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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
193
DOI: 10.4408/IJEGE.2011-03.B-023
STEADY DEBRIS FLOWS OVER ERODIBLE BEDS
d
ieGo
BERZI
(*)
, J
ames
t. JEMKINS
(**)
& e
nRiCo
LARCAN
(*)
(*)
Politecnico di Milano, Department of Environmental, Hydraulic, Infrastructure,
and Surveying Engineering - Milan, 20133, Italy
(**)
Cornell University, School of Civil and Environmental Engineering - Ithaca, NY 14853, USA
b
eRzi
& J
enkins
(2008a, b) developed a simple
theory based on a linear rheology for the particle in-
teractions, turbulent shearing of the fluid, buoyancy,
and drag. They assumed a constant concentration in
the particle-fluid mixture and the similarity of the par-
ticle and fluid velocity profiles to obtain a complete
analytical description of the steady, uniform flow of a
granular-fluid mixture over an inclined bed contained
between frictional sidewalls. The predictions of this
description compared favourably with the measure-
ments in experiments on steady, uniform granular-
fluid flows performed by a
Rmanini
et alii (2005) and
l
aRCHeR
et alii (2007) on mono-dispersed plastic
cylinders and water. b
eRzi
& J
enkins
(2009) used
their theory to solve for the propagation of a steady
granular-fluid wave over a rigid bed and were able
to reproduce the experiments performed by d
avies
(1988) on mono-dispersed plastic cylinders and water.
The theory was further simplified to obtain explicit
expressions for the particle and fluid friction slopes
as functions of the particle and fluid depth-averaged
velocities and depths to be employed in mathematical
models (b
eRzi
et alii, 2010).
Here, we extend the theory of b
eRzi
& J
enkins
(2008a, b; 2009) to deal with the steady propagation
of a granular-fluid wave over a previously deposited
erodible bed. We assume that the ratio of the parti-
cle shear to normal stress is distributed as in uniform
flows. This indeed allows to determine the position of
the interface between the flowing layer and the erod-
ABSTRACT
Recently, b
eRzi
& J
enkins
(2008a, b) proposed a
simple theory based on a linear rheology for the parti-
cle interactions, turbulent shearing of the fluid, buoy-
ancy, and drag. They provided a complete analytical
description of the steady, uniform flow of a granular-
fluid mixture over either an erodible or a rigid bed con-
tained between frictional sidewalls. They also used the
theory to solve for the propagation of a granular-fluid
wave moving at constant velocity over a rigid bed.
Here, we extend this theory to the case of a gran-
ular-fluid wave moving at constant velocity over an
erodible bed contained between frictional sidewalls.
This is indeed a natural step in view of a realistic
mathematical description of a real debris flow that
propagates over mobile surfaces, where erosion/depo-
sition phenomena are likely to occur. We make com-
parisons with the experiments performed with water
and gravel and show that the theory is able to repro-
duce the wave front and body.
K
ey
words
: steady wave, erodible bed, rheology
INTRODUCTION
Natural debris flows typically consist of unsteady,
non-uniform surges of heterogeneous mixtures of
muddy water and high concentrations of rock frag-
ments of different shapes and sizes, driven down a
slope by gravity. Despite that, our intent here is to em-
phasize steady flows of idealized composition.
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D. BERZI, J. T. JENkINS & E. LARCAN
194
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
particle velocity. The particle and fluid snouts are at x =
x
*
and x = X
*
, respectively. We consider here a steady
granular-fluid wave, so that h, H and b are functions
of position x, but not of time t. As in berzi & jenkins
(2009), we assume that the flow is dense and at approxi-
mately constant concentration, ĉ; we also assume the
yielding of the particles at the interface with the erod-
ible bed and that the ratio of the particle shear to normal
stress there is equal to ŭ.
