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Collision efficiency of cloud droplets in quiescent air considering lubrication interactions, mobility of interfaces, and noncontinuum molecular effects

Ahmad Ababaei* and Bogdan Rosa†

  • Institute of Meteorology and Water Management – National Research Institute, Podleśna 61, 01-673 Warsaw, Poland

  • *ahmad.ababaei@imgw.pl
  • †Also at the Department of Applied Mathematics, Warsaw University of Life Sciences, 02-776 Warsaw, Poland
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Phys. Rev. Fluids 8, 014102 – Published 17 January, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.014102

Abstract

The collision efficiency of cloud droplets settling under gravity in quiescent air is investigated by means of numerical simulations. In the developed model, the droplets are represented as either rigid spheres or nondeformable liquid particles of finite viscosity. For the latter, both the internal circulation of the fluid and the mobility of interfaces are accounted for. The aerodynamic interaction, resulting from relative motion of the particles in a viscous medium, is evaluated by making use of a Stokes flow solution. The effect of noncontinuum lubrication for the flow in the gap between surfaces of the droplets is also analyzed. This provides a more physical description of aerodynamic interactions valid for a wide range of the Knudsen number and gap sizes. In contrast to an earlier study by Rother et al. [Int. J. Multiphase Flow 146, 103876 (2022)], noncontinuum lubrication and internal circulation effects have been analyzed separately. Additionally, rotational motion is considered for rigid particles. An objective comparison of the obtained results with the reference data has been performed as well. Compared to the standard continuum description of aerodynamic interaction for nonrotating rigid spherical particles, noncontinuum lubrication and internal circulation effects both lead to a larger collision efficiency, whereas rotation reduces collision efficiency. In general, noncontinuum lubrication has a larger impact on the collision efficiency compared to the internal circulation of drops, which loses its influence as their inertia (size) increases. In numerical simulations, therefore, treating medium-sized cloud droplets as rigid particles is an accurate assumption, but considering noncontinuum effects in their aerodynamic interaction is expected to alter the results. In the limit of a large viscosity ratio, the values of collision efficiency for the liquid drops and freely rotating rigid particles are in a quantitative agreement. Numerical aspects are also discussed, with a focus on assessing computational complexity and algorithm parallelization. The simplified problem studied here is an important step towards improving the representation of aerodynamic interaction in systems with a large number of droplets interacting in a turbulent flow.

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References (55)

