Session P29: Focus Session: Granular Flows II

Beyond Navier-Stokes Order Effects in Granular Gases

The vast majority of continuum theories for rapid granular flows are based on Navier-Stokes order descriptions (up to first order in spatial gradients). In this effort, a simple system is used to illustrate the presence and impact of higher-order effects in both the Knudsen boundary layer and the domain interior. Specifically, a thermally-driven, zero mean flow system is considered via molecular dynamics (MD) simulations. The Knudsen boundary layer is identified via an abrupt mismatch in the simulation data for heat flux and predictions from Navier-Stokes order theories. When access to heat flux measurements is not available, a rule-of-thumb is established to estimate the thickness of the Knudsen boundary layer based on concentration measurements. The effect of boundary layer thickness on continuum predictions is assessed via MD simulations, and further illustrated via a comparison between predictions and experimental data for a vibro-fluidized bed. Next, the presence of higher-order effects in the domain interior is explored via MD simulations. The system displays a stress anisotropy, which can be traced to Burnett order effects. Furthermore, a surprisingly large mismatch is observed between Navier-Stokes order theory and MD values for the heat flux. Because there are no Burnett-order contributions to the heat flux, the responsible mechanisms appear to be beyond Burnett order.   

Power-law tail of the velocity distribution in granular gases

We use a two-dimensional event-driven molecular dynamics simulation to study the velocity distribution of a granular gas. We implement the high energy injection mechanism described in Ref. [1]. At a small rate $\gamma $ we boost randomly chosen particles to a high energy. The resulting driven steady state is found to have a power-law high-energy tail in the velocity distribution, f(v) $\sim $ v $^{- \sigma }$. The simulation results for the exponent $\sigma $ are in good agreement with the theoretical predictions of Ref. [1]. \newline \newline [1] E. Ben-Naim and J. Machta, Phys. Rev. Lett. 94, 138001 (2005).   

Translations and Rotations are correlated in Granular Gases

In a granular gas of rough spheres the axis of rotation is shown to be correlated with the translational velocity of the partciles. The average relative orientation of angular and linear velocities depends on the parameters which characterise the dissipative nature of the collision. We derive a simple theory for these correlations and validate it with numerical simulations for a wide range of normal and tangential restitution.The limit of smooth spheres is shown to be singular: even an arbitrarily small roughness of the particles gives rise to orientational correlations.   

On the possibility of aeolian dunes on a laboratory scale

Recent progress in modeling aeolian sand dunes in the field has resulted in the prediction of a critical linear length scale below which no shape stable dune can form. Under typical field conditions on earth, this length scale is around 10m or larger. Using small (0.05mm) lightweight (0.2g/cc) particles with a proper surface treatment to reduce cohesion we can demonstrate how the dune problem can be scaled down to a lab-size wind tunnel. We demonstrate (a) different transport properties of the particles upon variation of the wind speed, (b) the growth of a heap, (c) the formation of a crest, and (d) ripples on a smaller scale than the heap.   

Experiments on Washboard Road

Granular surfaces to develop lateral ripples (so-called ``washboard" or ``corrugated" road) under the action of rolling wheels. Similar ripples are observed on railroad tracks and many other rolling, load bearing surfaces. Our aim was to investigate this instability of the flat road surface from the point of view of driven, dissipative granular dynamics. We report the results of both laboratory experiments and soft-particle direct numerical simulations. The experiment consisted of a rotating table 60 cm in radius with a thick layer of sand forming a roadbed around the circumference. A 6 cm radius hard rubber wheel, with a support stationary in the lab frame, rolled on the sand layer. We varied the speed of the table, the details of the grains and the suspension of the wheel. The ripple pattern appears as small patches of travelling waves which eventually spread to the entire circumference. The ripples drift slowly in the driving direction. Interesting secondary dynamics of the saturated ripples were observed, as well as various ripple creation and destruction events. The wavelength of the ripples can be quantized by the finite circumference of the road. All of these effects are captured qualitatively by 2D soft particle simulations in which a disk rolls over a 2D bed of polydisperse particles in a periodic box.   

Probing Avalanche Dynamics using Speckle-Visibility Spectroscopy.

We apply a new light scattering technique called Speckle-Visibility Spectroscopy to the study of avalanches. By directly relating the rate of change of the scattered speckle pattern to the fluctuation dynamics of the flowing sand particles, we attain a precision of 0.1 mm/s. Running for 35 hours at 58 kHz, we simultaneously observe the microscopic short-time fluctuations of the sand particles and the long time behavior of thousands of avalanche events, and thus report avalanche frequency statistics and average shape. Interestingly, while all avalanches turn on in 0.3 s and in a similar way, there is a wide variation in how avalanches turn off. The fluctuation speed reaches a maximum just after the avalanche begins, it remains constant for a while, and then decays to zero. Power spectra of the full data set show that as avalanches slow the dynamics are self-similar ($\sim $1/f$^{ 2})$ and the normalized variance of different events diverge at the turning off time.   

