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Collision rates of small ellipsoids settling in turbulence

Published online by Cambridge University Press:  10 October 2014

C. Siewert ,
R. P. J. Kunnen  and
W. Schröder
C. Siewert*
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstrasse 5a, 52062 Aachen, Germany
R. P. J. Kunnen
Affiliation:
Fluid Dynamics Laboratory, Department of Physics and J.M. Burgers Centre for Fluid Mechanics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
W. Schröder
Affiliation:
Institute of Aerodynamics, RWTH Aachen University, Wüllnerstrasse 5a, 52062 Aachen, Germany
*
†Email address for correspondence: c.siewert@aia.rwth-aachen.de
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Abstract

We propose that the collision rates of non-spherical particles settling in a turbulent environment are significantly higher than those of spherical particles of the same mass and volume. The theoretical argument is based on the dependence of the particle drag force on the particle orientation, thus varying gravitational settling velocities, which can remain different until contact due to the particle inertia. Therefore, non-spherical particles can collide with large relative velocities. Direct numerical simulations (DNS) of streamwise decaying isotropic turbulence seeded with small, heavy, rotationally symmetric ellipsoids of five different aspect ratios are performed to confirm these arguments. The motion of 21 million ellipsoids is tracked by a Lagrangian particle solver assuming creeping flow conditions and neglecting the influence of the particles on the flow. We find that ellipsoids collide considerably more often than spherical particles of the same volume and mass due to a drastically increased mean relative velocity at contact.

JFM classification

Turbulent Flows: Isotropic turbulence Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 686 - 701
Copyright
© 2014 Cambridge University Press 

