review article towards a theoretical picture of dense...

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REVIEW ARTICLE

R. DELANNAY1*, M. LOUGE2, P. RICHARD1, N. TABERLET1 AND A. VALANCE1

1Groupe Matière Condensée et Matériaux, UMR CNRS 6626, Université Rennes I, Campus de Beaulieu, F-35042 Rennes, France2Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA

*e-mail: [email protected]

Dense fl ows of grains driven by gravity down inclines occur in nature and in industrial processes. Natural examples include rock avalanches and dune slides. Applications are found in the chemical, mining and pharmaceutical industries. Dense fl ows of macroscopic grains show features that are commonly observed in cooled or in transformed amorphous solids1, such as jamming and glass transition, yield, crystallization and fi rst-order phase transition from a disordered to an ordered state. For example, the inhomogeneous fl ow of bulk metallic glasses is localized in a few narrow shear bands2.

Individual grains are to granular mechanics what atoms are to material science. However, the complexity and distribution of the shape and size of granular materials make it daunting to predict their fl ow behaviour from material properties. Th us, tractable analyses of granular matter require grain properties to be represented by parameters, such as restitution or friction coeffi cients, that characterize overall grain behaviour, but not the detailed solid mechanics within a single grain. In general, fundamental understanding of granular mechanics must be rooted in simple model systems, such as the dense fl ows of monodisperse spheres considered in this review. Although signifi cant progress has been made in describing these fl ows, they still challenge our understanding and remain the subject of active research.

Kinetic theories for hard spheres that interact through uncorrelated, instantaneous, binary collisions exploit the analogies

between the colliding macroscopic particles and the molecules of a dense gas to derive constitutive laws for fl ows of inelastic grains. Th ey have had some success in describing relatively dilute fl ows or moderately dense fl ows in the absence of gravity3–5. However, the assumptions on which they are based can fail in the presence of strong volumetric forces. Repeated collisions or persistent contacts between multiple particles can then occur throughout much of the fl ow. Empirical constitutive equations are only beginning to be obtained in such situations, and none have yet been predicted from fi rst principles.

Recent physical experiments and numerical simulations have yielded a wealth of information on dense granular fl ows driven by gravity down inclined channels. Diff erent boundary conditions at the base and at the lateral walls are found to lead to remarkably distinct fl ow behaviour.

We restrict our attention to steady, fully developed (SFD) fl ows of identical spherical grains of diameter d and material density ρs driven by the gravitational acceleration g on fl at6, bumpy7 or erodible8 planar bases inclined at an angle α to the horizontal. By fully developed we mean that fl ow quantities, such as the fl ow height H perpendicular to the base or the mass fl ow rate Q, do not change in the fl ow direction. Th e fl ows are either unbounded in the direction normal to the plane of shearing9 or are channelled by fl at, vertical sidewalls8.

Our objective in the fi rst section is to identify indisputable observations from experiments and numerical simulations that a theory should reproduce. Th is summary of salient fl ow features complements the review of experiments that appears in the paper by the GDR MiDi group10, to which some of us contributed, and constitutes a more focused companion to the general review of dense fl ows written by Pouliquen and Chevoir11. Th e reviews of rapid granular fl ows by Campbell12 and Goldhirsch13 have little to say about dense, inclined fl ows.

Dense fl ows down fl at, frictional inclines are supported on a shear layer at their base in which collisional transfer of momentum

Towards a theoretical picture of dense granular fl ows down inclines

Unlike most fl uids, granular materials include coexisting solid, liquid or gaseous regions, which produce

a rich variety of complex fl ows. Dense fl ows down inclines preserve this complexity but remain

simple enough for detailed analysis. In this review we survey recent advances in this rapidly evolving

area of granular fl ow, with the aim of providing an organized, synthetic review of phenomena and a

characterization of the state of understanding. The perspective that we adopt is infl uenced by the

hope of obtaining a theory for dense, inclined fl ows that is based on assumptions that can be tested

in physical experiments and numerical simulations, and that uses input parameters that can be

independently measured. We focus on dense granular fl ows over three kinds of inclined surfaces:

fl at-frictional, bumpy-frictional and erodible. The wealth of information generated by experiments and

numerical simulations for these fl ows has led to meaningful tests of relatively simple existing theories.

nmat1813 Delannay Review.indd 99nmat1813 Delannay Review.indd 99 11/1/07 17:21:0911/1/07 17:21:09

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REVIEW ARTICLE

supports the less agitated material above. Dense fl ows over bumpy, frictional inclines are fl uidized throughout their height with a core of relatively uniform density that can occupy much of their interior. Sidewall friction, however, can infl uence this simple picture and can support fl ows over rough bases at increasingly high angles of inclination. Sidewall friction can also stabilize fl ows over the erodible surface of a heap at angles of inclination much greater than the angle of repose.

Th e second section focuses on theory. Th eories for dense inclined fl ows usually incorporate both rate-dependent collisional contributions and rate-independent frictional contributions to the shear and normal stresses. Th e former are associated with brief interaction between pairs of particles; the latter are associated with repeated or enduring contacts between multiple particles, oft en in a

chain. Most theories rely on incorporating some observed features of the inclined fl ow for closure.

