VOLUME 87, NUMBER 13
PHYSICAL REVIEW LETTERS
24 SEPTEMBER 2001
Structure Formation and Instability in a Tube of Sand
Eirik G. Flekkøy,1 Sean McNamara,2 Knut Jørgen Måløy,1 and Damien Gendron3
1
Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo 3, Norway
2
CECAM, Lyon, France
3
Groupe Matière Condensée et Matériaux, Université de Rennes 1, F-35042 Rennes Cedex, France
(Received 1 March 2001; revised manuscript received 2 August 2001; published 10 September 2001)
A new instability in the combined flow of fine grains and gas is investigated by means of experiments,
simulations, and analytic techniques. When a bubble of air rises through a granular packing in a tube,
a sequence of smaller bubbles spontaneously forms in front of it. The existence of this instability is
shown from the experiments, simulations, and theoretical considerations. Moreover, the simulations and
experiments agree on the quantitative level. In particular, when the tube is tilted away from the vertical
the experiments and the simulations show the same increase in the speed of the rising bubble.
DOI: 10.1103/PhysRevLett.87.134302
PACS numbers: 45.70.Mg, 46.55. +d, 47.11. +j
While a host of intriguing aspects of dry granular materials and flows have been revealed and extensively studied
over the last couple of decades, relatively little is known
about the behavior of granular flows where the interstitial gas plays a key role. Yet this is the case in such
diverse practical and industrial contexts that it is hard to
list them all. Practical examples include the fluidized flow
from cement or grain silos, landslides following heavy rain
falls, avalanches, and industrial powder transport processes
[1]. These examples all result from the characteristics of
fluid-grain flow. Like dry granular flows, these flows exhibit striking and intriguing effects that are uniquely granular. Such effects include the intermittent flow of powders in
hour glasses [2,3], the coupling between the granular flow,
the gas pressure and granular dilation [4,5], and the surprisingly complex dynamics of granular bubbles [6 –10],
including the granular Boycott effect [11,12]. Recently,
and much in the same spirit as the above studies, some interesting effects of gas-grain coupling in vibrated particle
systems have been studied [13].
The present Letter deals with an instability that arises
in gas-grain flows in tubes: In the regime where the particle sizes and background pressure are not too small the
granular flow around a single bubble of air rising in a tilted
tube under the influence of gravity is unstable to the formation of secondary bubbles or ripples in front of the main
bubble. Once the main bubble starts to rise a sequence of
precursor bubbles, as shown in Fig. 1, rapidly forms. This
hitherto unstudied and striking instability is here observed
and studied by means of both experiments and simulations.
These agree on all the main qualitative and most quantitative aspects. The instability presents itself as an intrinsically granular phenomenon (it could not have emerged,
say, in a liquid-gas system), and we devote the present Letter to the investigation of its origins and robustness.
The experimental setup consists of a glass tube of internal diameter 0.5 cm and length 100 cm filled with spherical glass particles of diameter d 苷 共180 6 15兲 mm. The
relative air humidity was kept within 27 6 3% during the
filling procedure. The initially horizontal tube has a 15 cm
FIG. 1. The two first images show the ripple instability as
it emerges from experiments and frictionless 2D simulations,
respectively. The angle between the tubes and gravity, which
acts downwards and to the right is Q 苷 70±. The two last
images show the structure of the main bubble. The main ticks
on the experimental scale are separated 1 cm.
134302-1
© 2001 The American Physical Society
0031-9007兾01兾 87(13)兾134302(4)$15.00
void space at one end. When the tube is tilted quickly to a
preset angle, the bubble rises and starts to form the precursor train of bubbles. Video recording and direct observation of the tube allow the determination of the propagation
speed. As the main bubble starts to rise a set of bubbles
form sequentially away from the main bubble. The train
134302-1
of bubbles rapidly selects a characteristic spacing between
them, and while there appears to be a certain effective repulsion between bubbles, merging events do occur. By
gently tapping the tube prior to the experiment it was possible to pass from a looser solid fraction c 苷 0.56 to a
more densely packed medium with c 苷 0.60 [11]. In a
separate experiment using silver coated, instead of pure
glass beads, it was checked that electrostatic effects play
no significant role. To check for possible artifacts in the
turning procedure we also performed experiments using a
mechanical shutter to release the bubble [14].
