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The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 1. Simulations without gravitational effects

Published online by Cambridge University Press:  11 May 2016

Peter J. Ireland ,
Andrew D. Bragg  and
Lance R. Collins
Peter J. Ireland
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA International Collaboration for Turbulence Research
Andrew D. Bragg
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA International Collaboration for Turbulence Research
Lance R. Collins*
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA International Collaboration for Turbulence Research
*
†Email address for correspondence: lc246@cornell.edu
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Abstract

In this study, we analyse the statistics of both individual inertial particles and inertial particle pairs in direct numerical simulations of homogeneous isotropic turbulence in the absence of gravity. The effect of the Taylor microscale Reynolds number, $R_{{\it\lambda}}$, on the particle statistics is examined over the largest range to date (from $R_{{\it\lambda}}=88$ to 597), at small, intermediate and large Kolmogorov-scale Stokes numbers $St$. We first explore the effect of preferential sampling on the single-particle statistics and find that low-$St$ inertial particles are ejected from both vortex tubes and vortex sheets (the latter becoming increasingly prevalent at higher Reynolds numbers) and preferentially accumulate in regions of irrotational dissipation. We use this understanding of preferential sampling to provide a physical explanation for many of the trends in the particle velocity gradients, kinetic energies and accelerations at low $St$, which are well represented by the model of Chun et al. (J. Fluid Mech., vol. 536, 2005, pp. 219–251). As $St$ increases, inertial filtering effects become more important, causing the particle kinetic energies and accelerations to decrease. The effect of inertial filtering on the particle kinetic energies and accelerations diminishes with increasing Reynolds number and is well captured by the models of Abrahamson (Chem. Engng Sci., vol. 30, 1975, pp. 1371–1379) and Zaichik & Alipchenkov (Intl J. Multiphase Flow, vol. 34 (9), 2008, pp. 865–868), respectively. We then consider particle-pair statistics, and focus our attention on the relative velocities and radial distribution functions (RDFs) of the particles, with the aim of understanding the underlying physical mechanisms contributing to particle collisions. The relative velocity statistics indicate that preferential sampling effects are important for $St\lesssim 0.1$ and that path-history/non-local effects become increasingly important for $St\gtrsim 0.2$. While higher-order relative velocity statistics are influenced by the increased intermittency of the turbulence at high Reynolds numbers, the lower-order relative velocity statistics are only weakly sensitive to changes in Reynolds number at low $St$. The Reynolds-number trends in these quantities at intermediate and large $St$ are explained based on the influence of the available flow scales on the path-history and inertial filtering effects. We find that the RDFs peak near $St$ of order unity, that they exhibit power-law scaling for low and intermediate $St$ and that they are largely independent of Reynolds number for low and intermediate $St$. We use the model of Zaichik & Alipchenkov (New J. Phys., vol. 11, 2009, 103018) to explain the physical mechanisms responsible for these trends, and find that this model is able to capture the quantitative behaviour of the RDFs extremely well when direct numerical simulation data for the structure functions are specified, in agreement with Bragg & Collins (New J. Phys., vol. 16, 2014a, 055013). We also observe that at large $St$, changes in the RDF are related to changes in the scaling exponents of the relative velocity variances. The particle collision kernel closely matches that computed by Rosa et al. (New J. Phys., vol. 15, 2013, 045032) and is found to be largely insensitive to the flow Reynolds number. This suggests that relatively low-Reynolds-number simulations may be able to capture much of the relevant physics of droplet collisions and growth in the adiabatic cores of atmospheric clouds.

JFM classification

Turbulent Flows: Isotropic turbulence Multiphase and Particle-laden Flows: Particle/fluid flow Turbulent Flows: Turbulent Flows
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Papers
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Journal of Fluid Mechanics , Volume 796 , 10 June 2016 , pp. 617 - 658
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© 2016 Cambridge University Press 

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Footnotes

‡

Present address: Applied Mathematics and Plasma Physics Group, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.

