Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-18T23:55:41.117Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative velocity of inertial particles in turbulent flows

Published online by Cambridge University Press:  27 July 2010

LIUBIN PAN  and
PAOLO PADOAN
LIUBIN PAN*
Affiliation:
Department of Physics, University of California, San Diego, CASS/UCSD 0424, 9500 Gilman Drive, La Jolla, CA 92093-0424, USA School of Earth and Space Exploration, Arizona State University, P.O. Box 871404, Tempe, AZ 85287-1404, USA
PAOLO PADOAN
Affiliation:
Department of Physics, University of California, San Diego, CASS/UCSD 0424, 9500 Gilman Drive, La Jolla, CA 92093-0424, USA ICREA & ICC, University of Barcelona, Marti i Franquès 1, E-08028 Barcelona, Spain
*
†Email address for correspondence: liubin.pan@asu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.

JFM classification

Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 661 , 25 October 2010 , pp. 73 - 107
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abrahamson, J. 1975 Collision rates of small particles in a vigorously turbulent fluid. Chem. Engng Sci. 30, 1371–1379.CrossRefGoogle Scholar
Ayala, O., Rosa, B. & Wang, L.-P. 2008 Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization. New J. Phys. 10, 075016(1–40).CrossRefGoogle Scholar
Batchelor, G. K. 1952 Diffusion in a field of homogeneous turbulence. Part II. The relative motion of particles. Math. Proc. Camb. Phil. Soc. 48, 345–362.CrossRefGoogle Scholar
Bec, J., Biferale, L., Boffetta, G., Cencini, M., Musacchio, S. & Toschi, F. 2006 Lyapunov exponents of heavy particles in turbulence. Phys. Fluids. 18, 091702.CrossRefGoogle Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. S. & Toschi, F. 2010 a Caustics and intermittency in turbulent suspensions of heavy particles. J. Fluid Mech. 646, 527–536.CrossRefGoogle Scholar
Bec, J., Biferale, L., Lanotte, A. S., Scagliarini, A., & Toschi, F. 2010 b Turbulent pair dispersion of inertial particles. J. Fluid Mech. 645, 497–528.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
Bec, J., Cencini, M. & Hillerbrand, R. 2007 Heavy particles in incompressible flows: the large Stokes number asymptotics. Physica D 226, 11–22.CrossRefGoogle Scholar
Berg, J., Luthi, B., Mann, J. & Ott, S. 2006 Backwards and forwards relative dispersion in turbulent flow: an experimental investigation. Phys. Rev. E 74, 016304 (1–7).CrossRefGoogle ScholarPubMed
Blum, J. & Wurm, G. 2008 The growth mechanisms of macroscopic bodies in protoplanetary disks. Annu. Rev. Astron. Astrophys. 46, 21–56.CrossRefGoogle Scholar
Boffetta, G. & Sokolov, I. M. 2002 Relative dispersion in fully developed turbulence: the Richardson's law and intermittency corrections. Phys. Rev. Lett. 88, 094501 (1–4)CrossRefGoogle ScholarPubMed
Borgas, M. S. 1993 The multifractal Lagrangian nature of turbulence. Phil. Trans.: Phys. Sci. Engng 342, 379–411.Google Scholar
Corrsin, S. 1959 Progress report on some turbulent diffusion research. In Advances in Geophysics(ed. Freinkel, F. N. & Sheppard, P. A.), vol. 6., pp. 161–164. Academic.Google Scholar
Cuzzi, J. N. & Hogan, R. C. 2003 Blowing in the wind. Part I. Velocities of chondrule-sized particles in a turbulent protoplanetary nebula. Icarus 164, 127–138.CrossRefGoogle Scholar
Derivich, I. V. 2006 Statistical modeling of particles relative motions in a turbulent gas flow. Intl J. Heat Mass Transfer 49, 4290–4304.CrossRefGoogle Scholar
Falkovich, G., Gawedzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975.CrossRefGoogle Scholar
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
Fouxon, I. & Horvai, P. 2008 Separation of heavy particles in turbulence. Phys. Rev. Lett. 100, 040601.CrossRefGoogle ScholarPubMed
Girimaji, S. S. & Pope, S. B. 1990 Material-element deformation in isotropic turbulence. J. Fluid. Mech. 220, 427–458.CrossRefGoogle 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.CrossRefGoogle Scholar
