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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
DIFFUSE AND STEEP JUMPS IN STEADY-STATE GRANULAR FLOWS
T. Faug
1,2,
*
(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 (Corresponding Author)
I. Einav
School of Civil Engineering, The University of Sydney
Sydney, NSW, 2006, Australia. itai.einav@sydney.edu.au
P. Childs
School of Civil Engineering, The University of Sydney
Sydney, NSW, 2006, Australia. pchi2783@uni.sydney.edu.au
E. Wyburn
School of Civil Engineering, The University of Sydney
Sydney, NSW, 2006, Australia. ewyb2833@uni.sydney.edu.au
ABSTRACT
The design of civil engineering structures such as protection dams and mitigation walls against the
impact of large-scale geophysical flows requires a fundamental knowledge of the underlying physics
of granular flows around obstacles. It is now well established that large discontinuities in depth and
velocity, namely standing waves, jumps and shocks, can be formed throughout the flowing granular
body, at the upstream side of the obstacle. The current paper describes a newly established small-scale
laboratory apparatus, a tank feeding a chute, designed for studying in detail the shape of the granular
jumps across a wide range of flow conditions. These flow conditions include variation of the input
flow rate, chute inclination and basal roughness. The preliminary tests presented here highlight the
existence of complex transitions from strong shocks to very diffuse jumps either two or three
dimensional in shape, and the possible development of a basal stagnant zone downstream of the jump.
KEYWORDS
Granular media, flows, obstacles, standing waves, energy dissipation.
INTRODUCTION
The interaction of a granular flow with an obstruction is ubiquitous in both nature and industry. The
collision of a rapid full-scale granular avalanche with a civil engineering structure may for instance
lead to the formation of a steep granular jump. This steep granular jump is a sharp discontinuity in
depth and velocity analogous to a hydraulic jump, and a thorough understanding is important for the
design of efficient protection structures able to break, deflect and stop avalanches. Some fundamental
aspects of the steady jumps have been earlier studied by Savage (1983) and Brennen et al. (1983).
More recently, the propagating steep jumps or bores (unsteady state) also received attention, as
illustrated among others by Gray et al. (2003). However, those studies were mostly focused on the
thickness of the jumps with very poor attention to their shape and length. These dimensions provide
crucial information for modelling of the granular jumps (Faug, 2013). More generally, a further study
ACMSM23 2014 2
of those patterns is expected to provide a better understanding of the rheology of granular flows
because the jump is likely to develop over various distances which strongly depend on the ability of
the granular medium to dissipate energy. This paper describes a newly established experimental set-up
that allows us to investigate the shape of granular standing waves under-steady state conditions.
Although the results are still preliminary, we highlight a complex underlying physics indicated by the
observation of very different patterns depending on the experimental conditions: input flow rate,
chute-flow inclination, basal flow conditions, etc. In particular, we observe (i) a transition from strong
shocks to diffuse jumps when the Froude number of the incident flow decreases and (ii) the
development, under some circumstances yet to be solved, of a quasi-static zone at the base of the flow
which is located downstream of the flow discontinuity. The paper describes the laboratory apparatus,
the associated experimental techniques, the preliminary results and conclusions.
LABORATORY DEVICE
The laboratory apparatus consists of a tank and a chute both made of clear acrylic (Figure 1). The
granular chute consists of a 1-m-long channel of width W=0.1m. The side-walls are 15-cm-high. The
chute can be inclined at an angle
ζ
, independent from the tank inclination. The tank can also be
inclined in order to avoid too much granular material remaining stuck inside itself. During the tests
described in this paper, the tank was systematically inclined at around 16°. The base of the chute can
be varied from a smooth base surface (clear acrylic) to a rougher base surface (sandpaper of varying
grade). The chute is fitted with two gates located at the connection with the tank. A first gate controls
the mass discharge into the entry of the chute (or exit of the tank) via an opening height H. The second
gate downward allows for the storage of the granular material inside the tank and can be suddenly
removed or inserted to respectively release or halt the flow. To ensure the flow is steady for the
duration of the experiment, the tank must initially be filled with a sufficient mass of grains, in the
typical range 25-50 kg for the tests addressed in this paper. A control steady flow is first established
that is entirely characterized by the couple (
ζ
, H ) and additionally by both the size d of the flowing
grains and the typical size
λ
of the asperities of the bottom surface, in accordance to the pioneering
work of Pouliquen (1999) and the recent well-documented discrete element simulations by Weinhardt
et al. (2012). Once a steady flow is established an obstacle is placed at the end of the chute, inserted
from the top, in order to generate a standing granular jump, as depicted in the inset in the upper right
corner of Figure 1. The granular material used for the tests are glass beads of mean diameter d=1.2
mm and of particle density
2500=
P
ρ
kg.m-3 (see inset in the bottom left corner of Figure 1).