References

Abrahamson, J. 1975 Collision rates of small particles in a vigorously turbulent fluid. Chem. Engng Sci. 30, 1371–1379.Google Scholar
Ashurst, W. T., Kerstein, A. R., Kerr, R. M. & Gibson, C. H. 1987 Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30 (8), 2343–2353.CrossRefGoogle Scholar
Ayala, O., Rosa, B., Wang, L.-P. & Grabowski, W. W. 2008 Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 1. Results from direct numerical simulation. New J. Phys. 10, 075015.CrossRefGoogle Scholar
Ayyalasomayajula, S., Warhaft, Z. & Collins, L. R. 2008 Modeling inertial particle acceleration statistics in isotropic turbulence. Phys. Fluids 20, 094104.CrossRefGoogle Scholar
Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111–133.CrossRefGoogle Scholar
Bec, J., Biferale, L., Boffetta, G., Celani, A., Cencini, M., Lanotte, A. S., Musacchio, S. & Toschi, F. 2006a Acceleration statistics of heavy particles in turbulence. J. Fluid Mech. 550, 349–358.CrossRefGoogle Scholar
Bec, J., Biferale, L., Boffetta, G., Cencini, M., Musacchio, S. & Toschi, F. 2006b Lyapunov exponents of heavy particles in turbulence. Phys. Fluids 18, 091702.Google Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. S., Musacchio, S. & Toschi, F. 2007 Heavy particle concentration in turbulence at dissipative and inertial scales. Phys. Rev. Lett. 98, 084502.Google Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. S. & Toschi, F. 2006c Effects of vortex filaments on the velocity of tracers and heavy particles in turbulence. Phys. Fluids 18, 081702.Google Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. S. & Toschi, F. 2010a Intermittency in the velocity distribution of heavy particles in turbulence. J. Fluid Mech. 646, 527–536.Google Scholar
Bec, J., Biferale, L., Lanotte, A. S., Scagliarini, A. & Toschi, F. 2010b Turbulent pair dispersion of inertial particles. J. Fluid Mech. 645, 497–528.Google Scholar
Biferale, L., Boffetta, G., Celani, A., Lanotte, A. & Toschi, F. 2005 Particle trapping in three-dimensional fully developed turbulence. Phys. Fluids 17, 021701.Google Scholar
Bragg, A. D. & Collins, L. R. 2014a New insights from comparing statistical theories for inertial particles in turbulence: I. Spatial distribution of particles. New J. Phys. 16, 055013.Google Scholar
Bragg, A. D. & Collins, L. R. 2014b New insights from comparing statistical theories for inertial particles in turbulence: II. Relative velocities. New J. Phys. 16, 055014.Google Scholar
Bragg, A. D., Ireland, P. J. & Collins, L. R. 2015 Mechanisms for the clustering of inertial particles in the inertial range of isotropic turbulence. Phys. Rev. E 92, 023029.CrossRefGoogle ScholarPubMed
Bragg, A. D., Ireland, P. J. & Collins, L. R. 2016 Forward and backward in time dispersion of fluid and inertial particles in isotropic turbulence. Phys. Fluids 28, 013305.Google Scholar
Calzavarini, E., Kerscher, M., Lohse, D. & Toschi, F. 2008 Dimensionality and morphology of particle and bubble clusters in turbulent flow. J. Fluid Mech. 607, 13–24.CrossRefGoogle Scholar
Chun, J., Koch, D. L., Rani, S., Ahluwalia, A. & Collins, L. R. 2005 Clustering of aerosol particles in isotropic turbulence. J. Fluid Mech. 536, 219–251.Google Scholar
Collins, L. R. & Keswani, A. 2004 Reynolds number scaling of particle clustering in turbulent aerosols. New J. Phys. 6, 119.Google Scholar
