^{1}and Donald L. Koch

^{1,a)}

### Abstract

Lattice-Boltzmann simulations are employed to determine the mean settling velocity and pair distribution function for spheres settling in a liquid. The Reynolds number based on the terminal velocity ranges from 1 to 20, the solid-to-fluid density ratio is , and the solid volume fraction is varied from 0.005 to 0.40. At volume fractions larger than about 0.05, the ratio of the mean settling velocity to the terminal velocity can be fit by a power-law expression , where and are functions of the Reynolds number based on the terminal velocity. The constant is typically about 0.86–0.92 and deviates from the power-law behavior in dilute suspensions. The extent of this deviation increases with increasing Reynolds number. We show that the hindered settling velocity follows a power law when the particle microstructure is similar to that in a hard-sphere suspension. The deviation from the power-law behavior can be correlated with an anisotropicmicrostructure resulting from wake interactions among the spheres. This microstructure, which occurs in dilute suspensions and is most pronounced at the higher Reynolds numbers explored in our study, consists of a decreased pair distribution function for pairs with vertical separation vectors and a peak in the pair distribution function for horizontal separations corresponding to about two particle diameters.

This work was supported by NASA Grant No. NAG3-1853 and DOE Grant No. DE-FG02-03-ER46073. The computational resources are provided by the Cornell Theory Center. The authors are grateful to R. Verberg and A. J. C. Ladd for many helpful discussions concerning the lattice-Boltzmann method.

I. INTRODUCTION

II. LATTICE-BOLTZMANN METHOD

III. SIMULATION RESULTS

A. Terminal velocities

B. Hindered settling velocities

C. Microstructure

IV. DISCUSSION

### Key Topics

- Suspensions
- 98.0
- Reynolds stress modeling
- 39.0
- Particle distribution functions
- 19.0
- Hydrodynamics
- 14.0
- Sedimentation
- 13.0

## Figures

The drag on a simple cubic array of spheres at moderately small Reynolds numbers. The drag is normalized by the Stokes drag and is compared to Sangani and Acrivos’s prediction (dashed line): .^{39} The squares represent , the upward triangles represent , the downward triangles represent .

The drag on a simple cubic array of spheres at moderately small Reynolds numbers. The drag is normalized by the Stokes drag and is compared to Sangani and Acrivos’s prediction (dashed line): .^{39} The squares represent , the upward triangles represent , the downward triangles represent .

Hindered settling velocities in small Re suspensions. (a) , ; (b) , ; (c) and (d) are the enlarged views of (a) and (b) in the dilute limit. The open symbols were obtained with and the filled symbols were obtained with . The solid lines correspond to Richardson and Zaki’s power-law Eq. (2); the dash-dot lines correspond to Garside and Al-Dibouni’s correlation Eq. (3); the dashed lines are the best power-law fits using and from Table IV. The dotted lines correspond to Batchelor’s asymptote of for dilute low Re suspensions.^{47}

Hindered settling velocities in small Re suspensions. (a) , ; (b) , ; (c) and (d) are the enlarged views of (a) and (b) in the dilute limit. The open symbols were obtained with and the filled symbols were obtained with . The solid lines correspond to Richardson and Zaki’s power-law Eq. (2); the dash-dot lines correspond to Garside and Al-Dibouni’s correlation Eq. (3); the dashed lines are the best power-law fits using and from Table IV. The dotted lines correspond to Batchelor’s asymptote of for dilute low Re suspensions.^{47}

Hindered settling velocities in suspensions with higher Reynolds numbers. (a) and ; (b) and ; (c) and ; (d) an enlarged view of the dilute regime. The meanings of the symbols and lines follow the definitions in Fig. 2.

Hindered settling velocities in suspensions with higher Reynolds numbers. (a) and ; (b) and ; (c) and ; (d) an enlarged view of the dilute regime. The meanings of the symbols and lines follow the definitions in Fig. 2.

Hindered settling velocities as functions of on a logarithmic scale. (a) [ and 40.0 ]; (b) [ ), 319 , and 815 ]. The lines are the best linear fits yielding the values of and presented in Table IV. All data were obtained from periodic unit cells with .

Hindered settling velocities as functions of on a logarithmic scale. (a) [ and 40.0 ]; (b) [ ), 319 , and 815 ]. The lines are the best linear fits yielding the values of and presented in Table IV. All data were obtained from periodic unit cells with .

and in power laws obtained from simulations. (a) the power-law exponent as a function of Re; (b) the prefactor as a function of Re. The solid and dashed lines in (a) correspond to based on Eq. (2) and based on Eq. (3).

and in power laws obtained from simulations. (a) the power-law exponent as a function of Re; (b) the prefactor as a function of Re. The solid and dashed lines in (a) correspond to based on Eq. (2) and based on Eq. (3).

(Color online) The pair probability density distributions and structure factors in suspensions with . (a), (c), and (e) show the pair probability in dilute , intermediate , and concentrated suspensions; (b), (d), and (f) show the corresponding structure factors, with error bars representing 90% confidence intervals. The triangles represent and the squares represent . The solid lines in the plots of structure factors are for hard-sphere suspensions.^{54} The results for dilute suspensions [(a) and (b)] were obtained in systems of size . The rest of the simulations were conducted with .

