^{1,a)}, Omar M. Knio

^{2,b)}, Joseph Katz

^{2,c)}and Olivier P. Le Maître

^{3,d)}

### Abstract

The rise of small oil droplets in water under isotropic turbulence conditions is analyzed computationally. The effort focuses on the puzzling behavior observed by Friedman and Katz [Phys. Fluids14, 3059 (2002)], namely, that the rise velocity of droplets smaller than in diameter is enhanced by turbulence, whereas the rise of larger droplets is suppressed. Specifically, the study explores whether these effects can be captured or explained using a simplified one-way coupling model that combines direct numerical simulation of the turbulent flow field with Lagrangian tracking of the droplets using a dynamical equation that accounts for buoyancy, virtual mass, pressure, drag, lift, and history forces. The computational method used is adapted from the model of Snyder *et al.* [Phys. Fluids19, 065108 (2007)], which showed excellent correlation between computational results and extensive experimental data for microbubbles in isotropic turbulence. The computed results indicate that, using the quasisteady, empirically determined drag and lift coefficients, one is unable to reproduce the experimentally observed droplet rise velocities. Numerical experiments on the effect of lift and history forces also indicate that, within a broad range of uncertainty, these forces do not account for the discrepancy between measured and computed trends. Guided by correlations obtained for the settling of heavy particles under high turbulence intensities, suppression of the drag and virtual mass coefficients for droplet diameters near ten times the Kolmogorov lengthscale was consequently postulated. Computed results indicate that, using this postulate, the simplified model is able to recover the observed enhancement of the mean rise of small droplets. These experiences underscore the difficulties in modeling the motion of small particles under high turbulence intensities, especially when the particle size is close to the turbulence microscale.

This research was funded by the Department of Energy and by the United States Naval Academy (via the Permanent Military Professor program).

I. INTRODUCTION

A. Overview of experimental results of Friedman and Katz

B. Outline

II. NUMERICAL APPROACH

A. Lagrangianequation of motion

B. Simulation of large faculty L3 turbulent field

III. BASELINE COEFFICIENTS AND BASELINE PREDICTIONS

A. Coefficient of drag

B. Coefficient of lift

C. Coefficient of virtual mass

D. Coefficient of history force

E. Baseline predictions

IV. NUMERICAL EXPERIMENTATION IN THE VARIATION OF THE LIFT COEFFICIENT

V. NUMERICAL EXPERIMENTS OF THE EFFECT OF THE HISTORY FORCE COEFFICIENT

A. Background

B. Parametrization of the history force

C. Effect of the history force on the mean rise

VI. NUMERICAL EXPERIMENTATION IN THE VARIATION OF THE COEFFICIENT OF DRAG

A. Behavior of at and low

B. Behavior of at and high

C. Behavior of at

D. Behavior of in oscillating flows

E. Variation in with

F. Summary of behavior of in turbulent flows

G. Effect of variation of on droplet mean rise

VII. NUMERICAL EXPERIMENTS WITH CONCURRENT VARIATION OF AND

VIII. CONCLUSIONS

### Key Topics

- Fluid drops
- 153.0
- Turbulent flows
- 93.0
- Isotropic turbulence
- 29.0
- Turbulence simulations
- 14.0
- Viscosity
- 14.0

## Figures

The Friedman and Katz (Ref. 1) experimentally determined quiescent rise velocity for slightly buoyant oil droplets. “Analysis fuel” refers to slightly buoyant fuel oil (used with permission).

The Friedman and Katz (Ref. 1) experimentally determined quiescent rise velocity for slightly buoyant oil droplets. “Analysis fuel” refers to slightly buoyant fuel oil (used with permission).

The Friedman and Katz (Ref. 1) large facility L3 experimentally determined turbulent rise velocity, normalized by quiescent rise velocity, for slightly buoyant oil droplets in isotropic turbulence. Histogram reference to right scale shows number of droplets in analysis bin of . Data points (◇) referenced to left scale show mean rise rate (used with permission).

