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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
503
DOI: 10.4408/IJEGE.2011-03.B-056
DAM-BREAK FLOWS OF DRY GRANULAR MATERIAL
ON GENTLE SLOPES
l
uCa
SARNO
(*)
, m
aRia
n
iColina
PAPA
(**)
& R
iCCaRdo
MARTINO
(*)
(*)
University of Naples “Federico II” - Dept. of Hydraulic, Geotechnical and Environmental Engineering - Via Claudio 21
80125 Naples (NA), Italy - E-mail: lucasarno@alice.it, riccardo.martino@unina.it
(**)
University of Salerno - Dept. of Civil Engineering - Via Ponte don Melillo - 84084 Fisciano (SA), Italy
E-mail: mnpapa@unisa.it
INTRODUCTION
Debris flows as well as snow and rock avalanch-
es are fast-moving flows that occur in many areas
of the world. They are particularly dangerous to life
and property because they move with high velocities,
destroy infrastructures in their paths, and often strike
without warning.
In some real world cases, geophysical flows can
be triggered by phenomena that are very similar to
a dam break. Water flows generated by a dam break
have been widely studied and mathematical models
for water dam-break waves are available in many text-
books and research papers. Compared to water dam-
break waves, debris flow waves display a wider vari-
ability and, for their mathematical description, require
models with a much greater complexity. As in the case
of clear water, particular attention must be given to
their numerical integration because of the frequent de-
veloping of steep gradients and shock waves.
Owing to this complexity, a number of simplifi-
cations are put in place and tested under laboratory
conditions: the results of the tests are then used to
improve the rheological models that underlie the nu-
merical simulations.
In the specific case of debris flows, flows arising
from a dam-break-like event of dry granular material
can help in the process of model validation. Moreo-
ver, a good understanding of the mechanics of dry
granular flows is also essential in order to set up two-
phase debris-flow models because two separate mod-
ABSTRACT
This work examines dam-break flows of dry
granular material and investigates the suitability of
the depth-averaged models with particular attention
being given to the description of the shear stresses and
pressure terms. The experimental results of dam-break
flows down a gently sloped channel have been report-
ed. Tests were carried out on both a smooth Plexiglas
bed as well as a rough one. Measurements of the flow
depth profiles and the front wave position were ob-
tained using two digital cameras. In order to compare
the prediction of the depth-averaged approach with
granular avalanche tests, a specific mathematical and
numerical model was implemented. The momentum
equation was modified in order to take into account
the resistances due to the side walls. The numerical
integration of the shallow water equations was carried
out through a TVD finite volume method. In order to
address the importance of a good estimate of the stress
distribution inside the pile, several numerical simula-
tions were performed, calculating with different for-
mulas the pressure coefficient that relates longitudinal
and vertical normal stresses in the momentum equa-
tion. The simulations present, in general, agree with
experimental data. The differences have been outlined
between the smooth and rough bed cases.
K
ey
words
: dry granular, dam-break, pressure coefficient,
avalanche, depth averaged model
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L. SARNO, M.N. PAPA & R. MARTINO
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less than 30°). The aim of this paper is to present
an experimental investigation into dry granular flows
down a gently inclined channel, with specific atten-
tion being given to the suitability of depth-averaged
models. Furthermore, an additional aim has been to
examine the commonly assumed hypotheses about
stress distributions and, in particular, some different
approaches to estimate the coefficient k were imple-
mented and compared.
EXPERIMENTAL APPARATUS
An 8-m-long chute (Fig. 1), designed for the study
of both mud and dry granular flows, was set up at the
LIDAM (Laboratory of Hydraulic, Environmental
and Maritime Engineering of the University of Saler-
no). The channel inclination, which is constant for its
whole length, can be varied between 0° and 23°, by
rotating the structure around its lower end through an
hydraulic ram controlled by a pumping system. The
channel width can be adjusted between 0 and 80 cm,
as the right side wall position can be moved by a screw
system. The 90-cm-high side walls and the bottom are
both made of Plexiglas and are suitably supported by
structural steels. At the upper end of the chute, there
is a wide tank, integral with the chute structure, with
a capacity of 3 m
3
. In addition, at the lower end the
channel ends in a collector tank, with about the same
capacity as the upper one.
