Session L24: Granular Flows: Fluctuations and Instabilities

Stability analysis of Couette flows of dry granular materials

In this talk, we investigate the stability of a unidirectional Couette flow of a dry granular material, as predicted by a continuum flow model, via a linear stability analysis. A classical normal-mode analysis is employed which results in a fourth-order polynomial eigenvalue equation for the modes of disturbance. The eigenvalue problem is solved numerically via a Chebyshev polynomial method and extensive parametric studies are performed. The results of this study suggest that the flow of interest is linearly unstable for all values of Froude, Reynolds and Galilei numbers of practical interest. Additionally, we discuss the relation between the magnitude of the predicted growth rates and the observability of this instability, as well as the connection between the shape of the predicted eigenfunctions and the formation of particle-clusters. Finally, we compare the results of the present study with those of earlier studies based on different flow models.   

Collapse of a brittle granular column: implications for rock fragmentation in a landslide

We investigate numerically the failure, collapse and flow of a brittle granular column over a horizontal surface. In our discrete element simulations, spherical particles are initially held together by tensile bonds, which can be irreversibly broken during the collapse. This leads to dynamic fragmentation within the material during the flow. Compared to what happens in the case of a non-cohesive granular column, the deposit is much rougher, and the stratigraphy of the column is not preserved during the collapse. As has been observed in natural rockslides, we find that the deposit consists of large blocks laying on a basal layer of fine fragments. The influence of the aspect ratio of the column on the run-out distance is roughly the same as in the granular case. Finally, we show that for a given aspect ratio of the column, the run-out distance is higher when the deposit is highly fragmented, which confirms previous hypotheses by Davies et al. (1999).   

From episodic avalanching to continuous flow in a granular drum

Experiments are performed with a rotating cylindrical drum half full of granular material in order to study the transition from episodic avalanching to continuous flow (slumping to rolling). To examine the effect of drum and particle geometry, drums with different radii and widths are used, and different granular materials, ranging from glass spheres with different radii to irregularly shaped sand. For the drums and materials used, it is found that the transition mostly takes the form of a blend of the characteristics of episodic avalanching and continuous flow, that gradually switches from slumping to rolling as rotation rate increases. Only for the sand in the narrower drums is there a hysteretic transition in which one can observe prolonged episodic avalanching or continuous flow at the same rotation rate, over a window of rotation speeds. The transition takes the form of intermittent switching driven by noisy fluctuations (a ``bifurcation by intermittency'') for sand in the widest drums and for the smallest ballotini (1mm diameter).   

Numerical study of axisymmetric collapses of submarine granular $>$ columns

In this talk, we report on the results of a numerical study of the axisymmetric collapse of subaqueous granular columns. Our study is based on a 2-pressure, 2-velocity continuum flow model for fluid-saturated granular materials. This model is integrated via a multi-phase projection method that incorporates a regularization method for the treatment of material interfaces. In our simulations, a dense column of a granular material immersed in water is placed on a horizontal plane and is allowed to collapse and spread due to its weight. Emphasis is placed on the run-out distance and the termination height and their correlation with the aspect ratio, the volume fraction and the diameter of the grains. Comparisons against experimental measurements and previous numerical predictions are also performed. Finally, in order to examine and quantify the role of the interstitial fluid, we compare our numerical predictions against experimental results from column collapses of dry granular materials.   

Computer simulations of the axisymmetric collapse of a granular column made of mixed grains

Measures of the final height and final run-out distance of deposits formed by collapses of granular columns formed by mixed grains, seem to follow a power law on the initial aspect ratio parameter, similar to the homogeneous case. We investigate this matter using numerical simulations in 3D, and present preliminary results of the collapse of a granular column formed by mixtures of grains with different shapes, considering the cases of spherical grains mixed with sticks and mixtures of sticks, keeping all other parameters fixed.   