Th e last section looks towards future work. Discrete numerical simulations of the individual particles of the fl ow have been developed to the point at which they can test the assumptions on which theories are based, and provide guidance on the additional modelling that is required. In addition, experimental techniques are now available that can measure quantities of interest within the fl ow. With the assistance of such simulations and experiments, truly predictive theories for inclined fl ows should be developed soon.

PHENOMENA

In this section we summarize salient results from laboratory experiments and numerical simulations.

Dense dry fl ows of spherical grains down inclines depend crucially on the nature of the base, the presence of sidewalls, and the height of the fl ow. To present a complete description of the corresponding eff ects, we focus on three generic fl ows, namely those on fl at, bumpy and erodible bases. Each kind has a diff erent behaviour, as well as a diff erent range of angles of inclination at which their velocity profi les and heights are SFD on time and length scales much longer than the inverse shear rate and the fl ow height. Recent experimental and numerical data have highlighted these diff erences, which have not always been fully understood.

For dense inertial fl ows of granular materials, a challenge in experiments is to exclude interactions that do not involve mechanical properties of the grains or the boundaries. Th us, adhesive, cohesive, electrostatic, van der Waals, capillary or magnetic forces are avoided, usually by controlling humidity, avoiding triboelectric or magnetic materials and selecting nearly spherical grains with diameters on the order of a millimetre. Brittle materials such as glass beads are forced to fl ow until their surface adopts steady frictional properties6.

With these precautions, the impulsive contacts of a wide variety of nearly spherical solids are captured by the simple model of Walton14, which involves a coeffi cient of sliding friction μ and coeffi cients of normal and tangential restitution e and β. Th ese parameters, which are averages over the microscopic properties of the grain material, can be measured independently15–17 or predicted in terms of material properties18,19. For relatively dense fl ows that involve impulsive contacts with little dissipation, these parameters provide a successful link between numerical simulations, microgravity experiments, and kinetic theory5. However, the presence of enduring contacts in dense, gravity-driven fl ows makes it more diffi cult to describe the mechanical properties of the grains so simply. In that case, normal restitution is less signifi cant, and simulations using diff erent contact models can yield diff erent numerical results, although the

Two types of simulation, both referred to as discrete element methods, are used for dense fl ows. Soft particle, or MD, simulations involve rigid, overlapping particles interacting at compliant elastic, frictional contacts with additional dissipative elements that provide a normal coeffi cient of restitution of less than unity14,25. Th e equations of motion are solved for the individual particles with a time step that is small enough to resolve the deformation of the contact elements during the course of the particle interactions. Contact dynamic simulations treat the particles as rigid, non-overlapping and frictional and, at each time step, obtain by iteration a solution to the under-determined algebraic system for the forces of interaction26.

Despite the fact that, to speed the simulation, the soft particles interact through normal springs that are more compliant than real contacts and that the solution to the iteration at each time step in the contact dynamics simulation is not unique, the methods seem to agree and, as far as can be determined, also with the physical experiments. For example, the simulations of Walton24 over a fl at, frictional base are consistent with profi les observed by Louge and Keast6. Th ose of Silbert et al.9 capture the essential behaviour recorded by Pouliquen7 for fl ows on a bumpy base. Th e numerical results of Bi et al.27 corroborate data of Taberlet et al.8 on an erodible heap bounded by sidewalls. Finally, Hanes and Walton23 report good qualitative agreement between physical experiments and numerical simulations for channelled fl ows down a bumpy incline.

Box 1 Numerical methods

Hopper

Gate

y

x

Sidewalls

α

Figure 1 A typical facility for producing dense granular fl ows down inclines. Once the inclination angle α is set to within 0.1°, the overall mass fl ow rate is typically found by weighing the output collected over a known time, and the height is recorded with a laser7. Other techniques include capacitance probes that measure the local volume fraction and the integrated volume fraction through the height6, fi bre-optic sensors at the sidewalls20 or in the interior23 that provide profi les of mean velocity u along the fl ow and velocity fl uctuations, and high-speed video imaging that gives similar information at the wall in three-dimensional fl ows23 or in the single layer of a two-dimensional fl ow of spheres or disks between lateral boundaries3,21,22.


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corresponding fl ow behaviours are qualitatively similar9 (see also Box 1). As will be described in our discussion of theories, present research involves attempts to bridge the gap between macroscopic fl ow behaviour and microscopic material properties when long-lasting contacts are present.

Figure 1 shows a typical setup that produces a dense, inclined fl ow. So far, the measurements that have been made in such devices (see, for example, refs 6, 7, 20–23) are limited.

Numerical simulations produce more detailed information on velocities, volume fraction and stresses9,24. Th ey reveal the role of contact parameters such as friction and normal and tangential restitution, and of the nature of the base and the lateral walls. Such numerical simulations of granular fl ows focus on individual particles of a fl ow as they travel between and interact in collisions25,26 (see Box 1).

We now describe the essential features of the three fl ows of interest. To illustrate and contrast their behaviours, we accompany our discussion with fi gures taken from numerical simulations of the three types of fl ow27.

