On the influence of forced homogeneous-isotropic turbulence on the settling and clustering of finite-size particles


We investigate the motion of heavy particles with a diameter of several multiples of the Kolmogorov length scale in the presence of forced turbulence and gravity, resorting to interface-resolved direct numerical simulation based on an immersed boundary method. The values of the particles’ relative density (1.5) and of the Galileo number (180) are such that strong wake-induced particle clustering would occur in the absence of turbulence (Uhlmann and Doychev in J Fluid Mech 752:310–348, 2014. https://doi.org/10.1017/jfm.2014.330). The forced turbulence in the two present cases (with Taylor-scale Reynolds number 95 and 140) would lead to mild levels of clustering in the absence of gravity (Uhlmann and Chouippe in J Fluid Mech 812:991–1023, 2017. https://doi.org/10.1017/jfm.2016.826). Here we detect a tendency to cluster with an intensity (quantified via the standard deviation of the distribution of Voronoï cell volumes) which is intermediate between these two limiting cases, meaning that forced background turbulence decreases the level of clustering otherwise observed under ambient settling. However, the clustering strength does not monotonically decay with the relative turbulence intensity. Various mechanisms by which coherent structures can affect particle motion are discussed. It is argued that the reduced interaction time due to particle settling through the surrounding eddy (crossing trajectories) has the effect of shifting upward the range of eddies with a time-scale matching the characteristic time-scale of the particle. In the present cases this shift might bring the particles into resonance with the energetic eddies of the turbulent spectrum. Concerning the average particle settling velocity, we find very small deviations (of the order of one percent) from the value obtained for an isolated particle in ambient fluid when defining the relative velocity as an apparent slip velocity (i.e., as the difference between the averages computed separately for the velocities of each phase). This is consistent with simple estimates of the nonlinear drag effect. However, the relative velocity based upon the fluid velocity seen by each particle (computed via local averaging over a particle-attached sphere) has on average a smaller magnitude (by 5–7%) than the ambient single-particle value.

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


  1. 1.

    Aliseda, A., Cartellier, A., Hainaux, F., Lasheras, J.: Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 468, 77–105 (2002)

    MATH  Article  Google Scholar 

  2. 2.

    Balachandar, S., Eaton, J.: Turbulent dispersed multiphase flow. Ann. Rev. Fluid Mech. 42, 111–133 (2010)

    MATH  Article  Google Scholar 

  3. 3.

    Bec, J., Biferale, L., Cencini, M., Lanotte, A., Musacchio, S., Toschi, F.: Heavy particle concentration in turbulence at dissipative and inertial scales. Phys. Rev. Lett. 98, 084502 (2007)

    Article  Google Scholar 

  4. 4.

    Bec, J., Homann, H., Ray, S.: Gravity-driven enhancement of heavy particle clustering in turbulent flow. Phys. Rev. Lett. 112, 184501 (2014)

    Article  Google Scholar 

  5. 5.

    Bragg, A., Ireland, P., Collins, L.: On the relationship between the non-local clustering mechanism and preferential concentration. J. Fluid Mech. 780, 327–343 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Bürger, K., Treib, M., Westermann, R., Werner, S., Lalescu, C.C., Szalay, A., Meneveau, C., Eyink, G.L.: Vortices within vortices: hierarchical nature of vortex tubes in turbulence (2012). arXiv preprint arXiv:1210.3325

  7. 7.

    Chouippe, A., Uhlmann, M.: Forcing homogeneous turbulence in DNS of particulate flow with interface resolution and gravity. Phys. Fluids 27(12), 123301 (2015)

    Article  Google Scholar 

  8. 8.

    Chun, J., Koch, D., Rani, S., Ahluwalia, A., Collins, L.: Clustering of aerosol particles in isotropic turbulence. J. Fluid Mech. 536, 219–251 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Cisse, M., Homann, H., Bec, J.: Slipping motion of large neutrally buoyant particles in turbulence. J. Fluid Mech. 735, R1 (2013)

  10. 10.

