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Forces experienced by the walls of a granular lid-driven cavity

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23
rd
Australasian Conference on the Mechanics of Structures and Materials (ACMSM23)
Byron Bay, Australia, 9-12 December 2014, S.T. Smith (Ed.)
1
FORCES EXPERIENCED BY THE WALLS OF A GRANULAR LID-DRIVEN
CAVITY
F. Kneib*
Irstea, UR ETGR, 2 rue de la papeterie, 38402 Saint Martin d’Heres, France.
francois.kneib@irstea.fr (Corresponding Author)
T. Faug
(1) Irstea, UR ETGR, 2 rue de la Papeterie BP76, 38402 Saint Martin d'Hères, France.
(2) School of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia.
thierry.faug@irstea.fr
F. Dufour
Université Grenoble Alpes, 3SR, F-38000 Grenoble, France.
CNRS, 3SR, F-38000 Grenoble, France.
frederic.dufour@3sr-grenoble.fr
M. Naaim
Irstea, UR ETGR, 2 rue de la papeterie, 38402 Saint Martin d’Heres, France.
mohamed.naaim@irstea.fr
ABSTRACT
Impact forces from full-scale granular flows on civil engineering structures exhibit huge fluctuations
mainly caused by the co-existence inside the granular medium between a quasi-static zone, in direct
contact with the structure, and a surrounding inertial zone. The aim of the current paper is to describe a
time-averaged study and to show first results about those force fluctuations with the help of discrete
numerical simulations on a two dimensional granular lid-driven cavity. A small-scale granular sample
is trapped between one bottom wall, two side walls and one upper wall shearing the sample at constant
speed under a given confinement pressure. We conducted a systematic study of the time-averaged
force on the walls under different velocities, pressures and lengths of the sample relative to its
thickness. Our study evidences a critical length above which the results are not influenced by the
length and we observe a recirculation zone whose center mainly depends on the shearing velocity.
Despite the strong inhomogeneities of pressure distribution, the steady forces experienced by the walls
are governed by the macroscopic inertial number. We also provide a preliminary analysis of the force
fluctuations.
KEYWORDS
Granular media, walls, force, fluctuations.
INTRODUCTION
The main motivation of the current paper is a better understanding of the forces experienced by rigid
walls in direct contact with a quasi-static granular medium that is subject to various shear rates and
confinement pressures. The two latter parameters are likely to model the effect of a surrounding layer
of flowing grains (see Gaume et al. (2011)). This situation is often encountered when a full-scale
avalanche-flow interacts with a civil engineering structure and is known to lead to large force
ACMSM23 2014 2
fluctuations (see Faug et al. (2010)). Our method is based on discrete numerical simulations. This
paper is focused on a lid-driven cavity with attention paid to the resulting forces on the wall. The lid-
driven cavity is often used in fluids mechanics to study turbulence (Shankar & Deshpande (2008)) but,
to the best of our knowledge, remains poorly investigated in the field of granular materials. A box
without gravity is subject to a constant horizontal shear velocity and a constant value (averaged over
the length of the box) of the vertical pressure at the top. Although the system is complex
(inhomogeneous force distribution at the top, recirculation inside the cavity), a systematic study of the
time-averaged force on the walls allows for the definition of the main regimes governing the system.
Once those regimes are defined, one can pay attention to the force fluctuations for each regime. The
numerical method and the simulated system are first presented. The phenomenology of the 2D
granular lid-driven cavity is then described. The next part describes a systematic study of the time-
averaged dynamics of the system and some preliminary results regarding the force fluctuations on the
walls. Finally, the future systematic studies to be conducted on the force fluctuations are discussed.
NUMERICAL METHOD AND SETUP
DEM General Formulation
The Discrete Element Method (DEM) is an extension of the Molecular Dynamics modeling method
introduced by Cundall & Strack (1979). A linear visco-elastic contact law, described in Pournin et al.
(2001) and well suited for dynamic processes in granular media, is used:
(1)

(2)
The above equations, expressed in the contact local coordinates, define the normal (
) and tangential


contact forces.
and
are the normal and tangential contact stiffnesses,
is the contact
normal viscosity coefficient,
and
are the contact normal and tangential relative displacements,
and
is the normal relative velocity. The tangential force is restricted to a threshold force via a
Coulomb friction coefficient , without any viscosity applied through this axis. As
,
and
are
not well-suited to directly drive the contact behaviour, we use a modulus/Poisson formulation coupled
to the formulation described in Pournin et al. (2001) which results from the integration of Newton's
equation of movement. Each sphere has a Young modulus
, a Poisson coefficient
and a friction
coefficient
. The contact properties for an interaction between a sphere (of mass
and diameter
) and a sphere (
,
) are computed as it follows:

