# IJEGE-11_BS-Berzi-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-023*

**STEADY DEBRIS FLOWS OVER ERODIBLE BEDS**

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

*et alii*(2005) and

*et alii*(2007) on mono-dispersed plastic

granular-fluid wave over a rigid bed and were able

to reproduce the experiments performed by d

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

*et alii*, 2010).

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**

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.

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**

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.

*D. BERZI, J. T. JENkINS & E. LARCAN*

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

*x*=

*x*

***

*x*=

*X*

***

*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*.

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

mation (H

aged particle and fluid velocities are equal and constant,

as in b

*et alii*(2010).

*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

tan [

*f*

*b*/ d

*x*)] ≈ tan

*f*

*b*/ d

*x*, valid if d

*b*/ d

*x*

the direction of the flow.

*j*and

*J*are the particle and

resistances due to internal shear stresses and the role

of the drag force. b

*et alii*(2010) express them as

*≡ h*-

*b*and H

*≡ H*-

*b*. The coefficients

*λ*

*λ*

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

*et alii*(2010), we show that the

file measured by t

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

*et alii*(2010); then we show the comparisons of

non-uniform flows of gravel and water over erodible

beds performed by t

**GOVERNING EQUATIONS AND CLOSURES**

*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*)

*η*is defined in terms

balances and constitutive relations in terms of dimen-

sionless variables, with lengths made dimensionless

by

*d*, velocities by (

*gd*)

*ρσgd*.

*x*and

*z*to be the coordinates parallel and

bed of inclination

*f*with respect to the horizontal;

*z*=

*b*is

the position of the erodible bed (with

*b*

*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*

*u*

*A*

*Fig. 1 - Sketch of the flow configuration with the frame of*

*reference*

**STEADY DEBRIS FLOWS OVER ERODIBLE BEDS**

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

*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

*et*

*alii*(2010), and

*χ*= 1

*and*

*μ*

*w*

*et alii*

ments in uniform flow conditions.

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*

*q*=

*ĉu*

*A*

*h*) as functions of tan

*f*and

*U*

*A*

*Q*= [(1 -

*ĉ*)

*α + β*- 1]

*U*

*A*

*h*). Figures 2 and 3 show the

measurements of

*q*and

*h*as functions of tan

*f*. 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.

method to see if the theory has the capability to repro-

duce also the wave front. t

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*

= 2.8

*t*, and we can compare the experimental measure-

ments with the results of the present theory.

*et alii*, 2010). There,

*χ*is a material coefficient

for the particles adopted by b

*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.

b

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*

*w*, in locally

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

*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*

***

*b*(max[

*x*

***

*X*

***

**COMPARISONS WITH EXPERIMENTS**

*Tab. 1 - Coefficients in Eq. (3).*

*Tab. 2 - Coefficients in Eq. (4).*

*D. BERZI, J. T. JENkINS & E. LARCAN*

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

**CONCLUDING REMARKS**

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.

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.

saturated, i.e. with the height of the particles over the

bed approximately equal to the height of the fluid.

*x*

***

*X*

***

*u*

*A*

*U*

*A*

(4), and

*f*= 17°. In Fig. 4, we show the predictions of

the theory against the experiment of Tt

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

*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°*

**STEADY DEBRIS FLOWS OVER ERODIBLE BEDS**

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

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