pure and applied geophysics

, Volume 115, Issue 4, pp 805–843 | Cite as

On the hydrodynamic behavior of supercooled water drops interacting with columnar ice crystals

  • R. J. Schlamp
  • H. R. Pruppacher


A numerical evaluation of the complete Navier-Stokes equations of motion for steady-state, incompressible flow past an infinite circular cylinder is given in terms of the stream function, vorticity, and pressure distribution past such bodies. A method is described which allows use of these flow characteristics: (1) to approximate the characteristics of air flow past hexagonal columnar ice crystals falling under gravity at terminal velocity in air, (2) to compute the trajectory of supercooled cloud drops relative to such ice crystals, and (3) to determine the efficiency with which short columnar ice crystals and needle shaped ice crystals collide with supercooled cloud drops. It is found that for all columnar type ice crystals riming is negligible if the cloud drop size is less than 5∼ μm, and that for riming to commence short columnar crystals must have diameters larger than ∼50 μm, while needle crystals must have diameters larger than ∼40 μm. It is further shown that the collision efficiency cut-offs at the small drop radius and at the large drop radius end of the collision efficiency diagram can be explained on the basis of the cloud drop trajectories for these drop size ranges.

Key words

Ice crystals Collision efficiency Cloud drop trajectories 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Auer, A. H. andVeal, D. L. (1970),The dimension of ice crystals in natural clouds, J. Atmos. Sci.27, 919–926.Google Scholar
  2. Beard, K. V. (1976),Terminal velocity and shape of cloud and precipitation drops aloft, J. Atmos. Sci.33, 851–864.Google Scholar
  3. Dennis, S. C. R. andChang, G. Z. (1970),Numerical solutions for steady flow past a circular cylinder at Reynolds number up to 100, J. Fluid Mech.,42, 471–489.Google Scholar
  4. Dennis, S. C. R. andShimshoni, M. (1964),The steady flow of a viscous fluid past a circular cylinder, Great Britain Astronaut. Res. Council, Current Papers, No. 797.Google Scholar
  5. Finn, R. K. (1953),Determination of the drag on a cylinder at low Reynolds numbers, J. Appl. Phys.24, 771–773.Google Scholar
  6. Griffin, F. O. (1972),The impaction of spherical particles on circular cylinders, M. S. Thesis (Dept. Chem. Eng., University of British Columbia, Vancouver, Canada).Google Scholar
  7. Hamielec, A. E. andRaal, J. D. (1969),Numerical studies of viscous flow around circular cylinders, Phys. Fluids12, 11–17.Google Scholar
  8. Harimaya, T. (1975),The riming properties of snow crystals, J. Meteorol. Soc. Japan,53, 384–392.Google Scholar
  9. Hobbs, P. V. (1971),Studies of winter cyclonic storms over the Cascade Mts., Res. Rept., No. 6, Dept. Atmos. Sci., University of Washington, Seattle, Wash.Google Scholar
  10. Hocking, L. M. (1959),The collision efficiency of small drops, Quart. J. Roy. Meteorol. Soc.85, 44–50.Google Scholar
  11. Homann, F. (1936),Einfluss grosser Zähigkeit bei Strömung um Zylinder, Forschg. Geb. Ingenieurwesen7, 1–9.Google Scholar
  12. Iwai, K. (1973),On the characteristic features of snow crystals developed along the c-axis, J. Meteor. Soc. Japan51, 458–465.Google Scholar
  13. Jayaweera, K. O. L. F. andCottis, R. E. (1969),Fall velocities of plate-like and columnaricecrystals, Quart. J. Roy. Meteorol. Soc.95, 703–709.Google Scholar
  14. Jayaweera, K. O. L. F. andRyan, B. F. (1972),Terminal velocities of ice crystals, Quart. J. Roy. Meteorol. Soc.98, 193–197.Google Scholar
  15. Kajikawa, M. (1971),A model experimental study on the fall velocity of ice crystals, J. Meteorol. Soc. Japan49, 367–375.Google Scholar
  16. Kaplun, S. (1957),Low Reynolds number flow past a circular cylinder, J. Math. Mech.6, 595–603.Google Scholar
  17. Keller, H. B. andTakami, H.,Numerical studies of steady viscous flow about cylinders, in Proc. Adv. Symp. on Numerical Solutions of Nonlinear Differential Equations (ed. D. Greenspan), May 9–11 (University of Wisconsin, Madison, 1966), pp. 115–140.Google Scholar
  18. Knight, C. A. andKnight, N. C. (1973),Conical graupel, J. Atmos. Sci.30, 118–124.Google Scholar
  19. Koenig, L. R. (1971),Numerical modeling of ice deposition, J. Atmos. Sci.28, 226–237.Google Scholar
