AN EXPERIMENTAL ANALYSIS ON THE RHEOLOGY OF A GRANULAR DEBRIS FLOW, PART I: COLLISIONAL STRESSES

Riccardo Martino, Carmine Sabatino and Lucio Taglialatela

Department of Hydraulic and Environmental Engineering “G. Ippolito”

University of Napoli (ITALY), Via Claudio, 21 – 80125 NAPOLI

E-mail: rmartino@unina.it - tel. +39 081 7683461 - fax: +39 081 5938936

Abstract: A local rheological analysis has been worked out to check the existence of collisional-dilatant regime as provided by a Bagnold type relationship. The results carried out on a uniform flow in an open channel (located at Hydraulic Laboratory of Civil and Environmental Engineering Department of the University of Trent, ITALY) of solid (PVC) and water mixture, have showed that the collisional regime takes place only when the Bagnold number is bigger than a critical value. This critical value is greater than that proposed from Bagnold.

Anyway the analysis confirms the validity of the Bagnold formulas, regarding both normal and shear stresses, even though the experimental parameters are different maybe because of the particular shape (cylindrical) of the particles constituting the solid phase.  
Finally the importance of the quasi-static stresses is showed. 

Keywords: debris flow, granular system, rheology, collisional stress

1    INTRODUCTION

The debris flow is an important geomorphologic process in many mountainous watersheds. Its occurrence is rather unpredictable and very destructive.

The debris flow occurs when, because of alluvial events, a large volume of sediments is mobilized forming a sediment flood that spreads downstream along preferential paths. It is characterized by a high mean velocity that can reach the order of tenths of meters per second. Its behaviour depends on the volumetric solid concentration, on the sediment size and on the size distribution. Finally the debris flow stops where the slope is low enough to cause the separation of water from the solids.

The sediment size (Seminara, 1993; Coussot, 1997) can change from clay to big boulders and depending on the granulometry of the material the involved flow is termed “mudflow” (fine fraction prevalent) or “granular debris flow” (coarse fraction prevalent).

The behaviour of a granular debris flow is typical of a granular mixture. A granular material is a collection of many individual solid particles that interact through short-range repulsive forces and frictional forces when in contact with other particles or boundaries.

Very important is the fact that during agitation, granular systems exhibit multiphase behavior. Fluid and solid-like regimes can exist simultaneously and the transitions between these may prove important in developing continuum methods to describe the dynamics (Julien & Lan, 1991).

When a granular mixture flows, the interactions between the particles and, in general with the interstitial fluid too, cause rapid dissipation of energy in the system. In many cases the fluids surrounding the particles affect the overall dynamics.

Granular matter flows like fluid, but do so in remarkably different way from fluid. It has distinctive features of solid, fluid or gaseous bodies depending on the kind of driving forces and on the exciting mechanisms.

2    COLLISIONAL STRESSES

During a granular mixture flow, when the solid particles are sufficiently distant (solid concentration not very high) and the shear rate is very high, the mechanical behaviour is called “collisional” or “grain-inertial” (Bagnold, 1954): the contacts between the particles are of short during, the momentum transfer occurs by the inter-particles collisions and the influence of the interstitial fluid can be neglected.

Bagnold (1954) verified the existence of the stresses associated to the granular interactions that he called “dispersive” stresses. In a special rheometer where the shearing surfaces are coaxial cylinders and in presence of a solid-liquid mixture with particles that had the same density of the water, he measured the dispersive (or collisional) stresses t_COLL and s_COLL and he found the relationships

                         (1)

                           (2)

where du/dy is the shear rate, r_S is the particles density, d_S is the particles diameter, F_D is an experimental parameter, called dynamic friction angle and equal to 17.75°, a is an other experimental parameter equal to 0.042, and l is the linear concentration (function of the solid concentration c and of maximum packing concentration c_¥)

                           (3)

These results are valid (Bagnold, 1954) when the number B (so called Bagnold number)

                             (4)

is bigger than 450 (m is the interstitial fluid viscosity).

More recent experimental studies (Savage & McKeown, 1983; Savage & Sayed 1984) on the collisional rheology of a granular mixture gave further contributions emphasizing more complex aspects even if they confirmed the mean characteristics of the Bagnold results.

3    DESCRIPTION OF THE EXPERIMENTS AND ANALYSIS OF THE RESULTS

The experiments were carried out with water and PVC particles with cylindrical shape (diameter of 3.2 mm, height of 2.8 mm, density r_S 1540 kg/m³) that has been used as the solid phase of the mixture. The experimental set-up, located at Hydraulic Laboratory of Civil and Environmental Engineering Department of the University of Trent (ITALY), consists of a flume with inclination that can be changed, 6 m long and 20 cm wide, with smooth glass 30 cm high (Fig. 1). Sticking a layer of PVC particles used in the experiments has roughened the channel bottom surface.

The channel was continuously fed with water and material granular by a properly designed close-circuit system: in this way it is possible to reach steady uniform flows. The flows were observed and filmed through the side walls. By an image analysis it is possible to determine a mean longitudinal velocity distribution and a mean solid concentration distribution in the transversal section (Martino, 1999).

In order to check the uniformity of the flow, velocity profiles in two different positions were compared.

The analysis of the tests carried out on this experimental apparatus (Armanini et al., 2000) showed that more different types of flow can occur:

(1) immature debris flow, where the sediments are not distributed on the entire flow depth;

(2) plug debris-flow, where a superficial layer of the flow is dry (depth of the sediment layer is bigger than that of the water);

(3) mature debris flow, where the sediments are distributed on the entire water flow.

