Modeling Evaporating Droplets in Complex Unsteady Flows

Affiliation(s)

Department of Mechanical Engineering, Northwestern University, Evanston, USA.

School of Mechanical Industrial and Manufacturing Engineering, Oregon State University, Corvallis, USA.

Department of Mechanical Engineering, Northwestern University, Evanston, USA.

School of Mechanical Industrial and Manufacturing Engineering, Oregon State University, Corvallis, USA.

ABSTRACT

In many applications, a moving fluid carries a suspension of droplets of a second phase which may change in size due to evaporation or condensation. Examples include liquid fuel drops in engines and raindrops or ice-crystals in a thunderstorm. If the number of such particles is very large, and, if further, the flow is inhomogeneous, unsteady or turbulent, it may be practically impossible to explicitly compute all of the fluid and particle degrees of freedom in a numerical simulation of the system. Under such circumstances Lagrangian Particle Tracking (LPT) of a small subset of the particles is used to reduce the computational effort. The purpose of this paper is to compare the LPT with an alternate method that is based on an approximate solution of the conservation equation of particle density in phase space by the method of moments (MOM). Closure is achieved by invoking the assumption that the droplet size distribution is locally lognormal. The resulting coupled transport equations for the local mean and variance of the particle size distribution are then solved in conjunction with the usual equations for the fluid and associated scalar fields. The formalism is applied to the test case of a uniform distribution of droplets placed in a non homogeneous temperature field and stirred with a decaying Taylor vortex. As a benchmark, we perform a direct numerical simulation (DNS) of high resolution that keeps track of all the particles together with the fluid flow.

In many applications, a moving fluid carries a suspension of droplets of a second phase which may change in size due to evaporation or condensation. Examples include liquid fuel drops in engines and raindrops or ice-crystals in a thunderstorm. If the number of such particles is very large, and, if further, the flow is inhomogeneous, unsteady or turbulent, it may be practically impossible to explicitly compute all of the fluid and particle degrees of freedom in a numerical simulation of the system. Under such circumstances Lagrangian Particle Tracking (LPT) of a small subset of the particles is used to reduce the computational effort. The purpose of this paper is to compare the LPT with an alternate method that is based on an approximate solution of the conservation equation of particle density in phase space by the method of moments (MOM). Closure is achieved by invoking the assumption that the droplet size distribution is locally lognormal. The resulting coupled transport equations for the local mean and variance of the particle size distribution are then solved in conjunction with the usual equations for the fluid and associated scalar fields. The formalism is applied to the test case of a uniform distribution of droplets placed in a non homogeneous temperature field and stirred with a decaying Taylor vortex. As a benchmark, we perform a direct numerical simulation (DNS) of high resolution that keeps track of all the particles together with the fluid flow.

Cite this paper

S. Ghosal and S. Apte, "Modeling Evaporating Droplets in Complex Unsteady Flows,"*Open Journal of Fluid Dynamics*, Vol. 2 No. 2, 2012, pp. 35-43. doi: 10.4236/ojfd.2012.22004.

S. Ghosal and S. Apte, "Modeling Evaporating Droplets in Complex Unsteady Flows,"

References

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[2] P. Moin and S. V. Apte, “LargeEddy Simulation of Realistic Gas TurbineCombustors,” AIAA Journal, Vol. 44, No. 4, 2006, pp. 698708. doi:10.2514/1.14606

[3] S. V. Apte and P. Moin, “Spray Modeling and Predictive Simulations in Realistic GasTurbine Engines,” Handbook of Atomization and Sprays, 2011, pp. 811835.

[4] R. A. Shaw, “ParticleTurbulence Interactions in Atmospheric Clouds,” Annual Review of Fluid Mechanics, Vol. 35, No. 1, 2003, pp. 183227. doi:10.1146/annurev.fluid.35.101101.161125

[5] F. Binkowski and U. Shankar, “The Regional Particulate Matter Model 1: Model Description and Preliminary Results,” Journal of Geophysical Research, Vol. 100, No. D12, 1995, pp. 2619126209.