FLAT, FRICTIONAL BASE

On a fl at, frictional base, fl ows of nearly spherical grains with H > 5d (ref. 24) feature a thin, agitated basal shear layer that supports a more passive, denser overburden at relatively low inclinations6 (Fig. 2a). In a shear layer of thickness less than 3d, grains roll around the axis perpendicular to the plane of shear but frustrate each other’s spin through collisions. Th ere, the mean velocity does not vanish at the base (Fig. 3a), the solid volume fraction ν is smaller than in the overburden (Fig. 3b), and the strength of the velocity fl uctuations grows with the thickness of the fl ow and the angle of inclination6. SFD fl ows occur in the range of tanα between the measured static and sliding friction coeffi cients between a grain and the base.

In shallow SFD fl ows with a height less than ten particle diameters, Louge and Keast6 measured a mass fl ow rate Q that was proportional to the product of H3/2 and an increasing function of α. Th eir results agreed at least qualitatively with those of Johnson et al.28 for fl ows of similar heights in a similar experimental facility. In such shallow fl ows, the lateral boundaries did not seem to be signifi cant6. However, these sidewalls aff ect the momentum balance for H > 10d, for example within the

long-period upward-moving waves observed by Louge and Keast6 or in the sidewall-stabilized heaps described in the section below.

Walton24 used molecular dynamics (MD) simulations that were periodic in the fl ow and lateral directions to observe SFD fl ows on a fl at, frictional base at a value of tanα that was lower than the coeffi cient of sliding friction μ that he adopted for collisions with the base and among grains. He confi rmed that Q was proportional to H3/2 and found that the shear layer grows thicker with increasing normal coeffi cient of restitution e, but that, surprisingly, Q also decreases with e as a greater proportion of the fl ow experiences collisional dissipation.

Because in Walton’s simulations all forces between the grains and the base are exerted at contacts, and the value of tanα is less than the coeffi cient of sliding friction, not all contacts are sliding at the base; instead, some have tangential contact forces less than the normal forces, consistent with Coulomb’s law24. However, because physical fl ows are observed to take place at angles of inclination with tangents greater than the coeffi cient of sliding friction6,15, the frictional forces in the experiments may involve static as well as sliding friction, or something more complicated29.

BUMPY, RIGID BASE

Dense SFD fl ows over rigid bases covered with bumps on the scale of order d require steeper inclinations than their counterparts on a fl at, frictional plane. As Fig. 4a shows, they exist in a range of thicknesses and inclination angles forming two regions, in which SFD fl ows have a diff erent character7,30–32.

Numerical simulations of dense, deep fl ows performed in cells that are periodic in the fl ow and lateral directions show that in the absence of lateral walls the fl ows are mobilized through their height9 and oft en have a negligible mean velocity at the base (Figs 2b and 3a). In region I of Fig. 4a, this complete mobilization results in Q proportional to H5/2. Th ere, fl ows consist of three zones: a thin top layer where ν decreases abruptly towards the free surface; a core, in which ν is remarkably constant but decreases with increasing α; and a basal layer with thickness less than 10d above the bottom boundary (Figs 2b and 3b). In the basal layer, the profi les of mean velocity and volume fraction depend on the nature of the base33,34.

If the fl ow is suffi ciently thick to establish a core, profi les in the latter are independent of the bottom boundary34, with the energy of the velocity fl uctuations proportional to the square of the shear

5

0

u(gd)–½

0

0.5

1.0

1.5

a b

u(gd)–½

Figure 2 Snapshots of periodic MD simulations of 10,000 grains in a domain bounded by frictionless sidewalls separated by W = 20d along z, streamwise length L = 25d, inclination α = 20°, a friction coeffi cient μ = 0.8 between grains, and other grain interaction parameters found in Bi et al.27. Colours represent the mean velocity made dimensionless with gravity and grain diameter. Shown are fl ows on a fl at, frictional base (a) and a bumpy base made of cylinders parallel to z (b).



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rate. Th is is the scaling expected from a collisional theory. As Fig. 3 shows, the core above a bumpy base has the same shear rate and volume fraction as the overburden above a fl at, frictional base. Th e MD simulations of Silbert et al.9 further reveal that profi les are independent of e, whereas μ aff ects the profi le of the mean velocity u only when μ is less than one, but it does not aff ect ν. Th is suggests that the value of ν in the core is independent of the dynamics of the fl ow and that the dissipation of fl uctuation energy is dominated by frictional losses.

Flows over bumpy boundaries are oft en infl uenced by the presence of sidewalls. Th e extensive measurements of Ancey35 provide a clear indication of this. Numerical simulations by Hanes and Walton23, performed in conjunction with physical experiments, showed that sidewalls can dissipate fl uctuation energy and produce profi les of fl uctuation energy in the interior with a completely diff erent character from that at the sidewalls. Th e lateral boundaries in the experiments of Hungr and Morgenstern36 on deep, very rapid, dense fl ows may be responsible for their observation that such fl ows were not sheared throughout their height. Th is localization of shearing at the base may also be due to their large H/d, which is much greater than in Pouliquen’s experiments7. In general, more experiments are needed at large H/d.