Computational and Information Systems Laboratory2012 Yellowstone: IBM iDataPlex System (University Community Computing). http://n2t.net/ark:/85065/d7wd3xhc.Google Scholar
Cuzzi, J. N., Hogan, R. C., Paque, J. M. & Dobrovolskis, A. R. 2001 Size-selective concentration of chrondrules and other small particles in protoplanetary nebula turbulence. Astrophys. J. 546, 496–508.CrossRefGoogle Scholar
Daitche, A. 2015 On the role of the history force for inertial particles in turbulence. J. Fluid Mech. 782, 567–593.Google Scholar
Dávila, J. & Hunt, J. C. R. 2001 Settling of small particles near vortices and in turbulence. J. Fluid Mech. 440, 117–145.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
Durham, W. M., Climent, E., Barry, M., Lillo, F. D., Boffetta, G., Cencini, M. & Stocker, R. 2013 Turbulence drives microscale patches of motile phytoplankton. Nat. Commun. 4 (2148), 1–7.CrossRefGoogle ScholarPubMed
Eaton, J. K. & Fessler, J. R. 1994 Preferential concentration of particles by turbulence. Intl J. Multiphase Flow 20, 169–209.Google Scholar
Elghobashi, S. E. & Truesdell, G. C. 1992 Direct simulation of particle dispersion in a decaying isotropic turbulence. J. Fluid Mech. 242, 655.Google Scholar
Elghobashi, S. E. & Truesdell, G. C. 1993 On the two-way interaction between homogeneous turbulence and dispersed particles. I. Turbulence modification. Phys. Fluids A 5, 1790–1801.CrossRefGoogle Scholar
ElMaihy, A. & Nicolleau, F. 2005 Investigation of the dispersion of heavy-particle pairs and Richardson’s law using kinematic simulation. Phys. Rev. E 71, 046307.Google Scholar
Falkovich, G., Fouxon, A. & Stepanov, M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154.Google Scholar
Falkovich, G. & Pumir, A. 2007 Sling effect in collisions of water droplets in turbulent clouds. J. Atmos. Sci. 64, 4497.Google Scholar
Good, G. H., Ireland, P. J., Bewley, G. P., Bodenschatz, E., Collins, L. R. & Warhaft, Z. 2014 Settling regimes of inertial particles in isotropic turbulence. J. Fluid Mech. 759, R3.Google Scholar
Goto, S. & Vassilicos, J. C. 2006 Self-similar clustering of inertial particles and zero-acceleration points in fully developed two-dimensional turbulence. Phys. Fluids 18, 115103.Google Scholar
Gotoh, T., Fukayama, D. & Nakano, T. 2002 Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation. Phys. Fluids 14, 1065–1081.Google Scholar
Grabowski, W. W. & Wang, L.-P. 2013 Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid Mech. 45, 293–324.CrossRefGoogle Scholar
Gustavsson, K. & Mehlig, B. 2011 Distribution of relative velocities in turbulent aerosols. Phys. Rev. E 84, 045304.CrossRefGoogle ScholarPubMed
Hill, R. J. 2002 Scaling of acceleration in locally isotropic turbulence. J. Fluid Mech. 452, 361–370.Google Scholar
van Hinsberg, M. A. T., Thije Boonkkamp, J. H. M., Toschi, F. & Clercx, H. J. H. 2012 On the efficiency and accuracy of interpolation methods for spectral codes. SIAM J. Sci. Comput. 34 (4), B479–B498.Google Scholar
van Hinsberg, M. A. T., ten Thije Bookkkamp, J. H. M., Toschi, F. & Clercx, H. J. H. 2013 Optimal interpolation schemes for particle tracking in turbulence. Phys. Rev. E 87, 043307.Google Scholar
IJzermans, R. H. A., Meneguz, E. & Reeks, M. W. 2010 Segregation of particles in incompressible random flows: singularities, intermittency and random uncorrelated motion. J. Fluid Mech. 653, 99–136.CrossRefGoogle Scholar
Ireland, P. J., Bragg, A. D. & Collins, L. R. 2016 The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects. J. Fluid Mech. 796, 659–711.Google Scholar