[6] J. Dukowicz, “A ParticleFluid Numerical Model for Liquid Sprays,” Journal of Computational Physics, Vol. 35, No. 2, 1980, pp. 229253. doi:10.1016/00219991(80)90087X

[7] H. M. Hulburt and S. Katz, “Some Problems in Particle Technology: A Statistical Mechanical Formulation,” Che mical Engineering Science, Vol. 19, No. 8, 1964, pp. 555574. doi:10.1016/00092509(64)850478

[8] P. Hill, “Condensation of Water Vapour during Supersonic Expansion in Nozzles,” Journal of Fluid Mechanics, Vol. 25, No. 3, 1966, pp. 593620. doi:10.1017/S0022112066000284

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[12] D. Terry, R. McGraw and R. Rangel, “Method of Moments Solutions for a Laminar Flow Aerosol Reactor Model,” Aerosol Science and Technology, Vol. 34, No. 4, 2001, pp. 353362. doi:10.1080/02786820118736

[13] S. Pratsinis, “Simultaneous Nucleation, Condensation, and Coagulation in Aerosol Reactors,” Journal of Colloid and Interface Science, Vol. 124, No. 2, 1988, pp. 416427. doi:10.1016/00219797(88)901804

[14] F. Laurent and M. Massot, “MultiFluid Modelling of Laminar Polydisperse Spray Flames: Origin, Assumptions and Comparison of Sectional and Sampling Methods,” Combustion Theory and Modelling, Vol. 5, No. 4, 2001, pp. 537572. doi:10.1088/13647830/5/4/303

[15] M. Smith and T. Matsoukas, “ConstantNumber Monte Carlo Simulation of Population Balances,” Chemical Engineering Science, Vol. 53, No. 9, 1998, pp. 17771786. doi:10.1016/S00092509(98)000451

[16] S. Turns, “An Introduction to Combustion: Concepts and Applications,” McGrawHill, New York, 1995.

[17] S. Apte and S. Ghosal, “A Presumed PDF Model for Drop let Evaporation/Condensation in Complex Flows,” 2004. http://www.stanford.edu/group/ctr/publications.html

[18] R. Reid, J. Prausnitz and B. Poling, “The Properties of Gases and Liquids,” McGrawHill, New York, 1987.

[19] S. Apte, K. Mahesh, P. Moin and J. Oefelein, “Large Eddy Simulation of Swirling ParticleLaden Flows in a CoaxialJet Combustor,” International Journal of Multiphase Flow, Vol. 29, No. 8, 2003, pp. 13111331. doi:10.1016/S03019322(03)001046

[20] S. Apte, M. Gorokhovski and P. Moin, “LES of Atomizing Spray with Stochastic Modeling of Secondary Breakup,” International Journal of Multiphase Flow, Vol. 29, No. 9, 2003, pp. 15031522. doi:10.1016/S03019322(03)001113

[1] R. Paoli, J. Helie, T. Poinsot and S. Ghosal, “Contrail Formation in Aircraft Wakes Using LargeEddy Simulations,” 2002. http://www.stanford.edu/group/ctr/publications.html

[2] P. Moin and S. V. Apte, “LargeEddy Simulation of Realistic Gas TurbineCombustors,” AIAA Journal, Vol. 44, No. 4, 2006, pp. 698708. doi:10.2514/1.14606

[3] S. V. Apte and P. Moin, “Spray Modeling and Predictive Simulations in Realistic GasTurbine Engines,” Handbook of Atomization and Sprays, 2011, pp. 811835.

