# IJEGE-11_BS-Sarno-et-alii

*Italian Journal of Engineering Geology and Environment - Book www.ijege.uniroma1.it © 2011 Casa Editrice Università La Sapienza*

*DOI: 10.4408/IJEGE.2011-03.B-056*

**DAM-BREAK FLOWS OF DRY GRANULAR MATERIAL**

**ON GENTLE SLOPES**

**INTRODUCTION**

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.

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.

conditions: the results of the tests are then used to

improve the rheological models that underlie the nu-

merical simulations.

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**

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*

*L. SARNO, M.N. PAPA & R. MARTINO*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

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**

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

capacity as the upper one.

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.

curacy is good, due to it being valued less than 0.1°

propagation of a dam break of dry granular material

can be modelled by a depth averaged s

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.

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-*

*σ*

*y*

*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.

(i.e. i

*et alii*(2005), the model

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

*et alii,*2005) show that

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*

**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*

*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.

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.

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.

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*

connected to a digital video recorder. The first camera

upper part of the channel was used to store the material.

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

RIAL

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.

the same initial condition in each test. A trapezoidal-

shaped initial deposit was used for all the runs and is

reported in Fig. 2.

In the second, on a bed with a lining of coarse sandpa-

per (grit P40 FEPA/ISO 6344).

*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*

*L. SARNO, M.N. PAPA & R. MARTINO*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

*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*

*σ*

*x*

*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

/ ∂x>0

*∂u /*

∂x<0(i.e. the pile is compressing).

∂x<0

the material, with the shear stress linearly varying with

depth as the normal stress

*σ*

*y*

*.δ≤ f .*

*δ > f*, i

by b

*et alii*. (1999). It could also be obtained from

*δ→0*. In this case, failure only occurs

inside the granular material and the

*τ*

*xv*

of (3) is that it is assumed that

*σ*

*x*

*σ*

*v*

*τ*

*xv*

*= 0*. Moreover, many studies have also

1999; P

*k=1*.

estimate the coefficient

*k*.

*β*was set

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

sheared flow expected.

*RESISTENCE DUE TO SIDE wALLS*

the bed friction. Nevertheless, in order to simulate a

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.

2 m downstream the gate. The same rectification of

the images was implemented using fixed reference

lines on the channel bed.

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**

use depth-averaged type models (s

tions over the flow depth.

perbolic partial differential equations, written in the

conservative form:

*S*

*0*

*α*represents the gravity force compo-

*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

**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*

*∆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.

space and time and can capture the shocks due to a

local decrease of the order of accuracy near the dis-

continuities.

*U=(h, hU)*

*T*

nation of the first order Godunov flux

*F*

*I*

*and the sec-*

*F*

*II*

*A=∂F/ ∂U*is the Jacobian matrix of the flux

F,

*|A|=R|Γ| R*

*-1*

*is obtained through eigen-decompo-*

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

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:

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

there is slip, therefore the Coulomb failure law holds

*τ =σ*

*z*

*, tanδ*

*lat*

*δ*

*lat*

like the channel bottom, so

*δ*

*lat*

*=18.5°*. Assuming that

*σ*

*z*

*σ*

*y*

*k*

*Z*

*τ*

*xz*

is the following:

*k*

*Z*

Jaky formula (J

member of the momentum equation results:

*δ*

*bed*

*w*the width of the

when the bed is made of the same material as the side

walls (i.e.

*δ*

*bed*

*= δ*

*lat*

*k*was set equal to 0.453.

**NUMERICAL METHOD**

spatial domain was divided by an equally spaced mesh

*L. SARNO, M.N. PAPA & R. MARTINO*

*5th International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment Padua, Italy - 14-17 June 2011*

*q*

*k*

*j+1/2*,

ues was applied (H

*j*and

*j+1*is the following:

(wet/dry zone transition). In particular, a very thin

layer of material (10

the calculation, in cells where h<10

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.

tion was applied by imposing the two following ghost

values at the right end of the domain:

*n*denotes the last cell at the right

end of the domain.

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

*∂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*

*k*

*act*

*k*

*bass*

ing

*∆x*= 0.02 m and a

*∆t*for each step such as CFL=0.2.

**EXPERIMENTAL DATA AND COMPARI-**

**SONS**

*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.

22.7°. Then, a set of runs on a sandpaper bed was car-

ried out with the same slopes.

*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.

to the Savage & Hutter formula (2), the second one

with

*k = 1*assuming the stress tensor is spherical, the

last one with

*k*

*act*

bed runs were the following:

*d*

*bed*

*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

value of

*k*

*z*

*= 0.554*

*was used.*

*SMOOTH BED TESTS*

**DAM-BREAK FLOWS OF DRY GRANULAR MATERIAL ON GENTLE SLOPES**

flow is faster than the observed one. In this window

time, the best fitting seems to be achieved using

*k*

*act*

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

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*

*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)*

*L. SARNO, M.N. PAPA & R. MARTINO*

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

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-

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*

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*

**DAM-BREAK FLOWS OF DRY GRANULAR MATERIAL ON GENTLE SLOPES**

**CONCLUDING REMARKS**

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

*& alii*,

a type of dry granular avalanche.

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.

material already deposited. At first, this suggests giv-

ing a more accurate estimate of the Boussinesq mo-

mentum coefficient.

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.

motion (0-1.5 s), as similarly observed in the smooth

runs, the best fitting is obtained using

*kr.*

*POSITION OF THE wAVE FRONT*

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)*

*L. SARNO, M.N. PAPA & R. MARTINO*

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**

signing and carrying out the tests.

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