(3)
The normal viscous coefficient is set from , the contact restitution coefficient:



with

. (4)
The contact law can be represented by the spring-dashpot model, as shown in Figure 1a.
Figure 1a. Linear spring-dashpot model. Figure 1b. Representation of the lid-driven cavity.
Simulation of a Two Dimensional Granular Lid-driven Cavity
The lid-driven cavity is modeled by a two dimensional assembly of disks trapped in a rectangular
gravity-free box (see Figure 1b). The two side walls are fixed, smooth and free slip while the bottom
wall is fixed and rough. The top wall is infinite, horizontal, rough and rigid. It applies the horizontal
velocity , and the constant vertical pressure is performed by its vertical displacement. Roughnesses
are made by glued grains. The sample height is and the grain parameters are chosen to match the
ACMSM23 2014 3
properties of glass beads (likely to be used in complementary laboratory tests not described in this
paper). The grain diameters are randomly chosen between 

and d

where is
the mean diameter and

the dispersion coefficient.
We first observe the system response through averaged values once steady state is reached. Time
averages are made over 10 to 20 seconds while space averages are made over the cavity height.
Recorded values are the time and space averaged normal force on the right (

) and left (

)
walls, the time averaged normal force on the top wall

and the time averaged grains velocity into
the cavity. The varying parameters are the shear velocity, top pressure and cavity length. The
whole parameters and their range of variation are given in Table 1. For convenience, dimensionless
numbers
and
are introduced:
 and  

. Though the simulations are
gravity free, the parameters are chosen to fit laboratory setups when
( ).
Symbol Value Unit Signification

mean diameter of grains



dispersion coefficient


particle density
estimated 2D volume fraction
restitution coefficient


normal stiffness
Poisson ratio
friction coefficient

[
] cavity height


cavity length

shear velocity coefficient


confinement pressure coefficient
Table 1. Parameters and notations used in this paper.
(a) Linear-linear plot (b) Log-linear plot
Figure 2. Time-averaged forces along the top wall for various shear velocities and for cavity lengths
( taken into ). All curves are drawn for the pressure coefficient
.
PHENOMENOLOGY OF THE 2D GRANULAR LID-DRIVEN CAVITY
The study of the time-averaged vertical force on the top wall

(Figure 2a) confirms an intuitive
phenomenon: the confinement pressure is highly heterogeneous along the box length. The log-linear
plot in Figure 2b shows an exponential force profile increasing in the shear direction. Under a constant
pressure and at a given cavity length, the curves from different shear velocities are superposed, which
means that the force profile is independent of the shear rate. In addition, one can notice that all rates of
rise are the same so the exponential increase is independent of the box length and the shear rate. Next
to the right wall the increase in force is higher than exponential, which can likely be explained by a
“wedge effect” due to the local jamming of grains in the top right corner. This wedge length relative to
the cavity length is small for large cavities while it is far from being negligible for shorter cavities. The
time-averaged velocity streamlines (Figure 3) reveal that the top shearing wall involves a circulation
ACMSM23 2014 4
of the grains into the box. Next to the top surface, the particle displacements are driven by the shearing
rough wall but near the bottom wall the grains are going back slowly to the low pressure zone. This
results in a thin horizontal layer of grains which is highly sheared, located between the top wall and a
quasi-static zone. Moreover, the x-position (through the cavity length) of the vortex center depends on
the shear velocity: high shear rates shift the vortex center to the right side of the cavity.
(a)
,
,   (b)
,
,  
Figure 3. Time averaged velocity streamlines inside the cavity for two distinct shear rates. Null
velocities are represented by thin light-grey lines while high velocities are represented by black thick
lines. The approximate x-position of the vortex center is represented by a vertical dashed line.
TIME-AVERAGED DYNAMICS AND FLUCTUATIONS
Cavity Length Relative to its Height
The boundary conditions of the system generate local phenomena affecting the global behavior. The
cavity size has to be taken into account in particular. Figure 4a shows the influence of the cavity length
to its height (aspect ratio ) on the mean force on the right wall rescaled by the top force,

.

) depends on  at low values of, confirming the importance of the wedge
effect for small cavities. However, for every shear rate,

 reaches a relatively constant
value at higher lengths, typically above a critical value of around. Figure 4b shows that the
results for the mean forces on the left wall, via the ratio

 as a function of , are
analogous. Furthermore, the mean force on the left wall tends to zero for high values of , which
makes it negligible in comparison with the mean force on the right wall. Since one wants to avoid left
side effects and in order to limit the calculation time, the results presented in the following were
obtained with a constant value of  equal to 5.
Flow Regimes
In the field of granular flows, it is now well-established that the dimensionless inertial number I is
relevant to qualify the different flow regimes (GDR MiDi (2004)): quasi-static (I smaller than 10
-4
),
dense inertial (10
-3
< I < 0.1) and dilute collisional (I > 0.1-1) regimes. This number is defined by the
ratio between the local shear velocity timescale and the local grain relaxation timescale under the
confinement pressure P:
, with
and
, (5)
where P
P
is the pressure applied on the grain and is the local shear rate. For this preliminary study
we used a macro-scale definition of I (via  and the mean pressure ):

