Multiscale clustering in granular surface flows

We investigate steady granular surface flows in a rotating drum and demonstrate the existence of rigid clusters of grains embedded in the flowing layer. These clusters appear to be fractal and their size is power law distributed from the grain size scale up to the thickness of the flowing layer. The implications of the absence of a characteristic length scale on available theoretical models of dense granular flows are discussed. Finally, we suggest a possible explanation of the difference between velocity profiles observed in surface flows and in flows down a rough inclined plane.     

Low-energy excitations in the three-dimensional random-field Ising model

General conservation equations are derived for 2D dense granular flows from the Euler equation within the Boussinesq approximation. In steady flows, the 2D fields of granular temperature, vorticity and stream function are shown to be encoded in two scalar functions only. We checked such prediction on steady surface flows in a rotating drum simulated through the Non-Smooth Contact Dynamics method even though granular flows are dissipative and therefore not necessarily compatible with Euler equation. Finally, we briefly discuss some possible ways to predict theoretically these two functions using statistical mechanics.     

From a grain to avalanches: on the physics of granular surface flows

Granular surface flows have still to be fully modelled. Analysing the motion of a single grain can already help us to understand the physical origin of several characteristic angles (starting, stopping, jumping angles). Then looking at layers of grains allows the inference of the velocity profile inside the flowing layer, and the physical origin of the flow depth selection. Then these results can be plugged into a general model of mass and momentum conservation integrated vertically (St-Venant). This model can be tested on stationary flows, but also on transients, such as avalanches. To cite this article: S. Douady et al., C. R. Physique 3 (2002) 177–186.     

Steady-state vectorial self-diffraction on a non-local photorefractive grating in a crystal of symmetry \overline{4}3m at symmetrical transmitting geometry

The steady-state two-wave interaction in a cubic crystal of the symmetry group 3m with the non-local photorefractive response in the absence of an external electric field is considered for the case of arbitrary interaction orientation with respect to the crystallographic coordinate system and for arbitrary intensities and polarization states of incident light waves. The self-diffraction problem is described on the basis of four coupled-wave equations in terms of the complex scalar amplitudes of components of the light waves with orthogonal linear polarization. The derived conservation laws are valid for the non-linear dependency of the photorefractive-grating amplitude on the modulation coefficient of the interference light pattern. It follows from these laws that the two non-unidirectional energy fluxes can form the total energy exchange between the two interacting light waves. A set of independent conservation laws allows us to decouple the coupled-wave equations and to obtain their analytical solution, at least, in the form of quadrature formulae. For example, such a solution is derived for the case of linearly polarized incident light waves and for the linearized dependency of the photorefractive-grating amplitude on the modulation coefficient. The explicit analytical expressions for the scalar amplitudes are obtained for the transversal electro-optic configuration of interaction. The possibility of polarization-state transformation of light waves without energy exchange between them is shown. Received: 30 July 2002 / Published online: 11 December 2002 RID="*" ID="*"Corresponding author. Fax: +7-3822/414321, E-mail: litvinov@ed.rk.tusur.ru     

The inertial dynamics of thin film flow of non-Newtonian fluids

Consider the flow of a thin layer of non-Newtonian fluid over a solid surface. I model the case where the viscosity depends nonlinearly on the shear-rate; power law fluids are an important example, but the analysis here is for general nonlinear dependence. The modelling allows for large changes in film thickness provided the changes occur over a relatively large enough lateral length scale. Modifying the surface boundary condition for tangential stress forms an accessible foundation for the analysis where flow with constant shear is a neutral critical mode, in addition to a mode representing conservation of fluid. Perturbatively removing the modification then constructs a model for the coupled dynamics of the fluid depth and the lateral momentum. For example, the results model the dynamics of gravity currents of non-Newtonian fluids when the flow is not creeping.     

II. Recirculation flow induced by a bubble stream rising in a viscous liquid

The recirculation flow induced by the rising motion of a bubble stream in a viscous fluid within an open-top rectangular enclosure is studied. The three-dimensional volume averaged conservation equations are solved by a control-volume method using a hybrid finite differencing scheme to describe the liquid phase hydrodynamics. The momentum exhange between the bubbles and the liquid phase is modeled with a source term equals to the volumetric buoyancy force acting on the gas in the bubble stream. The volumetric buoyancy force accounts for in line interactions between bubbles through the average gas volume fraction in the gas liquid column which depends on the size and the rising velocity of bubbles. The fluid flow within an open-top rectangular enclosure is further investigated by particle image velocimetry for a bubble stream rising in a water-glycerol solution. The measured fluid velocities in a vertical plane are compared with the predictions of the numerical model over a wide range of fluid viscosity (43 mPa s-800 mPa s) and gas flow rates. Finally, the recirculation flows resulting from the interaction of two neighbouring vertical bubble streams are studied. Received: 23 July 1997 / Revised: 19 December 1997 / Accepted: 11 May 1998     

