Ashok S. Sangani

Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed : kinetic theory and numerical simulations

A linear stability analysis is performed for the homogeneous state of a monodisperse gas-fluidized bed of spherical particles undergoing hydrodynamic interactions and solid-body collisions at small particle Reynolds number and finite Stokes number. A prerequisite for the stability analysis is the determination of the particle velocity variance which controls the particle-phase pressure. In the absence of an imposed shear, this velocity variance arises solely due to the hydrodynamic interactions among the particles. Since the uniform state of these suspensions is unstable over a wide range of values of particle volume fraction φ and Stokes number St , full dynamic simulations cannot be used in general to characterize the properties of the homogeneous state. Instead, we use an asymptotic analysis for large Stokes numbers together with numerical simulations of the hydrodynamic interactions among particles with specified velocities to determine the hydrodynamic sources and sinks of particle-phase energy. In this limit, the velocity distribution to leading order is Maxwellian and therefore standard kinetic theories for granular/hard-sphere molecular systems can be used to predict the particle-phase pressure and rheology of the bed once the velocity variance of the particles is determined. The analysis is then extended to moderately large Stokes numbers for which the anisotropy of the velocity distribution is considerable by using a kinetic theory which combines the theoretical analysis of Koch (1990) for dilute suspensions (φ [Lt ] 1) with numerical simulation results for non-dilute suspensions at large Stokes numbers. A linear stability analysis of the resulting equations of motion provides the first a priori predictions of the marginal stability limits for the homogeneous state of a gas-fluidized bed. Dynamical simulations following the detailed motions of the particles in small periodic unit cells confirm the theoretical predictions for the particle velocity variance. Simulations using larger unit cells exhibit an inhomogeneous structure consistent with the predicted instability of the homogeneous gas–solid suspension.

Transport processes in random arrays of cylinders. II. Viscous flow

The method described in Part I [Phys. Fluids 31, XXXX (1988)] is extended to treat the problem of determining the permeability of random arrays of infinitely long cylinders. The results for the transverse and longitudinal permeabilities averaged over several configurations of random arrays of cylinders are presented as a function of the area fraction of the cylinders. A detailed comparison is made with the estimates of the permeability obtained by various approximate and asymptotic theories to determine their range of validity.

Dynamic simulations of flows of bubbly liquids at large Reynolds numbers

Results of dynamic simulations of bubbles rising through a liquid are presented. The Reynolds number of the flow based on the radius and the terminal speed of bubbles is large compared to unity, and the Weber number, which is the ratio of inertial to surface tension forces, is small. It is assumed that the bubbles do not coalesce when they approach each other but rather bounce instantaneously, conserving the momentum and the kinetic energy of the system. The flow of the liquid is assumed to be irrotational and is determined by solving the many-bubble interaction problem exactly. The viscous force on the bubbles is estimated from the rate of viscous energy dissipation. It is shown that the random state of bubbly liquids under these conditions is unstable and that the bubbles form aggregates in planes transverse to gravity. These aggregates form even when the size distribution of the bubbles is non-uniform. While the instability results primarily from the nature of inertial interaction among pairs of bubbles, which causes them to be attracted toward each other when they are aligned in the plane perpendicular to gravity, it is shown that the presence of viscous forces facilitates the process.

A Method for Computing Stokes Flow Interactions Among Spherical Objects and its Application to Suspensions of Drops and Porous Particles

A method for computing Stokes flow interactions in suspensions of spherical objects is described in detail and applied to the suspensions of porous particles, drops, and bubbles to determine their hydrodynamic transport coefficients.

Measurements of the Average Properties of a Suspension of Bubbles Rising in a Vertical Channel

Experiments were performed in a vertical channel to study the behaviour of a monodisperse bubble suspension for which the dual limit of large Reynolds number and small Weber number was satisfied. Measurements of the liquid-phase velocity fluctuations were obtained with a hot-wire anemometer. The gas volume fraction, bubble velocity, bubble velocity fluctuations and bubble collision rate were measured using a dual impedance probe. Digital image analysis was performed to quantify the small polydispersity of the bubbles as well as the bubble shape. A rapid decrease in bubble velocity with bubble concentration in very dilute suspensions is attributed to the effects of bubble–wall collisions. The more gradual subsequent hindering of bubble motion is in qualitative agreement with the predictions of Spelt & Sangani (1998) for the effects of potential-flow bubble–bubble interactions on the mean velocity. The ratio of the bubble velocity variance to the square of the mean is O (0.1). For these conditions Spelt & Sangani predict that the homogeneous suspension will be unstable and clustering into horizontal rafts will take place. Evidence for bubble clustering is obtained by analysis of video images. The fluid velocity variance is larger than would be expected for a homogeneous suspension and the fluid velocity frequency spectrum indicates the presence of velocity fluctuations that are slow compared with the time for the passage of an individual bubble. These observations provide further evidence for bubble clustering.