While the simulations confirm most of the experimental
behavior, we stress that they are not carried out with the
aim of merely reproducing the experimental results. As the
simulations contain no Coloumb friction, are carried out in
2D, and coarse grain the fluid dynamics to a level above
the grain scale, they represent significant simplifications
[15]. For exactly that reason the understanding of the
experiments in terms of the simulations sheds significant
light on the processes at hand. This insight includes the
observation that at steep tube inclinations the rising bubble
behaves very much as if there were no sliding friction
between the grains or between the grains and the wall.
The simulation model combines a discrete particle description with a continuum description of the gas [15].
The grain packing is taken to define a porous medium
through which the gas flows according to a local Darcy
law. This model represents a simplification compared to
existing models for systems such as fluidized beds [16],
by virtue of the relatively simple fluid dynamics component. Mathematically, the fluid component of our model is
described by the pressure equation
µ
∂
µ
∂
≠P
k
f
1 u ? =P 苷 = ? fP
=P 2 P= ? u, (1)
≠t
m
where f is the porosity or volume fraction of the space
between grains, P is the pressure, m the viscosity of air,
and u the locally averaged velocity of the grains [15]. The
permeability is taken as the Carman-Kozeny relation [17]
k共f兲 苷 共a 2 兾45兲f 3 兾共1 2 f兲2 , where a 苷 d兾2 is the
(spherical) particle radius, and the number 45 is obtained
empirically for a random packing of spheres. Throughout
Eq. (1) we apply the transformation f ! 0.66f 1 0.33 so
that the volume fraction and porosity obtained from a 2D
closed packing coincide with the 3D closed packed value.
Equation (1) describes the mass conservation of an isothermal ideal gas with no inertia. It is solved in two dimensions by means of the Cranck-Nicholson scheme [18].
The Darcy law that is embedded in Eq. (1) requires the
particle Reynolds number Re to be small. When this is
no longer the case nonlinear corrections to the pressure
gradient –velocity relationship enter the picture. For moderate Re these corrections are contained in the Ergun equation [19], which reduces to the Darcy law when Re ! 0.
If we use the gas flow velocity in the region where the pressure forces are largest, i.e., in the immediate neighborhood
134302-2
24 SEPTEMBER 2001
PHYSICAL REVIEW LETTERS
of the bubbles, we obtain the estimate Re 艐 0.2. Then the
nonlinear correction term of the Ergun equation is of relative order 0.4%, and, for the smaller particles, even smaller.
The model is in general limited to flows where Re is small.
The particles obey the equation of motion
m
dv
=P
苷 mg 1 FI 2 m
,
dt
r
(2)
where m is the particle mass, v a single particle velocity,
g the acceleration of gravity, FI the interparticle force, and
r 苷 rg 共1 2 f兲 the granular mass density (rg is the mass
density of the material that makes up the grains). The force
FI prevents particle overlap and is calculated using contact
dynamics. This algorithm reduces to the standard event
driven method for dilute flows, and permits a consistent
treatment of Coulomb friction. It has been used successfully to investigate forces in granular packings [20]. While
Coulomb friction is absent in all the present simulations the
particles dissipate a fraction of their kinetic energy in each
collision. The restitution coefficient is 0.8.
The key dimensionless numbers that characterize the dynamics of Eqs. (1) and (2) are the Péclet number and the
Froude number. The Péclet number is defined as Pe 苷
U0 lm兾共P0 k0 兲, where l is a characteristic length, k0 苷
d 2 兾180 and P0 the background pressure. The Froude
number Fr has the form Fr ⬅ U02 兾共gl兲 苷 U0 兾g兾t, where
t 苷 l兾U0 . Here U0 is a characteristic velocity, which we
choose to be U0 苷 rgk共f0 兲兾m where f0 苷 0.4 is the
porosity of a 3D close packing of spheres. This implies
that Pe 苷 rg lg兾P0 . Taking t 苷 0.01 s it follows that
l 苷 0.25 cm. The experimental values used as input in
the simulations are Pe 苷 0.000 625, Fr 苷 2.5. The relative particle size variations are the same in the experiments
and simulations.
