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Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
331
DOI: 10.4408/IJEGE.2011-03.B-038
BOUNDARY SHEAR STRESS OF GRANULAR FLOWS
l
eslie
HSU
(*)
, R
oland
KAITNA
(**)
, w
illiam
E. DIETRICH
(***)
& l
eonaRd
S. SKLAR
(****)
(*)
Department of Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064, USA
(**)
Institute of Mountain Risk Engineering, University of Natural Resources and Life Sciences (BOKU), Vienna, Austria
(***)
Department of Earth and Planetary Science University of California, Berkeley, CA 94720, USA
(****)
Department of Geosciences, San Francisco State University, San Francisco, CA 94132, USA
MOTIVATION FROM THE FIELD AND
LABORATORY
Granular flows often scour their channels and
leave smooth surfaces, grooves, and scratches on the
channel bed and walls (e.g. s
toCk
et alii, 2005; H
su
,
2010). The polished surfaces and wear marks indicate
sliding motion at the base of the flow. This contact
between the flow and bed has implications for entrain-
ment and erosion of the bed (e.g. i
toH
et alii, 2003;
s
toCk
& d
ietRiCH
2006), for modeling runout dis-
tance or mobility of the flow (e.g. m
anGeney
et alii,
2007), and energy dissipation in the flow (e.g. b
aRtelt
et alii 2006). Although we have a growing catalog of
basal shear stress measurements in field and labora-
tory debris flows (e.g. m
C
a
Rdell
et alii, 2007; k
aitna
& R
iCkenmann
, 2007a, 2007b), we do not yet have a
complete understanding of the flow properties and
mechanisms that control local and average boundary
shear stress. Properties such as particle size, particle
shape, and fluid content may affect the internal veloc-
ity field and interaction with the boundary, and these
properties also vary greatly throughout and between
flows (i
veRson
, 1997; i
veRson
& v
allanCe
, 2001;
i
veRson
et alii, 2010;)
Understanding the relationships between bound-
ary shear stress and particle size, particle shape, and
fluid content, may explain what controls boundary
erosion or runout of granular flows..
There are measurements of basal and wall shear
stress for real debris flows in the field. m
C
a
Rdel
l et alii
ABSTRACT
The shear stress exerted at the bed and walls of
granular flows is an important quantity for modeling
and predicting runout, bulking up, channel erosion,
and entrainment. Although there are some measure-
ments of boundary shear stress for granular flows in
the field and laboratory, we lack systematic measure-
ment of shear stress for a range of flow properties such
as particle size, particle shape, and fluid content, es-
pecially with natural sediment or for long durations
where the flow may evolve. We used two vertically
rotating drums to study the boundary shear stress
of granular flows, a smaller drum that was 56 cm
diameter and 15 cm wide, and a larger drum that was 4
m diameter and 80 cm wide. The materials we used in-
cluded combinations of different sized glass marbles,
sand, gravel, fine sediment, and water. We compared
two ways of estimating the boundary shear stress us-
ing the force balance of the particle flow in the drum.
We varied particle size, particle shape, and fluid con-
tent. Larger and more angular particles increased the
total boundary shear stress, as did a decrease in fluid
content. This study illustrates some of the mechanisms
and particle dynamics that control the boundary shear
stress in natural geological flows, which has implica-
tions for debris flow modeling and hazard mitigation.
K
ey
words
: basal shear stress, experiments, particle size
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L. HSU, R. kAITNA , w.E. DIETRICH & L.S. SkLAR
332
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
flow behavior by relating shear resistance to flow ve-
locity and shear rate. They used three methods to esti-
mate shear stress in a 2.5 m vertically rotating drum:
derivation of shear stress from torque measurements
at the drum axis, from shear plates embedded in the
flume bottom, and from geometric considerations, sim-
ilar to the force balance analysis laid out by H
olmes
et alii (1993). Average shear stress was estimated by
assuming a uniform distribution of bottom shear stress
and a triangular shear stress distribution on the side
walls (R
iCkenmann
, 1990). Shear stress estimates from
torque measurements and from geometry were found
to be in good agreement. Due to their location in the
centre of the channel, the force plates tended to over-
estimate total shear stress systematically. The materials
used were a homogeneous transparent liquid polymer
(Carbopol Ultrez), PVC grains and a viscous natural
debris flow material.