ERODIBLE BASE

Grains fl owing at the surface of a quasi-static granular bed are frequently observed in industrial processes and in nature.

Interestingly, the gas, liquid and solid behaviours of granular materials are present at the same time in such fl ows. Th ey are strongly infl uenced by sidewalls27,37. Most studies concerning these fl ows have been conducted in two diff erent confi gurations. Th e fi rst is the rotating drum10,38,39, a cylinder, partly fi lled with granular material, rotating about its axis at a controlled speed. Because the rotating drum does not permit fully developed fl ows, except perhaps at its midpoint, we focus instead on the second confi guration, of which

a

b

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0.3 0.4 0.5 0.6

0

10

20

0.4 0.5 0.6

Tanα

Tanα

I

II

SFD flow Unsteady flow

No flow

Figure 4 Diagrams of dimensionless fl ow height against tanα, highlighting similarities between simulations and experiments. a, Symbols represent simulations of Silbert et al.30 at transitions between fl ow regimes. Lines are visual fi ts. Circles correspond to Hstop. The fl ow is unsteady to the right of the line joining the square symbols. The triangles and dashed line subdivide SFD fl ows in two regions, I and II, that show different behaviours. In region I the fl ow is thick enough to establish a core of invariant solid volume fraction. In region II thinner fl ows with H near Hstop or, equivalently, with α reduced to less than the starting angle, show more complex profi les that depend on the nature of the bottom boundary. As Fig. 3 shows, the boundary also induces layering of the particles in its neighbourhood. This is particularly pronounced in dense fl ows that involve a single layer of spheres or disks between vertical walls3,21,22. b, Similar fl ow diagram from Pouliquen’s experiments7. Circles represent Hstop and squares all other recorded heights. The solid and dashed curves are predictions for Hstop and for the minimum height to establish a core, respectively31. The right-hand vertical dashed line marks the maximum angle of inclination predicted for steady fl ows31.

0

1

2

3

4

5

6

7

5 10 15 20

a

0 5 10 15 20

0.2

0.4

0.6

0.8

v

b

y/d

u (g

d)– ½

y/d

Figure 3 Profi les along y in the simulations of Fig. 2 for a bumpy base (black lines) and a fl at base (red lines)27. a, Dimensionless mean velocity; for y > 9d, both curves have the same derivative. b, Solid volume fraction; for y > 9d, both fl ows have the same volume fraction.


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Taberlet et al.8 off er the simplest example. Th ese authors describe “side-wall stabilized heaps” (SSHs), where sidewalls of eff ective friction μw decrease the shear stress in the fl ow (Fig. 5), leading to a ratio of shear to normal stress at the base of the mobilized layer of approximate thickness H that is less than the tangent of the angle of repose αr of the pile (Fig. 6b, inset):

(S/N)base = tanα − μw(H/W) ≤ tanαr (1)

Sidewalls have a strong infl uence on both fl ow inclination and other characteristics. Th e typical velocity profi le consists of three distinct parts (Figs 5 and 6). Within the quasi-static pile, grains experience a slow intermittent motion with a convex velocity profi le, in agreement with earlier experiments on fl ows over an erodible base40. Above this pile, the velocity profi le becomes more linear. Near the free surface, the profi le levels off in a gaseous layer dominated by collisional interactions. Finally, the packing fraction increases progressively with distance from the free surface (Fig. 6). Th is is fundamentally diff erent from the behaviour of the volume fraction in deep, dense fl ows over a bumpy base in the absence of frictional sidewalls (Fig. 2b). Th e relation between mass fl ow rate and height in the SSH also diff ers from that on a bumpy, rigid base, for which the fl ow rate is proportional to H5/2. In the SSH, the fl ow rate is proportional to H − Hc, where Hc is the critical height at which the fl ow jams. Ancey35 had already noted the transition between the two kinds of fl ow in his experiments. He also observed volume fraction profi les similar to those in Fig. 6.

Recent numerical simulations27 have shown that a fl ow rate linear in H and with a variable volume fraction in the core can be obtained on a bumpy base with frictional sidewalls. Conversely, when frictionless sidewalls are introduced, the essential features of the SSH are replaced by those on a bumpy base27. Th e distinction between the two kinds of fl ow is therefore due to sidewall friction rather than to the nature of the base (erodible or bumpy). Moreover, it is the friction at the walls, not the confi nement and ordering that these walls impose, that rules the fl ow.

Recent experiments37 have shown that steady fl ows on piles at relatively small inclination are entirely controlled by sidewall eff ects, even for channels 600 particle diameters wide. Th e question remains whether it is possible to create rectilinear SFD fl ows on an erodible pile in the absence of confi ning sidewalls. Daerr and Douady32 reported that, without lateral confi nement, fl ows on an unstable heap spread laterally from the point at which they are mobilized. In contrast, Félix and Th omas41 observed how rectilinear SFD fl ows of polydisperse grains form natural levees that confi ne the slide in the lateral direction. Th us, rectilinear fl ows on an erodible base may require a channel to remain SFD.

