12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

LARGE-SCALE CLUSTERING OF INERTIAL PARTICLES INHOMOGENEOUS ISOTROPIC TURBULENCE

Keigo Matsuda
Japan Agency for Marine-Earth
Science and Technology (JAMSTEC),
Yokohama, Japank.matsuda@jamstec.go.jp

Katsunori Yoshimatsu
Institute of Materials and Systems
for Sustainability, Nagoya University
Nagoya, Japan
yoshimatsu@nagoya-u.jp

Kai Schneider
Institut de Mathématiques de Marseille (I2M)
Aix-Marseille Université, CNRS,Marseille, Francekai.schneider@univ-amu.fr

ABSTRACT

scale flows could affect the particle clustering at such highReynolds number. Matsuda et al. (2014) computed the particle number density spectrum, using direct numerical simulation (DNS) of particle-laden homogeneous isotropic turbulence. They observed a bump near the dissipation scale.
Moreover they showed that the number density spectrum isintensified at large scales, i.e. at small wavenumbers, as theReynolds number increases. This observation can arise a question whether large-scale clustering does occur or not. In thiswork, we aim to clarify the Reynolds-number dependence ofthe number density spectrum and, particularly, the large-scale
clustering, using three-dimensional DNS of particle-laden homogeneous isotropic turbulence at high Reynolds number.
In the following, we first briefly describe the governing
equations and the DNS approach. Numerical results showing
spectra of turbulent kinetic energy and particle number density
are discussed. Finally, some conclusions are given, and future
work is motivated.

The nonlinear dynamics of inertial particles in highReynolds number turbulence, and in particular clustering andvoid formation, are important fundamental processes, for instance, for droplets in atmospheric clouds and rain formation. Here we analyze particle data from three-dimensional
direct numerical simulations of particle-laden homogeneous
isotropic turbulence at high Reynolds number, up to Reλ = 678
and with up to 3.2 × 109 particles, computed at resolution
40963 . We show that the particle density spectrum of the analyzed data has a bump at large scales for sufficiently highReynolds number, in addition to the near-dissipation scale
bump observed so far. The large-scale bump is attributed tolarge-scale clustering of inertial particles. We find that this behavior is generic and independent of the forcing scheme used
to maintain a statistically stationary flow.

INTRODUCTION
Inertial particles suspended in high Reynolds number turbulence are ubiquitous in geophysical or industrial flows, suchas cloud droplets in atmospheric turbulence, dust particles inprotoplanetary disks, and fuel droplets in spray combustion.
Inertial particles show nonuniform distribution, i.e., clustering,in turbulence due to the centrifugal effect because particles are
swept out from turbulent vortices and concentrate in low vorticity regions (Maxey, 1987; Squires & Eaton, 1991). In atmospheric flows, turbulent clustering of cloud droplets can increase collision and coalescence frequency in raindrop formation process (Onishi & Vassilicos, 2014; Ireland et al., 2016).
The clustering can also affect the cloud radar reflectivity factor (Matsuda et al., 2014; Matsuda & Onishi, 2019). Thus,for improving cloud microphysics models, it is important tounderstand the clustering behavior in high Reynolds numberflow.
Even though the Reynolds-number dependence of inertial particle clustering has been studied extensively, the clustering behavior in high Reynolds number turbulence in atmospheric clouds (Reλ ∼ 103 –105 , where Reλ is the Taylor microscale Reynolds number) is not fully understood. Large-DIRECT NUMERICAL SIMULATION
A statistically homogeneous velocity field u (xx,t) of an incompressible fluid was obtained by the Navier–Stokes equation:1∂u+ (uu · ∇)uu = − ∇p + ν∇2 u + f ,∂tρ∇ · u = 0,(1)(2)

where f (xx,t) is an external solenoidal forcing, p(xx,t) is thepressure, ν is the constant kinematic viscosity, and ρ is theconstant fluid density.
The particles were treated as point particles and tracked
by the Lagrangian method. We assumed that the particle size
is sufficiently smaller than the Kolmogorov scale η and theparticle density ρ p is sufficiently larger than the fluid density
ρ. Then, Lagrangian motion of inertial heavy particles was1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
tracked by integrating the following equations:
νελ = 204νελ = 328/)νελ = 5315)νελ = 1555)νελ = 2515)νελ = 4025)νελ = 678/)/)

(3)



ηu (k)/(ν 5/4 ε 1/4 )dxx p= v,dt
v − u (xx p )
dvv
=−,dt
τp(4)

where x p and v are the position and velocity of a Lagrangianparticle, respectively, and τ p is the relaxation time of particle
motion.
Equations (1) and (2) were solved in a cubic computational domain with periodic boundary conditions. The domain
has side length of 2π and is discretized by Cartesian staggeredgrids. The fourth-order central-difference schemes were used
for the advection and viscous terms (Morinishi et al., 1998)
and the second-order Runge–Kutta scheme was used for timeintegration. The velocity and pressure were coupled by theHSMAC method (Hirt & Cook, 1972).
To clarify the influence of the external forcing for turbulence on the large-scale clustering, we applied two types offorcing schemes: large-scale linear forcing (LF) and largescale random forcing (RF). The large-scale linear forcing isapplied to the large-scale flow where the wavenumber k issmaller than 2.5. The forcing f is proportional to the largescale velocity u k<2.5 (Onishi et al., 2011), where u k<2.5 is thelow-pass filtered velocity field, retaining the energy of Fouriermodes only for k < 2.5. The large-scale random forcing is inaccordance with the velocity forcing proposed by Yoshida &Arimitsu (2007). In this case, the forcing f is similarly appliedfor k < 2.5 but changes randomly, having a correlation time ofT f . We set T f = 1 to be close to the large-eddy turnover time.
The important parameters in this study are the Taylormicroscale Reynolds number Reλ ≡ u0 λ /ν and the Stokes
number St ≡ τ p /τη , where u0 is the root-mean-square velocity fluctuation, λ is the Taylor microscale, and τη is the Kolmogorov time. The DNS with the linear forcing was performed for Reλ = 204, 328, and 531, using 5123 , 10243 , and20483 grid points, respectively. The DNS with the random
forcing was carried out for Reλ = 155, 251, 402, and 678, using 5123 , 10243 , 20483 , and 40963 grid points, respectively. In
all simulations, the Stokes number was set to 1.0. The numberof particles is in maximum 3.2 × 109 . The initial particle distribution is random and homogeneous. The flow velocity andparticle distribution data were sampled after the time integration of 10T f , and 10 snapshots of three-dimensional data were
saved at interval of T f . The statistical results are obtained byaveraging over 10 snapshots.

ííííkη



Figure 1. Turbulent kinetic energy spectra for different
Reynolds numbers using the large-scale linear forcing (LF)
and random forcing (RF) schemes as a function of kη. The
dotted line corresponds to k−5/3 .





Reλ = 2⟨4λt = ⟩Reλ = 328λt = ⟩/)Reλ = 53⟩λt = ⟩5)Reλ = ⟩55λt = ⟩5)Reλ = 25⟩λt = ⟩5)Reλ = 4⟨2λt = ⟩5)Reλ = 678λt = ⟩/)ηn (k)/⟨n⟩2 η/)íííí



kη







Figure 2. Particle number density spectra for St = 1 at different Reynolds numbers using the large-scale linear forcing (LF)
and random forcing (RF) schemes as a function of kη.

RESULTS AND DISCUSSIONS
Figure 1 shows time-averaged turbulent kinetic energy
spectra for different Reynolds numbers, for two different forcing mechanisms, either large-scale linear forcing (LF) or random forcing (RF). In the large Reynolds number case, Reλ =678, a k−5/3 power law behavior can be observed for about
one decade in the normalized wavenumber kη.
We also calculate the particle number density spectrum.
It should be noted that the number density spectrum is typically contaminated by Poisson noise. In this work, the Poissonnoise is removed from the spectrum by applying an analytical
Fourier transform directly to Lagrangian particle data (Matsuda et al., 2014; Matsuda & Onishi, 2019). Due to high computational cost of the direct evaluation of Fourier transform,
the spectrum values are calculated for 19 selected wavenumbers at k = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192,256, 384, 512, and 768. Figure 2 shows the time-averaged
number density spectra for St = 1.0. The number density
spectra for the linear forcing illustrate the influence of Reλ atsmall wavenumbers (kη . 0.03), as reported by Matsuda et al.
(2014). The spectra for the random forcing also show the similar Reλ dependence; i.e., for both forcing schemes, the particle density spectra exhibit two well pronounced bumps whenReλ is larger than 300. The bump at larger scale is attributed tolarge-scale clustering of inertial particles. Therefore, the largescale clustering is generic and independent of the two forcing

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(ANR), grant ANR-20-CE46-0010-01 and the French Research Federation for Fusion Studies within the frameworkof the European Fusion Development Agreement (EFDA). He
also acknowledges kind hospitality during his visit at JAMSTEC and financial support. The present DNS was performedon the Earth Simulator supercomputer system of JAMSTEC.

schemes used to maintain statistically stationary flows.
Recently, in Matsuda et al. (2021), we performed wavelet
analyses of inertial particle distribution, and a signature ofvoid-pronounced structure was observed at kη . 0.02 for
Reλ > 300. Thus, we conjecture that the large-scale clustering could be related to such large-scale voids. For illustration,a snapshot of the spatial distribution of particles in a slice ofthickness 4η is presented in figure 3 for Reλ = 678 using theRF scheme. The inhomogeneous particle distribution shows
large pronounced void regions and accumulated clusters. To
clarify the mechanism of the large-scale clustering, assessing
the convergence and divergence of particles is crucial. For instance, Oujia et al. (2020) introduced a finite time measure toquantify the convergence and divergence of the particle velocity using Voronoi tessellation. They showed that divergence isthe most prominent in cluster regions and less pronounced in
void regions. It was also found that for larger volumes, negative divergence values are observed which represent cluster
formation (i.e. particle convergence). For small volumes, positive divergence values are found which correspond to cluster destruction/void formation (i.e. particle divergence). Applying this technique to high Reynolds number particle-laden
flows will yield deeper insight into the dynamics of large-scale
clustering.

REFERENCES
Hirt, C. W. & Cook, J. L. 1972 Calculating three-dimensional
flow around structures and over rough terrain. J. Comput.
Phys. 10, 324–340.
Ireland, P. J., Bragg, A. D. & Collins, L. R. 2016 The effect ofReynolds number on inertial particle dynamics in isotropic
turbulence. Part 2. simulations with gravitational effects. J.
Fluid Mech. 796, 617–658.
Matsuda, K. & Onishi, R. 2019 Turbulent enhancement ofradar reflectivity factor for polydisperse cloud droplets. Atmos. Chem. Phys. 19, 1785–1799.
Matsuda, K., Onishi, R., Hirahara, M., Kurose, R., Takahashi,
K. & Komori, S. 2014 Influence of microscale turbulent
droplet clustering on radar cloud observations. J. Atmos.
Sci. 71, 3569–3582.
Matsuda, K., Schneider, K. & Yoshimatsu, K. 2021 Scaledependent statistics of inertial particle distribution in highReynolds number turbulence. Phys. Rev. Fluids 6, 064304.
Maxey, M. R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J.
Fluid Mech. 174, 441–465.
Morinishi, Y., Lund, T. S., Vasilyev, O. V. & Moin, P. 1998
Fully conservative higher order finite difference schemes
for incompressible flow. J. Comput. Phys. 143, 90–124.
Onishi, R., Baba, Y. & Takahashi, K. 2011 Large-scale forcing
with less communication in finite-difference simulations ofsteady isotropic turbulence. J. Comput. Phys. 230, 4088–4099.
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.
Oujia, T., Matsuda, K. & Schneider, K. 2020 Divergence andconvergence of inertial particles in high Reynolds number
turbulence. J. Fluid Mech. 905, A14.
Squires, K. D. & Eaton, J. K. 1991 Measurements of particle dispersion obtained from direct numerical simulationsof isotropic turbulence. J. Fluid Mech. 226, 1–35.
Yoshida, K. & Arimitsu, T. 2007 Inertial-subrange structuresof isotropic incompressible magnetohydrodynamic turbulence in the lagrangian renormalized approximation. Phys.
Fluids 19, 045106.

CONCLUDING REMARKS
Reynolds-number dependence of inertial heavy particle
clustering has been investigated by using three-dimensional
DNS of particle-laden homogeneous isotropic turbulence. The
DNS was performed for several Reynolds numbers up toReλ = 678 and two cases of large-scale forcing schemes (thelinear and random forcing schemes) to maintain statisticallystationary turbulence. The number of particles was in maximum 3.2 × 109 . The Stokes number was fixed to 1.0. The analyzed data show that for Reλ > 300 the particle density spectra
exhibit two well pronounced bumps. The bump at larger scale
is attributed to large-scale clustering of inertial particles. Thisbehavior is generic and independent of the forcing scheme.
Further investigation of the large-scale clustering mechanism
will be performed, the influence of the Stokes number will beassessed and elaborated analyses of the spectral bumps will be
published in forthcoming work.

ACKNOWLEDGEMENTS
K.M. acknowledges financial support from Grant-in-Aidfor Scientific Research (C) No. JP20K04298 from the Japan
Society for the Promotion of Science (JSPS). K.S. acknowledges funding from the Agence Nationale de la Recherche

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3.

Spatial distribution of inertial particles for Reλ = 678 using the RF scheme in a slice at 0 < x < 4η.

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