ESTIMATING
BOUNDARY
SHEAR
STRESS
Shear stress at the bed of a debris flow is often
calculated from the Coulomb model, with a linear
relationship between maximum shearing strength
and normal stress on the failure plane (e.g. H
owaRd
,
1998). More complete models separate the solid and a
fluid contribution to shear stress (i
veRson
1997; i
veR
-
son
et alii, 2010) where the solid contribution is mod-
eled with the Coulomb model under consideration of
the effective stress concept of t
eRzaGHi
(1936)
where σ
bed
is the total basal normal stress, p
bed
is
pore pressure at the bed, and Φ
bed
is the basal friction
angle (which may be a function of the Savage number,
S). Then fluid shear is estimated as
where τy is related to fluid yield stress, and μ is
fluid viscosity, v is depth averaged velocity, and h is
flow height (i
veRson
et alii, 2010).
In many entrainment models, there is a threshold
stress below which there is no erosion, and erosion
increases monotonically with total basal shear stress
above the threshold. Some numerical models assume
that shear stresses are proportional to the normal
(2007) measured shear stress at the base of a natural
debris flow in the Illgraben torrent, Switzerland. They
found that average normal and shear stresses varied in-
phase with flow depth, with a maximum basal shear
stress value of τb = 2.8 kPa for a flow height of 1 m.
Also at the Illgraben torrent, b
eRGeR
et alii (2010) used
a buried resistance chain to measure the timing of ero-
sion. Contrary to intuition, they found that most erosion
occurred during the watery body and not at the coarse
flow front where the shear stress was a factor of seven
larger than the watery body. This finding illustrates that
there may not be a simple relationship between basal
shear stress and erosion of underlying sediment, and
other factors such as the degree of sediment saturation
of the flow are important. Although polished surfaces
are common, grooves and scratches are not frequent
along recently scoured bedrock channels (s
toCk
et alii,
2005). However, grooves do occur, indicating local-
ized high shear stress from large boulders over small
areas, and these fluctuations from the mean stress may
also be a major factor in boundary erosion and energy
dissipation. The field observations so far show that we
still have an incomplete understanding of topics like
the controls on localized shear stresses caused by in-
dividual particles and the solid-fluid interactions that
affect boundary shear stress
Basal shear stresses have also been studied in
simple flows in the more controlled laboratory envi-
ronment. The influence of bed sediment size and flow
sediment concentration on erosion rate was studied in
a suite of chute flow experiments over an erodible bed
(e
GasHiRa
et alii, 2001, P
aPa
et alii, 2004). Egashira
developed a nondimensional effective bed shear stress
(total shear stress minus yield stress), similar to a criti-
cal Shields parameter for bed load movement, which
defines a critical condition for bed material entrain-
ment. In these experiments, some of the flows are
very dilute with very low solid particle concentrations
compared to natural debris flows.
Chute flows are short-lived, and for a longer pe-
riod of observation, vertically rotating drums are desir-
able because they allow observation of a quasi-station-
ary flow. In a small drum of 56 cm diameter and 15
cm width, H
su
et alii (2008) traced height profiles of
various granular flows to show that flows of different
particle size distribution and water content have differ-
ent surface slopes and flow front positions in the drum.
k
aitna
& R
iCkenmann
(2007a, 2007b) described bulk
(1)
(2)
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BOUNDARY SHEAR STRESS OF GRANULAR FLOWS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
333
13.8, and 25 mm while the gravel had diameters of 4,
10, and 13mm. The total mass of the flow was held
constant at 3 kg.