Figure 1. Photograph of the laboratory set-up: tank and granular chute in the smooth bed configuration
(without sandpaper). Inset in the bottom left corner: glass beads of diameter
d
=1.2 mm used in the
experiments. Inset of the upper right corner: example of a steep granular jump (see the details in the
results’ section) formed for
H/d
=18.3 and
°
=
25
ζ
over the smooth acrylic bed )0(
≈
λ
.
ACMSM23 2014 3
EXPERIMENTAL TECHNIQUES AND PROCEDURE
The following experimental techniques were designed to investigate the kinematics of the control
granular flows down the chute (
i.e.
flows without any obstruction placed at the chute exit) as well as
the length and shape of the jumps when the outgoing gate is partially obstructing the chute exit. The
experiments involved (i) measurement of the mass discharge at the chute exit with a weighing scale,
(ii) automatic detection of the flow free-surface thanks to recorded video images and (iii) granular
particle image velocimetry (granular PIV) aided by a high-speed camera in order to capture velocity
profiles and local strain rates. This paper is restricted to the results from the first two techniques.
Mass discharge, density and depth-averaged flow velocity
In the frame of depth-averaged equations applied to granular flows, one can define the depth-averaged
velocity:
∫
=
h
dzzu
h
u
0
)(
1. For control flows without any outgoing obstacle, the mean velocity at a
given position can be computed from the mass discharge and the flow thickness h at the same position
(see next subsection for the measurements of h): )/(/
hWqSqu
Pm
φρ
== . Some assumptions are
needed to estimate the volume fraction
φ
of the granular layer due to the difficulties associated with
accurately measuring the density of a granular flow (see detailed discussion at the end of this section).
Under steady-state conditions, the weighing scale provides us with the mass discharge 0
m
q
, as shown in
the typical discharge curve displayed in Figure 2 (left). Figure 2 (right) shows how the opening height
of the gate at the exit of the tank controls the mass discharge through linear scaling between 0
m
q
and
the typical mass discharge
gHWHq
PH
2
0
ρφ
=. The latter discharge was derived from the product
of the outlet area WH and the velocity gH2 produced by falling freely under gravity from a height H
(
0
φ
is the random close packing expected for monodisperse beads at rest in the tank).
Figure 2. Mass over time measured at the chute exit with the weighing scale:
H/d=17.5,
ζ
=25° and
λ
≈
0 (left). Mass discharge
0
m
q
versus gHWHq
PH
2
0
ρφ
=
with 64.0
0
=
φ
(right).
The Froude number of the incident flow is classically used in hydraulics and is defined as follows:
ζ
cosgh
u
Fr
=
(1)
The Froude number is often successfully used for granular flows, however the Froude number does
not account for the particle size involved in granular flows. It is now well established that the inertial
number, defined as the ratio between a microscopic time
P
Pd
ρ
// which represents the time it takes
for a particle to fall in a hole of size d (particle diameter) under a pressure P (typical time scale of
rearrangements), and a macroscopic time scale
γ
&
/1 linked to the mean deformation (where
γ
&is the
local strain shear rate), is a very good candidate to capture the transitions between various granular
regimes. These regimes include quasi-static, dense inertial and rapid dilute regimes, as discussed in the
ACMSM23 2014 4
review by Forterre & Pouliquen (2008) and references therein. In particular, a transition from dense
granular flows (at small I) to more rapid dilute granular flows (larger I) has been observed but this
transition is yet to be understood. Our preliminary tests confirmed the existence of this transition for
flows down an inclined chute. In practice, for a smooth chute bottom, much more rapid dilute and thin
flows are expected whatever the chute inclination
ζ
greater than the friction angle at the flow-bed
interface, meaning that the bulk internal friction angle is irrelevant. For a rougher chute bottom the
flow generated is dense at intermediate slopes (
maxmin
ζζζ
<<
), or accelerating and dilute at larger
slopes (
max
ζζ
>
), where
min
ζ
and
max
ζ
are the friction angles associated with quasi-static deformations
and rapid collisional flows, respectively (see details in GDRMidi (2004)).
