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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
1041
DOI: 10.4408/IJEGE.2011-03.B-113
DYNAMIC IMPACT OF A DEBRIS FLOW FRONT AGAINST
A VERTICAL WALL
a
Ronne
ARMANINI
(*)
, Michele LARCHER
(*)
& Michela ODORIZZI
(*)
(*) Università degli Studi di Trento, via Mesiano 77, 38050 Trento, Italy
E-mail: aronne.armanini@unitn.it, +39-0461-282612
INTRODUCTION
The correct estimation of the dynamic impact of
a debris flow front against a structure is a key phase
of its design procedure. It is evident that the dynamic
impact does not depend solely on the flow depth of
the incident front, but it depends mostly on its kinetic
characteristics. In spite of this, very often in the pro-
fessional praxis the dynamic impact is simply evalu-
ated as the hydrostatic pressure of the incident flow
multiplied by an arbitrary coefficient larger than one.
If this coefficient is not large enough, the impact force
of the debris or mud flow can be dramatically under-
estimated. Besides often the hydrostatic pressure is
referred to the density of the clear water, while the
density of a mudflow or of a debris flow can exceed
by a factor of two the density of water. Therefore, if
the coefficient is taken smaller than two, the design
pressure may be even exceeded by the effective hy-
drostatic load, which is clearly a wrong assumption.
In more favourable cases the impact force is evaluated
invoking a homogeneous fluid scheme accounting for
the formation of a completely reflected wave (Fig. 1),
as proposed by z
anuttiGH
& l
ambeRti
(2006), but we
will show in what follows that also this approach may
lead to the underestimation of the correct impact force
In fact the dynamic impact of a fluid surge
against a vertical wall can occur according to two
different mechanisms (a
Rmanini
& s
Cotton
, 1993;
a
Rmanini
, 1997): with the formation of a completely
reflected wave or with the formation of a vertical
ABSTRACT
Recent experimental results obtained at the Uni-
versity of Trento show that a liquid-granular wave
can impact against a vertical obstacle producing two
different mechanisms of reflection, depending on the
Froude number: if the front is sufficiently fast, the
flow is completely deviated in the vertical direction,
producing a vertical jet-like bulge, while if it is rela-
tively slow it can be totally reflected in direction nor-
mal to the obstacle.
The standard theoretical approaches for the analy-
sis of the dynamic impact of a granular front against
an obstacle take into account only the second mecha-
nism described above and are obtained from the mass
and momentum balances applied to the reflected bore
under the hypothesis of homogeneous fluid.
We extend this approach to the case of a two-phase
granular-liquid mixture, taking into account the pres-
ence of a deposit of granular material near the wall, as
observed in the experiments. Furthermore we propose
a theoretical analysis of the formation of the vertical
bulge, that is usually observed for Froude numbers
larger than one, and propose an original analytical ex-
pression to estimate the dynamic impact forces also in
this situation.The theoretical approaches we propose
are suitable to describe the experimental results with a
reasonable agreement.
K
ey
words
: dynamic impact, debris flow, vertical wall, Froude num-
ber
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A. ARMANINI, M. LARCHER & M. ODORIZZI
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
EXPERIMENTAL SETUP AND METHODS
The experiments were carried out in a 3.86 m long
flume (Fig. 2) with a 25.2 cm wide rectangular cross-
section, closed at the downstream end with a vertical
wall equipped with four piezoresistive pressure gaug-
es, positioned as depicted in Fig. 3.
Two supplementary pressure gauges were placed
at the bed of the flume, just upstream of the vertical
wall. In the upstream part the flume was equipped
with a removable gate, used to create a reservoir that
can be suddenly opened for the generation of a steep
wave. The experiments were carried out in a range of
slopes between 0 and 44%.
The experiments were filmed by means of two
synchronized high speed CCD cameras at a frame rate
of 500 frames per second and a pixel array of 1024
x 1024. The image sequences acquired with the two
cameras were synchronized also with the output of the
6 pressure gauges mentioned above.
The images were analyzed by means of imag-
ing techniques developed appositely for the study of
granular flows (C
aPaRt
et alii, 2002; s
Pinewine
et alii.
2003). The fluid was seeded with tracers and one ex-
ample for the results of the particle identification proc-
ess is depicted in Fig. 4, were an original image and a
filtered image with indication of the estimated particle
bulge (Fig. 1). In the first case, after the impact a
reflected wave is formed, propagating upstream with
a celerity a, which can be assumed constant at least
nearby the wall. In the second case, the flux is devi-
ated upwards parallelly to the vertical wall, with the
subsequent formation of a falling jet and then of a
hydraulic jump that propagates upstream.