After setting up the apparatus, dam-break tests
were carried out on dry granular material. The chan-
nel width was set to 24 cm, which seemed to be large
enough to somehow reduce the influence of the side
walls on the flow mechanics.
For each run, the slope was trigonometrically
calculated through geometric measurements. Its ac-
curacy is good, due to it being valued less than 0.1°
elling of the solid and fluid phases are needed. The
propagation of a dam break of dry granular material
can be modelled by a depth averaged s
aint
-v
enant
(1871) approach. The material is assumed to be in-
compressible, with the mass and momentum equa-
tions being written in a depth-averaged form. This
analysis is valid under the assumption that the flow-
ing layer is thin compared to its lateral extension,
as often occurs in the case of geophysical flows. A
depth averaged approach makes it possible to avoid
a complete three-dimensional description of the flow
field. In fact, it is only necessary to specify a sin-
gle term describing the frictional stress between the
flowing material and the boundary surface.
Depth-averaged equations were introduced in the
context of dry granular flows (S-H model) by s
avaGe
& H
utteR
(1989). In their model, the interaction be-
tween the granular flow and the boundary surface is
described by a simple Mohr-Coulomb yield criterion:
the shear stress at the bottom is proportional to the
normal stress by a constant friction coefficient. The
longitudinal normal stress σ
x
is considered propor-
tional to the normal stress σ
y
exerted on an element
normal to the bed, through a coefficient k suitably
calculated. Besides, because the aspect ratio ε= H/L
of the flow is most likely small (H and L are respec-
tively, the typical thickness and the typical length of
the avalanche), the terms of order ε are omitted in the
y-momentum equation (i.e. the momentum equation
projected along the normal direction to the flow). As
a consequence the the normal stress has an hydro-
static distribution over the flow depth, analogously
to the De Saint Venant equations.
It is worth to state that the S-H model has been
widely used in some two-phase debris flow models
(i.e. i
veRson
, 1997; i
veRson
& d
enlinGeR
, 2001) in
order to describe the mechanics of the solid phase.
As observed by H
utteR
et alii (2005), the model
works well when the surface of the plane is sufficient-
ly smooth. The model is able to predict the motion and
spreading of a granular mass on steep slopes in one
and two dimensions (s
avaGe
& H
utteR
, 1989).
Several experimental observations (P
ouliQuen
& f
oRteRRe
, 2002; H
utteR
et alii, 2005) show that
the Savage-Hutter approach is no longer suitable
for the flow of granular material on rough surfaces
(where the roughness of the bed is of the order of the
particle size) as well as moderate inclinations (i.e.
Fig. 1 - The experimental apparatus
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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
505
ure the bed friction angles δ between the two surfaces
and the granular material and the internal friction an-
gle φ of the granular material. A single layer of plastic
beads was glued onto a thin plywood sheet. Moreover,
three wooden plates were lined with Plexiglas, sandpa-
per and granular material by gluing a single layer. Then,
the plywood sheet was gently placed with the granular
layer downward onto the inclined surface with a fixed
slope. The friction angle, which depends on the granu-
lar material and the surface, is considered equal to the
angle at which the static equilibrium of the plywood
sheet on the inclined surface is no longer possible.
The measured internal angle is φ=26.5° and can
be considered a good estimate of the constant-volume
friction angle (i.e. the dynamical friction angle). In
fact, just prior to failure, the two overlapping layers
of material are weakly packed and thus the effect of
interlocking is negligible.
For the smooth bed of Plexiglas,δ= 19.5° was
measured. An angle much greater than the internal
friction angle of the granular material was observed
for the sandpaper bed (δ= 36.5°). Therefore, in this
latter case, failure occurs inside the pile and only the
friction angle of material instead of the bed friction
angle should be considered in order to calculate the
basal shear stress.
These last two angles are good estimates of the
static friction angles. Nevertheless, P
ouliQuen
& f
oR
-
teRRe
(2002) observed that the dynamic friction angle,
called “stop angle”, is roughly one degree below the
static friction value. Owing to this, it was decided to
lower the estimates by 1°. Thus, the friction angle of
the Plexiglas bed was set equal to 18.5°.
MEASURING INSTRUMENTS
The motion was recorded by two Sony Super
HAD CCD video cameras at 12 frames per second,
connected to a digital video recorder. The first camera
and results sufficient beyond parameter sensitivity.