    Coleman, S.W., Vassilicos, J.C.: A unified sweep-stick mechanism to explain particle clustering in two- and three-dimensional homogeneous, isotropic turbulence. Phys. Fluids 21(11), 113301 (2009)

    MATH  Article  Google Scholar 

  11. 11.

    Csanady, G.T.: Turbulent diffusion of heavy particles in the atmosphere. J. Atmos. Sci. 20(3), 201–208 (1963)

    Article  Google Scholar 

  12. 12.

    Doychev, T.: The dynamics of finite-size settling particles. PhD thesis, Karlsruhe Institute of Technology (2014)

  13. 13.

    Eswaran, V., Pope, S.: An examination of forcing in direct numerical simulations of turbulence. Comput. Fluids 16, 257–278 (1988)

    MATH  Article  Google Scholar 

  14. 14.

    Ferenc, J.-S., Neda, Z.: On the size distribution of Poisson Voronoi cells. Physica A 385, 518–526 (2007)

    Article  Google Scholar 

  15. 15.

    Ferrante, A., Elghobashi, S.: On the physical mechanisms of two-way coupling in particle-laden isotropic turbulence. Phys. Fluids 15(2), 315–329 (2003)

    MATH  Article  Google Scholar 

  16. 16.

    Fiabane, L., Zimmermann, R., Volk, R., Pinton, J.-F., Bourgoin, M.: Clustering of finite-size particles in turbulence. Phys. Rev. E 86, 035301(R) (2012)

    Article  Google Scholar 

  17. 17.

    Fornari, W., Picano, F., Brandt, L.: Sedimentation of finite-size spheres in quiescent and turbulent environments. J. Fluid Mech. 788, 640–669 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Fornari, W., Picano, F., Sardina, G., Brandt, L.: Reduced particle settling speed in turbulence. J. Fluid Mech. 808, 153–167 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Fortes, A., Joseph, D., Lundgren, T.: Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467–483 (1987)

    Article  Google Scholar 

  20. 20.

    Glowinski, R., Pan, T.-W., Hesla, T., Joseph, D.: A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 25, 755–794 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Good, G., Ireland, P., Bewley, G., Bodenschatz, E., Collins, L., Warhaft, Z.: Settling regimes of inertial particles in isotropic turbulence. J. Fluid Mech. 759, R3 (2014)

    Article  Google Scholar 

  22. 22.

    Goto, S., Vassilicos, J.: Self-similar clustering of inertial particles and zero-acceleration points in fully developed two-dimensional turbulence. Phys. Fluids 18(11), 115103 (2006)

    MATH  Article  Google Scholar 

  23. 23.

    Goto, S., Vassilicos, J.: Sweep-stick mechanism of heavy particle clustering in fluid turbulence. Phys. Rev. Lett. 100(5), 054503 (2008)

    Article  Google Scholar 

  24. 24.

    Grabowski, W., Wang, L.-P.: Growth of cloud droplets in a turbulent environment. Annu. Rev. Fluid Mech. 45(1), 293–324 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Gustavsson, K., Mehlig, B.: Statistical models for spatial patterns of heavy particles in turbulence. Adv. Phys. 65(1), 1–57 (2016)

    Article  Google Scholar 

  26. 26.

    Hogan, R., Cuzzi, J.: Stokes and Reynolds number dependence of preferential particle concentration in simulated three-dimensional turbulence. Phys. Fluids 13(10), 2938–2945 (2001)

    MATH  Article  Google Scholar 

  27. 27.

    Homann, H., Bec, J., Grauer, R.: Effect of turbulent fluctuations on the drag and lift forces on a towed sphere and its boundary layer. J. Fluid Mech. 721, 155–179 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Huisman, S.G., Barois, T., Bourgoin, M., Chouippe, A., Doychev, T., Huck, P., Bello Morales, C.E., Uhlmann, M., Volk, R.: Columnar structure formation of a dilute suspension of settling spherical particles in a quiescent fluid. Phys. Rev. Fluids 1, 074204 (2016)

    Article  Google Scholar 

  29. 29.

    Hunt, J., Wray, A., Moin, P.: Eddies, streams, and convergence zones in turbulent flows. In: Proceedings of the Summer Programm, pp. 193–208 (Center for Turbulence Research, Stanford) (1988)

  30. 30.