Ireland, P. J., Vaithianathan, T., Sukheswalla, P. S., Ray, B. & Collins, L. R. 2013 Highly parallel particle-laden flow solver for turbulence research. Comput. Fluids 76, 170–177.Google Scholar
Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high-Reynolds-number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.Google Scholar
Ishihara, T., Kaneda, Y., Yokokawa, M., Itakura, K. & Uno, A. 2007 Small-scale statistics in high-resolution direct numerical simualtion of turbulence: Reynolds number dependence of one-point velocity gradient statistics. J. Fluid Mech. 592, 335–366.CrossRefGoogle Scholar
de Jong, J., Salazar, J. P. L. C., Cao, L., Woodward, S. H., Collins, L. R. & Meng, H. 2010 Measurement of inertial particle clustering and relative velocity statistics in isotropic turbulence using holographic imaging. Intl J. Multiphase Flow 36, 324–332.CrossRefGoogle Scholar
Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K. & Uno, A. 2003 Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box. Phys. Fluids 15, L21–L24.Google Scholar
Kerr, R. M., Meneguzzi, M. & Gotoh, T. 2001 An inertial range crossover in structure functions. Phys. Fluids 13, 1985–1994.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
Lu, J., Nordsiek, H. & Shaw, R. A. 2010 Clustering of settling charged particles in turbulence: theory and experiments. New J. Phys. 12, 123030.Google Scholar
Maxey, M. R. 1987 The motion of small spherical particles in a cellular flow field. Phys. Fluids 30, 1915–1928.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889.Google Scholar
McQuarrie, D. A. 1976 Statistical Mechanics. Harper & Row.Google Scholar
Meneveau, C. 2011 Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows. Annu. Rev. Fluid Mech. 43, 219–245.Google Scholar
Monchaux, R., Bourgoin, M. & Cartellier, A. 2010 Preferential concentration of heavy particles: a Voronoï analysis. Phys. Fluids 22, 103304.CrossRefGoogle Scholar
Onishi, R., Takahashi, K. & Vassilicos, J. C. 2013 An efficient parallel simulation of interacting inertial particles in homogeneous isotropic turbulence. J. Comput. Phys. 242, 809–827.Google Scholar
Onishi, R. & Vassilicos, J. C. 2014 Collision statistics of inertial particles in two-dimensional homogeneous isotropic turbulence with an inverse cascade. J. Fluid Mech. 745, 279–299.Google Scholar
Orszag, S. A. & Patterson, G. S. 1972a Numerical simulation of three-dimensional homogeneous isotropic turbulence. Phys. Rev. Lett. 28, 76–79.Google Scholar
Orszag, S. A. & Patterson, G. S. 1972b Numerical Simulation of Turbulence. Springer.Google Scholar
Pan, L. & Padoan, P. 2010 Relative velocity of inertial particles in turbulent flows. J. Fluid Mech. 661, 73–107.Google Scholar
Pan, L. & Padoan, P. 2013 Turbulence-induced relative velocity of dust particles. I: identical particles. Astrophys. J. 776, 12.Google Scholar
Pan, L., Padoan, P., Scalo, J., Kritsuk, A. G. & Norman, M. L. 2011 Turbulent clustering of protoplanetary dust and planetesimal formation. Astrophys. J. 740, 6.CrossRefGoogle Scholar
Pekurovsky, D. 2012 P3DFFT: a framework for parallel computations of Fourier transforms in three dimensions. SIAM J. Sci. Comput. 34 (4), C192–C209.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pruppacher, H. R. & Klett, J. D. 1997 Microphysics of Clouds and Precipitation. Kluwer.Google Scholar