The Friedman and Katz (Ref. 1) large facility L3 experimentally determined turbulent rise velocity, normalized by quiescent rise velocity, for slightly buoyant oil droplets in isotropic turbulence. Histogram reference to right scale shows number of droplets in analysis bin of . Data points (◇) referenced to left scale show mean rise rate (used with permission).

The Friedman and Katz (Ref. 1) experimentally determined turbulence rise data for slightly buoyant oil droplets in large facility L3 compared with quiescent rise approximation using Feng and Michaelides (Ref. 9) with viscosity ratio of 6.41. Upper horizontal scale shows droplet diameter normalized by Kolmogorov microscale, .

The Friedman and Katz (Ref. 1) experimentally determined turbulence rise data for slightly buoyant oil droplets in large facility L3 compared with quiescent rise approximation using Feng and Michaelides (Ref. 9) with viscosity ratio of 6.41. Upper horizontal scale shows droplet diameter normalized by Kolmogorov microscale, .

Stirred tank settling velocity normalized by unstirred tank settling velocity, , vs the Kolmogorov microscale normalized by particle diameter, (Nocentini and Magelli, Ref. 10). Upper horizontal scale shows particle diameter normalized by the Kolmogorov microscale, .

Stirred tank settling velocity normalized by unstirred tank settling velocity, , vs the Kolmogorov microscale normalized by particle diameter, (Nocentini and Magelli, Ref. 10). Upper horizontal scale shows particle diameter normalized by the Kolmogorov microscale, .

The Friedman and Katz (Ref. 1) experimentally determined quiescent rise data for slightly buoyant oil droplets compared with numerical approximation using Feng and Michaelides (Ref. 9) .

The Friedman and Katz (Ref. 1) experimentally determined quiescent rise data for slightly buoyant oil droplets compared with numerical approximation using Feng and Michaelides (Ref. 9) .

Standard sphere and Feng and Michaelides (Ref. 9) for fuel oil in water vs Reynolds number.

Standard sphere and Feng and Michaelides (Ref. 9) for fuel oil in water vs Reynolds number.

The Kurose and Komori (adapted from Ref. 23) lift coefficient for a stationary sphere in linear shear flow: (△) , (◻) , (◇) , and (▽) . is dimensionless shear.

The Kurose and Komori (adapted from Ref. 23) lift coefficient for a stationary sphere in linear shear flow: (△) , (◻) , (◇) , and (▽) . is dimensionless shear.

The Friedman and Katz (Ref. 1) experimentally determined turbulence rise data for slightly buoyant oil droplets compared with baseline numerical approximation using Feng and Michaelides (Ref. 9) and Kurose and Komori (Ref. 23) . Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

The Friedman and Katz (Ref. 1) experimentally determined turbulence rise data for slightly buoyant oil droplets compared with baseline numerical approximation using Feng and Michaelides (Ref. 9) and Kurose and Komori (Ref. 23) . Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Simulation results for various combinations of lift and drag coefficients compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Simulation results for various combinations of lift and drag coefficients compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

pdfs of the droplet lift force in the vertical direction, , vs droplet radius.

pdfs of the droplet lift force in the vertical direction, , vs droplet radius.

Simulation results for Basset force with time history of , 0.1, 0.5, 1.0, and 2.0 compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). is the Kolmogorov time scale. All simulations used Feng and Michaelides (Ref. 9) , Kurose and Komori (Ref. 23) , and . Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Simulation results for Basset force with time history of , 0.1, 0.5, 1.0, and 2.0 compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). is the Kolmogorov time scale. All simulations used Feng and Michaelides (Ref. 9) , Kurose and Komori (Ref. 23) , and . Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Standard drag curve and selected experimental drag curves for relative turbulence intensity (adapted from Crowe *et al.*, Ref. 19).

Standard drag curve and selected experimental drag curves for relative turbulence intensity (adapted from Crowe *et al.*, Ref. 19).

based on relative and terminal droplet velocity vs oil droplet diameter for the simulation of the Friedman and Katz (Ref. 1) large facility L3 with . Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

based on relative and terminal droplet velocity vs oil droplet diameter for the simulation of the Friedman and Katz (Ref. 1) large facility L3 with . Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Variation in vs and following Uhlherr and Sinclair (Ref. 44).

