# IJEGE-11_BS-Armanini-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-113*

**DYNAMIC IMPACT OF A DEBRIS FLOW FRONT AGAINST**

**A VERTICAL WALL**

**INTRODUCTION**

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

lead to the underestimation of the correct impact force

different mechanisms (a

**ABSTRACT**

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.

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.

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*

*A. ARMANINI, M. LARCHER & M. ODORIZZI*

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

**EXPERIMENTAL SETUP AND METHODS**

section, closed at the downstream end with a vertical

wall equipped with four piezoresistive pressure gaug-

es, positioned as depicted in Fig. 3.

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%.

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.

granular flows (C

*et alii*, 2002; s

*et alii*.

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

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.

determination of the impact mechanism. When grav-

ity prevails over inertia, typically there is the forma-

tion of a reflected wave

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)*

**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*

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.

ergy is not conserved.

**HOMOGENEOUS FLUID SCHEME WITH**

**FORMATION OF A VERTICAL JET**

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).

height

*h*

*ro*

*u*

*wo*

assumed along the streamlines. The application of the

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**

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:

*h*

*r*

*and*

*h*

*i*

*are the depths of the reflected and in-*

*u*

*r*

*u*

*i*

momentum balance. At the moment of the impact the

velocity on the wall is zero, that is

*u*

*r*

*Y = h*

*r*

*/ h*

*i*

*Fr*

*i*

*= u*

*i*

*/ (gh*

*i*

*)*

*1/2*

*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)*

*A. ARMANINI, M. LARCHER & M. ODORIZZI*

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

*h*

which we put

*h*

*w*

*= h*

*wo*

*p*

*w*

*h*

*wo*

*/h*

*i*

*F*

*ri*

that the formation of a vertical bulge determinates a

larger force, that become significant for Froude num-

bers larger than 3.

(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*

to an underestimation of the pressures at the moment of

impact, as showed in Fig. 6.

lently, we assumed as valid the Bernoulli theorem:

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*

*w*o

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*

**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*

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.

*α*is not constant.

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

*α*in order to correct the

*u*

*i*

*can*

first order approximation, the time dependent term in

the momentum balance can be therefore estimate to

be proportional to

*ρu*

*i*

*α.*

tor, leading to:

as a design tool.

*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*

*A. ARMANINI, M. LARCHER & M. ODORIZZI*

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

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*.

from equation (5), it is possible to express a new for-

mulation for the impact force:

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**

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.

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).

a static deposit of solid material is generated upstream

of the wall with a repose concentration C

*z*

*0*

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*

*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*

**DYNAMIC IMPACT OF A DEBRIS FLOW FRONT AGAINST A VERTICAL WALL**

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-

*C*

*r*

profile that is less uniform than on a rigid bed (a

*et alii*, 2005; l

*et alii*, 2007; f

*et*

*alii,*2008; a

*et alii*, 2009).

that

*u*

*r*

*C*

*r*

impact force, similarly to what obtained for the homo-

geneous fluid case.

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.

stant depth

*h*

*i*,

*C*

*i*

*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*

*A. ARMANINI, M. LARCHER & M. ODORIZZI*

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**

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.

*ρ*.

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:

order of magnitude larger than what obtained for the

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