We compared two different methods for ob-
taining a metric related to the total boundary shear
stress. The first method is based on the force bal-
ance between the torque from the debris mass on the
wheel and the drag from the flume walls on the de-
bris mass. This method is similar to that of H
olmes
et alii (1993) and k
aitna
& R
iCkenmann
(2007b),
where the mass was divided into smaller elements.
However we found the torque from the total mass as
a point mass at the centroid of the flow to be equiva-
lent. To obtain the position of the centroid we traced
the outline of the mass from a side-view photograph
(e.g. Figure 2) and used ImageJ image processing
software (a
bRamoff
et alii, 2004). We then calcu-
lated torque using the equation
where r is the moment arm length, F is the force
on the point mass at the centroid (mass × acceleration
due gravity), and q is the angle between the vectors for
r and F. For a steady-state flow, this torque is balanced
by (and therefore correlated with) the sum of shearing
forces from boundary drag on the flow.
The second metric related to total boundary shear
stress was the force on the chain driving the drum ro-
tation. Figure 1 shows the set up of the system, where
a force transducer (Interface Force SML- 50) meas-
ured the chain tension while the drum was spinning.
The output was logged by a laptop connected to an
Interface Force 9820 Digital Indicator at 4 Hz for sev-
stress with constant proportionality, and also assume
that effective shear stress is proportional to flow depth
(e.g. l
e
& P
itman
, 2009). However, incomplete un-
derstanding of erosion mechanisms necessitates em-
pirical factors to describe quantitatively the erosion
of material (l
e
& P
itman
, 2009). In m
C
d
ouGall
&
H
unGR
(2005) the basal shear stress is constrained
by calibration using prototype events. Medina et al.
(2008) implemented a dynamic approach to entrain-
ment, where the newly incorporated material is ac-
celerated to the mean velocity of the flow so that the
quantity of additional mass depends on the availabil-
ity of momentum. These models lack incorporation of
rigorous mechanistic laws relating flow dynamics to
entrainment or erosion.
The existing field, laboratory, and numerical
studies illustrate that there are several remaining
questions about the relationship between boundary
shear stress and the solid particle and fluid matrix
properties. For example, how do particle size and
shape affect mechanisms at the base? What controls
the thickness of the shear zone? How do localized
interactions affect average and fluctuating bound-
ary shear stress? Through what mechanisms do fluid
content and viscosity affect the effective normal and
shear stresses? Finally, what happens when the flow
is not steady, uniform, and unidirectional?
Here, we begin to address these questions by
observing flows where we vary particle size, parti-
cle shape, and the amount of fluid. These properties
are valuable because some of them may be inferred
from a post-flow deposit in the field. To investigate
the questions above, we use two vertically rotating
drum flumes. We analyze two proxies for shear stress
based on the force balance of a granular mass in a ro-
tating drum. We show trends in the total shear stress
with particle size, shape, and fluid content, demon-
strating that predictions about boundary shear stress
can be made from these solid and fluid properties.
EXPERIMENTAL METHODS
The small drum is made from a section of PVC
pipe with an inner diameter of 56 cm. Both sidewalls
are composed of Plexiglass, bounding a 15 cm wide
channel. The drum speed was 12 RPM (0.35 m/s).
The flows were monosized and were composed of
either dry spherical glass marbles, dry gravel, dry
sand, or moist sand. The marbles had diameters of 5,
(3)
Fig. 1 - Set up of the small 56 diameter vertically rotating
drum
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L. HSU, R. kAITNA , w.E. DIETRICH & L.S. SkLAR
334
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
grain size distribution were evaluated. We tested wa-
tersaturated single-size gravel flows with mean diam-
eters of 4, 10, 13, and 21 mm (Table 2). The 21 mm
gravel were from a different source than the 4, 10, and
13 mm diameter river gravel, and were more angular
in shape. In the gravel-water flows, water was added
until it filled the pores, but did not run out in front of
the gravel mass (H
su
, 2010). Runs with a wide grain
size distribution of clay, silt, sand, gravel, and cobbles
were also performed (H
su
, 2010). Bulk density was
measured by sample tests immediately after experi-
ments. The density was calculated with the mass of
a sample of known volume of 270 cm
3
and included
clasts up to ~20 mm in diameter. The bulk density
range was 1.9 – 2.3 g/cm3 for the muddy mixed flows
and 1.9 g/cm3 for the water-saturated gravel flows.