THEORY

In this section we view the development of theories for dense granular fl ows down inclines from the perspective of the kinetic theory of dense gases. We adopt this perspective for several reasons.

First, although the simplest form of the kinetic theory, which assumes that particle interactions are instantaneous, binary and uncorrelated, fails in general to describe dense fl ows with repeated or enduring contacts between particles, there are indications that such a theory can predict some aspects of these fl ows. For example, in inclined fl ows that involve a dense, passive mass of grains riding on a basal shear layer24,36, the collisional fl ow in that layer is amenable to such a description42. In addition, simulations of dense fl ows34 that almost certainly involve multiple contacts show features of the constitutive relations for shear and normal stresses from the simplest kinetic theory, provided that a suitable dependence of the pair distribution on ν can be found (Fig. 7).

Second, the simplest form of the kinetic theory can be systematically extended to include repeated, correlated collisions with the BBGKY equations43, and enduring interactions between multiple particles can in principle be described by incorporating long-range forces into the expressions for the transport coeffi cients44. Th at is, a theoretical framework already exists within the kinetic theory for describing dense, granular fl ows down inclines.

Th ird, our view is that an understanding of the physics of the macroscopic phenomena requires an understanding of the physics at the particle scale and the formulation of the linkage between the micro and macro scales. An ab initio theory of this nature requires a minimum of assumptions. Consequently, when its predictions are tested, it is clear why the theory fails, although it may not be obvious how to change the assumptions to improve its performance.

Th us, in this section we fi rst consider ab initio microscopic constitutive theories based on the kinetic theory of colliding grains45, as well as variants in which pair distribution is adjusted to improve fl ow predictions46. We then describe mixed theories that assume coexisting collisional and persistent contacts31,47–51. We neither describe nor evaluate theories for dense inclined fl ows based on activation energy52,53. However, we mention constitutive relations with frictional and collisional contributions based on dimensional analysis and input from experiments10,54.

Kinetic theory for a dense gas of hard, smooth, elastic spheres has been extended to apply to granular materials by including small amounts of dissipation55–57. In this case, the constitutive relations

50

0

u(gd)–1/2

Figure 5 Snapshot of MD simulations with frictional sidewalls (μ = 0.8), 60,000 grains and α = 40° (refs 8, 27). Grains are placed in a horizontal cell that is periodic in the fl ow direction. Tilting the cell triggers fl ows that evolve towards a steady state over a wide range of inclination angles. The lower part of the granular assembly remains quasi-static, while a SFD fl ow parallel to the bottom develops on top. Such periodic simulations replicate the formation of SSHs in experiments, where grains are poured between confi ning walls at a given fl ow rate8,35,39. An SSH occurs when the fl ow rate exceeds a critical value that depends on channel width and material properties (mainly friction coeffi cients). Grains poured into the channel become progressively trapped, contributing to a slow increase in the free surface, until a rectilinear SFD fl ow occurs on top of the pile at an inclination that exceeds the angle of repose of the material.



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for stress τ are expressed in terms o f the mean velocity u as in a newtonian fl uid:

τ = [−p + (λ − 2η/3)∇∙u]1 + η[(∇u) + (∇u)T] (2)

In addition to equations of conservation of mass and linear momentum, a balance equation determines the evolution and spatial variation of the energy E per unit mass of the macroscopic velocity fl uctuations as it is conducted from point to point in the fl ow, produced by the mean shear rate and dissipated in collisions:

ρsν(∂E/∂t + u.∇E) = −∇∙q + τ : ∇u − γ (3)

where the energy fl ux q is

q = −κ∇T (4)

Th e pressure p = ρsν(1 + 4νg12)T, the viscosities λ and η and the conductivity κ are given in terms of ν, the ‘granular temperature’ T = 2E/3 and the pair distribution at contact g12(ν). For nearly elastic spheres, the rate of collisional dissipation of fl uctuation energy is

γ = (24/π)(ρsν2g12/d)(1 − e)T 3/2 (5)

Small amounts of friction in collisions can be incorporated into the theory, through an eff ective coeffi cient of restitution, without changing its structure58; however, signifi cant collisional friction requires the consideration of the balance equations for angular momentum and the energy of the fl uctuations in particle spin59. Increased dissipation due to friction or small restitution requires an even more complicated description that involves balance equations for higher moments of the velocity fl uctuations, higher spatial gradients and higher powers of lower gradients60–62.

In general, a granular fl ow will slip relative to a boundary; so, in addition to being dissipated in collisions at the boundaries, fl uctuation energy can be generated by slip63. Th at is, boundaries can either provide or remove fl uctuation energy from the fl ow.

Th e simplest kinetic theories have been applied to SFD fl ows down bumpy inclines45,51,64. Solutions in the absence of sidewalls that are relatively dense throughout their height are found to exist for bumpy bases that provide fl uctuation energy to the fl ow. In such solutions, the energy of the velocity fl uctuations typically decreases with distance from the base, and the solid volume fraction has a maximum in the interior of the fl ow. Flows with these features have been observed in experiments23,65 and simulations23 with sidewalls. However, in such fl ows there is signifi cant slip at the base, a variation in the volume fraction through the height, and sensitivity to the eff ective coeffi cient of restitution. None of these are features of the slow dense fl ows over bumpy inclines observed in numerical simulations of inclined fl ows without sidewalls9,30,66.