    Hwang, W., Eaton, J.K.: Homogeneous and isotropic turbulence modulation by small heavy (\({St \sim 50}\)) particles. J. Fluid Mech. 564, 361 (2006)

    MATH  Article  Google Scholar 

  31. 31.

    Ireland, P., Bragg, A., Collins, L.: The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects. J. Fluid Mech. 796, 659–711 (2016)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Jenny, M., Dušek, J., Bouchet, G.: Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 201–239 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Kajishima, T.: Influence of particle rotation on the interaction between particle clusters and particle-induced turbulence. Int. J. Heat Fluid Flow 25(5), 721–728 (2004)

    Article  Google Scholar 

  34. 34.

    Kajishima, T., Takiguchi, S.: Interaction between particle clusters and particle-induced turbulence. Int. J. Heat Fluid Flow 23, 639–646 (2002)

    Article  Google Scholar 

  35. 35.

    Kidanemariam, A., Chan-Braun, C., Doychev, T., Uhlmann, M.: Direct numerical simulation of horizontal open channel flow with finite-size, heavy particles at low solid volume fraction. New J. Phys. 15, 025031 (2013)

    Article  Google Scholar 

  36. 36.

    Lance, M., Bataille, J.: Turbulence in the liquid phase of a uniform bubbly air-water flow. J. Fluid Mech. 222, 95–118 (1991)

    Article  Google Scholar 

  37. 37.

    Maxey, M.: The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441–465 (1987)

    MATH  Article  Google Scholar 

  38. 38.

    Monchaux, R., Dejoan, A.: Settling velocity and preferential concentration of heavy particles under two-way coupling effects in homogeneous turbulence. Phys. Rev. Fluids 2, 104302 (2017)

    Article  Google Scholar 

  39. 39.

    Monchaux, R., Bourgoin, M., Cartellier, A.: Preferential concentration of heavy particles: a Voronoï analysis. Phys. Fluids 22(10), 103304 (2010)

    Article  Google Scholar 

  40. 40.

    Monchaux, R., Bourgoin, M., Cartellier, A.: Analyzing preferential concentration and clustering of inertial particles in turbulence. Int. J. Multiphase Flow 40, 1–18 (2012)

    Article  Google Scholar 

  41. 41.

    Nielsen, P.: Turbulence effects on the settling of suspended particles. J. Sed. Res. 63(5), 835–838 (1993)

    Google Scholar 

  42. 42.

    Obligado, M., Teitelbaum, T., Cartellier, A., Mininni, P., Bourgoin, M.: Preferential concentration of heavy particles in turbulence. J. Turbul. 15(5), 293–310 (2014)

    Article  Google Scholar 

  43. 43.

    Oesterlé, B.: A note on crossing-trajectory effects in gas-particle turbulent flows. In: Computational Methods in Multiphase Flow IV. WIT Press (2007)

  44. 44.

    Riboux, G., Risso, F., Legendre, D.: Experimental characterization of the agitation generated by bubbles rising at high Reynolds number. J. Fluid Mech. 643, 509–539 (2010)

    MATH  Article  Google Scholar 

  45. 45.

    Risso, F.: Theoretical model for \(k^{-3}\) spectra in dispersed multiphase flows. Phys. Fluids 23(1), 011701 (2011)

    Article  Google Scholar 

  46. 46.

    Rosa, B., Parishani, H., Ayala, O., Wang, L.-P.: Settling velocity of small inertial particles in homogeneous isotropic turbulence from high-resolution DNS. Int. J. Multiphase Flow 83, 217–231 (2016)

    MathSciNet  Article  Google Scholar 

  47. 47.

    Schiller, L., Naumann, A.: A drag coefficient correlation. VDI Zeitung 77(318), 51 (1935)

    Google Scholar 

  48. 48.

    Shaw, R.: Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35(1), 183–227 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  49. 49.

    Squires, K., Eaton, J.: Preferential concentration of particles by turbulence. Phys. Fluids A 3(5), 1169–1178 (1991)

    Article  Google Scholar 

  50. 50.