Both experiments and simulations are carried out at
varying tilt angles Q to the horizontal [11]. Figure 2 shows
the results of this. There is striking agreement between
50
40
Speed (cm/s)
VOLUME 87, NUMBER 13
30
20
10
0
20
40
60
Θ
80
100
FIG. 2. The speed of the main bubble as a function of tilt
angle Q to the horizontal. The simulations (䊐) represent the
frictionless results and have only one packing density. The
experimental measurements correspond to initially high- (±) and
low-packing densities (≤).
134302-2
VOLUME 87, NUMBER 13
5
35
4
30
3
2
1
25
20
15
0.2
0.4
0.6
P0(Bar)
0.8
1.0
FIG. 3. The number of precursor bubbles as a function of the
background pressure P0 .
134302-3
tion point 60 cm above the initial main bubble decay in a
roughly linear fashion down to the critical pressure P0 苷
0.07 bar at which the precursor bubbles are no longer
present. Figure 4 shows the speed UB of the main bubble as a function of P0 . Note that, while P0 varies by a
factor 16, UB changes by only a factor of 2. Note also that
while a correspondence between the existence of ripples
and the bubble speed is clearly visible under variations in
Q (Fig. 2), such a correspondence is not visible in Fig. 4.
When Q is reduced to the point where the ripples disappear UB reaches a maximum; when P0 is decreased to the
point where the ripples disappear no particular variation in
UB is observed.
In the simulations, which depend on the assumption that
pressure variations are much smaller than the background
pressure, the small P0 regime is currently not accessible.
At P0 苷 1 bar the numbers of bubbles per length are the
same in the experiments and the simulations within the
noise of the measurements (see Fig. 1). The absence of
friction, however, appears to cause a longer train of ripples.
In fact, the observation of the top of the ripple train has
been outside the scope of the computations.
Why does the ripple instability disappear when P0 or
d is decreased? To understand this, we need a theoretical picture of the forces that act in the granular packing
above the main bubble before the ripple instability sets in.
These forces are friction from the glass walls, gravity, the
interparticle granular stresses, and finally the force acting
from the gas on the particles. If Eq. (1) is taken to describe the pressure in the undeformed packing above the
main bubble in the frame of reference R that follows the
top of the main bubble, it becomes a simple advectiondiffusion equation with diffusivity D 艐 P0 k共f0 兲兾m. In
steady state this equation is solved by the pressure gradient =P 苷 共=P兲0 e2zV0 兾D , where V0 is the downwards
velocity of the sand in R and 共=P兲0 is the pressure gradient right above the bubble. Note that =P is not linear
in the distance z from the bubble. Since the gas inertia is
neglected =P represents the entire force per unit volume
acting on the grains from the gas. Taking the Q 苷 90±
Speed(cm/s)
Number of precursor bubbles
the frictionless simulations and the experiments, most so
at high angles. In the experiments the ripple instability
disappears when the angle becomes smaller than angles
of maximum velocity, Qm 苷 50± or 65± for the low- and
high-packing densities, respectively. At these angles there
is a clear change to the flow pattern. When Q , Qm the
flow above the bubble takes place in a surface layer only,
while the grains adjacent to the glass remain at rest. This
is clearly a friction effect. In the simulations there is no
friction to keep any part of the packing above the bubble at
a stationary position. Hence the maximum is at Q 苷 40±.
However, in the simulations too the ripples disappear at
the angle of maximum velocity; at Q 苷 30± there are no
ripples. These results demonstrate clearly that while static
and sliding friction play a role at small Q, friction plays
only a minor role at Q . Qm . The trailing eddy that forms
in the wake of the main bubble in the simulations is also
observed in experiments using smaller particles [11,13].
Both experiments and simulations exhibit the effect that
as Q is decreased the bubble velocity is increased [11,12].
In spite of some differences due the existence of friction
this effect has been named after Boycott who observed that
the sedimentation velocity of blood corpuscles in a tube
increased with decreasing Q [21] away from 90±. The tilt
causes the blood corpuscles and the fluid to separate into
different “lanes,” thus reducing the dissipation associated
with the process.