We will show in this paper that the Froude number
of the incoming surge plays an important role in the
determination of the impact mechanism. When grav-
ity prevails over inertia, typically there is the forma-
tion of a reflected wave
On the contrary, when inertia prevails there is the
formation of a vertical bulge. In this second mecha-
nism, the maximum pressures occur before the forma-
tion of the hydraulic jump, that is before the beginning
of the energy dissipation. Each mechanism can take
place both when the fluid is homogeneous and when it
is composed of two phases, e.g. a granular matrix sus-
pended in a liquid. In the first part of the present paper
we consider the flow of a homogeneous fluid and eval-
uate the impact force when the formation of a vertical
jet is observed after the collision. In the second part
of the paper, we introduce a two-phase approach. The
theory was verified by means of appositely designed
experiments carried out at the University ofTrento.
Fig. 1 - Possible schemes of reflec-
tion of a debris flow front
against a vertical wall ac-
cording to the homogene-
ous fluid case and control
volumes utilized for the
mass and momentum bal-
ance; a) formation of a
completely reflected wave;
b) formation of a vertical
jet
Fig. 2 - Experimental flume used for the experiments
Fig. 3 - Scheme of the position of the pressure gauges on
the vertical wall at the downstream end of the flume
(distances expressed in cm)
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DYNAMIC IMPACT OF A DEBRIS FLOW FRONT AGAINST A VERTICAL WALL
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
1043
An approximated solution for equation (2) was
proposed by a
Rmanini
(2009):
Under the hypothesis of hydrostatic pressure distri-
bution, using eq. (3), the impact force F = 1/2(g
ρ
mh
r
2
)
on the vertical wall can be evaluated in dimensionless
form, F, respect to the hydrostatic force of the incident
front under the hypothesis that at the impact the pres-
sure distribution on the wall is hydrostatic.
The above scheme is valid also for a reflected wave
of finite height. In this case however the mechanical en-
ergy is not conserved.
HOMOGENEOUS FLUID SCHEME WITH
FORMATION OF A VERTICAL JET
When the Froude number of the incident wave
exceeds unity, the impact mechanism consists in the
formation of a vertical bulge (Fig. 1b). The largest
pressure peaks on the vertical wall develop before the
formation of the falling jet and its consequent breaking
associated with energy dissipation (Fig. 5).
Before these dissipative phenomena take place
and, therefore, until the jet reaches the maximum
height h
ro
, corresponding to a raising velocity u
wo
= 0,
energy conservation (ideal fluid approximation) can be
assumed along the streamlines. The application of the
positions is shown.
In the present paper the imaging techniques were
utilized to derive the flow kinematics and the particle
trajectories, utilized for the interpretation of the vari-
ous possible impact schemes. A graphical example of
this type of analysis is given in Figure 4
HOMOGENEOUS FLUID SCHEME WITH
FORMATION OF A COMPLETELY RE-
FLECTED WAVE
We first consider a simplified homogeneous fluid
scheme with formation of a completely reflected wave.
Assuming a control volume translating upstream with
the reflected wave, as presented in Fig. 1a, the mass
and momentum balances of a homogenous mixture of
density ρm lead to the following expressions:
where h
r
and h
i
are the depths of the reflected and in-
cident wave and u
r
and u
i
are the respective flow ve-
locities. Wall friction and gravity are negligible in the
momentum balance. At the moment of the impact the
velocity on the wall is zero, that is u
r
= 0. In this hy-
pothesis the system becomes:
where Y = h
r
/ h
i
is the ratio between the flow depths
and Fr
i
= u
i
/ (gh
i
)
1/2
is the Froude number of the in-
cident wave.
Fig. 4 - Example of results of the particle tracking procedure: the fluid is seeded with tracers, that are identified and tracked
using particle tracking methods based on the Voronoï polygons (c
APArt
et alii, 2002; S
PiNewiNe
et alii, 2003)
(1)
(2)
(3)
(4)
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A. ARMANINI, M. LARCHER & M. ODORIZZI
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
where (h
wo
)
max
is the maximum height of the jet,
evaluated by means of the energy balance (eq. 5), in
which we put h
w
= h
wo
, p
w
= 0 and uw = 0, leading to
h
wo
/h
i
=1+ 0.5F
2
ri
.
the comparison (Fig. 6) of the dimensionless im-
pact force determined by the two mechanisms shows
that the formation of a vertical bulge determinates a
larger force, that become significant for Froude num-
bers larger than 3.
In our experiments we did not measure the forc-
es, but just the pressures at some points on the wall
(Fig. 2b). The experiments showed (Fig. 7) impact
pressures larger than that corresponding to the hy-
drostatic distribution (hypothesis utilized to obtain
equation 6). Besides, in order to derive equation
(6) we applied the Bernoulli theorem assuming a
stationary flow condition. This is just a rough ap-
proximation because the inertial term is presumably
important in the phenomenon that we are describing.