The rapid thin flows are characterized by a very dilute (gaseous) layer developing on top of a dense
(liquid) layer, which may cause difficulties in defining the flow thickness and the depth-averaged
density (beyond a reliable measurement of those quantities). The former denser flows are
characterized by a nearly constant density over the entire depth, which can be described by the
following constitutive law (Forterre & Pouliquen, 2008):
II
)()(
maxminmin
φφφφ
−+=
, where the typical
values for the constants for monodispersed glass beads (used in our tests) are 4.0
min
=
φ
and
6.0
max
=
φ
. As the velocity profiles are currently not measured in our tests (future tests involving
granular PIV will give us access to velocity profiles at the side-walls), we use a macroscopic inertial
number that can be linked to the Froude number and the grain diameter relative to the flow thickness:
h
d
FrI
M
2
5
=
(2)
The above equation is derived by assuming a Bagnold-like velocity profile for the dense liquid regime
over a rough base, which directly stems from the observation that the local inertial number is nearly
constant over the thickness in this dense granular regime (GDRMiDi, 2004). According to this latter
equation combined with the constitutive law )(
I
φ
and the definition of
0
m
q
, one can compute
u
as a
function of
h
and
0
m
q
for dense flows (physically acceptable solution of a second-order equation):
−
−
+
−
=
max
2
0
maxmin
2
max
maxmin
cos
)(10
)(5
cos2
φ
θρ φφ
φ
φφ θ
ghWh
dq
d
ghh
u
P
m
(3)
For much more rapid dilute and thin flows on the smooth bed, we systematically considered a constant
depth-averaged density equal to
min
φ
, which yields )/(
min
0
hWqu
Pm
ρφ
=.
Detection of the flow free-surface
A 25 fps video camera and image processing done using the software package ImageJ allow for the
reconstruction of the average free-surface of the granular flow along the channel as a function ),( txz .
For steady and uniform flows, it yields
0
),( htxz =where
0
hholds for the thickness of the control flow
without any gate obstructing the end of the chute. For steady but non-uniform flow conditions, it
yields either )(),(
0
xhtxz =for accelerating and decelerating control flows or )(),( xztxz = for the
shape of the granular jumps formed upstream of the outgoing gate (see example in Figure 3).
Figure 3. Free-surface of the flow under steady-state conditions: z(x)/h
0
as a function of x*=(x-x
0
)/L
(left graph). Local inclination
θ
fs
of the free-surface made with the horizontal versus x/h
0
(right graph).
Results obtained with ,25
°
=
ζ
3.18/
=
dH and 0
≈
λ
(steep jump shown in Figure 1).
ACMSM23 2014 5
Figure 3 (right graph) depicts the local inclination
θ
fs
of the free-surface (made with the horizontal and
defined in Figure 1), which allows us to describe the main features of the patterns generally observed
in our preliminary tests. In particular, two distinct zones systematically appear: (i) a zone for which the
angle
θ
fs
is an increasing function of x/h
0
(when x>x
0
, where x
0
is a well-defined value for which
θ
fs
reached a minimum value: inflection point) and (ii) a zone for which
θ
fs
is constant (close to 22° here),
which means that the free-surface can be represented by a straight line in this zone. The length of the
first zone defines the length L of the jump. It is important to note here that L is independent of the size
of the system, i.e. the location upstream at which the jump is formed, which can be tuned at will by the
operator. Then the normalized thickness z/h
0
of the jump can be represented as a function of the
normalized coordinate x*=(x-x
0
)/L (see left graph of Figure 3). This description appears to be
convenient to quantitatively compare the steepness of the various jumps observed for different
experimental conditions (see results in the following section).