For these tests, which involved a mass of material of
only 100 kg, the upper tank was not used. The 2-m-long
upper part of the channel was used to store the material.
At the beginning of each run, a granular pile was
restrained in the upper part of the chute by a small
wooden gate, which was placed at exactly 2 m from
the beginning of the chute. When closed, the gate is
perpendicular to the channel bed and able to release
the material when rapidly rotated counter-clockwise
due to a spring mechanism. This opening appara-
tus was designed to open quickly in order to avoid
any significant influence on the forming dam-break
wave. The total opening time results less than 2/12 s
and, only after 1/12 s, the captured frames show that
there is no contact between the gate and the upstream
material. Therefore, the influence on the flow seems
to be negligible.
CHARACTERSTCS OF THE GRANULAR MATE-
RIAL
The tests reported in this paper were carried out us-
ing lens-like shaped acetalic resin beads (HERAFORM
R900), with a maximum diameter and minimum diam-
eter respectively equal to 3.9 mm and 2.8 mm. In Table
1, the main features of the material are reported. At each
run, a mass of 100 kg of granular material was suddenly
released by opening the wooden gate.
Particular attention was given to set the initial po-
sition of the pile upstream the gate in order to impose
the same initial condition in each test. A trapezoidal-
shaped initial deposit was used for all the runs and is
reported in Fig. 2.
Two kinds of test were carried out. The first type
of flows were carried out on a smooth Plexiglas bed.
In the second, on a bed with a lining of coarse sandpa-
per (grit P40 FEPA/ISO 6344).
The following procedure was carried out to meas-
Tab. 1 - Features of the granular material
Fig. 2 - Initial deposit of the granular material. x-axis is
parallel to the sloping bottom of the chute
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
angle, δ the basal friction angle, β the Boussinesq mo-
mentum coefficient. k is the active/passive pressure
coefficient, i.e. the ratio of the normal stress σ
v
exerted
on an element normal to the bed and the normal stress
σ
x
exerted on an element parallel to it.
where f is the internal friction angle and the bed friction
angle. Eq. 2 should be taken with the minus sign if ∂u
/ ∂x>0
(i.e. the pile is elongating), the plus sign if ∂u /
∂x<0
(i.e. the pile is compressing).
This expression is obtained under the hypotheses
that failure simultaneously occurs at the bed and inside
the material, with the shear stress linearly varying with
depth as the normal stress σ
y
does. Moreover, it is worth
noting that the expression (2) is only valid when .δ≤ f .
If δ > f , i
veRson
(1997) proposed using Rankine
formula, instead of (2):
which is also used in the Voellmy-fluid model proposed
by b
aRtelt
et alii. (1999). It could also be obtained from
(2) by putting δ→0 . In this case, failure only occurs
inside the granular material and the τ
xv
distribution along
the y-direction is unlikely to be linear. The rationale
of (3) is that it is assumed that σ
x
and σ
v
are principal
stresses, as if τ
xv
= 0. Moreover, many studies have also
proposed a completely isotropic stress tensor (Bartelt,
1999; P
ouliQuen
& f
oRteRRe
, 2002), that is k=1.
In this work, the above-stated three different ap-
proaches were implemented and compared in order to
estimate the coefficient k.
For simplicity, the Boussinesq coefficient β was set
equal to 1. This is exact only if the velocity is constant
over the depth. It is therefore expected to be valid when
the flow is sheared in a thin basal layer, as when the bed
is sufficiently smooth (i.e δ< φ ). However, it was found
that eqs. (1) are quite insensitive to β changes (s
avaGe
&
H
utteR
, 1991), with further studies being required when
a large bed roughness is considered and a complete
sheared flow expected.
RESISTENCE DUE TO SIDE wALLS
The equations described above are for a 1D flow.
Therefore, the only resistance accounted for is due to
the bed friction. Nevertheless, in order to simulate a
was installed at the side of the channel and, thanks
to the side wall transparency, allowed for the view
of about 80 cm downstream the gate. The effective
resolution of the cameras was about 450 lines, with
a precision of 1 cm being assured in the chosen field
of view. In order to rectify the images from the first
camera, a 2 cm grid was put on the opposite side wall.
The second one was located over the chute and
was able to capture the front wave position in the first
2 m downstream the gate. The same rectification of
the images was implemented using fixed reference
lines on the channel bed.
