Skip to main content Accessibility help
×
Home

A population balance model for large eddy simulation of polydisperse droplet evolution

  • A. K. Aiyer (a1), D. Yang (a2), M. Chamecki (a3) and C. Meneveau (a1)

Abstract

In the context of many applications of turbulent multi-phase flows, knowledge of the dispersed phase size distribution and its evolution is critical to predicting important macroscopic features. We develop a large eddy simulation (LES) model that can predict the turbulent transport and evolution of size distributions, for a specific subset of applications in which the dispersed phase can be assumed to consist of spherical droplets, and occurring at low volume fraction. We use a population dynamics model for polydisperse droplet distributions specifically adapted to a LES framework including a model for droplet breakup due to turbulence, neglecting coalescence consistent with the assumed small dispersed phase volume fractions. We model the number density fields using an Eulerian approach for each bin of the discretized droplet size distribution. Following earlier methods used in the Reynolds-averaged Navier–Stokes framework, the droplet breakup due to turbulent fluctuations is modelled by treating droplet–eddy collisions as in kinetic theory of gases. Existing models assume the scale of droplet–eddy collision to be in the inertial range of turbulence. In order to also model smaller droplets comparable to or smaller than the Kolmogorov scale we extend the breakup kernels using a structure function model that smoothly transitions from the inertial to the viscous range. The model includes a dimensionless coefficient that is fitted by comparing predictions in a one-dimensional version of the model with a laboratory experiment of oil droplet breakup below breaking waves. After initial comparisons of the one-dimensional model to measurements of oil droplets in an axisymmetric jet, it is then applied in a three-dimensional LES of a jet in cross-flow with large oil droplets of a single size being released at the source of the jet. We model the concentration fields using $N_{d}=15$ bins of discrete droplet sizes and solve scalar transport equations for each bin. The resulting droplet size distributions are compared with published experimental data, and good agreement for the relative size distribution is obtained. The LES results also enable us to quantify size distribution variability. We find that the probability distribution functions of key quantities such as the total surface area and the Sauter mean diameter of oil droplets are highly variable, some displaying strong non-Gaussian intermittent behaviour.

