Session HM: Granular Matter

Density Measurements In a Vibro-Fluidized Deep Granular Bed

The local state of a granular bed is explored through capacitance measurements of granular density fluctuations. This talk will attempt to characterize the ``melting'' of granular beds through vibrofluidization. The granular bed is subjected to vibration in the frequency range of 10 Hz to 200 Hz with acceleration in the range 0 to 2.0g. The bed aspect ratio (effective bed height to bed diameter) is varied up to 3.6 with effective bed height range from 0 cm to 20 cm and bed diameter of 5.6 cm. We identify the transitions between three granular bed states, the static granular state, the quasi-static state and vibro-fluidized state. Lack of density fluctuations characterize the static granular state. Large density fluctuations during the vibration cycle are characteristic of the vibro-fluidized state. The entire height of the bed can reside in a static or vibro-fluidized state at low and high vibration levels, respectively. A quasi-static bed state occurs at intermediate vibration levels, characterized by vibro-fluidized state density fluctuations in the upper portion of the bed and static state lack of density fluctuations in the lower portion of the bed. The extent of density fluctuation penetration is a function of distance from the top of the bed, rather than total bed height. Density fluctuations during the vibration cycle scale with frequency at lower frequencies and are not present at higher frequency vibrations.   

Hydrodynamic Fluctuations Near the Onset of Patterns in Oscillated Granular Layers

We study the effects of noisy fluctuations near the onset of patterns in simulations of vertically oscillated granular layers. Above a critical acceleration of the cell, standing waves form stripe patterns. We study the onset of these patterns using continuum simulations of frictionless dissipative particles to Navier-Stokes order. The patterns formed in the continuum simulations reproduce wavelengths found in previous MD simulations and experiments for a range of oscillation frequencies. However, the critical acceleration for ordered standing waves in previous MD simulations is approximately 10\% higher than that found in our continuum simulations. These results are consistent with the presence of noise in the system. Adding Landau-Lifshitz hydrodynamic fluctuations to the continuum simulations raises the critical acceleration to a value consistent with the critical acceleration found in MD simulations. We compare the amplitude of patterns in continuum simulations with noisy fluctuations to patterns formed in the absence of such noise, and examine the effects of fluctuations on hysteresis exhibited near onset.   

Forcing and phase transitions in a thin granular layer

Recent experimental and computational studies of vibrated thin layers of identical spheres have shown transitions to ordered phases similar to those seen in equilibrium systems. Motivated by these results, we carry out simulations of hard inelastic spheres forced by homogenous white noise. We find a transition to an ordered state of the same symmetry as that seen in the experiments, but the clear phase separation observed in the experiments is absent. Simulations of purely elastic spheres also show no evidence for phase separation, suggesting that differential forcing from the vibrating plate is creating an effective surface tension. We do find, however, that inelasticity suppresses the onset of the ordered phase, as is observed in the vibrating system.   

Wave Driven Fluid-Sediment Interactions over Rippled Beds

Empirical investigations relating vortex shedding over rippled beds to oscillatory flows date back to Darwin in 1883. Observations of the shedding induced by oscillating forcing over fixed beds have shown vortical structures to reach maximum strength at 90 degrees when the horizontal velocity is largest. The objective of this effort is to examine the vortex generation and ejection over movable rippled beds in a full-scale, free surface wave environment. Observations of the two-dimensional time-varying velocity field over a movable sediment bed were obtained with a submersible Particle Image Velocimetry (PIV) system in two wave flumes. One wave flume was full scale and had a natural sand bed and the other flume had an artificial sediment bed with a specific gravity of 1.6. Full scale observations over an irregularly rippled bed show that the vortices generated during offshore directed flow over the steeper bed form slope were regularly ejected into the water column and were consistent with conceptual models of the oscillatory flow over a backward facing step. The results also show that vortices remain coherent during ejection when the background flow stalls (i.e. both the velocity and acceleration temporarily approach zero). These results offer new insight into fluid sediment interaction over rippled beds.   

Unsteady Granular Flows

The characteristics of steady granular flow in quasi-two-dimensional rotating tumblers have been thoroughly investigated and are fairly well understood. However, unsteady, time-varying flow has not been investigated in detail. Velocity measurements of granular flow in quasi-2D rotating tumblers are presented for periodic forcing protocols via sinusoidal variation in the rotational speed of the tumbler. Variations in the system level parameters of tumbler radius, particle size, and forcing frequency are explored. Similarities to steady flow include the fastest flow at the free surface of the flowing layer and an instantaneous linear velocity profile through the depth. The flowing layer depth varies between 9 and 13 particle diameters for minimum and maximum rotation rates. Unsteady periodic forcing also causes the flow to exhibit dynamic properties. The phase lag of the flow response increases linearly with increasing input forcing frequency to more than $0.6\pi$~radians over 0--20~cycles/revolution. The amplitude responses of the velocity and shear rate show a resonance unique to the system level parameters. The results indicate that unsteady granular flow analysis may be beneficial for characterizing the flowability or rheology of granular materials.   