For a dissipative bumpy base, the kinetic theory predicts that, in the absence of sidewalls, the fl uctuation energy increases and the volume fraction decreases with distance from the base. Th ese fl ows are not dense through their entire height, and, because of the high volume fraction and low fl uctuation energy near the base, it is unlikely that the collisions between particles are instantaneous and binary and that the positions and velocities of the colliding particles are uncorrelated. Th at is, the predictions of the simplest theory are incompatible with the assumptions on which it is based.

Th ere have been several attempts to extend the simple kinetic theory to include higher volume fractions, greater dissipation in collisions, and multiple or enduring contacts. Bocquet et al.46 adopt the dense limit of the simple kinetic theory but, by analogy with the behaviour of supercooled liquids in the neighbourhood of the glass transition, they introduce an anomalous density dependence of the viscosity close to random close-packed density. Th en the energy balance, subjected to zero energy fl ux at top and bottom, has solutions only for a fl ow thickness above a lower limit that varies with the inclination in a way similar to that observed in the experiments and simulations. When fl ow is possible, a given fl ow thickness can occur over a range of inclination angles, in contrast to the dense limit of the simple kinetic theory, in which the thickness is a unique function of inclination42. With a no-slip condition imposed at the base, the predicted profi les of velocity, energy and volume fraction have the qualitative features of those

0

0 1 2

50 1000

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0 25 50 75

y/d y/d

0

0.3

0.6

ν ν

H/W

1

2

a b

Tanα

V (gd) –½

Figure 6 Profi les along y highlighting similarities between SSH experiments and numerical simulations27. a, Experimental profi le of solid volume fraction ν for an SSH obtained by γ-densitometry with 500-μm glass beads with W = 10d, α = 45° and H = 30d. b, Profi les of solid volume fraction (black) and dimensionless streamwise mean velocity (red) for an SSH with conditions as in Fig. 5. Inset: tanα is plotted as a function of H /W, for different values of the channel width W shown as distinct colours, with polydisperse beach sand. The dashed line is a fi t to the data. Note that the relevant parameter capturing the effects of wall friction is H /W rather than H /d.


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measured in the simulations, including a maximum velocity that varies with H3/2(sinα)1/2 and, for specifi c conditions, a relatively uniform volume fraction through the height of the fl ow. However, the volume fractions predicted by the theory are much lower than those recorded in experiments or simulations.

Friction in enduring sliding contacts has been incorporated into kinetic theories by adding a heuristic frictional contribution to the collisional stress of the simple kinetic theory47. Johnson et al.28 and Louge and Keast6 use such a decomposition to interpret the results of their experiments on fl ow over a fl at, frictional base.

Johnson et al.28 relate the enduring part of the shear stress to the enduring part of the normal stress through a constant coeffi cient of internal friction. Th ey close the theory with a functional relationship between the enduring part of the normal stress and the volume fraction. When the resulting equations and boundary conditions are solved for SFD dense fl ows, they obtain rough agreement between the predicted and measured mass fl ow rate, mass per unit basal area, and inclination angle and velocity profi les that involve a region of localized shearing near the base.

Louge and Keast6 focus on the region of intense shearing near the fl at, frictional base, and phrase and solve a boundary-value problem for the fl ow velocity and energy. Th ey also decompose the stress into parts associated with collisional interactions and enduring frictional contacts, and take the ratio of the enduring shear stress to enduring normal stress to be a coeffi cient of internal friction. In contrast with Johnson et al.28, they use the balance of force normal to the fl ow to determine the normal component of the contact force, assume that the total shear stress works to produce fl uctuation energy, and close the theory by determining the volume fraction in the shear layer by using the balance of angular momentum appropriate to an

SFD fl ow. Th e theory captures the dependence of eff ective friction on Froude number (Fig. 8).

Louge31 proposes a similar theory in an analysis of relatively deep, dense fl ows down a bumpy incline. In considering the extended region of uniform volume fraction in the interior of the fl ow, he uses an algebraic balance between the production and dissipation of energy, and determines the fraction of the shear stress due to enduring contacts as a function of the inclination angle and the coeffi cient of internal friction. Th en he calculates profi les of the average and fl uctuation velocities from the constitutive relation for the collisional shear stress and the vertical momentum balance, respectively. When the values of the internal friction and coeffi cient of collisional restitution are appropriately specifi ed, the predicted relation between mass fl ow rate and angle of inclination agree with experimental data7, and the predicted profi les of average and fl uctuation velocity are very near those reported by Silbert and colleagues9. Ancey and Evesque49 operate in a similar context but relate the collisional and frictional contributions to the shear stress through an assumption about the uniformity of the energy dissipation through the height of the fl ow.

