^{1}and Donald L. Koch

^{1}

### Abstract

The velocity variance and the hydrodynamic diffusivity for a finite-Reynolds-number settling suspension are determined from lattice-Boltzmann simulations of many particles in cubic cells with periodic boundary conditions. The velocity variance is found to grow logarithmically with the size of the computational domain in contrast to the algebraic growth found in comparable Stokes-flow simulations. The growth rate and size of the velocity variance are found to be smaller than the theoretical prediction for a random suspension owing to a deficit in particle pair probability distribution in the wake of a test particle that screens the velocity disturbance felt by other particles. The particle velocity variance is smaller than the fluid velocity variance because a particle does not follow fluid motions on length scales comparable to or smaller than its own size. The hydrodynamic diffusivity of particles is proportional to the product of the root-mean-square velocity and the size of the computational domain.

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 thank Eric Climent for providing his simulation data and Chao Yang for aid with the comparison of the numerical solutions^{28} with the analytical Oseen-wake solutions.

I. INTRODUCTION

II. VELOCITY VARIANCE

III. HYDRODYNAMIC DIFFUSION

IV. SUMMARY AND DISCUSSION

### Key Topics

- Suspensions
- 52.0
- Particle velocity
- 24.0
- Hydrodynamics
- 12.0
- Reynolds stress modeling
- 11.0
- Particle fluctuations
- 7.0

## Figures

Vertical (∥) and horizontal (⊥) particle and fluid velocity variance in (squares) and (diamonds) suspensions. The volume fractions are 0.01 in (a) and (b) and 0.20 in (c) and (d). The open and filled symbols represent the velocity variances of the fluid and that of the particles, respectively. The error bars are based on the standard deviations among runs with different initial configurations. The dashed lines are the best logarithmic fits in the form of Eq. (2). The horizontal axes of were plotted using a natural log scale.

Vertical (∥) and horizontal (⊥) particle and fluid velocity variance in (squares) and (diamonds) suspensions. The volume fractions are 0.01 in (a) and (b) and 0.20 in (c) and (d). The open and filled symbols represent the velocity variances of the fluid and that of the particles, respectively. The error bars are based on the standard deviations among runs with different initial configurations. The dashed lines are the best logarithmic fits in the form of Eq. (2). The horizontal axes of were plotted using a natural log scale.

Contours of the magnitude of the vertical fluid velocity disturbance of a single particle (left) and the pair probability distribution in dilute suspensions with (right). (a) ; (b) .

Contours of the magnitude of the vertical fluid velocity disturbance of a single particle (left) and the pair probability distribution in dilute suspensions with (right). (a) ; (b) .

The deviation of the number of particles in a sphere of radius surrounding a test particle from the bulk [Eq. (4)] is plotted as a function of for and (a) ; (b) .

The deviation of the number of particles in a sphere of radius surrounding a test particle from the bulk [Eq. (4)] is plotted as a function of for and (a) ; (b) .

Initial development of average settling velocity and vertical particle and fluid velocity variance in a suspension with , , and . (a) Average settling velocity as a function of during ; (b) velocity variance of the particles (solid line) and of the fluid (dashed line) in the same time period. Both (a) and (b) are ensemble averaged from ten independent runs.

Initial development of average settling velocity and vertical particle and fluid velocity variance in a suspension with , , and . (a) Average settling velocity as a function of during ; (b) velocity variance of the particles (solid line) and of the fluid (dashed line) in the same time period. Both (a) and (b) are ensemble averaged from ten independent runs.

Plot of vertical velocity variance in a random dilute suspension with and showing the effects of random and nonrandom particle microstructures. The × symbols are calculated from superpositions of randomly distributed Oseen velocity disturbances, with the error bars representing the standard deviations among four runs with different irrational placements of and relative to reciprocal lattice space of the cubic periodic cell. The open triangles and squares correspond to the initial maxima of the fluid and particle velocity variances, respectively, found from the lattice-Boltzmann simulations. These initial maxima can be considered as rough estimates for the variance in a random suspension as they are reached before the nonrandom microstructure becomes fully developed. Finally, the fluid and particle velocity variance obtained after the sedimentation has reached the equilibrium state shown as filled triangles and squares, respectively, for comparison. The dashed lines are the best fits.