Afterwards, the recorded images were digitally
processed. At first, the Barrel deformations were mini-
mized by using a photo editing software. Then, the
frames were subjected to a perspective rectification.
Image rectification from the side camera was accom-
plished by exploiting the fixed spots of the grid behind
it. Strictly speaking, it was sufficient to choose 4 point
in order to rectify an image. Nevertheless, to mini-
mize any errors due to uncertainty, a set of 8 points
was taken and a residual evaluation was carried out.
The same procedure was carried out for the frames re-
corded by the front camera. However, any errors due
to rectification were less than 3 mm, with it being pos-
sible that global accuracy was within 1 cm.
MATHEMATICAL MODEL
The flow under study develops predominantly
in the longitudinal direction. It is therefore natural to
use depth-averaged type models (s
avaGe
& H
utteR
,
1989; i
veRson
, 1997; P
ouliQuen
, 1999; P
ouliQuen
&
f
oRteRRe
, 2002). These models can be obtained by in-
tegrating the mass and momentum conservation equa-
tions over the flow depth.
In this work, the Savage-Hutter 1D model was
used, which is described by the two following hy-
perbolic partial differential equations, written in the
conservative form:
where S
0
=sin α represents the gravity force compo-
nent in the flow direction per unit mass and S
f
=cos α
tan δ |u|/ u is the bed friction, h is the flow depth, u the
mean velocity, x and y are respectively the direction
parallel and normal to the bed, α the channel slope
(1)
(2)
(3)
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DAM-BREAK FLOWS OF DRY GRANULAR MATERIAL ON GENTLE SLOPES
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507
∆x. In addition, for each step, ∆t was calculated keep-
ing the maximum value of the CFL number constant.
The generic volume element is represented by the pair
(xj,ti) and the solution at (xj,ti) is the integral mean
value over the volume element.
An explicit Total Variation Diminishing scheme
was used, which is generally second order accurate in
space and time and can capture the shocks due to a
local decrease of the order of accuracy near the dis-
continuities.
The explicit formula to update the solution vector
U=(h, hU)
T
is the following:
where R
j
i
is the source term and F
TVD
are the numerical
fluxes. These fluxes are obtained as a convex combi-
nation of the first order Godunov fluxF
I
and the sec-
ond order Lax-Wendroff flux F
II
:
where A=∂F/ ∂U is the Jacobian matrix of the flux
F, |A|=R|Γ| R
-1
is obtained through eigen-decompo-
sition, R is the right-eigenvector matrix associated to
A and Γ the diagonal eigenvalues matrix. Both the ap-
proximated Jacobian matrix A and |A| at interfaces are
calculated through a local linearization of the system
of equations using the following Roe’s approxima-
tions (R
oe
, 1981):
The limiter Φ in eq. (11) is a 2x2 diagonal ma-
trix, whose elements can range between 0 and 1. Its
purpose is to lower the order of accuracy in high gra-
dient zones. The Minmod function was chosen as a
limiter function for the calculation of each element of
the matrix Φ, since it seems to be the best among the
available TVD limiters for this particular problem. Its
expression is reported as following:
dam-break flow down a chute, it is necessary to con-
sider the resistances due to side walls. In order to take
into account the side wall resistance, an approach sim-
ilar to the one reported in (s
avaGe
& H
utteR
, 1991)
was adopted. Along the side walls, it is assumed that
there is slip, therefore the Coulomb failure law holds
τ =σ
z
, tanδ
lat
with z being normal to the side walls and
δ
lat
the friction angle between the granular material
and side walls. The side walls are made of Plexiglass
like the channel bottom, so δ
lat
=18.5°. Assuming that
σ
z
is linearly distributed over the depth, as it is σ
y
,
there should be a pressure coefficient k
Z
so that:
Then, integrating τ
xz
over the depth, the resistance
per unit length of the channel due to a single side wall
is the following:
Since no pile elongation occurs along the z-direc-
tion during the flow, k
Z
should be close to the at-rest
pressure coefficient. In the simulations presented, the
Jaky formula (J
aky
, 1944) was used to estimate it:
Taking into account the effect of (5) for both the
walls in eq. (1), the following modified right-hand
member of the momentum equation results:
where δ
bed
is the bed friction and w the width of the
chute. In order to keep the same expression of eq. (2),
when the bed is made of the same material as the side
walls (i.e. δ
bed
= δ
lat
), an equivalent bed friction angle
could be defined as follows:
This expression agrees with the one used by s
av
-
aGe
& H
utteR
(1991), where for glass side walls and
rough bed, the coefficient k was set equal to 0.453.