Ray, B. & Collins, L. R. 2011 Preferential concentration and relative velocity statistics of inertial particles in Navier–Stokes turbulence with and without filtering. J. Fluid Mech. 680, 488–510.Google Scholar
Ray, B. & Collins, L. R. 2013 Investigation of sub-Kolmogorov inertial particle pair dynamics in turbulence using novel satellite particle simulations. J. Fluid Mech. 720, 192–211.Google Scholar
Reade, W. C. & Collins, L. R. 2000a Effect of preferential concentration on turbulent collision rates. Phys. Fluids 12, 2530–2540.Google Scholar
Reade, W. C. & Collins, L. R. 2000b A numerical study of the particle size distribution of an aerosol undergoing turbulent coagulation. J. Fluid Mech. 415, 45–64.Google Scholar
Rosa, B., Parishani, H., Ayala, O., Grabowski, W. W. & Wang, L. P. 2013 Kinematic and dynamic collision statistics of cloud droplets from high-resolution simulations. New J. Phys. 15, 045032.Google Scholar
Salazar, J. P. L. C. & Collins, L. R. 2012a Inertial particle acceleration statistics in turbulence: effects of filtering, biased sampling, and flow topology. Phys. Fluids 24, 083302.CrossRefGoogle Scholar
Salazar, J. P. L. C. & Collins, L. R. 2012b Inertial particle relative velocity statistics in homogeneous isotropic turbulence. J. Fluid Mech. 696, 45–66.Google Scholar
Salazar, J. P. L. C., de Jong, J., Cao, L., Woodward, S., Meng, H. & Collins, L. R. 2008 Experimental and numerical investigation of inertial particle clustering in isotropic turbulence. J. Fluid Mech. 600, 245–256.Google Scholar
Saw, E. W., Bewley, G. P., Bodenschatz, E., Ray, S. S. & Bec, J. 2014 Extreme fluctuations of the relative velocities between droplets in turbulent airflow. Phys. Fluids 26, 111702.Google Scholar
Saw, E. W., Shaw, R. A., Ayyalasomayajula, S., Chuang, P. Y. & Gylfason, A. 2008 Inertial clustering of particles in high-Reynolds-number turbulence. Phys. Rev. Lett. 100, 214501.Google 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, 105031.Google Scholar
Sawford, B. L., Yeung, P.-K., Borgas, M. S., La Porta, P. V. A., Crawford, A. M. & Bodenschatz, E. 2003 Conditional and unconditional acceleration statistics in turbulence. Phys. Fluids 15, 3478–3489.Google Scholar
Shaw, R. A. 2003 Particle–turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183–227.Google Scholar
Shaw, R. A., Kostinski, B. & Larsen, M. L. 2002 Towards quantifying droplet clustering in clouds. Q. J. R. Meteorol. Soc. 128, 1043–1057.CrossRefGoogle Scholar
Shen, X. & Warhaft, Z. 2002 Longitudinal and transverse structure functions in sheared and unsheared wind-tunnel turbulence. Phys. Fluids 14, 370–381.Google Scholar
Siebert, H., Lehmann, K. & Wendisch, M. 2006 Observations of small-scale turbulence and energy dissipation rates in the cloudy boundary layer. J. Atmos. Sci. 63, 1451–1466.Google Scholar
Soria, J., Sondergaard, R., Cantwell, B. J., Chong, M. S. & Perry, A. E. 1994 A study of the fine-scale motions of incompressible time-developing mixing layers. Phys. Fluids 6 (2), 871–884.Google Scholar
Spalart, P. R. 1988 Direct simulation of a turbulent boundary layer up to R 𝜃 = 1410. J. Fluid Mech. 187, 61–98.Google Scholar
Squires, K. D. & Eaton, J. K. 1991 Preferential concentration of particles by turbulence. Phys. Fluids A 3, 1169–1178.Google Scholar
Sundaram, S. & Collins, L. R. 1997 Collision statistics in an isotropic, particle-laden turbulent suspension I. Direct numerical simulations. J. Fluid Mech. 335, 75–109.Google Scholar