An O(N) algorithm for Stokes and Laplace interactions of particles

A method for computing Laplace and Stokes interactions among N spherical particles arbitrarily placed in a unit cell of a periodic array is described. The method is based on an algorithm by Greengard and Rokhlin [J. Comput. Phys. 73, 325 (1987)] for rapidly summing the Laplace interactions among particles by organizing the particles into a number of different groups of varying sizes. The far‐field induced by each group of particles is expressed by a multipole expansion technique into an equivalent field with its singularities at the center of the group. The resulting computational effort increases only linearly with N. The method is applied to a number of problems in suspension mechanics with the goal of assessing the efficiency and the potential usefulness of the method in studying dynamics of large systems. It is shown that reasonably accurate results for the interaction forces are obtained in most cases even with relatively low‐order multipole expansions.

Simple shear flows of dense gas-solid suspensions at finite Stokes numbers

We examine the problem of determining the particle-phase velocity variance and rhe-ology of sheared gas-solid suspensions at small Reynolds numbers and finite Stokes numbers. Our numerical simulations take into account the Stokes flow interactions among particles except for pairs of particles with a minimum gap width comparable to or smaller than the mean free path of the gas molecules for which the usual lubrication approximation breaks down and particle collisions occur in a finite time. The simulation results are compared to the predictions of two theories. The first is an asymptotic theory for large Stokes number St and nearly elastic collisions, i.e. St [Gt ] 1 and 0 ≤ 1 - e [Lt ] 1, e being the coefficient of restitution. In this limit, the particle velocity distribution is close to an isotropic Maxwellian and the velocity variance is determined by equating the energy input in shearing the suspension to the energy dissipation by inelastic collisions and viscous effects. The latter are estimated by solving the Stokes equations of motion in suspensions with the hard-sphere equilibrium spatial and velocity distribution while the shear energy input and energy dissipation by inelastic effects are estimated using the standard granular flow theory (i.e. St = ∞). The second is an approximate theory based on Grads moments method for which St and 1 – e are O (1). The two theories agree well with each other at higher values of volume fraction ϕ of particles over a surprisingly large range of values of St. For smaller ϕ however, the two theories deviate significantly except at sufficiently large St. A detailed comparison shows that the predictions of the approximate theory based on Grads method are in excellent agreement with the results of numerical simulations.

Inclusion of lubrication forces in dynamic simulations

A new method is described for incorporating close‐field, lubrication forces between pairs of particles into the multiparticle Stokes flow calculations. The method is applied to the suspensions of both spherical as well as cylindrical particles, and results computed by the method are shown to be in excellent agreement with the exact known results available in the literature.

Elastic coefficients of composites containing spherical inclusions in a periodic array

Abstract T he effective elasticity tensor of a composite medium consisting of spherical elastic particles firmly embedded in a periodic cubic arrangement into another elastic medium is characterized by three scalars α, β and γ. We employ a method of singularity distribution and determine these scalars for various values of volume fraction of the spheres and the Lameconstants of the particles and the matrix. The results, which are presented for the simple, body-centered and face-centered cubic arrays, are in agreement with the previously known results of Nunan and Keller (J. Mech. Phys. Solids32, 259, 1984) for the rigid particles and of Iwakuma and Nemat - Nasser (Computers and Structures16, 13, 1983) for the non-rigid particles.

The Added Mass, Basset, and Viscous Drag Coefficients in Nondilute Bubbly Liquids Undergoing Small-Amplitude Oscillatory Motion

The motion of bubbles dispersed in a liquid when a small‐amplitude oscillatory motion is imposed on the mixture is examined in the limit of small frequency and viscosity. Under these conditions, for bubbles with a stress‐free surface, the motion can be described in terms of added mass and viscous force coefficients. For bubbles contaminated with surface‐active impurities, the introduction of a further coefficient to parametrize the Basset force is necessary. These coefficients are calculated numerically for random configurations of bubbles by solving the appropriate multibubble interaction problem exactly using a method of multipole expansion. Results obtained by averaging over several configurations are presented. Comparison of the results with those for periodic arrays of bubbles shows that these coefficients are, in general, relatively insensitive to the detailed spatial arrangement of the bubbles. On the basis of this observation, it is possible to estimate them via simple formulas derived analytically for dilute periodic arrays. The effect of surface tension and density of bubbles (or rigid particles in the case where the no‐slip boundary condition is applicable) is also examined and found to be rather small.