However, we can estimate the time-derivative of the
velocity on the base of a scale analysis. In fact, dur-
ing the phenomenon the velocity decreases from u
i
,
at the incident front, to zero, at the moment when the
reflected-wave scheme in such a situation would lead
to an underestimation of the pressures at the moment of
impact, as showed in Fig. 6.
In a first approximation, we have neglected the
time variation of the velocity in the balance or, equiva-
lently, we assumed as valid the Bernoulli theorem:
where hw, pw and uw represent the distance from
the bed, the pressure and the velocity relevant to a ge-
neric point along the wall (Fig. 1b). We indicate with
hwo the distance of the top of the jet from the bed, and
with (h
wo
)
max
its maximum value. At the moment of
maximum height of the jet, in equation (5) we have u
w
= 0. Assuming the hydrostatic pressure distribution on
the vertical wall, the dimensionless maximum impact
force F ~ results to be:
F
ig. 6 - Impact force as a function of the Froude number of
the incident wave obtained for a homogeneous fluid
using the reflected-wave theory and the vertical-jet
theory. The parameter α is introduced in equation
(7) to take into account non-stationary phenomena
Fig. 5 - Particle trajectories at different time steps, expressed in seconds, obtained applying the imaging methods to an experi-
mental run with the formation of the vertical jet. The circle in the seventh panel gives evidence to the breaking of the jet
on the free surface of the incoming flow
(5)
(6)
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DYNAMIC IMPACT OF A DEBRIS FLOW FRONT AGAINST A VERTICAL WALL
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
1045
measured pressure exceeds the hydrostatic pressure
close to the bottom, where the maximum impact is
observed, while the effective pressure is smaller than
the hydrostatic one at the top, where nevertheless the
countermeasures do not usually present particular
static problems caused by the impact of a steep wave.
From Fig. 7 we observe that α is not constant.
Equation (6) underestimates systematically the
measurements, especially in the case of gauges n. 4
and n. 2, that are closer to the bed. The estimate rel-
evant to the higher gauges is more favourable since
the partial detachment of the jet from the wall, which
takes place when it approaches its maximum eleva-
tion, induces a reduction of the measured pressures
The results presented in Fig. 7 can be used to
calibrate the coefficient α in order to correct the
maximum height of the jet is reached, that is u
i
can
be considered the time scale of the phenomenon. In a
first order approximation, the time dependent term in
the momentum balance can be therefore estimate to
be proportional to ρu
i
2
through a constant α.
In this way eq. (5) is improved, accounting in an
approximate form for the variation of the velocity vec-
tor, leading to:
However, α is an unknown parameter and needs
to be estimated in order to make use of equation (7)
as a design tool.
A scheme of the picture given above is repre-
sented in Fig. 8 andFig. 9, where it is shown how the
(7)
Fig. 8 - Peak pressures measured with the pressure gauges
on the vertical wall (dashed red line) compared
with the hydrostatic pressure distribution (solid
black line)
Fig. 9 - Scheme of the jet and of the measured pressure dis-
tributions at the maximum height of the vertical jet
Fig. 7 - Experimental dimensionless maximum pressures compared with the theoretical expression. Equation (6) underestimates
systematically the measurements, especially in the case of gauges n. 4 and n. 2, that are closer to the bed
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A. ARMANINI, M. LARCHER & M. ODORIZZI
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
estimation of the impact pressures. Fig. 10 depicts
the behavior of α as a function of Froude number of
the incident wave in the different gauges. The figure
shows that α increases as the Froude number increas-
es, up to values of α ≈ 1. As a consequence we may
argue that for design purposes it is precautionary to
evaluate the impact force utilizing α = 1.
Starting from equation (7) and making the same
assumptions introduced for obtaining equation (6)
from equation (5), it is possible to express a new for-
mulation for the impact force:
Equation (8) improves the estimation of the im-
pact force given by equation (6), which could un-
derestimate the effective impact especially for large
Froude numbers of the incident flow. It should be
noted that equations (6) and (8) coincide if α = 0 and
that the magnitude of the correction increases as the
Froude number increase.
TWO-PHASE SCHEME
Actually debris flows consist of a mixture of wa-
ter and sediments and in many cases the homogeneous
fluid scheme represents a too strong idealization. If the
size of sediments is relatively large and the volume
concentration of the granular phase is big enough, as it
is often observed, the homogeneous fluid scheme pre-
sented above does not reproduce correctly what hap-
pens during the impact of the front against a vertical
obstacle. In fact the time scale ruling the segregation
between the water and the solids and the time scale of
the impact phenomenon may have the same order of
magnitude and therefore a static deposit of solid mate-
rial may develop upstream of the wall.