    Sumbekova, S., Cartellier, A., Aliseda, A., Bourgoin, M.: Preferential concentration of inertial sub-Kolmogorov particles. The roles of mass loading of particles, Stokes and Reynolds numbers. Phys. Rev. Fluids 2(2), 024302 (2016)

    Article  Google Scholar 

  51. 51.

    Tanaka, T., Eaton, J.K.: Classification of turbulence modification by dispersed spheres using a novel dimensionless number. Phys. Rev. Lett. 101(September), 1–4 (2008)

    Google Scholar 

  52. 52.

    Tunstall, E., Houghton, G.: Retardation of falling spheres by hydrodynamic oscillations. Chem. Eng. Sci. 23(9), 1067–1081 (1968)

    Article  Google Scholar 

  53. 53.

    Uhlmann, M.: An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209, 448–476 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Uhlmann, M.: Interface-resolved direct numerical simulation of vertical particulate channel flow in the turbulent regime. Phys. Fluids 20, 2008 (2013)

    Google Scholar 

  55. 55.

    Uhlmann, M., Chouippe, A.: Clustering and preferential concentration of finite-size particles in forced homogeneous-isotropic turbulence. J. Fluid Mech. 812, 991–1023 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  56. 56.

    Uhlmann, M., Doychev, T.: Sedimentation of a dilute suspension of rigid spheres at intermediate Galileo numbers: the effect of clustering upon the particle motion. J. Fluid Mech. 752, 310–348 (2014)

    Article  Google Scholar 

  57. 57.

    Uhlmann, M., Dušek, J.: The motion of a single heavy sphere in ambient fluid: a benchmark for interface-resolved particulate flow simulations with significant relative velocities. Int. J. Multiphase Flow 59, 221–243 (2014)

    Article  Google Scholar 

  58. 58.

    Wang, L.-P., Maxey, M.: Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 27–68 (1993)

    Article  Google Scholar 

  59. 59.

    Wells, M.R., Stock, D.E.: The effects of crossing trajectories on the dispersion of particles in a turbulent flow. J. Fluid Mech. 136, 31–62 (1983)

    Article  Google Scholar 

  60. 60.

    Wu, J., Manasseh, R.: Dynamics of dual-particles settling under gravity. Int. J. Multiphase Flow 24, 1343–1358 (1998)

    MATH  Article  Google Scholar 

  61. 61.

    Yang, T., Shy, S.: The settling velocity of heavy particles in an aqueous near-isotropic turbulence. Phys. Fluids 15(4), 868–880 (2003)

    MATH  Article  Google Scholar 

  62. 62.

    Yoshimoto, H., Goto, S.: Self-similar clustering of inertial particles in homogeneous turbulence. J. Fluid Mech. 577, 275–286 (2007)

    MATH  Article  Google Scholar 

  63. 63.

    Yudine, M.: Physical Considerations on Heavy-Particle Diffusion. Advances in Geophysics, vol. 6, pp. 185–191. Elsevier, Amsterdam (1959)

    Google Scholar 

  64. 64.

    Zaichik, L., Alipchenkov, V.: Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids 15(6), 1776–1787 (2003)

    MATH  Article  Google Scholar 

  65. 65.

    Zaichik, L., Alipchenkov, V.: Refinement of the probability density function model for preferential concentration of aerosol particles in isotropic turbulence. Phys. Fluids 19(11), 113308 (2007)

    MATH  Article  Google Scholar 

Download references


This work was supported by the German Research Foundation (DFG) under Project UH 242/1-2. The simulations were partially performed at LRZ München (under Grant pr83la) and at SCC Karlsruhe (Project DNSPARTHIT). The computer resources, technical expertise, and assistance provided by these centers are thankfully acknowledged.

Author information



Corresponding author

Correspondence to Agathe Chouippe.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chouippe, A., Uhlmann, M. On the influence of forced homogeneous-isotropic turbulence on the settling and clustering of finite-size particles. Acta Mech 230, 387–412 (2019). https://doi.org/10.1007/s00707-018-2271-7

Download citation