As in the small drum experiments, we calculated
a torque related to the total boundary shear stress us-
ing the position of the centroid of the mass, since the
smaller drum experiments supported that this was
linearly related to a direct torque measurement. The
outline of the debris mass was obtained from the la-
ser profiler. The height profile of the flow during the
experiment was measured by an Acuity AR4000 laser
scanner, which swept a laser line across the centerline
of the flow with a rotating mirror. The laser was con-
eral rotations and the average value was used. This
value, which measures the force necessary to drive the
drum at a given velocity, varied for flows of the same
mass but different particle compositions.
The large drum measured 4-meters in diameter
and 80-cm in width. The drum was driven by a 20-kil-
owatt induction motor and controlled by a variable
speed inverter drive. 32-mm thick Plexiglass windows
allowed a side view of the flow. A 6-mm thick rub-
ber liner with channel-spanning 25-mm tall by 25-mm
wide treads, spaced every 20 cm in the stream-wise
direction, prevented bulk sliding of the entire mass
on the flume bed. The drum velocity for these experi-
ments was held constant at 1.25 m/sec.
The experimental flows had between ~450 and
1200 kilograms of material, which created a shallow
flow with maximum height of ~25 cm. Both mono-
sized gravel-water flows and muddy flows with a wide
Fig. 2 - Outlines of the granular mass in the small drum at
a velocity of 12 RPM. (a) Experiment S02 – glass
marbles, 13.8 mm diameter, (b) experiment S06 –
dry gravel, 13 mm diameter, (c) experiment S08
– moist sand, 1 mm diameter
Tab. 1. Experiments in the small 56 cm diameter
Tab. 2 - Experiments in the large 4 m diameter drum
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BOUNDARY SHEAR STRESS OF GRANULAR FLOWS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
335
particle size plays a role, but it appears that particle
shape and moisture/cohesion are a larger influence
on boundary shear stress for the range of variables
that we tested in the smaller drum. This can be seen
in Table 1 as the natural sediment particles (all sizes
of gravel and sand) have similar values of the shear
stress metrics.
In the large drum, as the flume was started and
reached the target velocity, the mass quickly found a
near steady state position. There was some surging of
the front due to a non-constant bed roughness. Smooth
areas in the drum bed existed because the experiments
had a dual purpose to measure bedrock erosion, and in
each quadrant there was a 60 cm length of bed without
treads where the flow passed over an erodible sample
or a load plate (H
su
et alii, 2007). The same general
particle trajectory pattern of conveyor-belt motion
was established, as observed in the small drum. There
was a component of lateral (cross-stream) velocity in
the surface trajectories of the particles for many runs.
The muddy mixtures with wide grain size distribution
had a different particle trajectory velocity fields from
the homogeneous gravel flows, because the different
sized grains were affected by particle and wall interac-
tions to different extent. During the ~30 minute exper-
iment, there was some temporal variation in the bulk
properties of the flow, e.g. evaporation of fluids, loss
of some fines to the side walls, and bubble/foam de-
velopment in the fluid. For the water-saturated gravel
flows, a decrease in the centroid position was seen as
time progressed. This could be either due to particle
comminution and rounding, or changes in the fluid be-
cause of the bubble/foam development.