  1. W. W. Grabowski and L.-P. Wang, Growth of cloud droplets in a turbulent environment, Annu. Rev. Fluid Mech. 45, 293 (2013).
  2. L. Brandt and F. Coletti, Particle-laden turbulence: Progress and perspectives, Annu. Rev. Fluid Mech. 54, 159 (2022).
  3. K. V. Beard and H. T. Ochs III, Warm-rain initiation: An overview of microphysical mechanisms, J. Appl. Meteor. 32, 608 (1993).
  4. M. Voßkuhle, A. Pumir, E. Lévêque, and M. Wilkinson, Prevalence of the sling effect for enhancing collision rates in turbulent suspensions, J. Fluid Mech. 749, 841 (2014).
  5. X.-Y. Li, A. Brandenburg, G. Svensson, N. E. Haugen, B. Mehlig, and I. Rogachevskii, Effect of turbulence on collisional growth of cloud droplets, J. Atmos. Sci. 75, 3469 (2018).
  6. M. A. Rother, J. K. Stark, and R. Davis, Gravitational collision efficiencies of small viscous drops at finite Stokes numbers and low Reynolds numbers, Int. J. Multiphase Flow 146, 103876 (2022).
  7. A. M. Blyth, Entrainment in cumulus clouds, J. Appl. Meteor. 32, 626 (1993).
  8. Y. Yin, Z. Levin, T. G. Reisin, and S. Tzivion, The effects of giant cloud condensation nuclei on the development of precipitation in convective clouds—A numerical study, Atmos. Res. 53, 91 (2000).
  9. R. A. Shaw, W. C. Reade, L. R. Collins, and J. Verlinde, Preferential concentration of cloud droplets by turbulence: Effects on the early evolution of cumulus cloud droplet spectra, J. Atmos. Sci. 55, 1965 (1998).
  10. W. C. Reade and L. R. Collins, A numerical study of the particle size distribution of an aerosol undergoing turbulent coagulation, J. Fluid Mech. 415, 45 (2000).
  11. S. Sundaram and L. R. Collins, Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations, J. Fluid Mech. 335, 75 (1997).
  12. L.-P. Wang, O. Ayala, B. Rosa, and W. W. Grabowski, Turbulent collision efficiency of heavy particles relevant to cloud droplets, New J. Phys. 10, 075013 (2008).
  13. R. H. Davis, The rate of coagulation of a dilute polydisperse system of sedimenting spheres, J. Fluid Mech. 145, 179 (1984).
  14. R. H. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation, Atmospheric and Oceanographic Sciences Library, Vol. 18 (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997) .
  15. L.-P. Wang, O. Ayala, and W. W. Grabowski, Improved formulations of the superposition method, J. Atmos. Sci. 62, 1255 (2005).
  16. O. Ayala, W. W. Grabowski, and L.-P. Wang, A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles, J. Comput. Phys. 225, 51 (2007).
  17. A. D. Maude, End effects in a falling-sphere viscometer, Br. J. Appl. Phys. 12, 293 (1961).
  18. R. H. Davis, J. A. Schonberg, and J. M. Rallison, The lubrication force between two viscous drops, Phys. Fluids 1, 77 (1989).
  19. R. R. Sundararajakumar and D. L. Koch, Non-continuum lubrication flows between particles colliding in a gas, J. Fluid Mech. 313, 283 (1996).
  20. L. M. Hocking and P. R. Jonas, The collision efficiency of small drops, Q. J. Royal Met. Soc. 96, 722 (1970).
  21. B. Rosa, L.-P. Wang, M. R. Maxey, and W. W. Grabowski, An accurate and efficient method for treating aerodynamic interactions of cloud droplets, J. Comput. Phys. 230, 8109 (2011).
  22. J. Chun and D. L. Koch, Coagulation of monodisperse aerosol particles by isotropic turbulence, Phys. Fluids 17, 027102 (2005).
  23. J. Dhanasekaran, A. Roy, and D. L. Koch, Collision rate of bidisperse spheres settling in a compressional non-continuum gas flow, J. Fluid Mech. 910, A10 (2021).
  24. J. Dhanasekaran, A. Roy, and D. L. Koch, Collision rate of bidisperse, hydrodynamically interacting spheres settling in a turbulent flow, J. Fluid Mech. 912, A5 (2021).
  25. A. Z. Zinchenko, Calculation of hydrodynamic interaction between drops at low Reynolds numbers, J. Appl. Math. Mech. 42, 1046 (1978).
  26. S. Haber, G. Hetsroni, and A. Solan, On the low Reynolds number motion of two droplets, Int. J. Multiphase Flow 1, 57 (1973).
  27. A. Z. Zinchenko, The slow asymmetric motion of two drops in a viscous medium, J. Appl. Math. Mech. 44, 30 (1980).
  28. A. Z. Zinchenko, Calculation of the effectiveness of gravitational coagulation of drops with allowance for internal circulation, J. Appl. Math. Mech. 46, 58 (1982).
  29. M. K. Yau and R. R. Rogers, A Short Course in Cloud Physics (Elsevier, 1996).
  30. C. K. Batchelor and G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 2000).
  31. S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications (Courier Corporation, 2013).
  32. J. S. Hadamard, Mouvement permanent lent d'une sphère liquide et visqueuse dans un liquide visqueux, C. R. Acad. Sci. (Paris) 152, 1735 (1911).
  33. W. Rybczyński, Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium, Bull. Acad. Sci. Cracovie A, 40 (1911).
  34. D. J. Jeffrey and Y. Onishi, Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow, J. Fluid Mech. 139, 261 (1984).
  35. M. Stimson and G. B. Jeffery, The motion of two spheres in a viscous fluid, Proc. R. Soc. Lond. 111, 110 (1926).
  36. M. E. O'Neill and S. R. Majumdar, Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part I: The determination of exact solutions for any values of the ratio of radii and separation parameters, Z. Angew. Math. Phys. 21, 164 (1970).
  37. L. D. Reed and F. A. Morrison, Particle interactions in viscous flow at small values of Knudsen number, J. Aerosol Sci. 5, 175 (1974).
  38. E. Wacholder and D. Weihs, Slow motion of a fluid sphere in the vicinity of another sphere or a plane boundary, Chem. Eng. Sci. 27, 1817 (1972).
  39. A. J. Goldman, R. G. Cox, and H. Brenner, The slow motion of two identical arbitrarily oriented spheres through a viscous fluid, Chem. Eng. Sci. 21, 1151 (1966).
  40. M. E. O'Neill, Exact solutions of the equations of slow viscous flow generated by the asymmetrical motion of two equal spheres, Appl. Sci. Res. 21, 452 (1969).
  41. S. G. Jennings, The mean free path in air, J. Aerosol Sci. 19, 159 (1988).
  42. M. H. Davis, Collisions of small cloud droplets: Gas kinetic effects, J. Atmos. Sci. 29, 911 (1972).
  43. https://github.com/aababaei/Two-Sphere-Interaction.
  44. https://www.uwo.ca/apmaths/faculty/jeffrey/research/Resistance.html.
  45. H. Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface, Chem. Eng. Sci. 16, 242 (1961).
  46. G. B. Jeffery, On the steady rotation of a solid of revolution in a viscous fluid, Proc. London Math. Soc. S214, 327 (1915).
  47. E. Sternberg and M. A. Sadowsky, On the axisymmetric problem of the theory of elasticity for an infinite region containing two spherical cavities, J. Appl. Mech. 19, 19 (1952).
  48. S. I. Grashchenkov, The effect of slip on the motion of two droplets and of a droplet close to a plane surface of a liquid, Aerosol Sci. Technol. 25, 101 (1996).
  49. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media (Martinus Nijhof, 1983).
  50. B. D. Goddard, R. D. Mills-Williams, and J. Sun, The singular hydrodynamic interactions between two spheres in Stokes flow, Phys. Fluids 32, 062001 (2020).
  51. V. N. Beshkov, B. P. Radoev, and I. B. Ivanov, Slow motion of two droplets and a droplet towards a fluid or solid interface, Int. J. Multiphase Flow 4, 563 (1978).
  52. A. K. Townsend, The mechanics of suspensions, Ph. D. thesis, University College London, 2017.
  53. K. Ichiki, Errata on the papers for two-body exact solutions in Stokes flows, http://ryuon.sourceforge.net/twobody/errata.html.
  54. A. J. C. Ladd, Hydrodynamic transport coefficients of random dispersions of hard spheres, J. Chem. Phys. 93, 3484 (1990).
  55. M. A. Rother, Effects of incompressible surfactant on thermocapillary interactions of spherical drops, Int. J. Multiphase Flow 35, 417 (2009).

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Information

Phys. Rev. Fluids 8, 014102– Published 17 January, 2023
  • Received 19 August 2022
  • Accepted 22 December 2022
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