Erosion of a granular bed by laminar fluid flow

Motivated by examples of erosive incision of channels in sand, we investigate the motion of individual grains in a granular bed as a function of fluid flow rate to give us new insight concerning the relationship between hydrodynamic stress and surficial granular flow. A closed channel of rectangular cross section is partially filled with glass beads and a fluid and a constant flux $Q$ is circulated through the channel. The fluid has same refractive index as the glass beads and is illuminated with a laser sheet away from the sidewalls. The bed erodes quadratically in time to a height $h_c$ which depends on $Q$. The Shields criterion, which is proportional to the ratio of the viscous shear stress and gravitational normal stress, describes the observed $h_c \propto \sqrt {Q}$ when a height offset of approximately half a grain diameter is introduced. The offset can be interpreted as arising due to differences between the flow near a porous boundary and a smooth wall. Introducing this offset in the estimation of the shear stress yields a grain flux $q_x$ in the bed load regime proportional to $(\tau - \tau_c)^2$, where $\tau$ is the non-dimensional shear stress, and $\tau_c$ corresponds to the Shields criteria.   

Segregation in horizontal rotating cylinders: radial and axial band formation, band traveling and merging studied by Magnetic Resonance Imaging.

Radial and axial segregations are investigated by Magnetic Resonance Imaging (MRI). For the first time, full 3D structures and real-time 2D MRI movies showing the progress of segregation over many hours are reported. Data were acquired with high temporal (74 ms) and in-plane spatial resolutions (1 mm $\times $ 1 mm), giving new insights into the underlying mechanisms. The mixture composition can be quantified throughout segregation. The cylinder to be considered is 48 mm in diameter, up to 50 cm long and filled to 50 -- 82{\%} by volume with millet and poppy seeds at a 3:1 ratio. In particular, the effects of filling fraction, cylinder length and rotational speed on segregation are addressed. Radial segregation is found to be driven by both core diffusion and the free surface. The former is dominant in the cylindrical core buried under the avalanche layer in systems over 75{\%} full while the latter is significant at lower filling levels. Axial segregation is characterized by band formation, traveling, and merging. In all cases studied, the formation of poppy-rich bands is observed, after which individual bands start to travel at $\sim$3 $\mu $m s$^{-1}$ until they are within $\sim $3 cm of a stationary band. Adjacent bands then merge into a single, enlarged poppy band as millet seeds move out of the merging region.   

Granular Flow in a Rotating Drum: Dry vs. Submerged Flow

We have experimentally studied granular flows in a cylindrical rotating drum, half-filled with nearly monodisperse spherical glass particles in order to investigate the effect of interstitial fluid on these flows. We have conducted two classes of experiments under otherwise identical conditions: The first with air as interstitial fluid and the second where the empty space in the cylinder was completely filled with water. For varying rotation rates, we used a particle tracking method to measure particle velocities near the side wall as a function of distance from the flow surface and the surface velocity as a function of distance from the side wall. In all cases, the velocity (relative to rigid rotational motion) initially decreases linearly from its surface value, followed by exponential decay, as a function of increasing distance from the surface. At a given rotation angle (i.e. overall flux), subaqueous flows exhibit more dissipation and therefore result in steeper surface slopes, a lower strain rate and deeper flows. The effect of the interstitial fluid weakens as rotation rate is lowered, resulting in the same slope in the limit of no rotation, i.e., angle of repose.   

Velocity Profiles in a Rotating Drum: The Effects of Cohesion

The dynamics of granular media in a rotating drum is important in a wide range of applications in industry associated with mixing granular materials. The rotating drum also serves as a standard experimental geometry to observe continuous avalanching in the laboratory. We study the effect of interparticle cohesion on the velocity field of the rotating drum using large scale granular dynamics simulations. Such cohesion is easily introduced in the system by a wetting fluid that forms menisci at interparticle contacts. Previously, we have examined the effect of interparticle cohesion in gravity driven chute flows, and have shown that the cohesion has a dramatic effect on the granular rheology. For strong enough cohesion, these forces generate a coherently moving plug at the free surface. In this talk, we examine the velocity profile in the rotating drum geometry in this plug-flow regime. We compare our results for angle of the pile in the continuous flow regime to the experiments of Nowak et al. [\textit{Nature Physics}, \textbf{1} (2005)] and we examine the stress and velocity profile within the pile as well.   

Stability of Binary Granular Mixtures

We study stability of a binary granular mixture. The two grain types are spherical ball bearings, and hexagonal shapes created by welding seven of the spheres together. The shapes are confined to a two-dimensional drum, which rotates slowly enough for discrete avalanches to occur. On average homogeneous piles of hexagonal reach a higher angle before an avalanche than homogeneous piles of spheres, by nearly twenty degrees. As the concentration of spheres is increased in a pile of mostly hexagons, the stability angle decreases more than twice as fast as expected by a linear interpolation between the homogeneous values. The spheres also tend to clump in the middle of the drum, and this segregation appears to cause the nonlinearity in angle. This indicates that the central portion of the drum is the most important in triggering avalanches.   

Shape and Velocity Profile of the Core in a Radially Segregated Rotating Cylinder of Granular Particles

We experimentally investigate a 3D biparticulate system that segregates only radially, with no evidence of axial segregation either at or below the surface even after hours of rotation. We compare the location and shape of the core of smaller particles, as well as the location of the bottom of the flowing layer, at various rotation rates using magnetic resonance imaging (MRI) in a 5mm slice at the axial center of a 3D cylinder. MRI is used because of its ability to non-invasively measure bulk behavior as well as spatially resolve dynamic variables (e.g. velocity, diffusion) at any location within a 1-, 2- or 3D system. We also compare the velocity depth profile of the radially segregated system with that of pure small and pure large particle systems and provide an explanation for the observed differences. These investigations may help clarify not only what is occurring within a radially segregating system of particles, but also which mechanisms influence the development of axial segregation.