Skip to main content
SpringerLink
Log in

Turbulent Diffusion of Inertial Particle Pairs Such as in Pollen and Sandstorms

  • Conference paper
  • First Online:
Book cover Recent Advances in Mathematical and Statistical Methods (AMMCS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 259))

  • 858 Accesses

Abstract

We explore the concept of local and non-local diffusion processes [Malik N. A., PLoS ONE 12(12): e0189917 (2017)] in application to the diffusion of inertial particle pairs in the limit of Stoke’s drag. Inertial particles are arguably more important than fluid particles because most real world applications are related to inertial particle motion, from hail and pollen to sandstorms. The inertial pair diffusion regimes depend upon the local Stokes’ number St(l), where l is the pair separation distance. For \(St(l) \ll 1\), the inertia dominates and we observe ballistic motion for inertial pair separation. For \(St(l) \gg 1\), the turbulent energy dominates the diffusion process which asymptotes to the fluid non-local pair regime for very large inertial ranges. A numerical model, Kinematic Simulations, is used to generate inertia particle trajectories and we observe the predicted inertial pair diffusion regimes in the limit of large inertial subranges.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Calzavarini, E., Kerscher, M., Lohse, D., Toschi, F.: Dimensionality and morphology of particle and bubble clusters in turbulent flow. J. Fluid Mech. 607, 1324 (2008)

    Article  Google Scholar 

  2. Falkovich, G., Pumir, A.: Sling effect in collision of water droplet in turbulent clouds. J. Atm. Sci. 64, 44974505 (2007)

    Article  Google Scholar 

  3. Shaw, R.A.: Particle-turbulence interactions in atmospheric clouds. Ann. Rev. Fluid Mech. 35, 183227 (2003)

    Article  Google Scholar 

  4. Sofiev, M.: Airborne pollen transport. In: Sofiev, M., Bergmann, K.C. (eds.) Allergenic Pollen. Springer, Dordrecht (2013)

    Chapter  Google Scholar 

  5. Toschi, F., Bodenschatz, E.: Lagrangian properties of particles in turbulence. Ann. Rev. Fluid Mech. 41, 375404 (2009)

    Article  MathSciNet  Google Scholar 

  6. Maxey, M.R., Riley, J.R.: Equation of motion for a small rigid sophere in a non-uniform flow. Phys. Fluids 26, 883 (1983). https://doi.org/10.1063/1.864230

    Article  MATH  Google Scholar 

  7. Qureshi, N.M., Bourgoin, M., Baudet, C., Cartellier, A., Gagne, Y.: Turbulent transport of material particles: an experimental study of finite size effects. Phys. Rev. Lett. 99, 184502 (2007)

    Article  Google Scholar 

  8. Richardson, L.F.: Atmospheric diffusion shown on a distance-neighbour graph. Proc. Roy. Soc. Lond. A 100, 709–737 (1926)

    Article  Google Scholar 

  9. Obukhov, A.: Spectral energy distribution in a turbulent flow. Izv. Akad. Xauk. SSSR. Ser. Geogr. i Geojz 5, 453–466 (1941). (Translation : Ministry of Supply. p. 21 1097)

    Google Scholar 

  10. Malik, N.A.: Residual sweeping errors in turbulent pair diffusion in a Lagrangian diffusion model. PLoS ONE 12(12), e0189917 (2017). https://doi.org/10.1371/journal.pone.0189917

    Article  Google Scholar 

  11. Malik N.A.: Ninety years in the hunt for turbulent pair diffusion laws. IMA Conference on Turbulence, Waves and Mixing. 6–8 July, Cambridge (2016)

    Google Scholar 

  12. Malik, N.A.: Turbulent Particle Pair Diffusion: a theory based on local and non-local diffusional processes. Accepted on August 14 for publication in PLoS ONE, to appear soon (2018)

    Google Scholar 

  13. Malik, N.A.: Turbulent particle pair diffusion: numerical simulations. submitted to Plos One (2018)

    Google Scholar 

  14. Bec, J., Biferale, L., Lanotte, A., Scagliarini, A., Toschi, F.: J. Fluid Mech. 645, 497 (2010)

    Article  Google Scholar 

  15. Chang, K., Benedict, J.M., Shaw, R.A.: Turbulent pair dispersion in the presence of gravity. New J. Phys. 17(3), 033010 (2015)

    Article  Google Scholar 

  16. Bragg, A.D., Ireland, P.J., Collins, L.R.: Forward and backward in time dispersion of fluid and inertial particles in isotropic turbulence. Phys. Fluids 28(1) (2016)

