^{1}and Donald L. Koch

^{1,a)}

### Abstract

A lattice-Boltzmann method is used to probe the structure and average properties of suspensions of monodisperse, spherical, noncoalescing bubbles rising due to buoyancy with Reynolds numbers based on the bubble terminal velocities of 5.4 and 20. Unbounded suspensions subject to periodic boundary conditions exhibit a microstructure with a strong tendency toward horizontal alignment of bubble pairs even at volume fractions of as high as 0.2. This microstructure leads to a mean rise velocity whose dependence on the bubble volume fraction is not well fitted by a standard power-law function. Simulations with bounding vertical walls exhibit a deficit of bubbles near each wall and a peak of volume fraction approximately one bubble diameter from the wall. We attribute this structure to the effects of a repulsive wall-induced force and a lift force associated with the liquid flow driven by the variation in the buoyancy force with horizontal position. Weaker peaks of bubble volume fraction extend into the bulk of the suspension and these peaks are separated by a distance equal to the peak in the pair distribution function for bubble pairs in an unbounded fluid. This suggests that the layering is a result of hydrodynamic bubble-bubble interactions.

This research was supported by NASA under Grant No. NAG3–1853. The computational resources were provided by the Cornell Theory Center.

I. INTRODUCTION

II. LATTICE-BOLTZMANN METHOD

III. HINDERED RISE VELOCITY

IV. MICROSTRUCTURE

V. EFFECTS OF VERTICAL WALLS ON A BUBBLE-CHANNEL FLOW

VI. SUMMARY

### Key Topics

- Suspensions
- 134.0
- Reynolds stress modeling
- 46.0
- Anisotropy
- 20.0
- Boundary value problems
- 20.0
- Hydrodynamics
- 15.0

## Figures

The velocity fields around isolated bubbles. (a) and (b) correspond to and ; (c) and (d) correspond to and . Both simulations were done in a large computational domain whose size is , where bubble diameter . (a) and (c) show the distribution of fluid velocity, normalized by bubble terminal velocity, around the bubble. (b) and (d) show the contours of the fluid velocity in the direction of rising, also normalized by bubble terminal velocity.

The velocity fields around isolated bubbles. (a) and (b) correspond to and ; (c) and (d) correspond to and . Both simulations were done in a large computational domain whose size is , where bubble diameter . (a) and (c) show the distribution of fluid velocity, normalized by bubble terminal velocity, around the bubble. (b) and (d) show the contours of the fluid velocity in the direction of rising, also normalized by bubble terminal velocity.

Temporal evolution of average rise velocities in bubble suspensions. (a) and ; (b) and . The lines from top to bottom correspond to simulations with , 0.10, and 0.20, respectively.

Temporal evolution of average rise velocities in bubble suspensions. (a) and ; (b) and . The lines from top to bottom correspond to simulations with , 0.10, and 0.20, respectively.

Variation in the dimensionless relative velocity with the volume fraction . The open diamonds and squares represent the relative velocities in bubble suspensions with and 20. The filled diamonds and squares represent the relative velocities in solid particle suspensions with and 20. The lines correspond to the empirical formula in Ishii and Zuber (Ref. 47) for suspensions of nondistorted bubbles, i.e., Eq. (18) for (dash-dot) and Eq. (19) for (dashed). The data points for bubble suspensions were obtained with or . The “” and “” symbols represent results from validation runs using a higher lattice resolution , similar box size , and identical Reynolds numbers.

Variation in the dimensionless relative velocity with the volume fraction . The open diamonds and squares represent the relative velocities in bubble suspensions with and 20. The filled diamonds and squares represent the relative velocities in solid particle suspensions with and 20. The lines correspond to the empirical formula in Ishii and Zuber (Ref. 47) for suspensions of nondistorted bubbles, i.e., Eq. (18) for (dash-dot) and Eq. (19) for (dashed). The data points for bubble suspensions were obtained with or . The “” and “” symbols represent results from validation runs using a higher lattice resolution , similar box size , and identical Reynolds numbers.

The dimensionless relative velocity as a function of on a log-log scale. The symbols are as defined in Fig. 3. The dashed lines are the best power-law fits for in bubble suspensions; the values of are 2.9/0.91 for and 2.5/0.90 for . The solid lines are the best power-law fits for in solid particle suspensions, with being 3.0/0.86 and 2.6/0.88 for and 20.

The dimensionless relative velocity as a function of on a log-log scale. The symbols are as defined in Fig. 3. The dashed lines are the best power-law fits for in bubble suspensions; the values of are 2.9/0.91 for and 2.5/0.90 for . The solid lines are the best power-law fits for in solid particle suspensions, with being 3.0/0.86 and 2.6/0.88 for and 20.

The pair probability density distribution in bubble suspensions: (a) and ; (b) and ; (c) and ; and (d) and . In these simulations .

The pair probability density distribution in bubble suspensions: (a) and ; (b) and ; (c) and ; and (d) and . In these simulations .

Snapshots showing preferential alignment of bubble clusters in the horizontal direction. These two snapshots were taken from dilute suspensions . Left: . Right: . In both simulations .