Tables 1 and 2 list the average position (in drum
nected to a free-standing mount and slid into place in
the flume. Profiles were collected at 5 Hz. The cen-
troid was calculated for single profiles and for the av-
erage over the experimental run. The muddy natural
grain size distributions were not analyzed in the same
way due to changes in total volume and water con-
tent during the flow. These experiments are described
qualitatively in the results section.
The coordinate system for longitudinal position
of the centroid is the drum angle, which is 0 degrees
at the 6 o’clock position and increases to 90 degrees
at the 9 o’clock position (mirroring the tangent to the
circle for that position). In the large drum, at high
drum angle (~8 o’clock position) the laser profile is
ambiguous because there, the flow is thin and may be
falling vertically from the drum wall instead of being
a coherent part of the flow. Therefore we evaluate the
outline of the flow up to the drum angle of 60 degrees,
which is the transition to the detached part of the flow.
RESULTS
In the small drum, the material was placed in the
drum and the motor was started at the target velocity.
The mass quickly reached a near steady state position
in the drum. The particle trajectories were a function
of drag forces from the boundary, gravitational forces,
and contact forces from other particles. A velocity
field resembling a conveyor belt was established with
a shear zone in the center of the flow dividing down-
ward travelling particles at the surface and upstream
travelling particles at the wall, as described in H
su
et alii (2008). The maximum thickness of the flow
was ~5-7 cm. The shape of the particles affected the
particle movement and trajectories. Perfectly spheri-
cal particles had a large component of rolling, which
decreased the amount of jostling or velocity fluctua-
tions away from the mean velocity. Subangular gravel
particles had less rolling motion and more sliding and
collisional or glancing interactions with their neigh-
bors. The addition of a small amount of water to the
sand flows induced cohesion between sand grains and
greatly decreased internal shearing. Figure 2 shows
the outline of representative flows in the small drum.
The two methods for calculating bulk shear stress
correlated with each other linearly (Fig. 3). For both
methods, the lowest shear stress was measured for
the spherical marbles, followed by subangular gravel
and sand, and finally moist sand. Within these groups,
Fig. 3 - Linear trend between the centroid-derived
torque on the small drum and the force trans-
ducer value, showing the correlation between
the calculated and measured proxies for bulk
shear stress
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L. HSU, R. kAITNA , w.E. DIETRICH & L.S. SkLAR
336
5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011
angle degrees) of the centroid of the flows for different
experiments. For both the marbles and gravel flows
the position of the centroid is weakly correlated with
gravel size, showing either a slight or unclear increas-
ing trend with size. Particle shape has a more clear ef-
fect as all spherical marble flows have lower centroid
positions than sub angular gravel or sand flows. In the
large drum, increasing the particle size has a small
to negligible effect on bulk shear stress (~1 degree
change in the centroid for factor of three increase in
gravel diameter), a larger increase in centroid position
is seen for a change to a more angular 21 mm diameter
gravel (4 degrees). Also, the position of the centroid is
lower for water-saturated gravel in the large drum than
for the same sized dry gravel flows in the small drum
by 3-6 degrees, despite the additional roughness from
the treads in the large drum. Using the moment arm,
r, and the total mass of gravel and water, we calculate
the torque for the average profile of each run (Table
2), which increases with particle size and angularity.
The experiments in the large drum with the muddy,
wide grain size distributions had large clasts that were
nearly equal to the height of the flow. In these multi-size
flows, we observed the trajectories and wall drag effect
on different size particles (H
su
, 2010). The larger clasts
were affected by both the sidewall and bottom boundary
drag and with interactions with each other. Clasts that
were about the height of the flow almost always were
touching a boundary, and therefore bore a lot of drag.
The increased influence of boundary drag on the larger
particles resulted in an increase in the centroid position
of the entire flow. Over the ~30 minute experiment, the
muddy flows lost a significant amount of their fluid due
to evaporation. The decrease in the matrix fluid content
led to more collisional interactions between the parti-
cles, since there was less buffering by interstitial fluid.