Th e diffi culty with theories that decompose the stress into parts associated with collisions and enduring contacts is that information in addition to the coeffi cient of internal friction has to be specifi ed so as to close the theory. Aranson and Tsimring and Volfson et al.50,67,68 address this closure directly by introducing an evolution equation for the fraction of enduring contacts. By analogy with the transition between solid and liquid phases, they take the fi xed points of their evolution equation to correspond to all enduring contacts, all collisional contacts, and a fraction that is a function of inclination angle. Th is function varies between 0 and 1

0

10

20

30

40

50

0.7 0.80

10

20

30

40

0.7 0.8νν

a bN

/ρsd

T

S /ρ

sdT

½(d

u/dy

)Figure 7 Predictions of the kinetic theory for stresses on planes parallel to the fl ow compared with numerical simulations of a two-dimensional inclined fl ow of identical spheres34. Stresses are made dimensionless with the material surface density ρs defi ned here as the mass of a grain divided by its cross-section, the two-dimensional ‘granular temperature’ T = E, the grain diameter d and the mean strain rate du /dy; E is the fl uctuation kinetic energy divided by the grain mass and ν is the surface fraction. Lines are predictions of Jenkins and Richman86 for two different expressions of the pair distribution function at contact. The Verlet–Levesque distribution87 (dotted line) clearly fails at high surface fraction; Mitarai and Nakanishi34 replace it with a free-volume theory that Luding88 matched empirically to the Verlet–Levesque expression87 at low ν. Agreement is good in the ‘core’ of the fl ow where ν is invariant along y (open symbols), but the theory can fail near the base (solid symbols), particularly at low angles of inclination where enduring contacts are more prevalent. Angles of inclination are 20° (circles), 21° (triangles), 22° (squares) and 23° (diamonds). a, Two-dimensional normal stress N. b, Shear stress S having here the dimensions of a force per unit length.



REVIEW ARTICLE

as the inclination angle varies between a value below which a fl ow with purely collisional contacts is linearly unstable, and a value above which a static pile with purely enduring contacts is linearly unstable. However, although the theory has the capacity to predict the limiting thickness below which inclined fl ow is impossible, and other features of shallow and intermittent avalanches, it has yet to be applied to deep, dense SFD fl ows down inclines.

Th e existence of spatial correlations between particles in dense fl ows has been explicitly incorporated into a fl ow theory by Mills and his co-authors48,69–71. Th ey decompose the shear stress into a rate-independent frictional part associated with transient force chains, a local viscous part associated with mobile grains, and a non-local viscous part that accounts for the interactions between the mobile particles and the particles in the chains. Th e normal stress is also assumed to possess a static and a non-local part. Th e non-local parts are proportional to the integral of the viscous shear stress, normalized by a correlation length that is related to the spacing of the force chains along the fl ow. Th e static shear stress is related to the static normal stress by a coeffi cient of eff ective friction. Th e theory has the capacity to predict the variation in fl ow rate with fl ow thickness and angle of inclination, profi les of linear velocity and constant volume fraction, and the variation of the volume fraction with the angle of inclination seen in experiments on relatively shallow, dense, inclined fl ows, both over a rigid, bumpy base and over the surface of a heap. However, it makes no attempt to predict the fl uctuation energy profi le. In contrast, Rajchenbach72 emphasizes the energetics, and explains the linear velocity profi les observed in the dense, shallow fl ows as being due to the complete

non-local dissipation of fl uctuation energy in chains and clusters of particles.

Th e idea that a correlation length is important to the understanding and description of dense granular fl ows is also emphasized by other researchers. Inspired by da Cruz et al.73, the GDR MiDi group10 focuses on data from experiments and discrete simulation for rather rigid contacts and derives, from the data, dimensionless relations between the shear stress, normal stress, average fl ow velocity and mass density that apply to dense fl ows in several diff erent geometries. Th ese may be phrased as empirical constitutive relations for the shear stress and normal stress in steady, homogeneous shearing. Using these relations, the authors show to what extent their data on dense inclined fl ows can be interpreted in terms of a mixing length. Jop et al.74 generalized this approach to tensorial constitutive relations, although experimental justifi cation is at present limited to unidirectional fl ows of monodisperse spheres that do not slip at boundaries.

Motivated by numerical simulations of the run-out of deep, dense avalanches75, Campbell76 follows Babic et al.77 and characterizes the infl uence of the deformation of the contacts on the rheology of dense shearing fl ows. In complementary work, Zhang and Rauenzahn78 measure the temporal relaxation of contact duration and provide an evolution equation for the stress based on this. Finally, Ertas and Halsey79 interpret the rheology of dense, inclined fl ows as resulting from the formation of large-scale, coherent eddies of grains whose scale is determined by the second gradient of the velocity.

In short, theories for dense inclined fl ow all recognize the need to account for correlations between the velocities and/or locations of the particles associated with the enduring frictional contacts, but they diff er in the ways in which they incorporate them into the theory. Our view is that the existing constitutive relations of the kinetic theory need be modifi ed only by the incorporation of the correlations associated with the enduring and/or repeated contacts. However, the appropriate form of this modifi cation has yet to be determined.

