Channel flows of granular materials and their rheological implications

While the flow of a dry granular material down an inclined channel may seem at first sight to be a relatively simple flow, the experiments which have been conducted up to now suggest sufficient complexity which may be present in all but the very simplest granular material flows; consequently it is important to our general understanding of granular material rheology that these experimental observations be fully understood. This review of the current knowledge of channel flows will focus on the basic mechanics of these flows and the contributions the observations have made to an understanding of the rheology. In order to make progress in this objective, it is necessary to avoid some of the complications which can occur in practice. Thus we shall focus only on those flows in which the interstitial fluid plays very little role in determining the rheology. In his classic paper, Bagnold (1954) was able to show that the regime in which the rheology was dominated by particle/particle or particle/wall interactions and in which the viscous stresses in the interstitial fluid played a negligible role could be defined by a single, Reynolds-number-like parameter. It transpires that the important component in this parameter is a number which we shall call the Bagnold number, Ba, defined by Ba = p₈d²δ/µF where p₈,µF are the particle density and interstitial fluid viscosity, d is the particle diameter and δ is the principal velocity gradient in the flow. In the shear flows explored by Bagnold δ is the shear rate. Bagnold (1954) found that when Ba was greater than about 450 the rheology was dominated by particle/particle and particle/wall collisions. On the other hand, for Ba < 40, the viscosity of the interstitial fluid played the dominant role. More recently Zeininger and Brennen (1985) showed that the same criteria were applicable to the extensional flows in hoppers provided the extensional velocity gradient was used for δ. This review will focus on the simpler flows at large Ba where the interstitial fluid effects are small. Other important ancillary effects can be caused by electrical charge separation between the particles or between the particles and the boundary walls. Such effects can be essential in some flows such as those in electrostatic copying machines. Most experimenters have observed electrical effects in granular material flows, particularly when metal components of the structure are not properly grounded. The effect of such electrical forces on the rheology of the flow is a largely unexplored area of research. The lack of discussion of these effects in this review should not be interpreted as a dismissal of their importance. Apart from electrical and interstitial fluid effects, this review will also neglect the effects caused by non-uniformities in the size and shape of the particles. Thus, for the most part, we focus on flows of particles of spherical shape and uniform size. It is clear that while an understanding of all of these effects will be necessary in the long term, there remain some important issues which need to be resolved for even the simplest granular material flows

Experiments on Chute Flows of Granular Materials

Experiments on continuous, steady flows of granular materials down an inclined channel or chute were made with the object of acquiring information on the rheological properties of the granular material flow and the nature of the boundary condition on the base of the channel. Specifically measurements were made of the mean material velocities and velocity profiles on all boundaries of the flow using cross-correlation of two neighboring fibre-optic displacement probes. The output from these probes was used to obtain (1) the unsteady or random component of the particle velocity in the longitudinal direction and (2) a measure of the volume fraction of the flow in contact with the base by counting the frequency of passage of the particles. Measurement was also made of the depth of the flow, the mass flow rate and the shear stress on the base. The latter employed a strain-gauged shear force plate built into the base. The experiments are currently in progress and so further data will be presented at a later date. Nevertheless the preliminary data have yielded a number of interesting features

Some Observations of Flow Patterns and Statistical Properties of Three Component Flows

Vertical air-water flows, solids-water flows and three component air-solids-water flows were investigated in a Three Component Flow Facility. Visual observations of the flow patterns show that three component flows undergo transition and can exhibit strong unsteady vortical motions. Measurements of the fluctuations in cross-sectionally averaged volume fraction measurements were made. The statistical properties of the fluctuations are presented in terms of their amplitude and coherent time scale in the form of the Signal To Noise Ratio (STNR) and the Time Constant (symbol), respectively. Remarkably, the solids-water flows and the dispersed bubbly air-water flows exhibit almost identical values of STNR for the same volume fraction. Equally remarkable in the linear relationship between the Time Constant and the mean bubble or particle velocity; this relationship is found to have the same constant of proportionality for both species in the well behaved disperse regime. In the two-component churn-turbulent and the three-component agitated vortical regimes, the variables (symbol) and STNR significantly deviate from their dispersed regime values. The onset of large coherent structures characteristic of these regimes is reflected by a rise in the amplitude of the fluctuations and a marked increase in their coherent time scale. The results of this study demonstrate the large information content in the fluctuations of the measured quantity, both as a flow regime indicator and as a measure of flow quantities in two- and three-component flows