NUMERICAL METHOD
Savage-Hutter equations were numerically solved
using a shock-capturing Finite Volume method. The
spatial domain was divided by an equally spaced mesh
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11 a)
(11 b)
(12)
(13)
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L. SARNO, M.N. PAPA & R. MARTINO
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
where superscript k specifies the k-th component
and q
k
at the interface j+1/2 is the ratio:
being R
-1
i + 1/2
the left eigenvector matrix and
the k-th eigenvalue at interface j+1/2,
obtained using the Roe’s formulas
To avoid any convergence to a non physical weak
solution violating the entropy principle (l
eveQue
,
2002), a correction to absolute values of the eigenval-
ues was applied (H
aRten
& H
yman
, 1981). Firstly, let:
The entropy correction for k-th eigenvalue, at the
interface between j and j+1 is the following:
Special attention was given to managing the tran-
sition of the avalanche to the zones without material
(wet/dry zone transition). In particular, a very thin
layer of material (10
-5
m) was considered over the
dry zones as an initial condition. Furthermore, during
the calculation, in cells where h<10
-4
m, velocity was
kept at zero without solving the momentum equation.
In order to test the suitability of the chosen thresh-
old values, calculations were compared with the ones
obtained using lower threshold values of h, without
noting any significant differences.
Regarding the lower boundary condition, in order
to avoid any wave reflection, a non-reflective condi-
tion was applied by imposing the two following ghost
values at the right end of the domain:
where the subscript n denotes the last cell at the right
end of the domain.
Regarding the upper boundary condition, it is not
expected the flow to go upstream and so a management
of wave reflections should not be necessary. Anyway,
in order to preserve the mass balance, a solid wall con-
dition was imposed at the top boundary all the same,
because it is needed to correctly allow the reflection of
the small spurious waves, due to the thin layer located
in dry zones. To do so, the following ghost values were
imposed at the left end of the domain:
Due to the channel geometry (constant slope), it
results ∂u /∂x >0 in the whole domain and for any giv-
en time (i.e. the pile elongates). Therefore, in eqs. (1)
only the active pressure coefficient k
act
appears and nu-
merical issues due to the jump discontinuity between
k
act
and k
bass
do not arise.
Volume balance is always respected within an er-
ror below 0.05%. Simulations were carried out, impos-
ing ∆x= 0.02 m and a ∆t for each step such as CFL=0.2.
EXPERIMENTAL DATA AND COMPARI-
SONS
The aim of this work is to investigate how the
pressure coefficient k should be calculated in order to
have the best fitting of the experimental data. An addi-
tional aim is to explain the results obtained considering
the influence of the bed roughness on the flow.
At first, a set of tests with a smooth bed was car-
ried out at the following slopes of the chute: 19°, 20°,
22.7°. Then, a set of runs on a sandpaper bed was car-
ried out with the same slopes.
In order to compare the numerical solutions with
the experimental data, the zero time (t=0) was taken as
corresponding to the first frame where the gate is no
longer in contact with the upstream pile. This moment
is 1/12 s after the start of the movement of the gate.
Considering that at this point, the pile is practically still
motionless, the initial condition used in the numerical
simulations was the initial deposit depicted in Fig. 2.
For each run, three numerical simulations were
carried out: the first one with calculated according
to the Savage & Hutter formula (2), the second one
with k = 1 assuming the stress tensor is spherical, the
last one with k
act
calculated through the Rankine for-
mula (3). The other model parameters for the smooth
bed runs were the following: d
bed
=18.5°, k
z
= 1-sin(f)
=0.554. For the rough bed simulations, instead, the bed
friction angle is d
bed
= f = 18.5°, due to failure occurring
inside the granular material. For these runs, the same
value of k
z
= 0.554 was used.