Fig. 1    Experimental set-up

Moreover in the same paper, as also outlined by Egashira et al. (1997), the importance of the equilibrium of the flow with the bed (mobile bed condition) is showed. In this case the bottom does not represent an interface where profile of velocity changes abruptly or present cusps.

Fig. 2    Typical concentration and velocity profiles (Martino, 1999).

The tests that will be here analysed, refer to experiments in uniform flow condition and with a mobile bed characterised from an equilibrium condition. This means that the sediment velocity at the bed is zero (absence of a slip velocity).

The particles velocity and concentration profiles found in the experiments present very interesting properties (Fig. 2 ):

· the velocity distribution shows a concavity downward, contrary what happens in a velocity profiles with clear water, and it is possible to distinguish a superficial layer where the trend is linear or quasi linear;

· the solid concentration profile monotonically decreases upward assuming a maximum value (very close to that of maximum packing) in correspondence of the bed.

By the velocity and concentration distribution and with referring to a point at generic distance y from the bed, we have calculated the local shear velocity and the local value of l. Using a value of d_S equal to 3.15 mm (diameter of the volume equivalent sphere) we value the local Bagnold number.

In correspondence of the same point we can value the collisional shear stress t_COLL valuated by the Bagnold relationship and the acting shear stress t_ACT valuated by the relationship valid in a free surface flow of a granular mixture

                       (5)

where q is the slope of the sediment layer deposited on the flume bed, h is the depth of the flow and r is the density of the interstitial fluid (function of the point considered).

In order to understand the importance of collisional stresses we value the ratio, r_t,

                              (6)

between the acting shear stress t_ACT and the collisional shear stress t_COLL as provided by Bagnold formula, and the results against the Bagnold number are showed in Fig. .

Fig. 3    Ratio between the acting shear stress and the collisional shear stress valuated by the Bagnold formula, against the locally evaluated Bagnold number.

In a similar way we can value the acting inter-particles normal stress, difference between s_ACT and the pressure of the interstitial fluid s_INT, valuated by the relationship

              (7)

The ratio, r_s,

                          (8)

between the acting inter-particles normal stress and the collisional normal stress s_COLL, and Fig. shows its dependence on the Bagnold number.

Both r_t and r_s have an important meaning: if they are quite equal to 1 the acting stresses are only collisional and the Bagnold expression with the experimental value inside, is correct for the material used in the experiments.

Fig. 4    Ratio between the acting normal stress (between the particles) and the collisional normal stress valuated by the Bagnold formula, against the locally evaluated Bagnold number.

Fig.3 and Fig.4 show a certain dispersion of the experimental points. Probably it is caused by the technique used measuring the local solid concentration that is particularly delicate. Anyway we observe a clear trend that permits to affirm that for Bagnold values higher than 1500 both the ratios can be assumed as constant even if they are different from 1. Because of this we may argue that the Bagnold formula is satisfied for large values of the Bagnold number, but with different values of a and F_D. We found

 and

As mentioned above, Bagnold found the collisional regime in correspondence of B greater than 450. In our experiments we found an other value (1500) because our particle density is not equal to the interstitial fluid density: for this reason we have not only collisional stresses but also quasi-static stresses, that are absent in the Bagnold’s tests and that, for our mixture, are very important for B lower than 1500.

Finally we can justify different values of the experimental parameters a and F_D because the material used by Bagnold (paraffin wax and lead stearate) is different from that used in our experiments (PVC) and then the restitution coefficients that govern the collisional mechanic between two particles are not the same.

4    CONCLUSIONS

In this work we have presented some experimental results on the collisional rheology of a granular mixture, intended to confirm the validity of the Bagnold formulas in the grain-inertial regime. The analysis showed that for Bagnold values higher than a critical value the structure of the collisional Bagnold formulas is satisfied. For Bagnold values lower than this critical value it is not possible to neglect the quasi static stresses.

 Acknowledgements

The writers are grateful to prof. Aronne Armanini who gave useful suggestions and permitted to dr Riccardo Martino, during his PhD studies, to develop the experiments in the hydraulic laboratory of University of Trent.

References

Armanini A., Fraccarollo L., Guarino L., Martino R., Bin Y., 2000, “Experimental analysis of the general features of uniform flows over a loose bed”, Second International Conference on Debris-Flow Hazards Mitigation: Mechanics, Prediction, and Assessment, Taipei, August 16-18.

Bagnold, R.A. 1954. Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear, Proc. Roy. Soc, vol. 225, pp. 49-63.

Coussot, P. 1997. Mudflow rheology and dynamics, A.A. Balkema, Rotterdam, Brookfield.

Egashira, S., Miyamoto, K., & Itoh, T., 1997, “Constitutive equations of debris flow and their applicability”, First International Conference on debris flow hazards mitigation: mechanics, prediction, and assessment, ASCE.

Julien, P.Y. & Lan, Y. 1991. Rheology of hyperconcentrations, J. Hydraulic Engineering, vol. 117, n.3, pp. 346-353.

Martino, R. 1999. Aspetti di debris flow: analisi teorica e sperimentale. PhD Thesis, University of Naples, “Federico II” (in italian).

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Savage, S.B. & Sayed, M. 1984. Stresses developed by dry cohesionless materials sheared in an annular shear cell, J. Fluid Mech., vol. 142, pp. 391-430.

Seminara, G. & Tubino, M. 1993. Debris flow: meccanica, controllo e previsione. Monograph of the Gruppo Nazionale per la Difesa dalla Catastrofi Idrogeologiche (GNDCI), CNR (in italian).