Philippe Claudin

Jamming, Force Chains, and Fragile Matter

Consider a concentrated colloidal suspension of hard particles under shear [Fig. 1(a)]. Above a certain threshold of stress, this system may jam [1]. (To observe such an effect, stir a concentrated suspension of cornstarch with a spoon.) Jamming apparently occurs because the particles form “force chains” along the compressional direction [1]. Even for spherical particles the lubrication films cannot prevent contacts; once these arise, an array or network of force chains can support the shear stress indefinitely [2]. By this criterion, the material is a solid. In this Letter, we propose some simple models of jammed systems like this, whose solidity stems directly from the applied stress itself. We argue that such materials may show fundamentally new mechanical properties, very different from those of conventional (elastic or elastoplastic) bodies. We start from a simple model of a force chain: a linear string of rigid particles in point contact. Crucially, this chain can only support loads along its own axis[Fig. 2(a)]: successive contacts must be collinear, with the forces along the line of contacts, to prevent torques on particles within the chain [3]. (Neither friction at the contacts nor particle aspherity can obviate this.) Let us now model a jammed colloid by an assembly of such force chains, characterized by a director n ,i n a sea of “spectator” particles, and incompressible solvent. (We ignore for the moment any “collisions” between force chains or deflections caused by weak interaction with the spectators.) In static equilibrium, with no body forces acting, the pressure tensor pijs› 2sijd is then

Selection of dune shapes and velocities. Part 1: Dynamics of sand, wind and barchans

Abstract:Almost fifty years of investigations of barchan dunes morphology and dynamics is reviewed, with emphasis on the physical understanding of these objects. The characteristic quantities measured on the field (shape, size, velocity) and the physical problems they rise are presented. Then, we review the dynamical mechanisms explaining the formation and the propagation of dunes. In particular a complete and original approach of the sand transport over a flat sand bed is proposed and discussed. We conclude on open problems by outlining future research directions.

A scaling law for aeolian dunes on Mars, Venus, Earth, and for subaqueous ripples

Abstract The linear stability analysis of the equations governing the evolution of a flat sand bed submitted to a turbulent shear flow predicts that the wavelength λ at which the bed destabilises to form dunes should scale with the drag length L drag = ρ s ρ f d . This scaling law is tested using existing and new measurements performed in water (subaqueous ripples), in air (aeolian dunes and fresh snow dunes), in a high pressure CO2 wind tunnel reproducing conditions close to Venus atmosphere and in the low pressure CO2 Martian atmosphere (Martian dunes). The difficulty is to determine the diameter of saltating grains on Mars. A first estimate comes from photographs of aeolian ripples taken by the rovers Opportunity and Spirit, showing grains whose diameters are smaller than on Earth dunes. In addition we calculate the effect of cohesion and viscosity on the dynamic and static transport thresholds. It confirms that the small grains visualised by the rovers should be grains experiencing saltation. Finally, we show that, within error bars, the scaling of λ with Ldrag holds over almost five decades. We conclude with a discussion on the time scales and velocities at which these bed instabilities develop and propagate on Mars.

Field evidence for surface-wave-induced instability of sand dunes

Field studies of barchans—crescent-shaped dunes that propagate over solid ground under conditions of unidirectional wind—have long focused on the investigation of an equilibrium between sand transport by wind and the control of air flow by dune topography, which are thought to control dune morphology and kinematics. Because of the long timescale involved, however, the underlying dynamic processes responsible for the evolution of dune fields remain poorly understood. Here we combine data from a three-year field study in the Moroccan Sahara with a model study to show that barchans are fundamentally unstable and do not necessarily behave like stable solitary waves, as suggested previously. We find that dune collisions and changes in wind direction destabilize the dunes and generate surface waves on the barchans. Because the resulting surface waves propagate at a higher speed than the dunes themselves, they can produce a series of new barchans of elementary size by breaking the horns of large dunes. The creation of these new dunes provides a mechanism for sand loss that prevents dune fields from merging into a single giant dune and therefore plays a fundamental role in the control of size selection and the development of dune patterns.