In addition, the position of the centroid appeared to in-
crease with the loss of fluid, although it is unclear how
much of this trend is due to a decrease in mass and vol-
ume in general, a decrease in the fluid content only, or an
increase in the particle collisions with the bed and walls.
DISCUSSION
Natural debris flows vary spatially and temporally
along their flow path with particle size, shape, and wa-
ter content. Along their path, particles in the flow may
become more rounded and decrease in diameter due
to comminution, and the flow may become more di-
lute due to water input or drier due to evaporation and
water loss. Our experiments demonstrate how minor
changes in shape or fluid content may lead to a change
in shear stress at the boundaries of the debris flows
The bulk shear stress is an average of locally fluc-
tuating stresses which are highest at particle contacts
with the boundary. The number of contacts and total
contact area depend on the shape and size distribution
of the particles. Importantly, our observations show
how particle properties and dynamics play a part in
determining total bulk shear stress of the flow. This
role seems to be played mostly by particle shape,
which affects the amount of particle locking, rolling
versus sliding, and contact area and resultant drag
along the boundaries. Thus, deposit characteristics
can be useful for predicting boundary shear stress
even if the flow itself is not observed. The size, shape,
and relative amount of matrix leads to predictions
about the amount of particle locking, number and size
of contacts with the wall and bed, and disturbances
caused by flowheight sized particles..
Although the drum geometry is different than that
in nature, in these experiments most flows occupied
a similar position in the drum, so that the effect of
the drum geometry should be similar. The recircula-
tion effect may be the biggest difference from nature
(i.e. flows in the drum cannot deposit), although non-
depositing flows are also seen in nature. Centrifugal
force is a low fraction of the gravitational force at our
drum velocities (H
su
et alii, 2008).
Our experiments tested a small subset of variables
that vary in natural granular flows. Some next steps
are to evaluate the effect of flow velocity, bed rough-
ness, fluid viscosity, and pore fluid pressure on the to-
tal bulk shear stress. Also, high frequency fluctuations
in shear stress should be examined for information
about shear stress due to particle collisions. This will
help us to better understand the mechanisms causing
localized shear stress at the boundaries
Numerical experiments can provide detailed in-
formation about shear stress at a higher resolution and
larger spatial coverage than physical experiments. Us-
ing distinct element models to simulate granular flows
of different particle shape and size will help us to un-
derstand the role of particle interactions in determin-
ing average and total shear stresses (y
oHannes
, 2008).
These models are also useful for exploring the effects
of lateral movement and wall drag
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BOUNDARY SHEAR STRESS OF GRANULAR FLOWS
Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza
337
shear stress. These trends in boundary shear stress
with solid and fluid properties allow one to predict
trends in run out, energy dissipation, and bed entrain-
ment or erosion by natural granular flows.
ACKNOWLEDGEMENTS
The authors would like to thank Sigurd Anderson
of Engineering Laboratory Design for construction of
the flume. Stuart Foster, Jim Mullin, Chris Ellis, and
Terry Smith assisted with the experiments and data
analysis. Marisa Palucis provided helpful comments
on the manuscript. This work was supported by the
STC program of the National Science Foundation via
the National Center for Earth-surface Dynamics under
the agreement Number EAR- 0120914..
SUMMARY AND CONCLUSIONS
Using two vertically rotating drums, we illustrated
a change in the torque that balances the total boundary
shear stress with changing solid and fluid properties in
granular flows. We observed how particle size, parti-
cle shape, and fluid content affected the dynamics of
the flows, which in turn affect the total boundary shear
stress. In the experiments we conducted, we found the
particle shape to be a larger influence on total shear
stress than particle size. A particle shape that deviates
from spherical strongly reduces rolling motion and in-
creases collisions and velocity fluctuations from the
mean velocity, increasing boundary shear stress. The
addition of fluid decreased the effective normal stress
by the solid component of the flow, and decreased the
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