Copyright

Corresponding author

Email address for correspondence: aaiyer1@jhu.edu

References

Hide All
Albertson, J. D. & Parlange, M. B. 1999 Surface length scales and shear stress: implications for land-atmosphere interaction over complex terrain. Water Resour. Res. 35 (7), 21212132.
Azbel, D. 1981 Two Phase Flows in Chemical Engineering. Cambridge University Press.
Bandara, U. C. & Yapa, P. D. 2011 Bubble sizes, breakup, and coalescence in deepwater gas/oil plumes. J. Hydraul. Engng ASCE 137 (7), 729738.
Batchelor, G. K. 1951 Pressure fluctuations in isotropic turbulence. Proc. Camb. Phil. Soc. 47 (1950), 359374.
Bossard, J. A. & Peck, R. E. 1996 Droplet size distribution effects in spray combustion. Symposium (International) on Combustion 26 (1), 16711677.
Bou-Zeid, E., Meneveau, C. & Parlange, M. 2005 A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids 17 (2), 025105.
Brandvik, P. J., Johansen, Ø., Leirvik, F., Farooq, U. & Daling, P. S. 2013 Droplet breakup in subsurface oil releases. Part 1. Experimental study of droplet breakup and effectiveness of dispersant injection. Mar. Pollut. Bull. 73 (1), 319326.
Calabrese, R. V., Wang, C. Y. & Bryner, N. P. 1986 Drop breakup in turbulent stirred tank contactors. Part I. Effect of Dispersed-Phase Viscosity. AlChE J. 32 (4), 677681.
Chamecki, M., Meneveau, C. & Parlange, M. B. 2008 A hybrid spectral/finite-volume algorithm for large-eddy simulation of scalars in the atmospheric boundary layer. Boundary-Layer Meteorol. 128 (3), 473484.
Chatzi, E. & James, M. L. 1987 Analysis of interactions for liquid–liquid dispersions in agitated vessels. Ind. Engng Chem. Res. 26 (11), 22632267.
Chen, B., Yang, D., Meneveau, C. & Chamecki, M. 2018 Numerical study of the effects of chemical dispersant on oil transport from an idealized underwater blowout. Phys. Rev. Fluids 3, 083801.
Cortelezzi, L. & Karagozian, A. R. 2001 On the formation of the counter-rotating vortex pair in transverse jets. J. Fluid Mech. 446, 347373.
Coulaloglou, C. A. & Tavlarides, L. L. 1977 Description of interaction processes in agitated liquid–liquid dispersions. Chem. Engng Sci. 32 (11), 12891297.
Desjardins, O., Moureau, V. & Pitsch, H. 2008 An accurate conservative level set/ghost fluid method for simulating turbulent atomization. J. Comput. Phys. 227 (18), 83958416.
Duret, B., Luret, G., Reveillon, J., Menard, T., Berlemont, A. & Demoulin, F. X. 2012 Dns analysis of turbulent mixing in two-phase flows. Intl J. Multiphase Flow 40, 93105.
Eastwood, C. D., Armi, L. & Lasheras, J. C. 2004 The breakup of immiscible fluids in turbulent flows. J. Fluid Mech. 502, 309333.
Ferry, J. & Balachandar, S. 2001 A fast Eulerian method for disperse two-phase flow. Intl J. Multiphase Flow 27 (7), 11991226.
Gaskell, P. H. & Lau, A. K. C. 1988 Curvature compensated convective transport: SMART, a new boundedness preserving transport algorithm. Intl J. Numer. Meth. Fluids 8 (6), 617641.
Gopalan, B., Malkiel, E. & Katz, J. 2008 Experimental investigation of turbulent diffusion of slightly buoyant droplets in locally isotropic turbulence. Phys. Fluids 20 (9), 095102.
Gorokhovski, M. & Herrmann, M. 2008 Modeling primary atomization. Annu. Rev. Fluid Mech. 40 (1), 343366.
Herrmann, M. 2010 Detailed numerical simulations of the primary atomization of a turbulent liquid jet in crossflow. J. Engng Gas Turbines Power 132 (6), 061506.
Herrmann, M. 2013 A sub-grid surface dynamics model for sub-filter surface tension induced interface dynamics. Comput. Fluids 87, 92101.
Hinze, J. O. 1955 Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AlChE J. 1 (3), 289295.
Hulburt, H. M. & Katz, S. 1964 Some problems in particle technology. Chem. Engng Sci. 19 (8), 555574.
Hussein, H. J., Capp, S. P. & George, W. K. 1994 Velocity measurements in a high-Reynolds number, momentum-conserving, axisymmetric, turbulent jet. J. Fluid Mech. 258, 3175.
Igel, A. L. & van den Heever, S. C. 2016 The importance of the shape of cloud droplet size distributions in shallow cumulus clouds. Part I. Bin microphysics simulations. J. Atmos. Sci.; JAS–D–15–0382.1.
Jakobsen, H. A. 2014 The Population Balance Equation, pp. 9371003. Springer International Publishing.
Johansen, Ø., Brandvik, P. J. & Farooq, U. 2013 Droplet breakup in subsea oil releases. Part 2. Predictions of droplet size distributions with and without injection of chemical dispersants. Mar. Pollut. Bull. 73 (1), 327335.