As depicted in Fig. 1, we can distinguish a wave
front, where the flow is strongly non-uniform, and a
body, where h(x), H(x) and b(x) are approximately
rectilinear and parallel, as in uniform flows; this con-
figuration has been experimentally observed by d
av
-
ies
(1988) and t
ubino
& l
anzoni
(1993). We make use
of the so called uniformly progressive wave approxi-
mation (H
unGR
, 2000; P
ouliQuen
, 1999) to obtain the
shape of the front; i.e., we assume that the depth-aver-
aged particle and fluid velocities are equal and constant,
as in b
eRzi
& J
enkins
(2009) and b
eRzi
et alii (2010).
The depth-averaged momentum balances for the
particles and the fluid are
and
respectively, where α and β are functions of the degree
of saturation, so that, when the flow is under-saturat-
ed, α = H/h and β = 1, and, when the flow is over-
saturated, α = 1 and β = H/h. With respect to the corre-
sponding equations governing the motion of a steady
granular-fluid wave over a rigid bed (b
eRzi
& J
enkins
,
2009), we use the local inclination of the erodible bed,
tan [f
- arctan (db / dx)] ≈ tan f
- db / dx, valid if db / dx
is small, to account for the component of the weight in
the direction of the flow.
In Eqs. (1) and (2), j and J are the particle and
fluid friction slopes, respectively; they summarize the
resistances due to internal shear stresses and the role
of the drag force. b
eRzi
et alii (2010) express them as
and
where h ≡ h - b and H ≡ H - b. The coefficients λ
1
, λ
2
,
ible bed, where we assume that the granular material
is at yield. Using the set of model parameters appro-
priated for the uniform flow of 3 mm gravel and water,
as suggested by b
eRzi
et alii (2010), we show that the
theory is able to reproduce the experimental wave pro-
file measured by t
ubino
& l
anzoni
(1993).
The paper is organized as follows: first, we
present the depth-averaged equations governing the
motion of a steady granular-fluid wave over an erod-
ible bed and the closures for the particle and fluid re-
sistances and the location of the bed on the basis of
the theory of b
eRzi
& J
enkins
(2008a, b; 2009) and
b
eRzi
et alii (2010); then we show the comparisons of
the theory against experiments on steady uniform and
non-uniform flows of gravel and water over erodible
beds performed by t
ubino
& l
anzoni
(1993). Finally,
we point out some concluding remarks.
GOVERNING EQUATIONS AND CLOSURES
Figure 1 shows the sketch of the flow configu-
ration. We let ρ denote the fluid mass density, g the
gravitational acceleration, σ the particle specific mass,
d the particle diameter and η the fluid viscosity. The
Reynolds number R = ρd (gd)
1/2
/η is defined in terms
of these. In what follows, we phrase the momentum
balances and constitutive relations in terms of dimen-
sionless variables, with lengths made dimensionless
by d, velocities by (gd)
1/2
, and stresses by ρσgd.
We take x and z to be the coordinates parallel and
perpendicular, respectively, to the initially plane erodible
bed of inclination f with respect to the horizontal; z = b is
the position of the erodible bed (with
b
= 0 downstream
of the wave), while z = h and z = H are the top of the
particles and the fluid, respectively. The degree of satura-
tion, ζ ≡ H / h, is greater than unity in the over-saturated
flows and less than unity in the under-saturated. U
A
is the
depth-averaged fluid velocity, and u
A
the depth-averaged
Fig. 1 - Sketch of the flow configuration with the frame of
reference
(1)
(2)
(3)
(4)
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STEADY DEBRIS FLOWS OVER ERODIBLE BEDS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
195
t
ubino
& l
anzoni
(1993) with water and gravel, of spe-
cific mass σ = 2.65 and diameter d = 3 mm, flowing over
erodible beds in a rectangular channel of width w = 67
diameters, and contained between glass sidewalls. Here,
we use ŭ = 0.52 and ĉ = 0.60, as suggested by b
eRzi
et
alii (2010), and χ = 1 and μ
w
= 0.3. The latter two values
are slightly different than those adopted by b
eRzi
et alii
(2010), but they allow for a better fitting of the experi-
ments in uniform flow conditions.