Snapshots showing preferential alignment of bubble clusters in the horizontal direction. These two snapshots were taken from dilute suspensions . Left: . Right: . In both simulations .

The pair probability density distribution in solid particle suspensions: (a) and ; (b) and ; (c) and ; and (d) and . In these simulations .

The pair probability density distribution in solid particle suspensions: (a) and ; (b) and ; (c) and ; and (d) and . In these simulations .

The radial distribution function in bubble and solid particle suspensions: (a) ; (b) . The upward triangles represent ; the downward triangles represent . Dashed lines with open symbols are for bubble suspensions; solid lines with filled symbols are for solid particle suspensions.

The radial distribution function in bubble and solid particle suspensions: (a) ; (b) . The upward triangles represent ; the downward triangles represent . Dashed lines with open symbols are for bubble suspensions; solid lines with filled symbols are for solid particle suspensions.

The order parameter in bubble and solid particle suspensions: (a) ; (b) . The lines and symbols have the same meanings as in Fig. 8.

The order parameter in bubble and solid particle suspensions: (a) ; (b) . The lines and symbols have the same meanings as in Fig. 8.

The trajectories of a trailing bubble (dashed lines) or solid particle (solid lines) approaching a leading one and migrating to the side: (a) ; (b) . Length units are normalized by . Simulations were conducted in domains with size of (normalized by bubble size ), with gravity aligned with the longest dimension. The duration of the simulations is . The circle indicates the excluded volume of the leading bubble or particle.

The trajectories of a trailing bubble (dashed lines) or solid particle (solid lines) approaching a leading one and migrating to the side: (a) ; (b) . Length units are normalized by . Simulations were conducted in domains with size of (normalized by bubble size ), with gravity aligned with the longest dimension. The duration of the simulations is . The circle indicates the excluded volume of the leading bubble or particle.

The interaction between a pairs of bubbles (dashed lines) and a pair of solid particles (solid lines) rising side by side: (a) ; (b) . The horizontal axis is the horizontal center-to-center distance normalized by ; the vertical axis is the dimensionless time . Simulations were conducted in domains with size of (normalized by bubble size ), with gravity aligned with the longest dimension.

The interaction between a pairs of bubbles (dashed lines) and a pair of solid particles (solid lines) rising side by side: (a) ; (b) . The horizontal axis is the horizontal center-to-center distance normalized by ; the vertical axis is the dimensionless time . Simulations were conducted in domains with size of (normalized by bubble size ), with gravity aligned with the longest dimension.

Volume-fraction, fluid- and bubble-velocity profiles in a vertical channel with . (a), (c), (e), and (g) are the volume fraction profiles and (b), (d), (f), and (h) are the velocity profiles, where solid lines are for fluid velocities and dashed lines are for the bubble velocities. The figures on the left have ; the figures on the right have .

Volume-fraction, fluid- and bubble-velocity profiles in a vertical channel with . (a), (c), (e), and (g) are the volume fraction profiles and (b), (d), (f), and (h) are the velocity profiles, where solid lines are for fluid velocities and dashed lines are for the bubble velocities. The figures on the left have ; the figures on the right have .

Volume-fraction, fluid- and bubble-velocity profiles in a vertical channel with . (a), (c), (e), and (g) are the volume fraction profiles and (b), (d), (f), and (h) are the velocity profiles, where solid lines are for fluid velocities and dashed lines are for the bubble velocities. The figures on the left have ; the figures on the right have .

## Tables

Terminal velocity and Reynolds number for bubbles with and . The third column shows the terminal velocity in terms of , where is the unit lattice spacing and is the time between consecutive updates of the fluid molecular velocity distribution. The fourth column shows the Reynolds number based on , with the numbers in the brackets representing the standard deviations of successive runs with different orientations. The last column shows the Reynolds number calculated from the empirical relation given in Clift *et al.* (Ref. 18). The fluid viscosity is for and 0.36 for .

Terminal velocity and Reynolds number for bubbles with and . The third column shows the terminal velocity in terms of , where is the unit lattice spacing and is the time between consecutive updates of the fluid molecular velocity distribution. The fourth column shows the Reynolds number based on , with the numbers in the brackets representing the standard deviations of successive runs with different orientations. The last column shows the Reynolds number calculated from the empirical relation given in Clift *et al.* (Ref. 18). The fluid viscosity is for and 0.36 for .

The wavelength of the bubble volume fraction oscillations (or average distance between layers) in vertical channels, the distance from the primary layers to the walls , and the most probable distances between bubble pairs found from . All distances are normalized by bubble diameter . In the entries for and , the numbers before the slash “/” are from simulations with , and the numbers after “/” are from simulations with .

The wavelength of the bubble volume fraction oscillations (or average distance between layers) in vertical channels, the distance from the primary layers to the walls , and the most probable distances between bubble pairs found from . All distances are normalized by bubble diameter . In the entries for and , the numbers before the slash “/” are from simulations with , and the numbers after “/” are from simulations with .

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