SMOOTH BED TESTS
Comparisons between the numerical simulations
(14)
(15)
(16)
(17)
(18)
(19)
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DAM-BREAK FLOWS OF DRY GRANULAR MATERIAL ON GENTLE SLOPES
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
509
that at the very first times (below 1.5s), the simulated
flow is faster than the observed one. In this window
time, the best fitting seems to be achieved using k
act
calculated through the Rankine formula. However, it is
possible that the different behaviour of the simulations
at the first stages is linked more to the failure of many
of the hypotheses made (linear distribution of stresses,
rate independence of basal friction) rather than to a real
and the smooth bed tests are reported in Figs. 3, 4, 5.
The following notation applies to all the diagrams:
ksh=0.67 is the pressure coefficient calculated through
the Savage-Hutter formula; k=1 is calculated under
the hypothesis of spherical stress tensor, kr=0.38 cal-
culated through the Rankine formula. The simulations
using k
act
calculated according to the Savage & Hutter
formula gave the best fitting. It is important to state
Fig. 3 - Flow depth profiles, comparison among experimental data and numerical simulations (slope 19°, smooth bed).
Fig. 4 - Flow depth profiles, comparison among experimental data and numerical simulations (slope 20°, smooth bed)
Fig. 5 - Flow depth profiles, comparison among experimental data and numerical simulations (slope 22.7°, smooth bed)
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L. SARNO, M.N. PAPA & R. MARTINO
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
the momentum equation, representing the resistances
are much greater than the term representing the gravity
force. Therefore, varying the slope produces only small
differences on the dam-break waves. For these simula-
tions, it is worth noting that the numerical simulations
using k = 1 are in greater accordance with the empiri-
cal observation as was found also by P
ouliQuen
& f
oR
-
teRRe
(2002), even though there is a systematic light
underestimation of the thickness. On the other hand,
using calculated according to the Savage & Hutter pro-
duces a faster dam-break wave and a thinner deposit
profile. This could be due to the fact that flows on a
rough bed are more sheared than on a smooth bed and
therefore the hypothesis of linear distribution of stress-
fitting to the Rankine formula.
The numerical simulation cannot accurately repro-
duce the thickness profile at the end of the avalanche (8
s and 10 s) in the 20°-run. This could be due to an error
in taking into account the side wall stress.
ROUGH BED TESTS
Comparisons between the numerical simulations
and the rough bed tests are reported in Figs. 6, 7, 8. For
these runs ksh=1.49, because it is imposed δ=f=26.5 °
in expr. (7); kr=0.38, which depends only on f and not
on the bed friction. In this case, the thickness profiles
are very similar for different slopes. This happens be-
cause the source terms in the right-hand member (7) of
Fig. 6 - Flow depth profiles, comparison among experimental data and numerical simulations (slope 19°, rough bed)
.
Fig. 8 - Flow depth profiles, comparison among experimental data and numerical simulations (slope 22.7°, rough bed)
Fig. 7 - Flow depth profiles, comparison among experimental data and numerical simulations (slope 20°, rough bed
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DAM-BREAK FLOWS OF DRY GRANULAR MATERIAL ON GENTLE SLOPES
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511
CONCLUDING REMARKS
In this work the experimental data of a dam-break
on a gently sloped chute was compared to numerical
solutions obtained using the S-H model with various
expressions of the pressure coefficient K. Runs with
smooth and rough beds were carried out. Side wall
resistance was considered, assuming that the lateral
pressure coefficient is close to the at-rest pressure co-
efficient. Comparisons of smooth bed runs have con-
firmed, as many works have already stated (s
avaGe
&
H
utteR
, 1989; s
avaGe
& H
utteR
, 1989; H
utteR
& alii,
2005), the fitness of the S-H model in describing such
a type of dry granular avalanche.
On the other hand, comparisons of the rough bed
runs have shown that an isotropic stress tensor as-
sumption gives a better fitting of the experimental data
and suggests further investigations of the role of the
stresses and velocity distributions over the depth. In
fact, it is not clear if the best fitting obtained using k=1
is simply due to either an empirical compensation of
some other effects ignored by the model or to a truly
substantiated hypothesis about the stress distribution.
However, under these conditions, the flowing pile
is likely to have a wide fluidized layer above a layer of
material already deposited. At first, this suggests giv-
ing a more accurate estimate of the Boussinesq mo-
mentum coefficient.