(Color online) The pair probability density distributions and structure factors in suspensions with . (a), (c), and (e) show the pair probability in dilute , intermediate , and concentrated suspensions; (b), (d), and (f) show the corresponding structure factors, with error bars representing 90% confidence intervals. The triangles represent and the squares represent . The solid lines in the plots of structure factors are for hard-sphere suspensions.^{54} The results for dilute suspensions [(a) and (b)] were obtained in systems of size . The rest of the simulations were conducted with .

(Color online) The pair probability density distributions and structure factors in suspensions with . The volume fractions, from top to bottom, are , 0.05, and 0.20. For the definition of symbols and lines, as well as the information on the system size, see the caption of Fig. 6.

(Color online) The pair probability density distributions and structure factors in suspensions with . The volume fractions, from top to bottom, are , 0.05, and 0.20. For the definition of symbols and lines, as well as the information on the system size, see the caption of Fig. 6.

The radial distributions of settling spheres in suspensions with (a) and ; (b) and . The triangles represent , the diamonds represent , and the squares represent .

The radial distributions of settling spheres in suspensions with (a) and ; (b) and . The triangles represent , the diamonds represent , and the squares represent .

The order parameters in settling suspensions. (a) and ; (b) and . The symbols have the same meanings as in Fig. 8.

The order parameters in settling suspensions. (a) and ; (b) and . The symbols have the same meanings as in Fig. 8.

A qualitative view of the velocity field around a sphere settling with finite Re and the interaction between a pair of spheres. The fluid velocity due to sphere 1 in the absence of a disturbance due to the other sphere is sketched. It consists of a wake behind the sphere and a radial source flow in other directions. A sphere located at 2 would be in the wake of 1 and thus would be attracted toward 1. At the same time, a lift force would act to push sphere 2 horizontally outward. If the lift force is insufficient to avoid a collision, spheres 1 and 2 would experience a torque orienting them in the horizontal direction. Horizontally oriented spheres (such as at location 3) repel due to the source flow.

A qualitative view of the velocity field around a sphere settling with finite Re and the interaction between a pair of spheres. The fluid velocity due to sphere 1 in the absence of a disturbance due to the other sphere is sketched. It consists of a wake behind the sphere and a radial source flow in other directions. A sphere located at 2 would be in the wake of 1 and thus would be attracted toward 1. At the same time, a lift force would act to push sphere 2 horizontally outward. If the lift force is insufficient to avoid a collision, spheres 1 and 2 would experience a torque orienting them in the horizontal direction. Horizontally oriented spheres (such as at location 3) repel due to the source flow.

A sequence of images taken from a simulation showing the interaction between a pair of solid spheres, which are painted black, in a suspension of other particles painted gray. The images are displayed in a reference frame moving with the leading sphere. The simulation parameters are , , , and .

A sequence of images taken from a simulation showing the interaction between a pair of solid spheres, which are painted black, in a suspension of other particles painted gray. The images are displayed in a reference frame moving with the leading sphere. The simulation parameters are , , , and .

## Tables

The effective hydrodynamic sphere diameters used in this work and the corresponding input diameters and viscosities. The values are in terms of lattice units.

The effective hydrodynamic sphere diameters used in this work and the corresponding input diameters and viscosities. The values are in terms of lattice units.

The Archimedes numbers, terminal velocities, and Reynolds numbers based on terminal velocity studied in our simulations. The four middle columns give the terminal velocity and Reynolds number obtained from our simulations for a single sphere in a cubic periodic domain. The Reynolds numbers in the last column are based on an empirical drag law Eq. (11). For and , . For the higher Archimedes numbers, . In all simulations, the sphere diameter is . Each entry in this table is the average of 6–7 runs with the gravity oriented differently relative to the cell axes. The standard deviations in and are less than 2% of the means.

The Archimedes numbers, terminal velocities, and Reynolds numbers based on terminal velocity studied in our simulations. The four middle columns give the terminal velocity and Reynolds number obtained from our simulations for a single sphere in a cubic periodic domain. The Reynolds numbers in the last column are based on an empirical drag law Eq. (11). For and , . For the higher Archimedes numbers, . In all simulations, the sphere diameter is . Each entry in this table is the average of 6–7 runs with the gravity oriented differently relative to the cell axes. The standard deviations in and are less than 2% of the means.

Settling velocities in Stokesian suspensions. and . The simulations were carried out in domains with and . The viscosity of the fluid is .

Settling velocities in Stokesian suspensions. and . The simulations were carried out in domains with and . The viscosity of the fluid is .

Power-law exponent and prefactor that provide the best fits to the settling velocities in concentrated suspensions . The quality of the linear fits is indicated by values. The numbers after signs are the 95% confidence intervals. For comparison, we included the power-law exponents calculated from Eq. (2) and calculated from Eq. (3)

Power-law exponent and prefactor that provide the best fits to the settling velocities in concentrated suspensions . The quality of the linear fits is indicated by values. The numbers after signs are the 95% confidence intervals. For comparison, we included the power-law exponents calculated from Eq. (2) and calculated from Eq. (3)

Article metrics loading...

Full text loading...

Commenting has been disabled for this content