Low-frequency oscillations in vibrated granular media

We present simulations and a theoretical treatment of vertically vibrated granular media. The systems considered are confined in narrow quasi-two-dimensional and quasi-one-dimensional (column) geometries, where the vertical extension of the container is much larger than both horizontal lengths. The additional geometric constraint present in the column setup frustrates the convection state that is normally observed in wider geometries. This makes it possible to study collective oscillations of the grains with a characteristic frequency that is much lower than the frequency of energy injection. We observe that, in the quasi-two-dimensional setup, low-frequency oscillations are present even in the convective regime. This suggests that they may play a significant role in the transition from a density inverted state to convection. Our hydrodynamic model shows that a sufficient condition for the existence of the low-frequency mode is an inverted density profile with distinct low and high density regions, a condition that may apply to other systems. Lastly, we also present experimental results that confirm the presence of the oscillations in a vast region of phase-space. Theory, experiments and simulations are seen to be in high agreement, specially for high energy inputs.   

Hopping dynamics of granular kinks

We report on the experimental observation and theoretical characterization of the bifurcation diagram, dynamical properties and fluctuations of spatially modulated kinks in a shallow one-dimensional fluidized granular layer subjected to a periodic air flow. We show the appearance of these solutions as the layer undergoes a parametric instability. Due to the inherent fluctuations of the granular layer, the kink profile exhibits an effective wavelength, termed {\it precursor}, which modulates spatially the homogeneous states and drastically modifies the kink dynamics. We characterize the average and fluctuating properties of this solution. The long term evolution of these kinks is dominated by a hopping dynamics, related directly to the underlying spatial structure and inherent fluctuation. The properties of this motion can be described by a brownian particle in a symmetric periodic potential. Both the noise intensity of the brownian fluctuations and the amplitude and periodicity of the potential arise from the underlying precursor structure.   

Connections between the Boson peak and the Van Hove Singularity-- insights from the normal modes analysis of granular experiments

We have experimentally measured the density of states (DOS) from the hexagonal lattice to the disordered structures in 2D packing of granular materials, which are made of photo-elastic disks allowing a precise measurement of contact forces between disks to determine the dynamical matrix of the system. Two different analyses have been performed with and without the inclusion of the rotational degree of freedom. By varying the pressure of the disordered crystal, we find the strong evidence that the first Van Hove singularity gradually evolves into the Boson peak. In geometrically disordered packing, the position of the Boson peak is influenced by the degree of the geometric disorder. Incorporating the rotational degree of freedom, two peaks would appear at the vicinity of the original first Van Hove singularity in the hexagonal lattice and similarly at the vicinity of the original Boson peak in a disordered crystal; in a contrast, the two peaks are nearly merged in a geometrically disordered system. Moreover, further analysis shows that the first peak is only related to the rotational degree of freedom, whereas the second peak is due to the coupling between the rotational and translational degrees of freedom.   

Transient behavior of granular materials with symmetric conditions for tumbler shapes and fill fractions

The typical granular motion in circular tumblers is considered steady-state since there are no features to disrupt the top surface layer dimension. In polygon tumblers, however, the flowing layer is perpetually changing length, which creates unsteady conditions with corresponding change in the flow behavior. Prior work showed the minimization of free surface energy is independent of tumbler dimension, particle size, and rotation rate. This subsequent research reports on experiments where dimensional symmetry of the free surface in triangular and square tumblers with varying fill fractions do not necessarily produce the symmetric flow behaviors. Results of the quasi-2D tumbler experiment show that other dimensions aligned with gravity and the instantaneous free surface influence the phase when extrema for angle of repose and other flow features occur. The conclusion is that 50\% fill fraction may produce geometric symmetry of dimensions, but the symmetry point of flow likely occurs at a lower fill fraction.   

Simulations of secondary currents in rapid granular chute flow

The desire to understand granular flow as a fluid mechanical phenomena has long been the focus of theoreticians and experimentalists alike. Several analogies can be drawn with complex hydrodynamic behaviors at the continuum scale including Leidenfrost states, Rayleigh-Benard (R-B) convection, and Rayleigh-Taylor instability that allow for deeper understanding of collective granular motions. Here, we consider the case of longitudinal vortices in rapid granular flow down an inclined chute, and draw an analogy to hydraulic open channel flow. Previous experiments of rapid granular flow down inclined chutes have uncovered a unique regime where the flow exhibits stripes of slower and faster moving grains. We present results of molecular dynamics simulations that allowed us to study the full scale of the phenomena including smooth sidewalls and the rough bumpy bottom. We found that the secondary currents were intensified near the lateral walls.