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References

Abrahamson, J. 1975 Collision rates of small particles in a vigorously turbulent fluid. Chem. Engng Sci. 30, 1371–1379.CrossRefGoogle Scholar
Batten, P., Goldberg, U. & Chakravarthy, S. 2004 Interfacing statistical turbulence closures with large-eddy simulation. AIAA J. 42 (3), 485–492.CrossRefGoogle Scholar
Bec, J., Biferale, L., Boffetta, G., Celani, A., Cencini, M., Lanotte, A. S., Musacchio, S. & Toschi, F. 2006 Acceleration statistics of heavy particles in turbulence. J. Fluid Mech. 550, 349–358.CrossRefGoogle Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. S. & Toschi, F. 2010 Intermittency in the velocity distribution of heavy particles in turbulence. J. Fluid Mech. 646, 527–536.CrossRefGoogle Scholar
Bec, J., Celani, A., Cencini, M. & Musacchio, S. 2005 Clustering and collisions of heavy particles in random smooth flows. Phys. Fluids 17, 073301.CrossRefGoogle Scholar
Bernstein, O. & Shapiro, M. 1994 Direct determination of the orientation distribution function of cylindrical particles immersed in laminar and turbulent shear flows. J. Aerosol Sci. 25 (1), 113–136.CrossRefGoogle Scholar
Bewley, G. P., Saw, E.-W. & Bodenschatz, E. 2013 Observation of the sling effect. New J. Phys. 15 (8), 083051.CrossRefGoogle Scholar
Brenner, H. & Cox, R. G. 1963 The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers. J. Fluid Mech. 17, 561–595.CrossRefGoogle Scholar
Chester, W. 1990 A general theory for the motion of a body through a fluid at low Reynolds number. Proc. R. Soc. Lond. A 430, 89–104.Google Scholar
Choi, Y.-K., Chang, J.-W., Wang, W., Kim, M.-S. & Elber, G. 2009 Continuous collision detection for ellipsoids. IEEE Trans. Vis. Comput. Graphics 15 (2), 311–325.CrossRefGoogle ScholarPubMed
Cox, R. G. 1965 The steady motion of a particle of arbitrary shape at small Reynolds numbers. J. Fluid Mech. 23, 625–643.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. & Warhaft, Z. 2012 Droplet growth in warm turbulent clouds. Q. J. R. Meteorol. Soc. 138, 1401–1429.CrossRefGoogle Scholar
Dullemond, C. P. & Dominik, C. 2005 Dust coagulation in protoplanetary disks: a rapid depletion of small grains. Astron. Astrophys. 434 (3), 971–986.CrossRefGoogle Scholar
Durham, W. M., Climent, E., Barry, M., De Lillo, F., Boffetta, G., Cencini, M. & Stocker, R. 2013 Turbulence drives microscale patches of motile phytoplankton. Nat. Commun. 4, 2148.CrossRefGoogle ScholarPubMed
Falkovich, G., Fouxon, A. & Stepanov, M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154.CrossRefGoogle ScholarPubMed
Falkovich, G. & Pumir, A. 2007 Sling effect in collisions of water droplets in turbulent clouds. J. Atmos. Sci. 64, 4497–4505.CrossRefGoogle Scholar
Fan, F.-G. & Ahmadi, G. 1995 A sublayer model for wall deposition of ellipsoidal particles in turbulent streams. J. Aerosol Sci. 26 (5), 813–840.CrossRefGoogle Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994 Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95–134.CrossRefGoogle Scholar
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
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Hartmann, D., Meinke, M. & Schröder, W. 2008 An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods. Comput. Fluids 37, 1103–1125.CrossRefGoogle Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Kunnen, R. P. J., Siewert, C., Meinke, M., Schröder, W. & Beheng, K. D. 2013 Numerically determined geometric collision kernels in spatially evolving isotropic turbulence relevant for droplets in clouds. Atmos. Res. 127, 8–21.CrossRefGoogle Scholar
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12 (1), 435–476.CrossRefGoogle Scholar
Lundell, F., Söderberg, L. D. & Alfredsson, P. H. 2011 Fluid mechanics of papermaking. Annu. Rev. Fluid Mech. 43 (1), 195–217.CrossRefGoogle Scholar
Mao, W. & Alexeev, A. 2014 Motion of spheroid particles in shear flow with inertia. J. Fluid Mech. 749, 145–166.CrossRefGoogle Scholar
Marchioli, C., Fantoni, M. & Soldati, A. 2010 Orientation, distribution, and deposition of elongated, inertial fibers in turbulent channel flow. Phys. Fluids 22 (3), 033301.CrossRefGoogle Scholar