We first show the capability of the theory to re-
produce the experimental results in uniform flow
conditions. If we take the derivative with respect to
x to be zero in Eqs. (1), (2) and (5), we analytically
obtain h, H and u
A
(or equivalently, the particle flow
rate per unit width, q = ĉu
A
h) as functions of tanf and
U
A
(or equivalently the fluid flow rate per unit width,
Q = [(1 - ĉ) α + β - 1] U
A
h). Figures 2 and 3 show the
comparisons between the theory and the experimental
measurements of q and h as functions of tanf. Given
that the experiments are for a range of fluid flow rate
of 11.7 to 27.2, we use the average value, Q = 19.5, to
obtain the analytical results. The agreement is notable
and suggests that the theory can be used to predict the
characteristics of the body of the debris flow depicted
in Fig. 1, where the motion is approximately uniform.
We then numerically solve the full differential equa-
tions (1), (2) and (3) using a fourth-order Runge-Kutta
method to see if the theory has the capability to repro-
duce also the wave front. t
ubino
& l
anzoni
(1993) re-
ported measurements of the wave height as a function of
time t for one of their experiments. For that experiment,
where the inclination of the undisturbed erodible bed was
17°, they also measured the front velocity and found it
constant and equal to 0.476 m/s, corresponding to a non-
dimensional velocity of 2.8. If the flow is steady, then x
= 2.8t, and we can compare the experimental measure-
ments with the results of the present theory.
Λ
1
, Λ
2
and Λ
3
are reported in Tables 1 and 2 (from
b
eRzi
et alii, 2010). There, χ is a material coefficient
of order unity that characterizes the linear rheology
for the particles adopted by b
eRzi
& J
enkins
(2008a,
b; 2009), while k = 0.2 (half the Karman’s constant).
Equations (3) and (4) have been obtained in uniform
flow conditions, but, as usual in Hydraulics, we use
them also in the case of non-uniform motion.
To close the problem, we need an equation govern-
ing the evolution of the position of the erodible bed.
b
eRzi
& J
enkins
(2008b; 2009) derive the distribution of
the ratio of the particle shear to normal stress, along the
cross-section of the flow, in the case of steady, uniform
motion over erodible beds. They link the inclination of
the erodible bed to the particle and fluid heights above it
and emphasize the role of frictional sidewalls, character-
ized by their friction coefficient μ
w
and gap w, in locally
controlling the particle stress ratio. If we assume that. at
the erodible bed, the stress ratio is equal to the yielding
value, ŭ, and we use the local inclination of the bed, ,
the expression of b
eRzi
& J
enkins
(2008b; 2009) reads
Equations (1), (2) and (5) are three ordinary dif-
ferential equations governing the spatial evolution of
h, H and b. Their integration requires the knowledge
of three boundary conditions; a natural choice would
be the vanishing of the particle and fluid heights at the
snouts, h(x
*
) = H(X
*
) = 0, and the fact that the bed is
unperturbed downstream, b(max[x
*
,X
*
]) = 0.
COMPARISONS WITH EXPERIMENTS
We now make comparisons between the present
theoretical treatment and the experiments performed by
(5)
Tab. 1 - Coefficients in Eq. (3).
Tab. 2 - Coefficients in Eq. (4).
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D. BERZI, J. T. JENkINS & E. LARCAN
196
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
CONCLUDING REMARKS
We have extended the two-phase theory of b
eRzi
& J
enkins
(2008a, b; 2009) to deal with steady, non-
uniform debris flows over erodible beds contained
between frictional sidewalls. This flow configuration
represents a severe test to the practical applications
of the theory to real scale phenomena. The theory
can predict, beside the heights of the particles and the
fluid, also the spatial evolution of the position of the
interface between the flow and the erodible bed.