Variation in vs and following Uhlherr and Sinclair (Ref. 44).

Modified correlations used vs and mean turbulence intensity .

Modified correlations used vs and mean turbulence intensity .

Simulation results for variation in due to mean turbulence intensity compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Simulation results for variation in due to mean turbulence intensity compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Modified used vs droplet radius. Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Modified used vs droplet radius. Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Simulation results for variation in and due to mean turbulence intensity compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). Modified is that shown in Fig. 17. Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

Simulation results for variation in and due to mean turbulence intensity compared with experimental turbulent rise results of Friedman and Katz (Ref. 1). Modified is that shown in Fig. 17. Upper horizontal scale shows droplet diameter normalized by the Kolmogorov microscale, .

## Tables

Turbulent field modeled. is the side length of the domain, is the characteristic length scale, is the characteristic velocity, is the characteristic time scale, is the Kolmogorov time scale, is the integral time scale, is the Kolmogorov microscale, is the Taylor length scale, and is the dissipation rate.

Turbulent field modeled. is the side length of the domain, is the characteristic length scale, is the characteristic velocity, is the characteristic time scale, is the Kolmogorov time scale, is the integral time scale, is the Kolmogorov microscale, is the Taylor length scale, and is the dissipation rate.

Simulation results using the Feng and Michaelides (Ref. 9) and Kurose and Komori (Ref. 23) . , , , and are, respectively, minimum lift force in the vertical direction, maximum lift force in the vertical direction, mean lift force in the vertical direction, and total lift force (all in N). is mean lift force in the vertical direction normalized by magnitude of mean drag force. is mean lift force normalized by magnitude of mean drag force.

Simulation results using the Feng and Michaelides (Ref. 9) and Kurose and Komori (Ref. 23) . , , , and are, respectively, minimum lift force in the vertical direction, maximum lift force in the vertical direction, mean lift force in the vertical direction, and total lift force (all in N). is mean lift force in the vertical direction normalized by magnitude of mean drag force. is mean lift force normalized by magnitude of mean drag force.

Normalized time scales (value for calculated for radii droplets).

Normalized time scales (value for calculated for radii droplets).

Experimentally observed behavior and Stokes number St for heavy particles experiencing relative turbulence intensity . is the particle diameter, is the Kolmogorov microscale, is the particle density, and is the carrier fluid density.

Experimentally observed behavior and Stokes number St for heavy particles experiencing relative turbulence intensity . is the particle diameter, is the Kolmogorov microscale, is the particle density, and is the carrier fluid density.

Properties and observed behavior of plastic and marble spheres settling in a stirred tank under high (Schwartzberg and Treybal, Ref. 57). is the particle diameter, is the particle density, is the carrier fluid density, is the relative turbulence intensity, is the settling velocity in a stirred tank, and is the settling velocity in a still tank.

Properties and observed behavior of plastic and marble spheres settling in a stirred tank under high (Schwartzberg and Treybal, Ref. 57). is the particle diameter, is the particle density, is the carrier fluid density, is the relative turbulence intensity, is the settling velocity in a stirred tank, and is the settling velocity in a still tank.

Experimentally observed behavior and Stokes number St for heavy particles in oscillating flows. is the particle diameter, is the oscillation amplitude, is the particle density, is the carrier fluid density, and is the relative oscillation intensity.

Experimentally observed behavior and Stokes number St for heavy particles in oscillating flows. is the particle diameter, is the oscillation amplitude, is the particle density, is the carrier fluid density, and is the relative oscillation intensity.

Experimentally observed behavior for heavy particles vs or . is the particle diameter, is the particle density, is the carrier fluid density, is the relative turbulence intensity, is the stirred tank settling velocity, is the unstirred tank settling velocity, and is the Kolmogorov microscale. (Note: estimated from .)

Experimentally observed behavior for heavy particles vs or . is the particle diameter, is the particle density, is the carrier fluid density, is the relative turbulence intensity, is the stirred tank settling velocity, is the unstirred tank settling velocity, and is the Kolmogorov microscale. (Note: estimated from .)

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