Journal of Fluid Mechanics


Three-dimensional Lagrangian Voronoï analysis for clustering of particles and bubbles in turbulence

Yoshiyuki Tagawaa1a3 c1, Julián Martínez Mercadoa1a3, Vivek N. Prakasha1a3, Enrico Calzavarinia2a3, Chao Suna1a3 c1 and Detlef Lohsea1a3 c1

a1 Physics of Fluids Group, Faculty of Science and Technology, J.M. Burgers Center for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands

a2 Laboratoire de Mécanique de Lille CNRS/UMR 8107, Université Lille 1 and Polytech’Lille, Cité Scientifique Av. P. Langevin, 59650 Villeneuve d’Ascq, France

a3 International Collaboration for Turbulence Research


Three-dimensional Voronoï analysis is used to quantify the clustering of inertial particles in homogeneous isotropic turbulence using data sets from numerics in the point particle limit and one experimental data set. We study the clustering behaviour at different density ratios, particle response times (i.e. Stokes numbers $\mathit{St}$) and two Taylor–Reynolds numbers (${\mathit{Re}}_{\lambda } = 75$ and 180). The probability density functions (p.d.f.s) of the Voronoï cell volumes of light and heavy particles show different behaviour from that of randomly distributed particles, i.e. fluid tracers, implying that clustering is present. The standard deviation of the p.d.f. normalized by that of randomly distributed particles is used to quantify the clustering. The clustering for both light and heavy particles is stronger for higher ${\mathit{Re}}_{\lambda } $. Light particles show maximum clustering for $\mathit{St}$ around 1–2 for both Taylor–Reynolds numbers. The experimental data set shows reasonable agreement with the numerical results. The results are consistent with previous investigations employing other approaches to quantify the clustering. We also present the joint p.d.f.s of enstrophy and Voronoï volumes and their Lagrangian autocorrelations. The small Voronoï volumes of light particles correspond to regions of higher enstrophy than those of heavy particles, indicating that light particles cluster in higher vorticity regions. The Lagrangian temporal autocorrelation function of Voronoï volumes shows that the clustering of light particles lasts much longer than that of heavy or neutrally buoyant particles. Due to inertial effects arising from the density contrast with the surrounding liquid, light and heavy particles remain clustered for much longer times than the flow structures which cause the clustering.

(Received July 21 2011)

(Reviewed September 28 2011)

(Accepted November 14 2011)

(Online publication January 06 2012)

Key Words:

  • homogeneous turbulence;
  • multiphase and particle-laden flows;
  • turbulent flows


c1 Email addresses for correspondence:,,