The comparisons of the theoretical results with
the experiments performed using natural gravel and
water is remarkably good. The theory is able to quan-
titatively predict both the front and the body of the
steady waves; it also confirms the experimentally ob-
served tendency of debris flows to be depositional at
the front, and erosional upstream.
The erodible bed in the experiments of t
ubino
&
l
anzoni
(1993) was initially saturated with water;
moreover, they describe the debris flow as being fully
saturated, i.e. with the height of the particles over the
bed approximately equal to the height of the fluid.
We therefore assume here that the particle and fluid
snouts coincide, x
*
= X
*
, and we solve Eqs. (1), (2) and
(3) with u
A
= U
A
=2.8, to obtain, at every step of integra-
tion, the values of the friction slopes from Eqs. (3) and
(4), and f = 17°. In Fig. 4, we show the predictions of
the theory against the experiment of Tt
ubino
& l
an
-
zoni
(1993). The free surface of the wave is well re-
produced by our numerical solution. Also, the position
b of the interface with the erodible bed is positive in the
wave front and is negative upstream; this indicates that
the debris flow tends to deposit material at the front and
to erode it at its upstream end, in accordance with the
experimental observations of t
ubino
& l
anzoni
(1993).
Fig. 2 - Theoretical (solid line) and experimental (circles,
from t
uBiNo
& l
ANZoNi
, 1993) particle flow rate
against the angle of inclination of the erodible
bed. The theoretical results are for Q = 19.5
Fig. 3 - Same as in Fig. 2, but for the particle depth
against the angle of inclination of the erodible bed
Fig. 4 - Theoretical prediction of the spa-
tial evolution of the top of the parti-
cles (solid line), the top of the fluid
(dot-dashed line) and the position
of the erodible bed (dashed line)
against the experimental meas-
urements (circles, from t
uBiNo
&
l
ANZoNi
, 1993) of the profile of a
steady wave over an erodible bed,
for u
A
= U
A
= 2.8 and f = 17°
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STEADY DEBRIS FLOWS OVER ERODIBLE BEDS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
197
REFERENCES
a
Rmanini
a., C
aPaRt
H., f
RaCCaRollo
l. & l
aRCHeR
m. (2005) - Rheological stratification in experimental free-surface flows of
granular-liquid mixtures. J. Fluid Mech., 532: 269-319.
b
eRzi
d. & J
enkins
J.t. (2008a) - A theoretical analysis of free-surface flows of saturated granular-liquid mixtures. J. Fluid
Mech., 608: 393-410.
b
eRzi
d. & J
enkins
J.t. (2008b) - Approximate analytical solutions in a model for highly concentrated granular-fluid flows. Phys.
Rev. E, 78: 011304.
b
eRzi
d. & J
enkins
J.t. (2009) - Steady inclined flows of granular-fluid mixtures. J. Fluid Mech., 641: 359-387.
b
eRzi
d., J
enkins
J.t. & l
aRCHeR
m. (2010) - Debris Flows: Recent Advances in Experiments and Modeling. Adv. Geophys., in press.
d
avies
t.R.H. (1988) - Debris Flow Surges -A Laboratory Investigation. Nr. 96, Mittellungen der Versuchsanstalt fur Wasserbau,
Hydrologie und Glaziologie.
H
unGR
O. (2000) - Analysis of debris flow surges using the theory of uniformly progressive flow. Earth Surf. Proc. Lndfrms., 25: 483-495.
l
aRCHeR
m., f
RaCCaRollo
l., a
Rmanini
a. & C
aPaRt
H. (2007) - Set of measurement data from flume experiments on steady,
uniform debris flows. J. Hydraul. Res., 45: 59-71.
P
ouliQuen
o. (1999) - On the shape of granular fronts down rough inclined planes. Phys. Fluids, 11: 1956-1958.
t
ubino
m.a. & l
anzoni
s. (1993) - Rheology of debris flows: experimental observations and modeling problems. Excerpta Ital.
Contrib. Field Hydraul. Engng., 7: 201-236.
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