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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”).

REFERENCES

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G.D.R. MiDi (2004) “On dense granular flows”, The European Physical Journal E, Vol. 14, No. 4, pp.

341-365.

Gray, J.M.N.T., Tai Y.C. and Noelle S. (2003) “Shock waves, dead-zones and particle-free regions in

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Faug, T. (2013) “Jumps and bores in bulky frictional granular flows”, AIP Conference Proceedings,

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