Selection of dune shapes and velocities Part 2: A two-dimensional modelling

Abstract:We present in this paper a simplification of the dune model proposed by Sauermann et al. which keeps the basic mechanisms but allows analytical and parametric studies. Two kinds of purely propagative two dimensional solutions are exhibited: dunes and domes. The latter, by contrast to the former, do not present a slip face. Their shape and velocity can be predicted as a function of their size. We recover that dune profiles are not scale invariant (small dunes are flatter than the large ones), and that the inverse of the velocity grows almost linearly with the dune size. We furthermore get the existence of a critical mass below which no dune solution exists. It rises the problem of dune nucleation: how can dunes appear if any bump below this minimal mass gets eroded and disappears? The linear stability analysis of a flat sand bed shows that it is unstable at large wavelengths: dune can in fact nucleate from a small sand mass if the proto-dune is sufficiently long.

Stress Propagation and Arching in Static Sandpiles

We present a new approach to the modelling of stress propagation in static gran- ular media, focussing on the conical sandpile constructed from a point source. We view the medium as consisting of cohesionless hard particles held up by static frictional forces; these are subject to microscopic indeterminacy which corresponds macroscopically to the fact that the equations of stress continuity are incomplete no strain variable can be defined. We propose that in general the continuity equations should be closed by means of a constitutive relation (or relations) between different components of the (mesoscopically averaged) stress tensor. The primary constitutive relation relates radial and vertical shear and normal stresses (in two di- mensions, this is all one needs). We argue that the constitutive relation(s) should be local, and should encode the construction h~story of the p~te: this history determines the organization of the grains at a mesoscopic scale, and thereby the local relationship between stresses. To the accuracy of published experiments, the pattern of stresses beneath a pile shows a scaling between piles of different heights (RSF scalingj which severely limits the form the constitutive relation can take; various asymptotic features of the stress patterns can be predicted on the basis of this scaling alone. To proceed further, one requires an explicit choice of constitutive relation; we review sonie from the literature and present two new proposals. The first, the FPA (fixed principal axes) model, assumes that the eigendirections (but not the eigenvalues) of the stress tensor are determined forever when a material element is first buried. (This assumes. among other things, that subsequent loadings are not so large as to produce slip deep inside the pile.) A macroscopic consequence of this mesoscopic assumption is that the principal axes have fixed orientation in space: the major axis everywhere bisects the vertical and the free surface. As a result of this, stresses propagate along a nested set of archlike structures within the pile, resulting in a m~T~imum of the vertical normal stress beneath the apex of the pile, as seen ex- perimentally (the dip). This experiment has not been explained within previous continuum approaches; the appearance of arches within our model corroborates earlier physical arguments (of S-F- Edwards and others) as to the origin of the dip, and places them on a more secure math- ematical footing. The second model is that of oriented stress linearity (OSL) which contains an adjustable parameter lone value of which corresponds to FPA). For the general OSL case, the simple interpretation in terms of nested arches does not apply, though a dip is again found over a finite parameter range. In three dimensions, the choice for the primary constitutive relation must be supplemented by a secondary one; we have tried several, and find that the results for

Giant aeolian dune size determined by the average depth of the atmospheric boundary layer

Depending on the wind regime, sand dunes exhibit linear, crescent-shaped or star-like forms resulting from the interaction between dune morphology and sand transport. Small-scale dunes form by destabilization of the sand bed with a wavelength (a few tens of metres) determined by the sand transport saturation length. The mechanisms controlling the formation of giant dunes, and in particular accounting for their typical time and length scales, have remained unknown. Using a combination of field measurements and aerodynamic calculations, we show here that the growth of aeolian giant dunes, ascribed to the nonlinear interaction between small-scale superimposed dunes, is limited by the confinement of the flow within the atmospheric boundary layer. Aeolian giant dunes and river dunes form by similar processes, with the thermal inversion layer that caps the convective boundary layer in the atmosphere acting analogously to the water surface in rivers. In both cases, the bed topography excites surface waves on the interface that in turn modify the near-bed flow velocity. This mechanism is a stabilizing process that prevents the scale of the pattern from coarsening beyond the resonant condition. Our results can explain the mean spacing of aeolian giant dunes ranging from 300u2009m in coastal terrestrial deserts to 3.5u2009km. We propose that our findings could serve as a starting point for the modelling of long-term evolution of desert landscapes under specific wind regimes.