  20. Lamb, H. (1911),On the uniform motion of j a sphere in a viscous fluid, Phil. Mag.21, 112–119.Google Scholar
  21. Langmuir, I. (1948),The production of rain by a chain reaction in cumulus clouds at temperatures above freezing, J. Meteorol.5, 175–192.Google Scholar
  22. le Clair, B. P., Hamielec, A. E. andPruppacher, H. R. (1970),A numerical study of the drag on a sphere at low and intermediate Reynolds numbers, J. Atmos. Sci.27, 308–315.Google Scholar
  23. Lin, C. L. andLee, S. C. (1975),Collision efficiency of water drops in the atmosphere, J. Atmos. Sci.32, 1412–1418.Google Scholar
  24. Nieuwstadt, F. andKeller, H. B. (1973),Viscous flow past circular cylinders, Comput. Fluids1, 59–71.Google Scholar
  25. Nishioka, M. andSato, H. (1974),Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers, J. Fluid Mech.65, 97–112.Google Scholar
  26. Ono, A. (1969),The shape and riming properties of ice crystals in natural clouds, J. Atmos. Sci.26, 138–147.Google Scholar
  27. Pitter, R. L., Pruppacher, H. R. andHamielec, A. E. (1973),A numerical study of the viscous flow past a thin oblate spheroid at low and intermediate Reynolds numbers, J. Atmos. Sci.30, 125–134.Google Scholar
  28. Pitter, R. L. andPruppacher, H. R. (1974),A numerical investigation of collision efficiencies of of simple ice plates colliding with supercooled water drops, J. Atmos. Sci.31, 551–559.Google Scholar
  29. Pruppacher, H. R., Le Clair, B. P. andHamielec, A. E. (1970),Some relations between the drag and flow patterns of viscous flow past a sphere and a cylinder at low and intermediate Reynolds numbers. J. Fiuid Mech.44, 781–790.Google Scholar
  30. Reinking, R. F.,The onset and early growth of snow crystals by riming, preprints,Internat. Conf. on Cloud Physics, July 26–30, 1976, Boulder, Colorado (Am. Meterorol. Soc., Boston, Mass., 1976), pp. 207–214.Google Scholar
  31. Roshko, A. (1954),On the development of turbulent wakes from vortex streets. Rept. Nat. Advisory Comm. on Aeronautics (NACA), No. 1191.Google Scholar
  32. Schlamp, R. J., Pruppacher, H. R. andHamielec, A. E. (1975),A numerical investigation of the efficiency with which simple columnar ice crystals collide with supercooled water drops. J. Atmos. Sci.32, 2330–2337.Google Scholar
  33. Schlamp, R. J., Grover, S. N., Pruppacher, H. R. andHamielec, A. E. (1976),A numerical investigation of the effect of electric charges and vertical external electric fields on the collision efficiency of cloud drops, J. Atmos. Sci.33, 1747–1755.Google Scholar
  34. Shafrir, U. andNeiburger, M. (1963),Collision efficiency of two spheres falling in a viscous medium, J. Geophys. Res.68, 4141–4147.Google Scholar
  35. Shafrir, U. andGal-Chen, T. (1971),A numerical study of collision efficiencies and coalescence parameters for droplet pairs with radii up to 300 μm, J. Atmos. Sci.28, 741–751.Google Scholar
  36. Takami, H. andKeller, H. B. (1969),Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder, Phys. Fluids12, Suppl. II, 51–56.Google Scholar
  37. Taneda, S. (1956),Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers, J. Phys. Soc., Japan11, 302–307.Google Scholar
  38. Tritton, D. J. (1959),Experiments on the flow past a circular cylinder at low Reynolds numbers, J. Fluid Mech.6, 547–567.Google Scholar
  39. Underwood, R. L. (1969),Calculation of incompressible flow past a circular cylinder at moderate Reynolds numbers, J. Fluid Mech.37, 95–114.Google Scholar
  40. Wieselsberger, C. (1922),Weitere Festellungen über die Gesetze des Flüssigkeits-und Luftwiderstandes, Physik. Zeitschr.23, 219–224.Google Scholar
  41. Wilkins, R. I. andAuer, A. H.,Riming properties of hexagonal ice crystals, preprintsConf. Cloud Physics, Fort Collins, colorado (Amer. Meteor. Soc., 1970), pp. 81–82.Google Scholar
  42. Woo, S. (1971),Simultaneous free and forced convection around submerged cylinders and spheres, Ph. D. Thesis (Dept. Chem. Eng., McMaster University, Hamilton, Canada).Google Scholar
  43. Zikmunda, J. andVali, G. (1972),Fall patterns and fall velocities of rimed ice crystals, J. Atmos. Sci.29, 1334–1347.Google Scholar

Copyright information

© Birkhäuser Verlag 1977

Authors and Affiliations

  • R. J. Schlamp
    • 1
  • H. R. Pruppacher
    • 1
  1. 1.Department of Atmospheric SciencesUniversity of CaliforniaLos AngelesUSA

Personalised recommendations