Solid-fluid transition in a granular shear flow

The rheology of a granular shear flow is studied in a quasi-2D rotating cylinder. Measurements are carried out near the midpoint along the length of the surface flowing layer where the flow is steady and nonaccelerating. Streakline photography and image analysis are used to obtain particle velocities and positions. Different particle sizes and rotational speeds are considered. We find a sharp transition in the apparent viscosity (eta) variation with rms velocity (u). Below the transition depth we find that the rms velocity decreases with depth and eta proportional to u(-1.5) for all the different cases studied. The material approaches an amorphous solidlike state deep in the layer. The velocity distribution is Maxwellian above the transition point and a Poisson velocity distribution is obtained deep in the layer. The results indicate a sharp transition from a fluid to a fluid + solid state with decreasing rms velocity.     

Solitary waves in the Madelung's fluid: Connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation

An investigation to deepen the connection between the family of nonlinear Schr?dinger equations and the one of Korteweg-de Vries equations is carried out within the context of the Madelung's fluid picture. In particular, under suitable hypothesis for the current velocity, it is proven that the cubic nonlinear Schr?dinger equation, whose solution is a complex wave function, can be put in correspondence with the standard Korteweg-de Vries equation, is such a way that the soliton solutions of the latter are the squared modulus of the envelope soliton solution of the former. Under suitable physical hypothesis for the current velocity, this correspondence allows us to find envelope soliton solutions of the cubic nonlinear Schr?dinger equation, starting from the soliton solutions of the associated Korteweg-de Vries equation. In particular, in the case of constant current velocities, the solitary waves have the amplitude independent of the envelope velocity (which coincides with the constant current velocity). They are bright or dark envelope solitons and have a phase linearly depending both on space and on time coordinates. In the case of an arbitrarily large stationary-profile perturbation of the current velocity, envelope solitons are grey or dark and they relate the velocity u₀ with the amplitude; in fact, they exist for a limited range of velocities and have a phase nonlinearly depending on the combined variable x-u_{0 s} (s being a time-like variable). This novel method in solving the nonlinear Schr?dinger equation starting from the Korteweg-de Vries equation give new insights and represents an alternative key of reading of the dark/grey envelope solitons based on the fluid language. Moreover, a comparison between the solutions found in the present paper and the ones already known in literature is also presented. Received 20 February 2002 and Received in final form 22 April 2002 Published online 6 June 2002     

Core precession and global modes in granular bulk flow

We report a novel transition to core precession for granular flows in a split-bottomed shear cell. This transition is related to a qualitative change in the 3D flow structure: For shallow layers of granular material, the shear zones emanating from the split reach the free surface, while for deep layers the shear zones meet below the surface, causing precession. The surface velocities reflect this transition by a change of symmetry. As a function of layer depth, we find that three qualitatively different smooth and robust granular flows can be created in this simple shearing geometry.     

CONSERVATION LAWS FOR HEATED LAMINAR RADIAL LIQUID AND FREE JETS

The conserved quantities for the heated radial liquid jet and the heated radial free jet are established by using conservation laws. The flow in a heated radial jet is described by Prandtl's momentum boundary layer equation, the continuity equation and the energy equation. Viscous dissipation is neglected. The multiplier approach is used to derive the conservation laws for the system of three equations for the velocity components and the temperature and three conserved vectors are obtained. The conservation laws for the system of two partial differential equations for the stream function formulation are also computed by the multiplier approach and three conserved vectors are obtained. One of these is a non-local conserved vector for the system. The conserved quantities for the heated radial liquid jet and the heated radial free jet, emitted into a stationary fluid of uniform temperature θ_∞, are derived by integrating the conservation laws across the jet.     

Velocity correlations in dense granular flows

Velocity fluctuations of grains flowing down a rough inclined plane are experimentally studied. The grains at the free surface exhibit fluctuating motions, which are correlated over a few grain diameters. The characteristic correlation length is shown to depend on the inclination of the plane and not on the thickness of the flowing layer. This result strongly supports the idea that dense granular flows are controlled by a characteristic length larger than the particle diameter.     

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.     