Measurement of Friction Pressure Drops in Vertical Slurry and Bubbly Flows

A Three Component Flow Facility (TCFF) was used to study friction pressure drops in vertical two component flows of both air bubbles in water and polyester particle-water mixtures. Friction factors of up to two orders in magnitude higher than those at zero volume fraction were observed for both bubbly and slurry flows. This deviation is shown to decrease with increased liquid Reynolds number. Bubbly and slurry flow friction factors were comparably large in magnitude and displayed the same decreasing trend as a function of Reynolds number. The two phase friction multiplier for bubbly flow was shown to attain values up to one order of magnitude higher than the prediction given by Lockhart and Martinelli. Two phase multiplier data is presented for the dispersed flow regime

Some Observations of Flow Patterns and Statistical Properties of Three Component Flows

Air-water flows, solids-water flows and three component air-solids-water flows in a vertical pipe have been investigated in a Three Component Flow Facility. Visual observations of the patterns show that the three component flow exhibits strong unsteady vertical motions which do not occur in the two phase flows studied. Quantitative results of the fluctuating component of the cross-sectionally averaged volume fraction measurements are presented, and related to the nature of the flows. The ratio of the steady component to the r.m.s of the fluctuating component of the volume fraction measurement (Signal To Noise Ratio) is found to be a good flow structure indicator. Remarkably, the solids-water flows and the bubbly air-water flows exhibit almost identical signal to noise ratios for the same volume fraction. However, the corresponding values for the three component flows reflect greater fluctuations corresponding to the vertical structures

Shear Flows of Rapidly Flowing Granular Materials

Shear flows of granular materials are studied in an open channel. The wall shear is calculated from an open channel momentum equation which includes the density variations in the flow. An experimental technique was developed that allowed the measurement of the average density of the flow at different longitudinal locations in the channel. Two sizes of glass beads are examined and results show the variations in the wall shear as a function of various dimensionless parameters

Dynamics of cavitating cascades

Brief accounts of the theoretical research conducted on the unsteady cavitation characteristics of liquid rocket engine turbopumps are reported. The objective is to produce estimates of the cavitation compliance and other unsteady characteristics which could then be used in analysis of the pogo instability. Blade cavitation is the particular pheonomenon which is investigated and line arized free streamline methods were employed in both quasistatic and complete dynamic cascade analyses of the unsteady flow. The simpler quasistatic analysis was applied to particular turbopumps but yielded values of compliances significantly smaller than those indirectly obtained from experiments. Reasons for this discrepancy are discussed. The complete dynamic analysis presents a new problem in fundamental hydrodynamics and, though the basic solution is presented, numerical results have not as yet been obtained

Cavitation Inception in Spool Valves

Cavitation has been investigated in directional control valves in order to identify damage mechanisms characteristic of components of aircraft hydraulic systems. Tests have been conducted in a representative metal spool valve and in a model three times larger. Data taken under noncavitating conditions with both valves showed that the position of the high-velocity annular jet shifts orientation, depending upon valve opening and Reynolds number. By means of high-frequency response pressure transducers strategically placed in the valve chamber cavitation could be sensed by the correlation of noise with a cavitation index. The onset of cavitation can be detected by comparing energy spectra for a fixed valve opening and a constant discharge. Another sensitive indicator of cavitation inception is the ratio of cavitating to noncavitating spectral densities. The incipient cavitation number as defined in this investigation is correlated with the Reynolds number for both valves

Cavitation Inception in Spool Valves

Cavitation has been investigated in directional control valves in order to identify damage mechanisms characteristic of components of aircraft hydraulic systems. Tests have been conducted in a representative metal spool valve and in a model three times larger. Data taken under non-cavitating conditions with both valves showed that the position of the high-velocity annular jet shifts orientation depending upon valve opening and Reynolds number. By means of high-frequency response pressure transducers strategically placed in the valve chamber cavitation could be sensed by the correlation of noise with a cavitation index. The onset of cavitation can be detected by comparing energy spectra for a fixed valve opening and a constant discharge. Another sensitive indicator of cavitation inception is the ratio of cavitating to non-cavitating spectral densities. The incipient cavitation number as defined in this investigation is correlated with the Reynolds number for both valves

Quantum Logic for Trapped Atoms via Molecular Hyperfine Interactions

We study the deterministic entanglement of a pair of neutral atoms trapped in an optical lattice by coupling to excited-state molecular hyperfine potentials. Information can be encoded in the ground-state hyperfine levels and processed by bringing atoms together pair-wise to perform quantum logical operations through induced electric dipole-dipole interactions. The possibility of executing both diagonal and exchange type entangling gates is demonstrated for two three-level atoms and a figure of merit is derived for the fidelity of entanglement. The fidelity for executing a CPHASE gate is calculated for two 87Rb atoms, including hyperfine structure and finite atomic localization. The main source of decoherence is spontaneous emission, which can be minimized for interaction times fast compared to the scattering rate and for sufficiently separated atomic wavepackets. Additionally, coherent couplings to states outside the logical basis can be constrained by the state dependent trapping potential.Comment: Submitted to Physical Review