The stress tensor distribution over the depth is
likely to be too complex to be described by a simple
pressure coefficient k. In order to overcome this dif-
ficulty, it could be interesting to develop a multi-layer
depth-averaged model which joins the simplicity of the
depth-averaged approach and a better capability to de-
scribe this variation.
es appears to be less realistic. At the first stages of the
motion (0-1.5 s), as similarly observed in the smooth
runs, the best fitting is obtained using kr.
POSITION OF THE wAVE FRONT
A comparison between the observed positions of
the wave front and the simulated ones is depicted in
Fig. 9. Only the comparison for the run at 22.7° for
both the smooth and rough beds is reported, since the
other profiles are very similar at these first stages.
The numerical simulation chosen for the comparison
is calculated using kr=0.38, which allows for the
best fitting. The experimental position of the wave
front is determined by analyzing the frames from the
camera located over the chute. Unfortunately, the
front camera was only able to capture a limited field
of view, i.e. the first 2 m downwards the position of
the gate. For the numerical simulations, determining
the wave front position using a criterion based on
the front depth could be ambiguous because of the
thin layer of material used to describe the dry/wet
transitions. Therefore, the front has been defined as
the point corresponding to the first cell in the mesh,
upstream from the right boundary, where the velocity
is zero. As shown in the graph (Fig. 9), the numeri-
cal model is able to predict fairy well the velocity of
the wave front, as the two diagrams have approxi-
mately the same slope. Nevertheless, it is worth not-
ing that the numerical simulations are faster than the
observed avalanches. However, this disagreement
could be partially due to either a time shift error in
determining the zero-time or to the particular tech-
nique used to determine the position of the wave
front in the numerical simulations.
Fig. 9 - Position of the wave front, comparison among ex-
perimental data and numerical simulations(slope
22.7°,smooth and rough bed)
background image
L. SARNO, M.N. PAPA & R. MARTINO
512
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
In this perspective, interesting experimental data
may be obtained by measuring the velocity profiles as
well as a wider window of flow depth profiles. In order
to achieve these results the test equipment needs to be
modified and completed.
ACKNOWLEDGEMENTS
The writers wish to thank eng. N. Immediata and
eng. A. Russo for their useful help and advice in de-
signing and carrying out the tests.
REFERENCES
b
aRtelt
P., s
alm
. b. & G
RubeR
. u. (1999) – Calculating dense-snow avalanche runout using a Voellmy-fluid model with active/
passive longitudinal straining. J. Glaciology, 45 (150): 243-254.
d
e
s
aint
-v
enant
a. J. C. (1871) – Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à
l’introduction des marées dans leur lit. C.R. Acad. Sci. Paris, 73: 147-154.
H
aRten
a & H
yman
P. (1983) – Self adjusting grid methods for one dimensional hyperbolic conservation laws. J. Comp. Phys.,
50: 235-269.
H
utteR
k., w
anG
y. & P
udasaini
s.P. (2005) – The Savage-Hutter avalanche model: how far can it be pushed? Philosophical
Trans. of The Royal Society.
i
veRson
R.m. (1997) – The physics of debris flows. Reviews of Geophysics, 35:245-296.
i
veRson
R.m. & d
enlinGeR
R.P. (2001) – Flow of variably fluidized granular masses across three-dimensional terrain 1. Cou-
lomb mixture theory. J. Geophys. Res., 106(B1): 537-552.
J
aky
J. (1944) – The coefficient of earth pressure at rest. (in Hungarian). Journal for Society of Hungarian Architects and Engi-
neers, Budapest, 7: 355-358.
l
eveQue
R.J. (2002) – Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press.
P
ouliQuen
o. (1999) – Scaling laws in granular flows down rough inclined planes. Physics of Fluids, 11: 542-548.
P
ouliQuen
o. & f
oRteRRe
y. (2002) – Friction law for dense granular flows: application to the motion of a mass down a rough
inclined plane. J. Fluid Mech., 453: 133-151.
R
oe
P.l. (1981) – Approximate Riemann solvers, parameter vectors and difference schemes. J. Comp. Phys., 43: 357-372.
s
avaGe
s.b. & H
utteR
k. (1989) – The motion of a finite mass of granular material down a rough incline. J. Fluid Mech., 199:
177-215.
s
avaGe
s.b. & H
utteR
k. (1991) – The dynamics of avalanches of granular materials from initiation to runout. Part I: Analysis.
Acta Mechanica 86: 201-223.
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