Mortensen, P. H., Andersson, H. I., Gillissen, J. J. J. & Boersma, B. J. 2008 Dynamics of prolate ellipsoidal particles in a turbulent channel flow. Phys. Fluids 20, 093302.CrossRefGoogle Scholar
Newsom, R. K. & Bruce, C. W. 1994 The dynamics of fibrous aerosols in a quiescent atmosphere. Phys. Fluids 6, 521–530.CrossRefGoogle Scholar
Oberbeck, A. 1876 Ueber stationäre Flüssigkeitsbewegungen mit Berücksichtigung der inneren Reibung. Crelle’s J. 81, 62–80.Google Scholar
Parsa, S., Calzavarini, E., Toschi, F. & Voth, G. A. 2012 Rotation rate of rods in turbulent fluid flow. Phys. Rev. Lett. 109, 134501.CrossRefGoogle ScholarPubMed
Pinsky, M. B. & Khain,   1998 Some effects of cloud turbulence on water–ice and ice–ice collisions. Atmos. Res. 47–48, 69–86.CrossRefGoogle Scholar
Pinsky, M. B., Khain, A. P. & Shapiro, M. 2007 Collisions of cloud droplets in a turbulent flow. Part IV: droplet hydrodynamic interaction. J. Atmos. Sci. 64, 2462–2482.CrossRefGoogle Scholar
Pruppacher, H. & Klett, J. 1997 Microphysics of Clouds and Precipitation. Kluwer.Google Scholar
Pumir, A. & Wilkinson, M. 2011 Orientation statistics of small particles in turbulence. New J. Phys. 13 (9), 093030.CrossRefGoogle Scholar
Saffman, P. G. & Turner, J. S. 1956 On the collision of drops in turbulent clouds. J. Fluid Mech. 1, 16–30.CrossRefGoogle Scholar
Saw, E.-W., Shaw, R. A., Salazar, J. P. L. C. & Collins, L. R. 2012 Spatial clustering of polydisperse inertial particles in turbulence: II. Comparing simulation with experiment. New J. Phys. 14 (10), 105031.Google Scholar
Schneiders, L., Hartmann, D., Meinke, M. & Schröder, W. 2013 An accurate moving boundary formulation in cut-cell methods. J. Comput. Phys. 235, 786–809.CrossRefGoogle Scholar
Shin, M., Koch, D. L. & Subramanian, G. 2006 A pseudospectral method to evaluate the fluid velocity produced by an array of translating slender fibers. Phys. Fluids 18 (6), 063301.CrossRefGoogle Scholar
Siewert, C., Kunnen, R. P. J., Meinke, M. & Schröder, W. 2014a On the collision detection for ellipsoidal particles in turbulence. In ASME 2014 4th Joint US-European Fluids Engineering Division Summer Meeting, Chicago, Paper FEDSM2014-21982 (accepted for publication).Google Scholar
Siewert, C., Kunnen, R. P. J., Meinke, M. & Schröder, W. 2014b Orientation statistics and settling velocity of ellipsoids in decaying turbulence. Atmos. Res. 142, 45–56 (the 16th International Conference on Clouds and Precipitation).CrossRefGoogle Scholar
Squires, K. D. & Eaton, J. L. 1991 Preferential concentration of particles by turbulence. Phys. Fluids A 3, 1169–1178.CrossRefGoogle Scholar
Sundaram, S. & Collins, L. R. 1997 Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations. J. Fluid Mech. 335, 75–109.CrossRefGoogle Scholar
Wang, L.-P., Ayala, O., Rosa, B. & Grabowski, W. W. 2008 Turbulent collision efficiency of heavy particles relevant to cloud droplets. New J. Phys. 10, 075013.CrossRefGoogle Scholar
Wang, L.-P. & Maxey, M. R. 1993 Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 27–68.CrossRefGoogle Scholar
Wang, L.-P., Wexler, A. S. & Zhou, Y. 2000 Statistical mechanical description and modelling of turbulent collision of inertial particles. J. Fluid Mech. 415, 117–153.CrossRefGoogle Scholar
Wilkinson, M., Mehlig, B. & Bezuglyy, V. 2006 Caustic activation of rain showers. Phys. Rev. Lett. 97, 048501.CrossRefGoogle ScholarPubMed
Williams, J. P. & Cieza, L. A. 2011 Protoplanetary disks and their evolution. Annu. Rev. Astron. Astrophys. 49 (1), 67–117.CrossRefGoogle Scholar
Xue, Y., Wang, L.-P. & Grabowski, W. W. 2008 Growth of cloud droplets by turbulent collision-coalescence. J. Atmos. Sci. 65 (2), 331–356.CrossRefGoogle Scholar
Zhan, C., Sardina, G., Lushi, E. & Brandt, L. 2014 Accumulation of motile elongated micro-organisms in turbulence. J. Fluid Mech. 739, 22–36.CrossRefGoogle Scholar
Zhang, H., Ahmadi, G., Fan, F.-G. & McLaughlin, J. B. 2001 Ellipsoidal particles transport and deposition in turbulent channel flows. Intl J. Multiphase Flow 27 (6), 971–1009.CrossRefGoogle Scholar
Zheng, X., Iglesias, W. & Palffy-Muhoray, P. 2009 Distance of closest approach of two arbitrary hard ellipsoids. Phys. Rev. E 79, 057702.CrossRefGoogle ScholarPubMed