RESULTS AND DISCUSSIONS
In the following section we briefly analyze the preliminary results showing that the granular pattern
can dramatically change when the various experimental parameters (
ζ
, H ,
λ
) are varied, as shown in
Figure 4 (left) summarizing the tests conducted to date.
Smooth (clear acrylic) chute base
Our preliminary tests under a smooth chute configuration highlight the transition from diffuse jumps at
low values (but still supercritical) of the incoming Froude number, i.e. Fr ≈ 2 (or low values of the
macroscopic inertial number: I
M
≈ 1-2) to very steep jumps at larger Fr
≈
8 (or I
M
≈ 4-5), as shown in
Figure 4. It is worth noting that the granular jumps can have a complex three dimensional structure for
some of the tests, specifically the thickness and the steepness of the jump are not constant across the
width of the chute and the jump developing in the centerline is steep while a more diffuse, longer jump
develops at the sidewalls. This variation across the chute width, possibly caused by disturbances
(caused by some imperfect input conditions) that propagate down the chute, will be studied.
Figure 4. Preliminary tests summarized in the )/,( dH
ζ
diagram for two distinct values of the
roughness of the chute (left). Cross-comparison of the shape of granular jumps on a smooth bed for
three opening heights of the gate at the tank exit )25(
°
=
ζ
: see notation in Figure 3 (right).
Rough (sandpaper of grade P60) chute base
Preliminary tests on rougher beds (sandpaper of grit P60 glued on the chute bottom) show a crucial
effect where lower values of the incoming Froude number are measured. The macroscopic inertial
number is systematically below unity. The resulting jump is very diffuse (Figure 5) and purely two-
dimensional, presenting no variation across the chute width. The minimum in local slope
θ
fs
is nearly
inexistent (see inset in Figure 5) and we do not observe a recirculation zone in contrast to the typical
curve shown in the right graph of Figure 3 for steep jumps over a smooth bed at much higher Froude
numbers. Moreover, we observe the formation of a quasi-static dead zone co-existing with the flowing
zone above whereas the layer downstream of the jump is entirely sheared when the chute bottom is
ACMSM23 2014 6
smooth whatever the steepness of the jump. The pattern shown in Figure 5, obtained for Fr = 0.75 (I
M
= 0.2), resembles more a flow over a static pile than a granular shock (so diffuse that one might not be
able to detect this in latter experiments).
Figure 5. Picture obtained by superposing 100 images (4 s of steady flow) that shows the granular
pattern obtained from H/d=10,
°
=
5.33
ζ
and 22.0
≈
λ
; the jump is very diffuse and a stagnant zone is
formed downstream of the jump. The boundary of the dead zone (dotted line as a guide for the eyes)
and an expected (not measured) velocity profile are shown. Inset: corresponding
θ
fs
(x)-curve.
CONCLUSIONS
The preliminary tests conducted on the newly established granular chute showed evidence of (i)
complex transitions from steep to very diffuse jumps, (ii) jumps either two or three dimensional in
shape and (iii) with or without the presence of a downstream quasi-static zone. Future tests will
systematically investigate the shape of the granular jumps across the flow conditions developed from
variation of input flow rate, chute inclination and base roughness for a range of particle sizes.
Increasing the scope of flow conditions will endeavour to build a complete phase diagram governing
these jumps. Our laboratory device is fully operational and will also be used to observe the segregation
patterns of mixtures of distinct grain sizes that are expected to develop inside the standing jumps.
ACKNOWLEDGMENTS
The authors are very grateful to Todd Budrodeen and Garry Towell for the design of the granular
chute, as well as to Ross Barker and Bogumil Eichstaedt for their assistance for the calibration of the
weighing scale. The authors also acknowledge Nico Gray for the initial discussion they had with him,
which enhanced their motivation to start setting the laboratory tests. TF is grateful to the financial
support by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme under REA grant agreement No. 622899 (FP7-PEOPLE-2013-IOF, “GRAINPACT”).
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