautomated circle fitting program thereby removing any human bias and massively increasing the number of sample points for both the perspective correction and edge detection. This also allows the radius to be measured precisely
throughout the the duration of the collapse, as illustrated by the examples in
Figure 2. In addition, I have used a laser scanner to provide accurate height
profiles during the collapse, an example of which can be seen in Figure 3. Unfortunately the maximum measuring range of the laser scanner is insufficient to
obtain full profiles for the larger deposits.
I have also begun to use particle imaging velocimetry (PIV) to measure the
surface velocity of the flow, but, at present, it is not exact due to the oblique
camera angle combined with the varying height of the deposit. It does, however,
allow features like the location of any stationary material to be easily observed.
Future Plans
I am now in the process of collecting data, examples of which can be seen in
Figures 2 to 4. Over the summer, I plan to vary the mass (and hence aspect)
ratio and initial column radius, whilst keeping the mass ratio between the layers
constant. Depending on how this progresses I may also vary the mass ratio
or possibly the particles used. Towards the beginning of my second year I
hope to work on modelling the collapse, both with DEM and with simplified
3
210
Base
Deposit
Height/mm
205
200
195
190
185
180
−50
0
50
100
Distance/mm
Figure 3: Radial profile of the deposit 350 ms into a collapse.
0.0
Speed/ms-1
1.0
Figure 4: Velocity field from 400 ms into the collapse as calculated using PIV.
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mathematical models. Looking further ahead I could either choose to extend
this work to different geometries or more complex set-ups, or I could instead
consider a different problem in the general field of granular flows.
References
Balmforth, N. J. & Kerswell, R. R. 2005 Granular collapse in two dimensions. Journal of Fluid Mechanics 538 (-1), 399.
Caplan, Joshua 2012 Segregated Granular Collapse .
Doyle, Emma E., Huppert, Herbert E., Lube, Gert, Mader, Heidy M.
& Sparks, R. Stephen J. 2007 Static and flowing regions in granular collapses down channels: Insights from a sedimenting shallow water model.
Physics of Fluids 19 (10), 106601.
Drahun, JA A & Bridgwater, J 1983 The mechanisms of free surface segregation. Powder technology 36 (1983), 39–53.
Hogg, Andrew J 2007 Two-dimensional granular slumps down slopes. Physics
of Fluids 19 (9), 093301.
Hungr, Oldrich 2008 Simplified models of spreading flow of dry granular
material. Canadian Geotechnical Journal 45 (8), 1156–1168.
Kerswell, R. R. 2005 Dam break with Coulomb friction: A model for granular
slumping? Physics of Fluids 17 (5), 057101.
Lagrée, P.-Y., Staron, L & Popinet, S 2011 The granular column collapse
as a continuum: validity of a two-dimensional NavierStokes model with a
µ(I)-rheology. Journal of Fluid Mechanics 686, 378–408.
Lajeunesse, E., Mangeney-Castelnau, A. & Vilotte, J. P. 2004 Spreading of a granular mass on a horizontal plane. Physics of Fluids 16 (7), 2371.
Lajeunesse, E., Monnier, J. B. & Homsy, G. M. 2005 Granular slumping
on a horizontal surface. Physics of Fluids 17 (10), 103302.
Larrieu, E., Staron, L. & Hinch, E. J. 2006 Raining into shallow water as
a description of the collapse of a column of grains. Journal of Fluid Mechanics
554 (-1), 259.
Lube, Gert, Huppert, Herbert, Sparks, R. & Freundt, Armin 2005
Collapses of two-dimensional granular columns. Physical Review E 72 (4),
41301.
Lube, Gert, Huppert, Herbert E, Sparks, R Stephen J & Freundt,
Armin 2007 Static and flowing regions in granular collapses down channels.
Physics of Fluids 19 (4), 043301.
Lube, Gert, Huppert, Herbert E., Sparks, R. Stephen J. & Freundt,
Armin 2011 Granular column collapses down rough, inclined channels. Journal of Fluid Mechanics 675, 347–368.
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Lube, Gert, Huppert, Herbert E., Sparks, R. Stephen J. & Hallworth, Mark A. 2004 Axisymmetric collapses of granular columns. Journal
of Fluid Mechanics 508, 175–199.
Mangeney, A, Roche, O, Hungr, O, Mangold, N, Faccanoni, G &
Lucas, A 2010 Erosion and mobility in granular collapse over sloping beds.
Journal of Geophysical Research 115 (F3), F03040.
Mangeney-Castelnau, A 2005 On the use of Saint Venant equations to
simulate the spreading of a granular mass. Journal of Geophysical Research
110 (B9), B09103.
Rondon, L., Pouliquen, O. & Aussillous, P. 2011 Granular collapse in a
fluid: Role of the initial volume fraction. Physics of Fluids 23 (7), 073301.
Staron, L. & Hinch, E. J. 2005 Study of the collapse of granular columns
using two-dimensional discrete-grain simulation. Journal of Fluid Mechanics
545 (-1), 1.
Staron, L. & Hinch, E. J. 2006 The spreading of a granular mass: role of
grain properties and initial conditions. Granular Matter 9 (3-4), 205–217.
Trepanier, M & Franklin, Scott 2010 Column collapse of granular rods.
Physical Review E 82 (1), 11308.
Tunyasuvunakool, Saran 2011 Collapse of a Stratified Granular Column.
Zenit, Roberto 2005 Computer simulations of the collapse of a granular column. Physics of Fluids 17 (3), 031703.
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