Kaneda, Y. & Gotoh, T. 1991 Lagrangian velocity autocorrelation in isotropic turbulence. Phys. Fluids. A 3, 1924–1933.CrossRefGoogle Scholar
Kruis, F. E. & Kusters, K. A. 1997 The collision rate of particles in turbulent flow. Chem. Engng Commun. 158, 201–230.CrossRefGoogle Scholar
Lundgren, T. S. 1981 Turbulent pair dispersion and scalar diffusion. J. Fluid Mech. 111, 27–57.CrossRefGoogle Scholar
Markiewicz, W. J., Mizuno, H. & Voelk, H. J. 1991 Turbulence induced relative velocity between two grains. Astron. Astrophys. 242, 286–289.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2. MIT Press.Google Scholar
Ormel, C. W. & Cuzzi, J. N. 2007 Closed-form expressions for particle relative velocities induced by turbulence. Astron. Astrophys. 466, 413–420.CrossRefGoogle Scholar
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A 110, 709–737.Google Scholar
Saffman, P. G. & Turner, J. S. 1956 On the collision of drops in turbulent clouds. J. Fluid Mech. 1, 16–30.CrossRefGoogle Scholar
Salazar, J. P. L. C. & Collins, L. R. 2009 Two-particle dispersion in isotropic turbulent flows. Annu. Rev. Fluid Mech. 41, 405–432.CrossRefGoogle Scholar
Sawford, B. L. 1991 Reynolds number effects in Lagrangian stochastic models of turbulent dispersion. Phys. Fluids 3, 1577–1586.CrossRefGoogle Scholar
Sawford, B. L., Yeung, P. K. & Borgas, M. S. 2005 Comparison of backwards and forwards relative dispersion in turbulence. Phys. Fluids 17, 095109 (1–9).CrossRefGoogle Scholar
Sawford, B. L., Yeung, P. K. & Hackl, J. F. 2008 Reynolds number dependence of relative dispersion statistics in isotropic turbulence. Phys. Fluids 20, 065111 (1–13).CrossRefGoogle Scholar
She, Z.-S. & Leveque, E. 1994 Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72, 336–339.CrossRefGoogle ScholarPubMed
Shlien, D. J. & Coorsin, S. 1974 A measurement of Lagrangian velocity auto-correlation in approximately isotropic turbulence. J. Fluid Mech. 62, 255–271.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
Volk, H. J., Jones, F. C., Morfill, G. E. & Roeser, S. 1980 Collisions between grains in a turbulent gas. Astron. Astrophys. 85, 316–325.Google Scholar
Voth, G. A., Satyanarayan, K. & Bodenschatz, E. 1998 Lagrangian acceleration measurements at large Reynolds numbers. Phys. Fluids 10, 2268–2280.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. 2005 Caustics in turbulent aerosols. Europhys. Lett. 71, 186–192CrossRefGoogle Scholar
Wilkinson, M., Mehlig, B. & Bezuglyy, V. 2006 Caustic activation of rain showers. Phys. Rev. Lett. 97, 048501.CrossRefGoogle ScholarPubMed
Weidenschilling, S. J. 1980 Dust to planetesimals: settling and coagulation in the solar nebula. Icarus 44, 172–189.CrossRefGoogle Scholar
Williams, J. J. E. & Crane, R. I. 1983 Particle collision rate in turbulent flow. Intl J. Multiph. Flow 9, 421–435.CrossRefGoogle Scholar
Yeung, P. K. & Pope, S. B 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid. Mech. 207, 531–586.CrossRefGoogle Scholar
Yeung, P. K., Pope, S. B., Lamorgese, A. G. & Donzis, D. A. 2006 a Acceleration and dissipation statistics of numerically simulated isotropic turbulence. Phys. Fluids 18, 065103 (1–14).CrossRefGoogle Scholar
Yeung, P. K., Pope, S. B. & Sawford, B. L. 2006 b Reynolds number dependence of Lagrangian statistics in large numerical simulations of isotropic turbulence. J. Turbul. 7, 58 (1–12).CrossRefGoogle Scholar
Yuu, S. 1984 Collision rate of small particles in a homogeneous and isotropic turbulence. AIChE J. 30, 802–807.CrossRefGoogle Scholar
Zaichik, L. I. & Alipchenkov, V. M. 2003 Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids 15, 1776–1787.CrossRefGoogle 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.CrossRefGoogle Scholar
Zaichik, L. I., Simonin, O. & Alipchenkov, V. M. 2006 Collision rates of bidisperse inertial particles in isotropic turbulence. Phys. Fluids 18, 035110 (1–13).CrossRefGoogle Scholar
Zhou, Y., Wexler, A. S. & Wang, L.-P. 2001 Modelling turbulent collision of bidisperse inertial particles. J. Fluid Mech. 433, 77–104.CrossRefGoogle Scholar