CONCLUSIONS AND OUTLOOK

Numerical simulations have now been tested against physical experiments on dense granular fl ows down inclines, and, despite the relative simplicity of the particle interaction models, they have predicted the global features of such fl ows. Figures 4 and 6 are examples. With the confi dence that the simulations are faithful to the physics, the detailed information provided by them can serve as the basis for ab initio models based on the particle interactions and appropriate statistical characterizations of velocity and spatial distributions.

In general, simulations and experiments have highlighted the importance of sidewalls. Th ey infl uence relatively thick fl ows and are responsible for maintaining fl ows on steep erodible heaps. One important challenge is to understand their infl uence on the profi les of solid volume fraction. For example, no theory yet explains why this quantity varies with height in an SSH but remains essentially constant in the core of a mobilized fl ow in the absence of sidewalls.

Basal boundary morphology infl uences the fl ow in the neighbourhood of the base in ways that remain poorly understood. At the base, several assumptions of the simplest kinetic theory no longer hold. Force can be transmitted between grains through contacts that are not impulsive, layering is associated with anisotropy in the spatial distribution of pairs of particles, and the balance of angular momentum may have a role. Although the infl uence of boundaries can diminish with distance as, for example, in the core of fl ows on a bumpy base, theories and simulations should pay greater attention to the features of the fl ow near the base.

0.2

0.6

Effe

ctiv

e fri

ctio

n

0 2 4 6

0.22

0.43

0.63

Fr

μE

μI

Figure 8 Ratio of shear and normal stress (effective friction) on a fl at, frictional wall versus Froude number Fr based on mean velocity and weight6. For SFD fl ows, the effective friction is tanα. Dashed lines represent theoretical predictions at constant solid volume fractions (shown) within the thin agitated basal shear layer. If this volume fraction remained constant with α, then effective friction would decrease with Fr between the friction μE for long-lasting contacts and its counterpart μI for short impacts on the base, thus producing unstable fl ows. A balance of angular momentum (solid curve) for grains in the basal layer resolves this paradox by setting the basal volume fraction, thus predicting an effective friction consistent with data (circles). For identical spheres, this theory forbids fl ows in the shaded region, where basal volume fraction exceeds random jammed packing.


REVIEW ARTICLE


Th e role of friction is pervasive but unresolved. For example, it is unknown why the eff ective friction exerted by the walls on the SSH is so high, and how its magnitude is related to the friction between individual grains and the sidewalls. It is unclear whether sliding friction on a fl at base is suffi cient to describe the interaction with the base. Here, numerical simulations should be used to produce fl ows on fl at frictional boundaries with sidewalls. From these it could be determined why fl ows exist at angles of inclinations with tangents greater than the coeffi cient of sliding friction at the base. In the interior of dense fl ows, theories should explain why the friction between individual grains is much more important than normal restitution, and should derive new constitutive relations for the stresses and for the transport and dissipation of fl uctuation energy that account for the multiple, correlated, long-lasting contacts that particles oft en experience.

Other specifi c questions remain. Can SFD fl ows exist over an erodible base without lateral boundaries? In narrow channels of width on the order of a few grain diameters, does arching lead to deviations from the behaviour of a wider SSH? What causes fl ows on a bumpy base to develop a thinner basal shear layer at very high speeds?

We expect that further progress will be facilitated by the introduction of new experimental techniques and the additional exploitation of some existing techniques. Th ese all have the common feature of permitting measurements in the interior of the fl ow. Th ey include positron emission particle tracking80, magnetic resonance imaging81, diff usive wave scattering82, fl uid index matching83, confocal imaging84 and X-ray tomography85.

We have seen that several theories have the capability of reproducing the observed features of relatively shallow, dense inclined fl ows. Now is the time to distinguish between them. So far, numerical simulations have demonstrated the phenomena over various ranges of parameter space. Th ey should now be used to test the essential features of the proposed theories, to eliminate as many as possible from consideration, and to focus on those that seem to be securely founded on the appropriate physics.

Finally, although we have focused this review on relatively shallow, slow and dense granular fl ows over bumpy, fl at or erodible bases, other regimes with greater heights and speed deserve future attention, despite the experimental and numerical challenges they elicit. We expect that rich physics have yet to be discovered on the stability, transitions and multiplicity among granular fl ow regimes with greater scale and speed.

doi:10.1038/nmat1813

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AcknowledgementsWe thank Daniel Bideau, Gérard Le Caër, Luc Oger, Nathalie Thomas, and our colleagues in

the Groupement de Recherche Milieux Divises (GDR MiDi) for valuable discussions. We thank

James T. Jenkins for contributing several paragraphs on merits of the kinetic theory, and Namiko

Mitarai for providing data shown in Fig. 7. The preparation of this review was assisted by fi nancial

support from the GDR MiDi and US–France Cooperative Research grant INT-0233212. Our research

in dense, inclined fl ows is sponsored by the French Ministry of Education and Research (ACI PCN

(INSU): Écoulements gravitaires: modélisation des processus), the CNRS (PNRN: Programme National

des Risques Naturels, écoulements gravitaires), and NASA grants NCC3-468, NAG3-2705, NCC3-797

and NAG3-2353.

Correspondence and requests for materials should be addressed to R.D.

Competing fi nancial interestsThe authors declare that they have no competing fi nancial interests.


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