[4] R. A. Shaw, “ParticleTurbulence Interactions in Atmospheric Clouds,” Annual Review of Fluid Mechanics, Vol. 35, No. 1, 2003, pp. 183227. doi:10.1146/annurev.fluid.35.101101.161125

[5] F. Binkowski and U. Shankar, “The Regional Particulate Matter Model 1: Model Description and Preliminary Results,” Journal of Geophysical Research, Vol. 100, No. D12, 1995, pp. 2619126209.

[6] J. Dukowicz, “A ParticleFluid Numerical Model for Liquid Sprays,” Journal of Computational Physics, Vol. 35, No. 2, 1980, pp. 229253. doi:10.1016/00219991(80)90087X

[7] H. M. Hulburt and S. Katz, “Some Problems in Particle Technology: A Statistical Mechanical Formulation,” Che mical Engineering Science, Vol. 19, No. 8, 1964, pp. 555574. doi:10.1016/00092509(64)850478

[8] P. Hill, “Condensation of Water Vapour during Supersonic Expansion in Nozzles,” Journal of Fluid Mechanics, Vol. 25, No. 3, 1966, pp. 593620. doi:10.1017/S0022112066000284

[9] S. Adam and G. Schnerr, “Instabilities and Bifurcation of NonEquilibrium TwoPhase Flows,” Journal of Fluid Me chanics, Vol. 348, No. 1, 1997, pp. 128.

[10] R. McGraw, “Description of Aerosol Dynamics by the Quadrature Method of Moments,” Aerosol Science and Technology, Vol. 27, No. 2, 1997, pp. 255265. doi:10.1080/02786829708965471

[11] D. L. Marchisio, J. T. Pikturna, R. O. Fox, R. D. Vigil and A. A. Barresi, “Quadrature Method of Moments for PopulationBalance Equations,” AIChE Journal, Vol. 49, No. 5, 2003, pp. 12661276. doi:10.1002/aic.690490517

[12] D. Terry, R. McGraw and R. Rangel, “Method of Moments Solutions for a Laminar Flow Aerosol Reactor Model,” Aerosol Science and Technology, Vol. 34, No. 4, 2001, pp. 353362. doi:10.1080/02786820118736

[13] S. Pratsinis, “Simultaneous Nucleation, Condensation, and Coagulation in Aerosol Reactors,” Journal of Colloid and Interface Science, Vol. 124, No. 2, 1988, pp. 416427. doi:10.1016/00219797(88)901804

[14] F. Laurent and M. Massot, “MultiFluid Modelling of Laminar Polydisperse Spray Flames: Origin, Assumptions and Comparison of Sectional and Sampling Methods,” Combustion Theory and Modelling, Vol. 5, No. 4, 2001, pp. 537572. doi:10.1088/13647830/5/4/303

[15] M. Smith and T. Matsoukas, “ConstantNumber Monte Carlo Simulation of Population Balances,” Chemical Engineering Science, Vol. 53, No. 9, 1998, pp. 17771786. doi:10.1016/S00092509(98)000451

[16] S. Turns, “An Introduction to Combustion: Concepts and Applications,” McGrawHill, New York, 1995.

[17] S. Apte and S. Ghosal, “A Presumed PDF Model for Drop let Evaporation/Condensation in Complex Flows,” 2004. http://www.stanford.edu/group/ctr/publications.html

[18] R. Reid, J. Prausnitz and B. Poling, “The Properties of Gases and Liquids,” McGrawHill, New York, 1987.

[19] S. Apte, K. Mahesh, P. Moin and J. Oefelein, “Large Eddy Simulation of Swirling ParticleLaden Flows in a CoaxialJet Combustor,” International Journal of Multiphase Flow, Vol. 29, No. 8, 2003, pp. 13111331. doi:10.1016/S03019322(03)001046

[20] S. Apte, M. Gorokhovski and P. Moin, “LES of Atomizing Spray with Stochastic Modeling of Secondary Breakup,” International Journal of Multiphase Flow, Vol. 29, No. 9, 2003, pp. 15031522. doi:10.1016/S03019322(03)001113