. (6)
The Figure 4c displays

 as a function of

for different couples of velocity and
pressure (each set of data is obtained by varying
at a given
). All curves are nearly
superimposed so the couple  used to obtain the macro-scale flow regime is not significant while
looking at the mean force on the right wall. This means that

is a pertinent indicator to define a
certain flow regime. The linear-log plot of the curves (Figure 4d) shows a saturation of

at low
ACMSM23 2014 5
values (around 10
-3
)
as well as at high values (around 0.1), while it has a logarithmic behavior for
intermediate values. Even though this conclusion deserves a wider range of

values to clearly
identify those transitions, it is in accordance with results of GDR MiDi (2004) and the analysis of the
force fluctuations will be conducted by taking the regime transitions into consideration.
(a) Force on the right wall (b) Force on the left wall
(c) Linear-linear plot (d) Linear-log plot
Figure 4. (a),(b): mean force on the two side walls, both scaled by the mean total force  applied to
the top wall, as a function of the cavity length ratio for different shear velocities and
. (c),(d):
mean force on the right wall, scaled by the force applied on the top , as a function of the
macroscopic inertial number

. Each set of data corresponds to one shear velocity U and the
variations of

are obtained by varying the top pressure .
Force Fluctuations
In order to analyze the fluctuations of the normal force on the right wall, a discrete Fourier transform
algorithm is applied on the force on the right wall signal. The resulting spectra depends on the top
pressure and velocity conditions, so the preliminary results discussed thereafter are mainly based on
two representative simulations corresponding respectively to the intermediate and collisional regime.
We were not able at this time to conclude about the quasi-static regime. The roughness frequency U/d
is defined as the frequency at which the top glued grains are moving horizontally over a distance equal
to their diameter. In order to avoid the top right corner wedge effect affecting our analysis of time
force series, only the normal force between and    will be accounted for.
A very thin and intense peak appears at the roughness frequency for the dense flow regime (Figure 5a),
which means that the force applied on the top horizontal grain layer at the roughness frequency is well
transmitted to the right wall through the sample. This phenomenon might be explained by a periodic
succession of loading and unloading of the strong force network (see Radjaï et al. (1998)).
In the inset of Figure 5a, one can observe that in the collisional flow regime the above high peak is no
longer detectable and the right wall does not experience directly the top roughness. It might be
explained by force chains that are unable to establish because of large spatial gaps between particles in
this very dilute collisional regime. However, a high peak at a very low frequency (period about 1.4 s in
Figure 5a) is observed and corresponds to the periodic force signal obtained from the filtering of high
frequencies and depicted in Figure 5b. It can be explained by a wave-like effect detectable in the
ACMSM23 2014 6
collisional regime: grains from the top layers are periodically driven against the right wall, but the
resulting heterogeneous confinement pressure drive them back to the left toward a more homogeneous
state. This phenomenon has not been observed yet in the dense inertial flow regime.
Figure 5a. Power spectrum from the time series
of
force intensity for an intermediate flow regime
simulation (I
macro
8.10
-2
). Inset: same spectrum
for a collisional flow regime (I
macro
0.6).
Figure 5b. Force on the right wall versus time
for a collisional regime
: raw signal and
averages
over three different time windows (0.1
s, 0.2 s and 14 s) to filter high frequencies.
CONCLUSIONS
The time-averaged forces on the total height of the side walls of a lid-driven cavity were well
characterized with the evidence of a critical box size and a transitional regime at intermediate values
of

. The spatial distribution of the force on the right wall and the force at the bottom will be
tackled in the future. A preliminary study of the temporal force fluctuations on the right wall
highlighted one peak (observed in the dense regime) controlled by the frequency at which the grains of
the upper plate roughness move horizontally and another peak at much lower frequency (detectable in
the collisional regime) which reveals an oscillating wave-like regime. The latter findings will be
further studied in relation to how the network of force chains evolves inside the box.
ACKNOWLEDGMENTS
Thanks to Guillaume Chambon for the fruitful discussions we had. This work has been partially
supported by the LabEx Tec 21 (Investissements d’Avenir - grant agreement n°ANR-11-LABX-0030).
REFERENCES
Gaume, J., Chambon, G., Naaim, M. (2011) “Quasistatic to inertial transition in granular materials and
the role of fluctuations”, Phys. Rev. E, 84, 051304.
Faug, T., Chanut, B., Beguin, R., Naaim, M., Thibert, M., Baroudi, D. (2010) “A simple analytical
model for pressure on obstacles induced by snow avalanches”, Annals of Glaciology, 51 (54), 1-8.
Shankar, P. N. and Deshpande, M. D. (2008) “Fluid mechanics in the driven cavity”, Annual Review of
Fluid Mechanics, Vol. 32, pp. 93-136.
Cundall, P.A., Strack, O.D.L. (1979) “A discrete numerical model for granular assemblies”,
Géotechnique, Vol. 29, No. 1, pp. 47-65.
Pournin, L., Liebling, T. M., Mocellin, A. (2001) “Molecular-dynamics force models for better control
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Physical Society, 65, 011302.
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Radjai, F., Dietrich, E.W., Michel, J., Moreau, J.J. (1998) “Bimodal Character of Stress Transmission
in Granular Packings”, Physical Review Letters , Vol. 80, No. 1, pp. 61-64.
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