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Statistical steady state in turbulent droplet condensation

Published online by Cambridge University Press:  25 November 2016

Christoph Siewert ,
Jérémie Bec  and
Giorgio Krstulovic
Christoph Siewert*
Affiliation:
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, 06300 Nice, France
Jérémie Bec
Affiliation:
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, 06300 Nice, France
Giorgio Krstulovic
Affiliation:
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, 06300 Nice, France
*
†Email address for correspondence: christoph.siewert@oca.eu
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Abstract

Motivated by systems in which droplets grow and shrink in a turbulence-driven supersaturation field, we investigate the problem of turbulent condensation in a general manner. Using direct numerical simulations, we show that the turbulent fluctuations of the supersaturation field offer different conditions for the growth of droplets which evolve in time due to turbulent transport and mixing. Based on this, we propose a Lagrangian stochastic model for condensation and evaporation of small droplets in turbulent flows. It consists of a set of stochastic integro-differential equations for the joint evolution of the squared radius and the supersaturation along the droplet trajectories. The model has two parameters fixed by the total amount of water and the thermodynamic properties, as well as the Lagrangian integral time scale of the turbulent supersaturation. The model reproduces very well the droplet size distributions obtained from direct numerical simulations and their time evolution. A noticeable result is that, after a stage where the squared radius simply diffuses, the system converges exponentially fast to a statistical steady state independent of the initial conditions. The main mechanism involved in this convergence is a loss of memory induced by a significant number of droplets undergoing a complete evaporation before growing again. The statistical steady state is characterized by an exponential tail in the droplet mass distribution. These results reconcile those of earlier numerical studies, once these various regimes are considered.

JFM classification

Phase change: Condensation/evaporation Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 810 , 10 January 2017 , pp. 254 - 280
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Copyright
© 2016 Cambridge University Press 