Sundaram, S. & Collins, L. R. 1999 A numerical study of the modulation of isotropic turbulence by suspended particles. J. Fluid Mech. 379, 105–143.Google Scholar
Tagawa, Y., Mercado, J. M., Prakash, V. N., Calzavarini, E., Sun, C. & Lohse, D. 2012 Three-dimensional Lagrangian Voronoï analysis for clustering of particles and bubbles in turbulence. J. Fluid Mech. 693, 201–215.Google Scholar
Tavoularis, S., Bennett, J. C. & Corrsin, S. 1978 Velocity-derivative skewness in small Reynolds number, nearly isotropic turbulence. J. Fluid Mech. 88, 63–69.Google Scholar
Voßkuhle, M., Pumir, A., Lévêque, E. & Wilkinson, M. 2014 Prevalence of the sling effect for enhancing collision rates in turbulent suspensions. J. Fluid Mech. 749, 841–852.Google Scholar
Voth, G. A., La Porta, A., Crawford, A. M., Alexander, J. & Bodenschatz, E. 2002 Measurement of particle accelerations in fully developed turbulence. J. Fluid Mech. 469, 121–160.CrossRefGoogle Scholar
Wang, L.-P. & Grabowski, W. W. 2009 The role of air turbulence in warm rain initiation. Atmos. Sci. Lett. 10, 1–8.Google 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.Google Scholar
Wang, L.-P., Wexler, A. S. & Zhou, Y. 1998 Statistical mechanical descriptions of turbulent coagulation. Phys. Fluids 10, 2647–2651.Google Scholar
Wang, L.-P., Wexler, A. S. & Zhou, Y. 2000 Statistical mechanical description and modeling of turbulent collision of inertial particles. J. Fluid Mech. 415, 117–153.Google Scholar
Wilkinson, M. & Mehlig, B. 2005 Caustics in turbulent aerosols. Europhys. Lett. 71, 186–192.Google Scholar
Wilkinson, M., Mehlig, B. & Bezuglyy, V. 2006 Caustic activation of rain showers. Phys. Rev. Lett. 97, 048501.Google Scholar
Witkowska, A., Brasseur, J. G. & Juvé, D. 1997 Numerical study of noise from isotropic turbulence. J. Comput. Acoust. 5, 317–336.Google Scholar
Xue, Y., Wang, L.-P. & Grabowski, W. W. 2008 Growth of cloud droplets by turbulent collision-coalescence. J. Atmos. Sci. 65, 331–356.Google Scholar
Yeung, P. K., Donzis, D. A. & Sreenivasan, K. R. 2012 Dissipation, enstrophy, and pressure statistics in turbulence simulations at high Reynolds numbers. J. Fluid Mech. 700, 5–15.Google Scholar
Yeung, P. K. & Pope, S. B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586.Google Scholar
Yeung, P. K., Pope, S. B., Lamorgese, A. G. & Donzis, D. A. 2006 Acceleration and dissipation statistics of numerically simulated isotropic turbulence. Phys. Fluids 18 (6), 065103.Google Scholar
Yoshimoto, H. & Goto, S. 2007 Self-similar clustering of inertial particles in homogeneous turbulence. J. Fluid Mech. 577, 275–286.Google Scholar
Yudine, M. I. 1959 Physical considerations on heavy-particle dispersion. Adv. Geophys. 6, 185–191.Google Scholar
Zaichik, L. I. & Alipchenkov, V. M. 2003 Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids 15, 1776–1787.Google Scholar
Zaichik, L. I. & Alipchenkov, V. M. 2008 Acceleration of heavy particles in isotropic turbulence. Intl J. Multiphase Flow 34 (9), 865–868.Google Scholar
Zaichik, L. I. & Alipchenkov, V. M. 2009 Statistical models for predicting pair dispersion and particle clustering in isotropic turbulence and their applications. New J. Phys. 11, 103018.Google Scholar
Zaichik, L. I., Simonin, O. & Alipchenkov, V. M. 2003 Two statistical models for predicting collision rates of inertial particles in homogeneous isotropic turbulence. Phys. Fluids 15, 2995–3005.Google Scholar