Surface flows of granular mixtures: II. Segregation with grains of different size

We present the generalization of a theoretical model for segregation of granular mixtures due to surface flows, published in J. Phys. I France 6, 1295 (1996). Our generalized model is valid for grains differing by their size and/or their surface properties; in the present paper, we describe the case of two species with the same surface properties but two different sizes. The rolling stream is assumed to be homogeneous. Exchanges between the grains at rest and the rolling stream are modelized via binary collisions. The model predicts that during the filling of a two-dimensional silo, continuous segregation appears inside the static phase: small (respectively large) grains tend to stop uphill (respectively downhill), although both species remain present everywhere. This fits the observations when the size difference between the species is small. When the size difference is large, a different regime is observed. We argue that in this case, segregation occurs directly inside the rolling stream. Received: 25 February 1998 / Received in final form and Accepted: 6 July 1998     

Instantaneous velocity profiles during granular avalanches

We perform experimental measurements of the instantaneous velocity profile of the flowing layer during granular avalanches. In the pile depth, the velocity profile follows a pure exponential decrease in contrast with steady flows that are known to exhibit a well developed upper linear part. The velocity profile in the pile width is a plug flow with two exponential boundary layers at the walls. Even though no steady state is observed during the avalanche, these velocity profiles are self-similar and build up almost instantaneously, with time independent characteristic lengths.     

Influence of roughness and dilatancy for dense granular flow along an inclined wall

In a previous work [#!ref1!#], the flow velocity of a steady two-dimensional granular flow along an inclined wall was investigated. The scaling law for the velocity field was found in good agreement with recent experimental results. The purpose of the present paper is to reformulate in more systematic manner and in a somewhat more general context the equations of mass and momentum conservation for dense granular flow, and also to present some new results with particular emphasis on roughness influence and dynamic dilatancy. Theoretical results are found in good agreement with experiments. Received 19 July 1999 and Received in final form 14 October 1999     

Giant electrical fluctuations in metallic disordered packings

We have experimentally studied granular arches through electrical measurements. The packing is composed of 2d metallic pentagons and is submitted to small taps. Large electrical fluctuations are observed and they are distributed along power laws. This indicates the presence of long-time memory effects even the packing density remains constant around a value ρ = 0.72±0.02. Large electrical fluctuations should be associated with the breaking/creation of granular arches. Received 3 October 2000     

On dense granular flows

The behaviour of dense assemblies of dry grains submitted to continuous shear deformation has been the subject of many experiments and discrete particle simulations. This paper is a collective work carried out among the French research group Groupement de Recherche Milieux Divisés (GDR MiDi). It proceeds from the collection of results on steady uniform granular flows obtained by different groups in six different geometries both in experiments and numerical works. The goal is to achieve a coherent presentation of the relevant quantities to be measured i.e. flowing thresholds, kinematic profiles, effective friction, etc. First, a quantitative comparison between data coming from different experiments in the same geometry identifies the robust features in each case. Second, a transverse analysis of the data across the different configurations, allows us to identify the relevant dimensionless parameters, the different flow regimes and to propose simple interpretations. The present work, more than a simple juxtaposition of results, demonstrates the richness of granular flows and underlines the open problem of defining a single rheology.Received: 12 December 2003, Published online: 31 August 2004PACS: 45.70.-n Granular systems     

Tracer dispersion in power law fluids flow through porous media: Evidence of a cross-over from a logarithmic to a power law behaviour

An analytical model is presented to describe the dispersion of tracers in a power-law fluid flowing through a statistically homogeneous and isotropic porous medium. The model is an extension of Saffman's approach to the case of non-Newtonian fluids. It is shown that an effective viscosity depending on the pressure gradient and on the characteristics of the fluid, must be introduced to satisfy Darcy's law. An analytical expression of the longitudinal dispersivity is given as a function of the Peclet number Pe and of the power-law index n that characterizes the dependence of the viscosity on the shear rate . As the flow velocity increases the dispersivity obeys an asymptotic power law: . This asymptotic regime is achieved at moderate Peclet numbers with strongly non-Newtonian fluids and on the contrary at very large values when n goes to 1 ( for n=0.9). This reflects the cross-over from a scaling behaviour for towards a logarithmic behaviour predicted for Newtonian fluids (n=1). Received: 22 July 1997 / Revised and Accepted: 2 July 1998     

A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows

We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. This equation is necessary to close the overall system. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. We propose robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. It allows the simulation of interface problems between pure fluids as well as multiphase mixtures. Several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible. The method provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.