Bedforms in a turbulent stream: formation of ripples by primary linear instability and of dunes by nonlinear pattern coarsening

It is widely accepted that both ripples and dunes form in rivers by primary linear instability; the wavelength of the former scaling on the grain size and that of the latter being controlled by the water depth. We revisit here this problem in a theoretical framework that allows to give a clear picture of the instability in terms of dynamical mechanisms. A multi-scale description of the problem is proposed, in which the details of the different mechanisms controlling sediment transport are encoded into three quantities: the saturated flux, the saturation length and the threshold shear stress. Hydrodynamics is linearized with respect to the bedform aspect ratio. We show that the phase shift of the basal shear stress with respect to the topography, responsible for the formation of bedforms, appears in an inner boundary layer where shear stress and pressure gradients balance. This phase shift is sensitive to the presence of the free surface, and the related effects can be interpreted in terms of standing gravity waves excited by topography. The basal shear stress is dominated by this finite depth effect in two ranges of wavelength: when the wavelength is large compared to the flow depth, so that the inner layer extends throughout the flow, and in the resonant conditions, when the downstream material velocity balances the upstream wave propagation. Performing the linear stability analysis of a flat sand bed, the relation between the wavelength at which ripples form and the flux saturation length is quantitatively derived. It explains the discrepancy between measured initial wavelengths and predictions that do not take this lag between flow velocity and sediment transport into account. Experimental data are used to determine the saturation length as a function of grain size and shear velocity. Taking the free surface into account, we show that the excitation of standing waves has a stabilizing effect, independent of the details of the flow and sediment transport models. Consequently, the shape of the dispersion relation obtained from the linear stability analysis of a flat sand bed is such that dunes cannot result from a primary linear instability. We present the results of field experiments performed in the natural sandy Leyre river, which show the formation of ripples by a linear instability and the formation of dunes by a nonlinear pattern coarsening limited by the free surface. Finally, we show that mega-dunes form when the sand bed presents heterogeneities such as a wide distribution of grain sizes.

Numerical simulation of turbulent sediment transport, from bed load to saltation

Sediment transport is studied as a function of the grain to fluid density ratio using two phase numerical simulations based on a discrete element method for particles coupled to a continuum Reynolds averaged description of hydrodynamics. At a density ratio close to unity (typically under water), vertical velocities are so small that sediment transport occurs in a thin layer at the surface of the static bed, and is called bed load. Steady, or “saturated” transport is reached when the fluid borne shear stress at the interface between the mobile grains and the static grains is reduced to its threshold value. The number of grains transported per unit surface is therefore limited by the flux of horizontal momentum towards the surface. However, the fluid velocity in the transport layer remains almost undisturbed so that the mean grain velocity scales with the shear velocity u*. At large density ratio (typically in air), the vertical velocities are large enough to make the transport layer wide and dilute. Sedime...

Barchan dune corridors: Field characterization and investigation of control parameters

[1]xa0The structure of the barchan field located between Tarfaya and Laayoune (Atlantic Sahara, Morocco) is quantitatively investigated and compared to that in La Pampa de la Joya (Arequipa, Peru). On the basis of field measurements, we show how the volume, the velocity, and the output sand flux of a dune can be computed from the value of its body and horn widths. The dune size distribution is obtained from the analysis of aerial photographs. It shows that these fields are in a statistically homogeneous state along the wind direction and present a “corridor” structure in the transverse direction, in which the dunes have a rather well selected size. Investigating the possible external parameters controlling these corridors, we demonstrate that none among topography, granulometry, wind, and sand flux is relevant. We finally discuss the dynamical processes at work in these fields (collisions and wind fluctuations) and investigate the way they could regulate the size of the dunes. Furthermore, we show that the overall sand flux transported by a dune field is smaller than the maximum transport that could be reached in the absence of dunes, i.e., in saltation over the solid ground.