Johnson, P. L. & Meneveau, C. 2018 Predicting viscous-range velocity gradient dynamics in large-eddy simulations of turbulence. J. Fluid Mech. 837, 80114.
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. C. R. Acad. Sci. USSR 30, 301.
Kolmogorov, A. N. 1949 On the breakage of drops in a turbulent flow. Dokl. Akad. Nauk SSSR 66, 825828.
Konno, M., Aoki, M. & Saito, S. 1982 Scale effect on breakup process in liquid–liquid agitated tanks. J. Chem. Engng Japan 16, 312319.
Kumar, J., Peglow, M., Warnecke, G., Heinrich, S. & Mörl, L. 2006 Improved accuracy and convergence of discretized population balance for aggregation. The cell average technique. Chem. Engng Sci. 61 (10), 33273342.
Kumar, S. & Ramakrishna, D. 1996 On the solution of population balance equations by discretization. I. A fixed pivot technique. Chem. Engng Sci. 51 (8), 13111332.
Lasheras, J. C., Eastwood, C., Martínez-Bazán, C. & Montaes, J. L. 2002 A review of statistical models for the break-up an immiscible fluid immersed into a fully developed turbulent flow. Intl J. Multiphase Flow 28 (2), 247278.
Lehr, F. & Mewes, D. 1999 A transport equation for the interfacial area density applied to bubble columns. Chem. Engng Sci. 56, 11591166.
Li, C., Miller, J., Wang, J., Koley, S. S. & Katz, J. 2017 Size distribution and dispersion of droplets generated by impingement of breaking waves on oil slicks. J. Geophys. Res. Oceans 122 (10), 79387957.
Liao, Y. & Lucas, D. 2009 A literature review of theoretical models for drop and bubble breakup in turbulent dispersions. Chem. Engng Sci. 64 (15), 33893406.
Liu, Y. G., You, L. G., Yang, W. N. & Liu, F. 1995 On the size distribution of cloud droplets. Atmos. Res. 35 (2-4), 201216.
Luo, H. & Svendsen, H. F. 1996 Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J. 42 (5), 12251233.
Marchal, P., David, R., Klein, J. P. & Villermaux, J. 1988 Crystallization and precipitation engineering-I. An efficient method for solving population balance in crystallization with agglomeration. Chem. Engng Sci. 43 (1), 5967.
Martínez-Bazán, Rodríguez-rodríguez, J., Deane, G. B., Montañés, J. L. & Lasheras, J. C. 2010 Considerations on bubble fragmentation models. J. Fluid Mech. 661, 159177.
Martínez-Bazán, C., Montañés, J. L. & Lasheras, J. C. 1999a On the breakup of an air bubble injected into a fully developed turbulent flow. Part 1. Breakup frequency. J. Fluid Mech. 401, 157182.
Martínez-Bazán, C., Montañés, J. L. & Lasheras, J. C. 1999b On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size pdf of the resulting daughter bubbles. J. Fluid Mech. 401, 183207.
Murphy, D. W., Xue, X., Sampath, K. & Katz, J. 2016 Crude oil jets in crossflow: effects of dispersant concentration on plume behavior. J. Geophys. Res. Oceans 121 (6), 42644281.
Narsimhan, G., Gupta, J. P. & Ramkrishna, D. 1979 A model for transitional breakage probability of droplets in agitated lean liquid–liquid dispersions. Chem. Engng Sci. 34 (2), 257265.
Narsimhan, G., Ramkrishna, D. & Gupta, J. P. 1980 Analysis of drop size distributions in lean liquid–liquid dispersions. AlChE J. 26 (6), 9911000.
Neuber, G., Kronenburg, A., Stein, O. T. & Cleary, M. J. 2017 MMC-LES modelling of droplet nucleation and growth in turbulent jets. Chem. Engng Sci. 167, 204218.
Nissanka, I. D. & Yapa, P. D. 2016 Calculation of oil droplet size distribution in an underwater oil well blowout. J. Hydraul Res. 54 (3), 307320.
Nissanka, I. D. & Yapa, P. D. 2018 Calculation of oil droplet size distribution in ocean oil spills: a review. Mar. Pollut. Bull. 135, 723734.
North, E. W., Adams, E. E., Thessen, A. E., Schlag, Z., He, R., Socolofsky, S. A., Masutani, S. M. & Peckham, S. D. 2015 The influence of droplet size and biodegradation on the transport of subsurface oil droplets during the deepwater horizon spill: a model sensitivity study. Environ. Res. Lett. 10 (2), 024016.
Pedel, J., Thornock, J. N., Smith, S. T. & Smith, P. J. 2014 Large eddy simulation of polydisperse particles in turbulent coaxial jets using the direct quadrature method of moments. Intl J. Multiphase Flow 63, 2338.
Pope, S. B. 2011 Turbulent Flows. Cambridge University Press.