Hydrodynamic Correlation Functions of a Driven Granular Fluid in Steady State

We study a homogeneously driven granular fluid of hard spheres at intermediate volume fractions and focus on time-delayed correlation functions in the stationary state. The results of computer simulations using an event driven algorithm are compared to the predictions of generalized fluctuating hydrodynamics. The incoherent scattering function ($F_{\rm incoh}(q,t)$) follows time-superposition and is well approximated by a Gaussian $F_{\rm incoh}=\exp \left ( - \frac{q^2}{6} \langle \Delta r^2(t) \rangle \right )$. For sufficiently small wavenumber $q$ we observe sound waves in the coherent scattering function $S(q,\omega)$ and determine their dispersion and damping. Temperature fluctuations are predicted to be either diffusive or nonhydrodynamic, depending on wavenumber and inelasticity as characterized by incomplete normal restitution.   

Study of the melting of the square ordered phase in a thin vibrated granular layer

We present an experimental and computational study of a vibrating thin, dense granular layer of identical metal spheres. In a recent study, experiments and molecular dynamics simulations have shown that for high enough vibration amplitude there is a transition from an ordered phase of square symmetry to a disordered liquid-like phase, and that this transition occurs earlier in more inelastic materials. We investigate now the mechanisms causing this melting, as a function of input acceleration, inelasticity, and geometry of the system. In order to look for similarities/differences with the equilibrium-analogues of this transition, we investigate the mean square displacement of the particles in the ordered phase. Preliminary results indicate that this magnitude increases close to crystal melting, as in equilibrium melting. We also analyze the role of the system size in the melting transition, showing that for small enough system the square phase is absent.   

Solid-liquid-like transition in vibrated granular monolayers

The theory of non-ideal gases in thermodynamic equilibrium, for instance the van der Waals gas model, has played a central role in the understanding of coexisting phases. Here, we report a combined experimental, numerical and theoretical study of a liquid-solid-like phase transition which takes place in a vertically vibrated fluidized granular monolayer. The first experimental setup is a long, narrow channel, with a width of the order of a few particle diameters, hence the dynamics is quasi-one-dimensional. We have considered this configuration to characterize the dynamic behavior of the phase transition. The second setup is used to measure the pressure as function of particle density in order to clarify the physical mechanism behind this phase transition. We demonstrate that the transition is mediated by waves and that it is triggered by a negative compressibility as in van der Waals phase coexistence, although the system does not satisfy the hypotheses used to understand atomic systems. Finally, in order to further characterize this phase transition, we study static and dynamic correlation functions, and bond-orientational order parameters.   

Buoyancy driven convection in vertically shaken granular matter

Buoyancy driven granular convection is studied for a shallow, vertically shaken granular bed in a quasi-two-dimensional container. At sufficiently strong shaking counter-rotating convection rolls form with pronounced density variations. These rolls are also found in molecular dynamics simulations. The onset of convection is analyzed through a linear stability analysis of the hydrodynamic continuum model presented in Phys. Rev. Lett. 95, 258001 (2005). There is very good agreement between experiments, simulations, and theory.   

Dynamics of Barchan dunes in a turbulent boundary layer

When a fluid flow transports a small amount of solid heavy particles on a non-erodible ground, particles form isolated dunes which slowly propagate downstream. Such dunes have been studied experimentally in a channel. Strikingly, particle heaps always form dunes with crescentic shape, similar to that of Barchan dunes in deserts at a much larger scale. Varying the fluid flow and particle properties, it was found that the dune velocity scales as $V \sim 1/L$ where $L$ is the dune length, as expected, but does not follow Bagnold's prediction $V \sim u_*^3$ where $u_*$ is the friction velocity; some dependence on the particle Reynolds number, and perhaps relaxation effects in the particle flux on the dune surface, have to be considered. PIV measurements show that the fluid velocity does not increase on the lee side of the dune, as predicted by Hunt and co-workers, but slightly decreases because of the sudden increase of roughness. The roughness change also appears to be of particular importance for understanding the variation of the turbulent stresses $-\rho \overline{u'v'}$ along the dune.