    Google Scholar 

  17. Bragg, A.D.: Analysis of the forward and backward in time pair-separation probability density functions for inertial particles in isotropic turbulence. J. Fluid Mech. 830, 63–92 (2017)

    Article  MathSciNet  Google Scholar 

  18. Bec, J., Biferale, L., Cencini, M., Lanotte, A.S., Toschi, F.: Intermittency in the velocity distribution of heavy particles in turbulence. J. Fluid Mech. 646, 527536 (2010)

    Article  Google Scholar 

  19. Gustavsson, K., Mehlig, B.: Distribution of relative velocities in turbulent aerosols. Phys. Rev. E 84, 045304 (2011)

    Article  Google Scholar 

  20. Gustavsson, K., Vajedi, S., Mehlig, B.: Clustering of particles falling in a turbulent flow. Phys. Rev. Lett. 112, 214501 (2014)

    Article  Google Scholar 

  21. Bragg, A.D., Collins, L.R.: New insights from comparing statistical theories for inertial particles in turbulence: II. relative velocities of particles. New J. Phys. 16, 055014 (2014)

    Article  Google Scholar 

  22. Kraichnan, R.H.: Diffusion by a random velocity field. Phys. Fluids 13, 22–31 (1970)

    Article  Google Scholar 

  23. Fung, J.C.H., Hunt, J.C.R., Malik, N.A., Perkins, R.J.: Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes. J. Fluid Mech. 236, 281 (1992)

    Article  MathSciNet  Google Scholar 

  24. Turfus, C., Hunt, J.C.R.: A stochastic analysis of the displacements of fluid element in inhomogeneous turbulence using Kraichnan’s method of random modes. In: Comte-Bellot, G., Mathieu, J. (eds.) Advances in Turbulence, pp. 191–203. Springer, Berlin (1987)

    Chapter  Google Scholar 

  25. Murray, S., Lightstone, M.F., Tullis, S.: Single-particle Lagrangian and structure statistics in kinematically simulated particle-laden turbulent flows. Phys. Fuids 28, 033302 (2016)

    Google Scholar 

  26. Malik N.A. :Structural diffusion in 2D and 3D random flows. In: S. Gavrilakis et al. (eds.) Adv. in Turbulence vol. VI, pp. 619–620 (1996)

    Google Scholar 

  27. Fung, J.C.H., Vassilicos, J.C.: Two-particle dispersion in turbulent-like flows. Phys. Rev. E 57, 1677 (1998)

    Article  MathSciNet  Google Scholar 

  28. Malik, N.A., Vassilicos, J.C.: A Lagrangian model for turbulent dispersion with turbulent-like flow structure: comparison with direct numerical simulation for two-particle statistics. Phys. Fluids 11, 1572–1580 (1999)

    Article  MathSciNet  Google Scholar 

  29. Nicolleau, F.C.G.A., Nowakowski, A.F.: Presence of a Richardson’s regime in kinematic simulations. Phys. Rev. E 83, 056317 (2011)

    Article  Google Scholar 

  30. Meneguz, E., Reeks, M.W.: Statistical properties of particle segregation in homogeneous isotropic turbulence. J. Fluid Mech. 686, 338–351 (2011)

    Article  Google Scholar 

  31. Farhan, M., Nicolleau, F.C.G.A., Nowakowski, A.F.: Effect of gravity on clustering patterns and inertial particle attractors in kinematic simulations. Phys. Rev. E 91, 043021 (2001)

    Article  Google Scholar 

  32. Thomson, D.J., Devenish, B.J.: Particle pair separation in kinematic simulations. J. Fluid Mech. 526, 277–302 (2005)

    Article  MathSciNet  Google Scholar 

  33. Eyink, G.L., Benveniste, D.: Suppression of particle dispersion by sweeping effects in synthetic turbulence. Phys. Rev. E 87, 023011 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Syed M. Usama & Nadeem A. Malik

Authors
  1. Syed M. Usama
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nadeem A. Malik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nadeem A. Malik .

Editor information

Editors and Affiliations

  1. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    D. Marc Kilgour

  2. Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada

    Herb Kunze

  3. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    Roman Makarov

  4. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    Roderick Melnik

  5. Department of Mathematics & MS2Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada

    Xu Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Usama, S.M., Malik, N.A. (2018). Turbulent Diffusion of Inertial Particle Pairs Such as in Pollen and Sandstorms. In: Kilgour, D., Kunze, H., Makarov, R., Melnik, R., Wang, X. (eds) Recent Advances in Mathematical and Statistical Methods . AMMCS 2017. Springer Proceedings in Mathematics & Statistics, vol 259. Springer, Cham. https://doi.org/10.1007/978-3-319-99719-3_22

Download citation

Publish with us

Policies and ethics

Search

Navigation