Remarkably, the ripple instability disappears when the
particle size is decreased. For d 苷 65 mm particles, the
instability no longer exists, and the main bubble rises alone
without the company of precursor bubbles. The bubble
also disappears in the simulations at this particle size when
care is taken to allow the particles to settle well. Practically, the small particle regime is investigated in the simulations by changing k0 , thus changing U0 and Fr. In a
similar way the instability disappears when P0 is reduced.
By connecting the tube of sand to a vacuum pump and
a pressure sensor we repeated the Q 苷 80±, d 苷 180 mm
experiment at different pressures P0 . Figure 3 shows that
the number of precursor bubbles that pass an observa-
0
0.0
24 SEPTEMBER 2001
PHYSICAL REVIEW LETTERS
10
0.0
0.2
0.4
0.6
P0(Bar)
0.8
1.0
FIG. 4. The speed of the main bubble as a function of the
background pressure P0 .
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VOLUME 87, NUMBER 13
PHYSICAL REVIEW LETTERS
case for simplicity and noting that the friction force from
the tube always acts upwards, it is then possible to write
the force balance on a slab of height dz and cross-sectional
area pr 2 as
kjPg 2prdz 2 =共P 1 Pg 兲pr 2 dz 苷 rgpr 2 dz ,
(3)
where Pg is the local average across the tube of the zz
component of the stress tensor, and r the granular mass
density as before. The constant k is the averaged ratio
of the horizontal to the vertical granular stress, and j the
dynamic friction coefficient between the grains and the
glass walls.
Equation (3) may be solved as an ordinary differential
equation for Pg when substitution is made for =P and
the boundary conditions Pg 共0兲 苷 Pg 共H兲 苷 0, where H
is the height of the granular packing above the bubble, are
applied. The result is
2共=P兲0
共ez兾hf 2 e2z兾hD 兲 2 rghf 共ez兾hf 2 1兲 ,
21
hD
1 hf21
(4)
f共H兾hf 兲
,
共=P兲0 苷 rg
f共H兾hD 1 H兾hf 兲
Pg 共z兲 苷
where f共x兲 苷 共1 2 e2x 兲兾x, the screening length for
the friction hf 苷 r兾共2kj兲, and the diffusion length
hD 苷 P0 k0 兾共V0 m0 兲. The gas pressure decays exponentially away from the bubble over a distance hD . In our
case, hD 苷 250 cm for the large 共d 苷 180 mm兲 particles,
and hD 苷 75 cm for the small 共d 苷 65 mm兲 particles.
Interestingly, the Pg solution in Eq. (4) may be used to
work out the total average friction force that acts from the
tube on the grains: When H 苷 0.5 m (and Q 苷 90±)
this friction is only about 10% of the grain weight. This
observation gives support to the neglect of friction made
in the simulations.
Now, can Eqs. (4) explain why the bubble formation
goes away when d or P0 is decreased? Note that the
velocity V0 艐 UB 艐 10 cm兾s does not vary drastically
with pressure (Fig. 4), nor does it vary strongly with particle size. Since the permeability k0 ~ d 2 we therefore
have that hD ~ P0 d 2 . This implies that when P0 or d
becomes small H兾hD becomes large and, since f共x兲 is a
monotonously decreasing function, 共=P兲0 becomes large
compared to rg. Then the force from the gas on the
grains acts more like a piston, i.e., it is localized close
to the bubble. This will pack the granular material at
a higher granular pressure, thus inhibiting the initial dilation that is needed for the secondary bubbles to form.
(When the granular density is at its closed packed value
there can be no bubbles.) When P0 or d is large, on the
134302-4
24 SEPTEMBER 2001
other hand, 共=P兲0 艐 rg everywhere, the granular weight
is nearly balanced by the gas pressure forces, and Pg becomes small. This argument holds both for the experimental case with friction and for the simulations without
friction as f共H兾hf 兲 ! 1 when hf ! `, corresponding to
the frictionless limit.
In conclusion, we have investigated a ripple instability
that arises in the flow of (not too-) small grain systems with
hydrodynamic interactions. The simulations reproduce the
experimentally observed ripple formation, qualitatively as
well as quantitatively for the speed of the bubbles as a
function of the tilt angle. The absence of friction in these
simulations and the agreement with the experiments have
stimulated theoretical investigations that shed light on the
existence of the crossover effect that ripples disappear at a
sufficiently small background pressure or particles size.
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