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Lattice-Boltzmann simulation of finite Reynolds number buoyancy-driven bubbly flows in periodic and wall-bounded domains
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10.1063/1.3001728
/content/aip/journal/pof2/20/10/10.1063/1.3001728
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/10/10.1063/1.3001728

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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

Image of FIG. 6.
FIG. 6.

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

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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.

Image of FIG. 12.
FIG. 12.

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 .

Image of FIG. 13.
FIG. 13.

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

Generic image for table
Table I.

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 .

Generic image for table
Table II.

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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/content/aip/journal/pof2/20/10/10.1063/1.3001728
2008-10-24
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Lattice-Boltzmann simulation of finite Reynolds number buoyancy-driven bubbly flows in periodic and wall-bounded domains
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/10/10.1063/1.3001728
10.1063/1.3001728
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