The experimental analysis has shown that also in
presence of two phase flow the two schemes of im-
pact occur. The experiments showed, in fact, that the
reflection of the wave takes place with the formation
of a completely reflected wave for Froude numbers
roughly smaller than unity (Fig. 11), while for faster
flows the impact is generally followed by the forma-
tion of a vertical jet (Fig. 12).
In the first case the flow is proximity of the wall is
substantially different than that of pure water, because
a static deposit of solid material is generated upstream
of the wall with a repose concentration C
*
. The deposit
propagates upstream at a constant thickness z
0
with the
same celerity a of the reflected wave (Fig. 11). The flow
depth, the velocity and the granular concentration of the
reflected wave are represented with the symbols h
r
, u
r
Fig. 10 - Experimental calibration of coefficient α in the case
of a clear water steep wave
Fig.11 - Possible mechanism of reflection of a debris flow
front against a vertical wall according to the two-
phase scheme and control volume utilized for the
mass and momentum balance in the case of forma-
tion of a completely reflected wave
(8)
Fig.12 - Possible mechanism of reflection of a debris flow
front against a vertical wall according to the two-
phase scheme in the case of formation of a vertical
jet
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DYNAMIC IMPACT OF A DEBRIS FLOW FRONT AGAINST A VERTICAL WALL
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
1047
where
and
For incident waves characterized by a Froude
number larger than one, the formation of a vertical
bulge is generally observed. In this case the phenom-
enon does not show segregation between liquid and
solids and the impact dynamics resemble the homo-
geneous fluid case. The impact force can therefore be
evaluated using the same method presented above, if
the density of the mixture ρm is used instead of the den-
and C
r
. It should be noted that on the deposit the flow
takes place in mobile bed conditions, with a velocity
profile that is less uniform than on a rigid bed (a
Rmani
-
ni
et alii, 2005; l
aRCHeR
et alii, 2007; f
RaCCaRollo
et
alii, 2008; a
Rmanini
et alii, 2009).
From a mass and momentum balance applied to the
control volume represented on Fig. 11 and assuming
that u
r
= 0 and C
r
= 0 at the impact moment, it is pos-
sible to obtain an expression for the non-dimensional
impact force, similarly to what obtained for the homo-
geneous fluid case.
On the contrary, in case of vertical jet we have not
observed any deposit during the first stage of the im-
pact, during which the maximum of the action against
the wall occurs. In this case, the same approach of the
pure water can be applied to the two phase flow.
In case of reflected wave, the incident flow is sup-
posed to move at a constant velocity ui and with a con-
stant depth h
i,
with a solid concentration C
i
assumed to
be constant throughout the flow depth
Fig. 13 - Experimental calibration of coefficient α in the case of a two-phase wave on a smooth bed for gauges n. 4 and n. 2
Fig. 14 - Experimental calibration of coefficient α in the case of a two-phase wave on a rough bed for gauges n. 4 and n. 2
(9)
(10)
(11)
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A. ARMANINI, M. LARCHER & M. ODORIZZI
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5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
homogeneous fluid case (Fig. 13 and Fig. 14). Prob-
ably this is partially due to the collision of isolated
grains against the pressure gauges, however this as-
pect will beobject of future investigations
CONCLUDING REMARKS
We presented a scheme for the estimation of
the impact force of a steep wave against a vertical
wall, which is a key phase of the design procedure
of hydraulic countermeasures. We showed that the
approaches commonly used in the design praxis may
lead to a significant underestimation of the impact
and that the new scheme we propose in this paper can
overcome this problem. Our simpler scheme is based
on the hypothesis of homogeneous fluid, but in fact,
it can be easily extended to two-phase flows. In some
experiments done with granular wet material we ob-
served that in some conditions a deposit may form
and propagate in upstream direction after the impact
of a debris flow against an obstacle. We showed that
to treat this problem properly the hypothesis of ho-
mogeneous fluid should be removed.
sity of clear water ρ.
Also in this case, we have neglected the time
variation of the velocity in the balance or, equiva-
lently, we assumed as valid the Bernoulli theorem
and, as a consequence, the experimental results were
underestimated by the theoretical expression. Apply-
ing the same procedure obtained to derive equation
(8), equation (13) can be corrected as follows:
However in this case we obtained values of the
adjusting parameter α that in some cases may be one
order of magnitude larger than what obtained for the
(12)
(13)
(14)
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Rmanini
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aPaRt
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RaCCaRollo
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aRCHeR
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DYNAMIC IMPACT OF A DEBRIS FLOW FRONT AGAINST A VERTICAL WALL
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
1049
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