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References

Bartlett, J. T. & Jonas, P. R. 1972 On the dispersion of the sizes of droplets growing by condensation in turbulent clouds. Q. J. R. Meteorol. Soc. 98, 150–164.Google Scholar
Bec, J., Homann, H. & Krstulovic, G. 2014 Clustering, fronts, and heat transfer in turbulent suspensions of heavy particles. Phys. Rev. Lett. 112, 234503.CrossRefGoogle ScholarPubMed
Celani, A., Falkovich, G., Mazzino, A. & Seminara, A. 2005 Droplet condensation in turbulent flows. Europhys. Lett. 70 (6), 775.Google Scholar
Celani, A., Lanotte, A., Mazzino, A. & Vergassola, M. 2001 Fronts in passive scalar turbulence. Phys. Fluids 13 (6), 1768–1783.CrossRefGoogle Scholar
Celani, A., Mazzino, A., Seminara, A. & Tizzi, M. 2007 Droplet condensation in two-dimensional Bolgiano turbulence. J. Turbul. 8, N17.Google Scholar
Celani, A., Mazzino, A. & Tizzi, M. 2008 The equivalent size of cloud condensation nuclei. New J. Phys. 10 (7), 075021.Google Scholar
Celani, A., Mazzino, A. & Tizzi, M. 2009 Droplet feedback on vapor in a warm cloud. Intl J. Mod Phys. B 23 (28n29), 5434–5443.CrossRefGoogle Scholar
Devenish, B. J., Bartello, P., Brenguier, J.-L., Collins, L. R., Grabowski, W. W., IJzermans, R. H. A., Malinowski, S. P., Reeks, M. W., Vassilicos, J.-C., Wang, L.-P. et al. 2012 Droplet growth in warm turbulent clouds. Q. J. R. Meteorol. Soc. 138, 1401–1429.CrossRefGoogle Scholar
Devenish, B. J., Furtado, K. & Thomson, D. J. 2016 Analytical solutions of the supersaturation equation for a warm cloud. J. Atmos. Sci. 73 (9), 3453–3465.CrossRefGoogle Scholar
Field, P. R., Hill, A. A., Furtado, K. & Korolev, A. 2014 Mixed-phase clouds in a turbulent environment. Part 2: analytic treatment. Q. J. R. Meteorol. Soc. 140 (680), 870–880.Google Scholar
Gotoh, T. & Watanabe, T. 2015 Power and nonpower laws of passive scalar moments convected by isotropic turbulence. Phys. Rev. Lett. 115, 114502.CrossRefGoogle ScholarPubMed
Grabowski, W. W. & Wang, L.-P. 2013 Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid Mech. 45 (1), 293–324.CrossRefGoogle Scholar
Homann, H., Dreher, J. & Grauer, R. 2007 Impact of the floating-point precision and interpolation scheme on the results of DNS of turbulence by pseudo-spectral codes. Comput. Phys. Commun. 177 (7), 560–565.Google Scholar
Ingersoll, A. P., Dowling, T. E., Gierasch, P. J., Orton, G. S., Read, P. L., Sánchez-Lavega, A., Showman, A. P., Simon-Miller, A. A. & Vasavada, A. R. 2004 Dynamics of Jupiter’s atmosphere. In Jupiter: The Planet, Satellites and Magnetosphere (ed. Bagenal, F., Dowling, T. E. & McKinnon, W. B.), chap. 6. Cambridge University Press.Google Scholar
Khvorostyanov, V. I. & Curry, J. A. 1999 Toward the theory of stochastic condensation in clouds. Part I: a general kinetic equation. J. Atmos. Sci. 56 (23), 3985–3996.2.0.CO;2>CrossRefGoogle Scholar
Kulmala, M., Rannik, Ü., Zapadinsky, E. L. & Clement, C. F. 1997 The effect of saturation fluctuations on droplet growth. J. Atmos. Sci. 28 (8), 1395–1409.Google Scholar
Kumar, B., Janetzko, F., Schumacher, J. & Shaw, R. A. 2012 Extreme responses of a coupled scalar–particle system during turbulent mixing. New J. Phys. 14 (11), 115020.Google Scholar
Lanotte, A. S., Seminara, A. & Toschi, F. 2009 Cloud droplet growth by condensation in homogeneous isotropic turbulence. J. Atmos. Sci. 66 (6), 1685–1697.Google Scholar
Lasher-Trapp, S. G., Cooper, W. A. & Blyth, A. M. 2005 Broadening of droplet size distributions from entrainment and mixing in a cumulus cloud. Q. J. R. Meteorol. Soc. 131 (605), 195–220.CrossRefGoogle Scholar
Lehmann, K., Siebert, H. & Shaw, R. A. 2009 Homogeneous and inhomogeneous mixing in cumulus clouds: dependence on local turbulence structure. J. Atmos. Sci. 66 (12), 3641–3659.Google Scholar
McGraw, R. & Liu, Y. 2006 Brownian drift–diffusion model for evolution of droplet size distributions in turbulent clouds. Geophys. Res. Lett. 33 (3), L03802.Google Scholar
Paoli, R. & Shariff, K. 2009 Turbulent condensation of droplets: direct simulation and a stochastic model. J. Atmos. Sci. 66 (3), 723–740.Google Scholar
Pinsky, M., Mazin, I. P., Korolev, A. & Khain, A. 2013 Supersaturation and diffusional droplet growth in liquid clouds. J. Atmos. Sci. 70 (9), 2778–2793.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pruppacher, H. & Klett, J. 1997 Microphysics of Clouds and Precipitation. Kluwer Academic Publishers.Google Scholar
Reveillon, J. & Demoulin, F.-X. 2007 Effects of the preferential segregation of droplets on evaporation and turbulent mixing. J. Fluid Mech. 583, 273–302.Google Scholar
Sancho, J. M., San Miguel, M. & Dürr, D. 1982 Adiabatic elimination for systems of Brownian particles with nonconstant damping coefficients. J. Stat. Phys. 28 (2), 291–305.CrossRefGoogle Scholar
Sardina, G., Picano, F., Brandt, L. & Caballero, R. 2015 Continuous growth of droplet size variance due to condensation in turbulent clouds. Phys. Rev. Lett. 115, 184501.Google Scholar
Seifert, A. & Beheng, K. D. 2006 A two-moment cloud microphysics parameterization for mixed-phase clouds. Part 1: model description. Meteorol. Atmos. Phys. 92 (1–2), 45–66.Google Scholar
Shaw, R. A. 2003 Particle–turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183–227.Google Scholar
Sidin, R. S., IJzermans, R. H. & Reeks, M. W. 2009 A Lagrangian approach to droplet condensation in atmospheric clouds. Phys. Fluids 21 (10), 106603.Google Scholar
Squires, P. 1952 The growth of cloud drops by condensation. I. General characteristics. J. Sci. Res. A 5 (1), 59–86.Google Scholar
Srivastava, R. C. 1989 Growth of cloud drops by condensation: a criticism of currently accepted theory and a new approach. J. Atmos. Sci. 46 (7), 869–887.Google Scholar
Twomey, S. 1959 The nuclei of natural cloud formation part II: the supersaturation in natural clouds and the variation of cloud droplet concentration. Geofis. Appl. 43 (1), 243–249.Google Scholar
Vaillancourt, P. A., Yau, M. K., Bartello, P. & Grabowski, W. W. 2002 Microscopic approach to cloud droplet growth by condensation. Part II: turbulence, clustering, and condensational growth. J. Atmos. Sci. 59 (24), 3421–3435.Google Scholar
Vaillancourt, P. A., Yau, M. K. & Grabowski, W. W. 2001 Microscopic approach to cloud droplet growth by condensation. Part I: model description and results without turbulence. J. Atmos. Sci. 58 (14), 1945–1964.Google Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32 (1), 203–240.Google Scholar
Wetchagarun, S. & Riley, J. J. 2010 Dispersion and temperature statistics of inertial particles in isotropic turbulence. Phys. Fluids 22 (6), 063301.CrossRefGoogle Scholar
Yeung, P. K. 2001 Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations. J. Fluid Mech. 427, 241–274.CrossRefGoogle Scholar