Plot of vertical velocity variance in a random dilute suspension with and showing the effects of random and nonrandom particle microstructures. The × symbols are calculated from superpositions of randomly distributed Oseen velocity disturbances, with the error bars representing the standard deviations among four runs with different irrational placements of and relative to reciprocal lattice space of the cubic periodic cell. The open triangles and squares correspond to the initial maxima of the fluid and particle velocity variances, respectively, found from the lattice-Boltzmann simulations. These initial maxima can be considered as rough estimates for the variance in a random suspension as they are reached before the nonrandom microstructure becomes fully developed. Finally, the fluid and particle velocity variance obtained after the sedimentation has reached the equilibrium state shown as filled triangles and squares, respectively, for comparison. The dashed lines are the best fits.

Dependence of particle velocity variance on the Reynolds number. In these simulations we fixed and and varied Re between 0.2 and 20. (a) Vertical fluid (upward triangles) and particle (squares) velocity variance. The circle corresponds to the theoretical estimate (Ref. 15) for the fluid velocity variance in a dilute Stokes suspension of point particles. (b) Horizontal fluid (downward triangles) and particle (diamonds) velocity variance. In (c), the vertical fluid and particle velocity variance are normalized by the prefactor in Eq. (1) and are plotted as a function of . The × symbols are calculated from superpositions of Oseen velocity disturbances. The horizontal line shows the previous theoretical prediction for Stokesian suspensions (Ref. 15), and the dashed lines have the slope of predicted for high Reynolds numbers, cf. Eq. (1).

Dependence of particle velocity variance on the Reynolds number. In these simulations we fixed and and varied Re between 0.2 and 20. (a) Vertical fluid (upward triangles) and particle (squares) velocity variance. The circle corresponds to the theoretical estimate (Ref. 15) for the fluid velocity variance in a dilute Stokes suspension of point particles. (b) Horizontal fluid (downward triangles) and particle (diamonds) velocity variance. In (c), the vertical fluid and particle velocity variance are normalized by the prefactor in Eq. (1) and are plotted as a function of . The × symbols are calculated from superpositions of Oseen velocity disturbances. The horizontal line shows the previous theoretical prediction for Stokesian suspensions (Ref. 15), and the dashed lines have the slope of predicted for high Reynolds numbers, cf. Eq. (1).

Normalized vertical (∥) and horizontal (⊥) particle diffusivities in (squares) and (diamonds) suspensions. The volume fractions are 0.01 in (a) and (b) and 0.20 in (c) and (d). The filled symbols and open symbols are obtained using the mean-square displacement and velocity correlation function of the particles, respectively. The error bars are the standard deviations.

Normalized vertical (∥) and horizontal (⊥) particle diffusivities in (squares) and (diamonds) suspensions. The volume fractions are 0.01 in (a) and (b) and 0.20 in (c) and (d). The filled symbols and open symbols are obtained using the mean-square displacement and velocity correlation function of the particles, respectively. The error bars are the standard deviations.

The particle diffusivities normalized by the product of the root-mean-square velocity and the system size . (a) ; (b) . The error bars are the standard deviations. The solid lines are a guide to the eye, indicating that the normalized diffusivities approach constant values with increasing .

The particle diffusivities normalized by the product of the root-mean-square velocity and the system size . (a) ; (b) . The error bars are the standard deviations. The solid lines are a guide to the eye, indicating that the normalized diffusivities approach constant values with increasing .

## Tables

The values of and in Eq. (2) that provide the best fits to the vertical (∥) and horizontal (⊥) variance found from simulations. The numbers before and after the slash represent particle velocity variance and fluid velocity variance, respectively. The coefficients from Eq. (1) for vertical particle/fluid velocity variance in a dilute suspension are included in the last column for comparison.

The values of and in Eq. (2) that provide the best fits to the vertical (∥) and horizontal (⊥) variance found from simulations. The numbers before and after the slash represent particle velocity variance and fluid velocity variance, respectively. The coefficients from Eq. (1) for vertical particle/fluid velocity variance in a dilute suspension are included in the last column for comparison.

Anisotropy of the root-mean-square velocity and particle diffusivity. These ratios are obtained in the largest computational domains ( for and for ).

Anisotropy of the root-mean-square velocity and particle diffusivity. These ratios are obtained in the largest computational domains ( for and for ).

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