Porté-Agel, F., Meneveau, C. & Parlange, M. B. 2000 A scale-dependent dynamic model for large-eddy simulation: application to a neutral atmospheric boundary layer. J. Fluid Mech. 415, 261284.
Prince, M. J. & Blanch, H. W. 1990 Bubble coalescence and break-up in air-sparged bubble columns. AlChE J. 36 (10), 14851499.
Ramkrishna, D. 1985 The status of population balances. Rev. Chem. Engng 3 (1), 4995.
Randolph, A. D. 1964 A population balance for countable entities. Can. J. Chem. Engng 42 (6), 280281.
Salehi, F., Cleary, M. J. & Masri, A. R. 2017 Population balance equation for turbulent polydispersed inertial droplets and particles. J. Fluid Mech. 831, 719742.
Sathyagal, A. N., Ramkrishna, D. & Narsimhan, G. 1996 Droplet breakage in stirred dispersions. Breakage functions from experimental drop-size distributions. Chem. Engng Sci. 51 (9), 13771391.
Sato, Y. & Yamamoto, K. 1987 Lagrangian measurement of fluid-particle motion in an isotropic turbulent field. J. Fluid Mech. 175, 183199.
Seubert, N., Kronenburg, A., Stein, O. T., Ge, Y. & Cleary, M. J. 2012 Large eddy simulation-probability density function modelling of nucleation and condensation of DBP droplets in a turbulent jet. In ICLASS 2012, 12th Triennial Int. Conference on Liquid Atomization and Spray Systems, Heidelberg, Germany, 2012, pp. 18. Institute for Liquid Atomization and Spray Systems.
Sewerin, F. & Rigopoulos, S. 2017 An LES-PBE-PDF approach for modeling particle formation in turbulent reacting flows. Phys. Fluids 29 (10), 105105.
Skartlien, R., Sollum, E. & Schumann, H. 2013 Droplet size distributions in turbulent emulsions: breakup criteria and surfactant effects from direct numerical simulations. J. Chem. Phys. 139 (17), 174901.
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Mon. Weath. Rev. 91 (3), 99164.
Smoluchowski, M. V. 1916 Drei Vorträge über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Zeitschrift fur Physik 17, 557585.
Solsvik, J. & Jakobsen, H. A. 2016 Development of fluid particle breakup and coalescence closure models for the complete energy spectrum of isotropic turbulence. Ind. Engng Chem. Res. 55 (5), 14491460.
Solsvik, J., Maaß, S. & Jakobsen, H. A. 2016 Definition of the single drop breakup event. Ind. Engng Chem. Res. 55 (10), 28722882.
Tseng, Y. H., Meneveau, C. & Parlange, M. B. 2006 Modeling flow around bluff bodies and predicting urban dispersion using large eddy simulation. Environ. Sci. Technol. 40 (8), 26532662.
Tsouris, C. & Tavlarides, L. L. 1994 Breakage and coalescence models for drops in turbulent dispersions. AlChE J. 40 (3), 395406.
Wang, T. & Wang, J. 2007 Numerical simulations of gas–liquid mass transfer in bubble columns with a CFD-PBM coupled model. Chem. Engng Sci. 62 (24), 71077118.
Xu, G. & Antonia, R. 2002 Effect of different initial conditions on a turbulent round free jet. Exp. Fluids 33 (5), 677683.
Yang, D., Chen, B., Chamecki, M. & Meneveau, C. 2015 Oil plumes and dispersion in langmuir, upper-ocean turbulence: large-eddy simulations and k-profile parameterization. J. Geophys. Res. Oceans 120 (7), 47294759.
Yang, D., Chen, B., Socolofsky, S. A., Chamecki, M. & Meneveau, C. 2016 Large-eddy simulation and parameterization of buoyant plume dynamics in stratified flow. J. Fluid Mech. 794, 798833.
Yang, D., Meneveau, C. & Shen, L. 2014 Effect of downwind swells on offshore wind energy harvesting a large-eddy simulation study. Renewable Energy 70, 1123.
Zhao, L., Boufadel, M. C., Socolofsky, S. A., Adams, E., King, T. & Lee, K. 2014a Evolution of droplets in subsea oil and gas blowouts: development and validation of the numerical model VDROP-J. Mar. Pollut. Bull. 83 (1), 5869.
Zhao, L., Shaffer, F., Robinson, B., King, T., DAmbrose, C., Pan, Z., Gao, F., Miller, R. S., Conmy, R. N. & Boufadel, M. C. 2016 Underwater oil jet: hydrodynamics and droplet size distribution. Chem. Engng J. 299, 292303.
Zhao, L., Torlapati, J., Boufadel, M. C., King, T., Robinson, B. & Lee, K. 2014b VDROP: a comprehensive model for droplet formation of oils and gases in liquids – incorporation of the interfacial tension and droplet viscosity. Chem. Engng J. 253, 93106.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

A population balance model for large eddy simulation of polydisperse droplet evolution

  • A. K. Aiyer (a1), D. Yang (a2), M. Chamecki (a3) and C. Meneveau (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed