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

ROLE OF THE HIERARCHY OF COHERENT STRUCTURES IN THE
TRANSPORT OF SOLID PARTICLES IN TURBULENT CHANNEL FLOW
AT HIGH REYNOLDS NUMBERS

Yutaro Motoori
Graduate School of Engineering Science
Osaka University
1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japany.motoori.es@osaka-u.ac.jp

Susumu Goto
Graduate School of Engineering Science
Osaka University
1-3 Machikaneyama, Toyonaka, Osaka, 560-8531, Japans.goto.es@osaka-u.ac.jp

ABSTRACT

subjected only to the Stokes drag with the nonlinear correctionk p = 1 + 0.15Re0.687, where Re p is the particle Reynolds nump
ber. We emphasize that the motion of pointwise particles ischaracterized only by the particle velocity relaxation time τ p .
In wall-bounded turbulence, it is known that inertial particles governed by (1) tend to accumulate in the near-wall region. Many studies (e.g. Soldati & Marchioli, 2009) on inertial
particles dispersed in low-Reynolds-number wall-turbulence
have revealed the role of the buffer-layer coherent structuresin the wall-accumulation process. However, as the Reynolds
number increases, larger-size structures emerge in addition tothe near-wall ones. Their roles in the transport of inertial particles remain unclear. Meanwhile, our recent study (Motoori& Goto, 2021) revealed the generation mechanism of the hierarchy of multiscale vortices in turbulent channel ﬂow at thefriction Reynolds number Reτ = 4179. Recently, we investigated how inertial particles are transported in turbulent channel
ﬂow based on the hierarchy of ﬂow coherent structures (Motoori et al., 2022). The key to these studies is the extraction of
the hierarchical structures in real space.
In the present study, we examine the details of the transport of inertial particles based on the hierarchy of vortices
for the higher Reynolds number turbulence. For this purpose,
we newly conduct the direct numerical simulations (DNS) of
the dispersion of inertial particles in turbulent channel ﬂow at
Reτ = 2000. The Reynolds number is higher than in the turbulent channel ﬂow (Reτ = 1000) examined in our previous study(Motoori et al., 2022). In the present paper, we visualize thespatial distributions of particles and the hierarchy of vortices,
and also we quantitatively show the relation between them.

To investigate the transport of heavy small particles (inertial particles) in high-Reynolds-number wall turbulence, weconduct direct numerical simulations of inertial particles inturbulent channel ﬂow at the friction Reynolds number Reτ =2000. The Reynolds number is higher than the one analysedin our previous study (Motoori et al., 2022). In the statisticallysteady state, particles distribute inhomogeneously; particles
with different relaxation times form voids and clusters withdifferent sizes in the bulk of the ﬂow. To explore the origin of
the multiscale voids and clusters of particles, we objectively
identify the axes of tubular vortices. These identiﬁcations enable us to quantitatively show that vortices sweep out the particles of the relaxation time comparable with their turnover time,
irrespective of their size and existing height. This explains
the reason why the multiscale clusterings are well describedin terms of the local Stokes number deﬁned by the turnovertime of multiscale vortices. The results obtained by the present
study are consistent with our previous study (Motoori et al.,2022), and the description of the particle transport leads to understanding velocity statistics and wall-deposition mechanismof inertial particles in high-Reynolds-number wall turbulence.

INTRODUCTION
We investigate the motion of rigid spherical particles inturbulent channel ﬂow at a high Reynolds number. In particular, we focus on heavy small particles, which are sometimes
called inertial particles. When the particles are sufﬁciently
smaller than the Kolmogorov length, and the mass density ofthe particles is much larger than that of ﬂuid, the particles aregoverned by)
duu pkp (dxx p= u p and=−up − u .
dt
dt
τp

DIRECT NUMERICAL SIMULATIONS
We numerically simulate turbulent channel ﬂow at Reτ =2000 driven by a constant pressure gradient by solving theNavier-Stokes equations of an incompressible ﬂuid. Since thetarget of the present study is a dilute dispersion of small particles, we neglect their effect on ﬂuid motion. The temporal(1)

Here, x p and u p are the position and velocity of particles, andu is the ﬂuid velocity at the particle position. Particles are1

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

(a)

(b)

y6

Wall-zFigure 1. Two-dimensional (y–z) spatial distributions of inertial particles with (a) St+ = 25 and (b) 250 within a layer of thickness
200 wall units. The clusters and voids of particles are larger for larger St+ .

(a)

(b)

y6 x*QszQ

Figure 2. (a) Smallest-scale vortices identiﬁed by the isosurfaces of the second invariant Q of the velocity gradient tensor. (b)Hierarchy of vortices identiﬁed by the isosurfaces of the second invariant Q(σ ) of the scale-decomposed velocity gradient tensor atscales σ + = 480 (red), 120 (yellow) and 30 (blue). We visualize a subdomain (half in the wall-normal direction) of the computationaldomain.
correlation function in turbulent channel ﬂow at Reτ = 1000
(Motoori et al., 2022). To understand the formation mechanism of the multiscale clustering, we quantitatively investigatethe relationship between the clustering of particles and the hierarchy of vortices. For this purpose, in the next section, weidentify the hierarchy of vortices in turbulent channel ﬂow.

integrations of the viscous and convection terms are made byusing the second-order Crank-Nicolson method and the thirdorder Adams-Bashforth method, respectively. For the spatial
discretization of the terms in the governing equations, we use
a sixth-order central difference scheme on a staggered gridin the streamwise (x) and spanwise (z) directions and a nonuniform ﬁnite difference scheme in the wall-normal (y) direction. The numbers of grid points are (Nx × Ny × Nz ) =(3072 × 1536 × 1536). The numerical simulation method is
the same as that used by our previous study (Motoori et al.,2022) but the present Reynolds number is higher than in theprevious one (Reτ = 1000).
We track 107 particles by numerically integrating (1) by
the second-order Adams-Bashforth method. We assume thatthe diameter of particles is sufﬁciently small so that Re p = 0.
As mentioned in the introduction, the most important parameter is the particle relaxation time τ p , which is expressedin nondimensional form as the Stokes number. We use theStokes number St+ = τ p /τ+ nondimensionalized by the wallunit time-scale τ+ (= ν /u2τ ), where uτ is the friction velocity
and ν is the kinematic viscosity. We examine seven cases with
St+ = 1, 10, 25, 50, 100, 250 and 1000, which are the same parameters in Bernardini (2014) and Motoori et al. (2022). For
the following analyses, we take the average in the statisticallysteady state.

IDENTIFICATION OF MULTISCALE VORTICESTo extract vortical structures, we show in ﬁgure 2(a) theisosurface of the second invariant Q of the velocity gradienttensor. In this ﬁgure, we can only observe ﬁne tubular vortices. This is because the smallest-scale vortices determine thevelocity gradients. Therefore, to capture the hierarchy of vortices, we need a scale decomposition.
To extract the hierarchy of vortices in turbulence, we use
a band-pass ﬁlter, which was also used in Motoori & Goto(2021) and Motoori et al. (2022). In this method, we ﬁrst apply
the Gaussian ﬁlter with the ﬁlter scale σ to the ﬂuctuating ve(σ )locity ǔi . Since the obtained velocity ui low contains the information of all the scales larger than σ , the ﬁlter corresponds toa low-pass ﬁlter of the Fourier modes of the velocity. We thentake the difference between the low-pass ﬁltered ﬁelds at two(σ )
(2σ )
(σ )
different scales, i.e. ui (xx) = ui low (xx) − ui low (xx). Thisﬁlter corresponds to a band-pass ﬁlter of the Fourier modes in(σ )the sense that ui has the contributions only from around thescale σ .
To identify the hierarchy of vortices, we evaluate scaledecomposed second invariant Q(σ ) of the velocity gradient ten(σ )sor from ui . Figure 2(b) shows positive isosurfaces of Q(σ )at scales σ + = 30, 120 and 480, where ·+ denotes the wall
units nondimensionalized by uτ and ν . The isosurfaces indeed capture vortices with the ﬁlter scale σ (see also ﬁgure 2

CLUSTERING OF PARTICLESWe show in ﬁgure 1 the two-dimensional (y–z) spatial distributions of particles with (a) St+ = 25 and (b) 250 in thesteady state. We visualize only particles in a thin (200 wallunits) layer. The important observation is that particles formclusters and that the size of the voids is larger for larger St+ .
This multiscale nature was quantitatively veriﬁed by the pair2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022(a)(b) St+ = 25
 St+ = 10)

vortex size

 St+ = 250XyX St+ = 100

σ + = 30vortex size

σ + = 480

Figure 3. Average number N p of inertial particles existing at distance r from an axis of vortices with sizes (a) σ + = 30 and (b) 480
in the layers (a) 500 ≲ y+ < 1000 and (b) 1500 ≲ y+ < 2000. The values are normalized by those at r = 3σ . From the thinner (anddarker) to the thicker (and lighter) lines, St+ = 1, 10, 25, 50, 100, 250 and 1000. (a) Vortices with size σ + = 30 tend to sweep out
particles with St+ = 10, whereas (b) those with σ + = 480 sweep out particles with St+ = 100 and 250.

(a)(b)

Figure 4. (a) Average number N p (r = 0) of inertial particles on the axes of vortices, namely, the values at r = 0 shown in ﬁgure 3.
In (b), we show N p (r = 0) for the several cases: the different sizes [σ + = 30 (blue), 60 (green), 120 (yellow), 240 (orange) and 480(red)], and different layers [500 ≲ y+ < 1000 (triangles), and 1500 ≲ y+ < 2000 (squares)]. The values are shown as functions of (b)StQ instead of (a) St+ . Here, we use the value of τQ at the centre of each layer.
in Motoori & Goto, 2021). Here, σ + = 30 is of the order ofsize of the smallest-size vortices, whereas σ + = 480 is approximately the largest-size vortices. Moreover, by applying
the low-pressure method (Miura & Kida, 1997) to the bandpass ﬁltered ﬁelds, we objectively identify the axes of tubular
vortices (see ﬁgure 3 in Motoori et al., 2022). In the following,
we use the axes for the examination of the spatial distributionof particles around vortices at different sizes.

ratio takes the minimum at r = 0, whereas it takes the maximum around for σ ≲ r ≲ 2σ . This tendency is also observedfor particles with another Stokes number (St+ = 25) close to
St+ = 10. On the other hand, the number of particles withlarger St+ (≳ 100) is almost constant, irrespective of r. Hence,in the layer 500 ≲ y+ ≲ 1000, the small-size (σ + = 30) vortices most likely to sweep out particles with St+ ≈ 10. This isconsistent with the observation (ﬁgure 1a) that particles form
small-size clusters in the instantaneous ﬁeld.
Figure 3(b) shows N p around vortices with a larger size
σ + = 480 within the layer 1500 ≲ y+ < 2000(= h+ ). We cansee that particles with St+ = 100 and 250 are swept out fromthe large (σ + = 480) vortices to form a cluster around them.
We replot, in ﬁgure 4(a), N p at r = 0 as a function of St+ .
Note that the blue triangles show the results for σ + = 30 in
the layers 500 ≲ y+ ≲ 1000, whereas the red squares show
those for σ + = 480 in the layers 1500 ≲ y+ ≲ 2000. In otherwords, these symbols show the values at r = 0 shown in ﬁgure 3. The lines do not collapse. This means that different-sizeand different-height vortices do contribute to the clustering ofparticles in different ranges of St+ . More precisely, vortices atlarger sizes or those further from the wall tend to sweep out
particles with larger St+ ; note that the well of red line shifts tolarger St+ . This suggests that vortices sweep out particles ofPARTICLE DISTRIBUTION AROUND VORTICESTo quantitatively show the origin of clusters of particles,we use the identiﬁed vortex axes. This enables us to evaluate
the number of particles around the axis of each vortex. For
example of the results obtained by the present study, we show
the number density N p of particles as a function of the distancer for y+ ≳ 500.
We ﬁrst show, in ﬁgure 3, the normalized N p for particles
around the axes of small-scale vortices (σ + = 30; see blue vortices in ﬁgure 2b) existing in the layer 500 ≲ y+ ≲ 1000. Here,
we normalize N p so that N p < 1 when particles are sparsely
distributed; whereas N p > 1, when they form clusters. Thethinner (and darker) lines indicate particles with smaller St+ .
Looking at the second thinnest line (St+ = 10), the number3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
conducted DNS of those with seven relaxation times (St+ = 1–1000) in the turbulence at Reτ = 2000. The key ingredient of
our analysis is the objective identiﬁcation of the hierarchy ofcoherent vortices. We have shown the distribution of particles around the axes of vortices at each level of the hierarchy. The analyses lead to the main conclusion: vortices sweep
predominantly particles with the relaxation times comparablewith their turnover time. Hence, the clustering of particles isdetermined by the local Stokes number (StQ ≈ 0.2) irrespectiveof the size and height of coherent vortices (ﬁgure 4b).

the relaxation time similar to their eddy turnover time, because
larger-size vortices have longer eddy turnover time.
To verify this, we deﬁne the eddy turnover time τQ (σ , y)of vortices with size σ at height y in terms of the root-mean(σ )square Qrms (y) of the second invariant tensor of the velocity
gradient tensor at the size σ . We then deﬁne the local Stokes
number by
StQ (σ , y) =τp,
τQ (σ , y)

(2)

√(σ )
where τQ = 1/ Qrms (y). The vertical values in ﬁgure 4(b) are
the same as in (a) but are shown as functions of StQ instead of
St+ . In addition to the two results shown in ﬁgure 4(a), we investigate the cases for vortices with different sizes and heights,
and we show in ﬁgure 4(b) all the results together. It is mostimportant to observe that the well of all the lines collapses
around StQ ≈ 0.2. We can therefore conclude that, irrespectiveof the height and size, vortices preferentially sweep out particles of the relaxation time τ p comparable with the turnovertime τQ . This observation around wall-detached (σ < y) vortices is similar to that in homogeneous turbulence (Yoshimoto& Goto, 2007; Oka & Goto, 2021).
In summary, in the log and outer layers of turbulent channel ﬂow, the origin of the multiscale clustering is the hierarchy of vortices. More concretely, vortices most effectively
sweep out particles of the relaxation time comparable with
their turnover time irrespective of their size, their existingheight and the Reynolds number. Incidentally, we showed theclustering near the wall is also explained in terms of the hierarchy of wall-attached quasi-streamwise vortices and lowspeed streaks (Motoori et al., 2022). From the present results,
irrespective of the Reynolds number, we may infer that wallattached structures contribute most signiﬁcantly to transporting particles with the relaxation time comparable with their
turnover time of the vortices.

REFERENCES
Bernardini, M. 2014 Reynolds number scaling of inertial particle statistics in turbulent channel ﬂows. Journal of Fluid
Mechanics 758, R1.
Miura, H. & Kida, S. 1997 Identiﬁcation of tubular vortices
in turbulence. Journal of the Physical Society of Japan 66,1331–1334.
Motoori, Y. & Goto, S. 2021 Hierarchy of coherent structuresand real-space energy transfer in turbulent channel ﬂow.
Journal of Fluid Mechanics 911, A27.
Motoori, Y., Wong, C. & Goto, S. 2022 Role of the hierarchyof coherent structures in the transport of heavy small particles in turbulent channel ﬂow. Journal of Fluid Mechanics(in press).
Oka, S. & Goto, S. 2021 Generalized sweep-stick mechanismof inertial-particle clustering in turbulence. Physical Review
Fluids 6, 044605.
Soldati, A. & Marchioli, C. 2009 Physics and modelling ofturbulent particle deposition and entrainment: Review of asystematic study. International Journal of Multiphase Flow35, 827–839.
Yoshimoto, H. & Goto, S. 2007 Self-similar clustering of inertial particles in homogeneous turbulence. Journal of Fluid
Mechanics 577, 275–286.

CONCLUSIONS
To investigate the transport mechanism of inertial particles in high-Reynolds-number turbulent channel ﬂow, we have4